:: TAYLOR_2 semantic presentation

begin

theorem :: TAYLOR_2:1
for x being ( ( ) ( V11() real ext-real ) Real)
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) holds abs (x : ( ( ) ( V11() real ext-real ) Real) |^ n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) = (abs x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) |^ n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ;

definition
let f be ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) PartFunc of ,) ;
let Z be ( ( ) ( V75() V76() V77() ) Subset of ( ( ) ( ) set ) ) ;
let a be ( ( real ) ( V11() real ext-real ) number ) ;
func Maclaurin (f,Z,a) -> ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Real_Sequence) equals :: TAYLOR_2:def 1
 Taylor (f : ( ( ) ( ) set ) ,Z : ( ( V21() V30(f : ( ( ) ( ) set ) ,f : ( ( ) ( ) set ) ) ) ( Relation-like f : ( ( ) ( ) set ) -defined f : ( ( ) ( ) set ) -valued V21() total V30(f : ( ( ) ( ) set ) ,f : ( ( ) ( ) set ) ) ) Element of K6(K7(f : ( ( ) ( ) set ) ,f : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V61() V74() V75() V76() V77() V78() V79() V80() V81() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ,a : ( ( real ) ( V11() real ext-real ) set ) ) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ;
end;

theorem :: TAYLOR_2:2
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) PartFunc of ,)
for r being ( ( ) ( V11() real ext-real ) Real) st 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V61() V74() V75() V76() V77() V78() V79() V80() V81() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) & ].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) c= dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) PartFunc of ,) : ( ( ) ( ) set ) & f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) PartFunc of ,) is_differentiable_on n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ,].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) holds
for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in ].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) holds
ex s being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V61() V74() V75() V76() V77() V78() V79() V80() V81() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) < s : ( ( ) ( V11() real ext-real ) Real) & s : ( ( ) ( V11() real ext-real ) Real) < 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) & f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) = ((Partial_Sums (Maclaurin (f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) PartFunc of ,) ,].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( V11() real ext-real ) Real) )) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) + (((((diff (f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) PartFunc of ,) ,].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) -valued V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . (n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . (s : ( ( ) ( V11() real ext-real ) Real) * x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) * (x : ( ( ) ( V11() real ext-real ) Real) |^ (n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) / ((n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ;

theorem :: TAYLOR_2:3
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) PartFunc of ,)
for x0, r being ( ( ) ( V11() real ext-real ) Real) st 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V61() V74() V75() V76() V77() V78() V79() V80() V81() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) & ].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) c= dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) PartFunc of ,) : ( ( ) ( ) set ) & f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) PartFunc of ,) is_differentiable_on n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ,].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) holds
for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in ].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) holds
ex s being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V61() V74() V75() V76() V77() V78() V79() V80() V81() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) < s : ( ( ) ( V11() real ext-real ) Real) & s : ( ( ) ( V11() real ext-real ) Real) < 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) & abs ((f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) - ((Partial_Sums (Taylor (f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) PartFunc of ,) ,].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) ,x : ( ( ) ( V11() real ext-real ) Real) )) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) = abs (((((diff (f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) PartFunc of ,) ,].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) -valued V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . (n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . (x0 : ( ( ) ( V11() real ext-real ) Real) + (s : ( ( ) ( V11() real ext-real ) Real) * (x : ( ( ) ( V11() real ext-real ) Real) - x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) * ((x : ( ( ) ( V11() real ext-real ) Real) - x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) |^ (n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) / ((n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ;

theorem :: TAYLOR_2:4
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) PartFunc of ,)
for r being ( ( ) ( V11() real ext-real ) Real) st 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V61() V74() V75() V76() V77() V78() V79() V80() V81() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) & ].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) c= dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) PartFunc of ,) : ( ( ) ( ) set ) & f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) PartFunc of ,) is_differentiable_on n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ,].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) holds
for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in ].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) holds
ex s being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V61() V74() V75() V76() V77() V78() V79() V80() V81() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) < s : ( ( ) ( V11() real ext-real ) Real) & s : ( ( ) ( V11() real ext-real ) Real) < 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) & abs ((f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) - ((Partial_Sums (Maclaurin (f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) PartFunc of ,) ,].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( V11() real ext-real ) Real) )) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) = abs (((((diff (f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) PartFunc of ,) ,].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) -valued V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . (n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . (s : ( ( ) ( V11() real ext-real ) Real) * x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) * (x : ( ( ) ( V11() real ext-real ) Real) |^ (n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) / ((n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ;

theorem :: TAYLOR_2:5
for Z being ( ( open ) ( open V75() V76() V77() ) Subset of ( ( ) ( ) set ) ) holds
( exp_R : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V75() V76() V77() ) Subset of ( ( ) ( ) set ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) = exp_R : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V75() V76() V77() ) Subset of ( ( ) ( ) set ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) & dom (exp_R : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V75() V76() V77() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = Z : ( ( open ) ( open V75() V76() V77() ) Subset of ( ( ) ( ) set ) ) ) ;

theorem :: TAYLOR_2:6
for Z being ( ( open ) ( open V75() V76() V77() ) Subset of ( ( ) ( ) set ) )
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) holds (diff (exp_R : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V75() V76() V77() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) -valued V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) = exp_R : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V75() V76() V77() ) Subset of ( ( ) ( ) set ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: TAYLOR_2:7
for Z being ( ( open ) ( open V75() V76() V77() ) Subset of ( ( ) ( ) set ) )
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) )
for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V75() V76() V77() ) Subset of ( ( ) ( ) set ) ) holds
((diff (exp_R : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V75() V76() V77() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) -valued V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) = exp_R : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ;

theorem :: TAYLOR_2:8
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) )
for r, x being ( ( ) ( V11() real ext-real ) Real) st 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V61() V74() V75() V76() V77() V78() V79() V80() V81() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) holds
(Maclaurin (exp_R : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ,].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( V11() real ext-real ) Real) )) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) = (x : ( ( ) ( V11() real ext-real ) Real) |^ n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) / (n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ;

theorem :: TAYLOR_2:9
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) )
for r, x, s being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in ].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) & 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V61() V74() V75() V76() V77() V78() V79() V80() V81() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) < s : ( ( ) ( V11() real ext-real ) Real) & s : ( ( ) ( V11() real ext-real ) Real) < 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) holds
abs (((((diff (exp_R : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ,].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) -valued V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . (n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . (s : ( ( ) ( V11() real ext-real ) Real) * x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) * (x : ( ( ) ( V11() real ext-real ) Real) |^ (n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) / ((n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) <= ((abs (exp_R : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . (s : ( ( ) ( V11() real ext-real ) Real) * x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) * ((abs x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) |^ (n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) / ((n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ;

theorem :: TAYLOR_2:10
for Z being ( ( open ) ( open V75() V76() V77() ) Subset of ( ( ) ( ) set ) )
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) holds exp_R : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ,Z : ( ( open ) ( open V75() V76() V77() ) Subset of ( ( ) ( ) set ) ) ;

theorem :: TAYLOR_2:11
for r being ( ( ) ( V11() real ext-real ) Real) st 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V61() V74() V75() V76() V77() V78() V79() V80() V81() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) holds
ex M, L being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V61() V74() V75() V76() V77() V78() V79() V80() V81() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) <= M : ( ( ) ( V11() real ext-real ) Real) & 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V61() V74() V75() V76() V77() V78() V79() V80() V81() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) <= L : ( ( ) ( V11() real ext-real ) Real) & ( for n being ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) )
for x, s being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in ].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) & 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V61() V74() V75() V76() V77() V78() V79() V80() V81() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) < s : ( ( ) ( V11() real ext-real ) Real) & s : ( ( ) ( V11() real ext-real ) Real) < 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) holds
abs (((((diff (exp_R : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ,].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) -valued V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . (s : ( ( ) ( V11() real ext-real ) Real) * x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) * (x : ( ( ) ( V11() real ext-real ) Real) |^ n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) / (n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) <= (M : ( ( ) ( V11() real ext-real ) Real) * (L : ( ( ) ( V11() real ext-real ) Real) |^ n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) / (n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ) ;

theorem :: TAYLOR_2:12
for M, L being ( ( ) ( V11() real ext-real ) Real) st M : ( ( ) ( V11() real ext-real ) Real) >= 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V61() V74() V75() V76() V77() V78() V79() V80() V81() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) & L : ( ( ) ( V11() real ext-real ) Real) >= 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V61() V74() V75() V76() V77() V78() V79() V80() V81() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) holds
for e being ( ( ) ( V11() real ext-real ) Real) st e : ( ( ) ( V11() real ext-real ) Real) > 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V61() V74() V75() V76() V77() V78() V79() V80() V81() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) holds
ex n being ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) st
for m being ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(M : ( ( ) ( V11() real ext-real ) Real) * (L : ( ( ) ( V11() real ext-real ) Real) |^ m : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) / (m : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) < e : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: TAYLOR_2:13
for r, e being ( ( ) ( V11() real ext-real ) Real) st 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V61() V74() V75() V76() V77() V78() V79() V80() V81() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) & 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V61() V74() V75() V76() V77() V78() V79() V80() V81() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) < e : ( ( ) ( V11() real ext-real ) Real) holds
ex n being ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) st
for m being ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) holds
for x, s being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in ].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) & 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V61() V74() V75() V76() V77() V78() V79() V80() V81() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) < s : ( ( ) ( V11() real ext-real ) Real) & s : ( ( ) ( V11() real ext-real ) Real) < 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) holds
abs (((((diff (exp_R : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ,].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) -valued V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . m : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . (s : ( ( ) ( V11() real ext-real ) Real) * x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) * (x : ( ( ) ( V11() real ext-real ) Real) |^ m : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) / (m : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) < e : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: TAYLOR_2:14
for r, e being ( ( ) ( V11() real ext-real ) Real) st 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V61() V74() V75() V76() V77() V78() V79() V80() V81() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) & 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V61() V74() V75() V76() V77() V78() V79() V80() V81() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) < e : ( ( ) ( V11() real ext-real ) Real) holds
ex n being ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) st
for m being ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) holds
for x being ( ( real ) ( V11() real ext-real ) number ) st x : ( ( real ) ( V11() real ext-real ) number ) in ].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) holds
abs ((exp_R : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) - ((Partial_Sums (Maclaurin (exp_R : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ,].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . m : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) < e : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: TAYLOR_2:15
for x being ( ( ) ( V11() real ext-real ) Real) holds x : ( ( ) ( V11() real ext-real ) Real) rExpSeq : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) is absolutely_summable ;

theorem :: TAYLOR_2:16
for r, x being ( ( ) ( V11() real ext-real ) Real) st 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V61() V74() V75() V76() V77() V78() V79() V80() V81() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) holds
( Maclaurin (exp_R : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ,].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Real_Sequence) = x : ( ( ) ( V11() real ext-real ) Real) rExpSeq : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) & Maclaurin (exp_R : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ,].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Real_Sequence) is absolutely_summable & exp_R : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) = Sum (Maclaurin (exp_R : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ,].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( V11() real ext-real ) Real) )) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Real_Sequence) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ;

theorem :: TAYLOR_2:17
for Z being ( ( open ) ( open V75() V76() V77() ) Subset of ( ( ) ( ) set ) ) holds
( sin : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V75() V76() V77() ) Subset of ( ( ) ( ) set ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) = cos : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V75() V76() V77() ) Subset of ( ( ) ( ) set ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) & cos : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V75() V76() V77() ) Subset of ( ( ) ( ) set ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) = (- sin : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V75() V76() V77() ) Subset of ( ( ) ( ) set ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) & dom (sin : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V75() V76() V77() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = Z : ( ( open ) ( open V75() V76() V77() ) Subset of ( ( ) ( ) set ) ) & dom (cos : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V75() V76() V77() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = Z : ( ( open ) ( open V75() V76() V77() ) Subset of ( ( ) ( ) set ) ) ) ;

theorem :: TAYLOR_2:18
for f being ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) PartFunc of ,)
for Z being ( ( ) ( V75() V76() V77() ) Subset of ( ( ) ( ) set ) ) st f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) PartFunc of ,) is_differentiable_on Z : ( ( ) ( V75() V76() V77() ) Subset of ( ( ) ( ) set ) ) holds
(- f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) PartFunc of ,) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( ) ( V75() V76() V77() ) Subset of ( ( ) ( ) set ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) = - (f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) PartFunc of ,) `| Z : ( ( ) ( V75() V76() V77() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: TAYLOR_2:19
for Z being ( ( open ) ( open V75() V76() V77() ) Subset of ( ( ) ( ) set ) )
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( (diff (sin : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V75() V76() V77() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) -valued V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . (2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) = ((- 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real non positive ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) |^ n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) (#) (sin : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V75() V76() V77() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) & (diff (sin : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V75() V76() V77() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) -valued V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . ((2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) = ((- 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real non positive ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) |^ n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) (#) (cos : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V75() V76() V77() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) & (diff (cos : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V75() V76() V77() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) -valued V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . (2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) = ((- 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real non positive ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) |^ n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) (#) (cos : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V75() V76() V77() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) & (diff (cos : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V75() V76() V77() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) -valued V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . ((2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) = ((- 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real non positive ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) |^ (n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) (#) (sin : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V75() V76() V77() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: TAYLOR_2:20
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) )
for r, x being ( ( ) ( V11() real ext-real ) Real) st r : ( ( ) ( V11() real ext-real ) Real) > 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V61() V74() V75() V76() V77() V78() V79() V80() V81() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( (Maclaurin (sin : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ,].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( V11() real ext-real ) Real) )) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Real_Sequence) . (2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) = 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V61() V74() V75() V76() V77() V78() V79() V80() V81() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) & (Maclaurin (sin : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ,].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( V11() real ext-real ) Real) )) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Real_Sequence) . ((2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) = (((- 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real non positive ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) |^ n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) * (x : ( ( ) ( V11() real ext-real ) Real) |^ ((2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) / (((2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) & (Maclaurin (cos : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ,].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( V11() real ext-real ) Real) )) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Real_Sequence) . (2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) = (((- 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real non positive ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) |^ n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) * (x : ( ( ) ( V11() real ext-real ) Real) |^ (2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) / ((2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) & (Maclaurin (cos : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ,].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( V11() real ext-real ) Real) )) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Real_Sequence) . ((2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) = 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V61() V74() V75() V76() V77() V78() V79() V80() V81() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: TAYLOR_2:21
for Z being ( ( open ) ( open V75() V76() V77() ) Subset of ( ( ) ( ) set ) )
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( sin : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ,Z : ( ( open ) ( open V75() V76() V77() ) Subset of ( ( ) ( ) set ) ) & cos : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ,Z : ( ( open ) ( open V75() V76() V77() ) Subset of ( ( ) ( ) set ) ) ) ;

theorem :: TAYLOR_2:22
for r being ( ( ) ( V11() real ext-real ) Real) st r : ( ( ) ( V11() real ext-real ) Real) > 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V61() V74() V75() V76() V77() V78() V79() V80() V81() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) holds
ex r1, r2 being ( ( ) ( V11() real ext-real ) Real) st
( r1 : ( ( ) ( V11() real ext-real ) Real) >= 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V61() V74() V75() V76() V77() V78() V79() V80() V81() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) & r2 : ( ( ) ( V11() real ext-real ) Real) >= 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V61() V74() V75() V76() V77() V78() V79() V80() V81() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) & ( for n being ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) )
for x, s being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in ].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) & 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V61() V74() V75() V76() V77() V78() V79() V80() V81() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) < s : ( ( ) ( V11() real ext-real ) Real) & s : ( ( ) ( V11() real ext-real ) Real) < 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( abs (((((diff (sin : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ,].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) -valued V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . (s : ( ( ) ( V11() real ext-real ) Real) * x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) * (x : ( ( ) ( V11() real ext-real ) Real) |^ n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) / (n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) <= (r1 : ( ( ) ( V11() real ext-real ) Real) * (r2 : ( ( ) ( V11() real ext-real ) Real) |^ n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) / (n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) & abs (((((diff (cos : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ,].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) -valued V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . (s : ( ( ) ( V11() real ext-real ) Real) * x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) * (x : ( ( ) ( V11() real ext-real ) Real) |^ n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) / (n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) <= (r1 : ( ( ) ( V11() real ext-real ) Real) * (r2 : ( ( ) ( V11() real ext-real ) Real) |^ n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) / (n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ) ) ;

theorem :: TAYLOR_2:23
for r, e being ( ( ) ( V11() real ext-real ) Real) st 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V61() V74() V75() V76() V77() V78() V79() V80() V81() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) & 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V61() V74() V75() V76() V77() V78() V79() V80() V81() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) < e : ( ( ) ( V11() real ext-real ) Real) holds
ex n being ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) st
for m being ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) holds
for x, s being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in ].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) & 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V61() V74() V75() V76() V77() V78() V79() V80() V81() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) < s : ( ( ) ( V11() real ext-real ) Real) & s : ( ( ) ( V11() real ext-real ) Real) < 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( abs (((((diff (sin : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ,].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) -valued V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . m : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . (s : ( ( ) ( V11() real ext-real ) Real) * x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) * (x : ( ( ) ( V11() real ext-real ) Real) |^ m : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) / (m : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) < e : ( ( ) ( V11() real ext-real ) Real) & abs (((((diff (cos : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ,].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) -valued V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . m : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . (s : ( ( ) ( V11() real ext-real ) Real) * x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) * (x : ( ( ) ( V11() real ext-real ) Real) |^ m : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) / (m : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) < e : ( ( ) ( V11() real ext-real ) Real) ) ;

theorem :: TAYLOR_2:24
for r, e being ( ( ) ( V11() real ext-real ) Real) st 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V61() V74() V75() V76() V77() V78() V79() V80() V81() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) & 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V61() V74() V75() V76() V77() V78() V79() V80() V81() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) < e : ( ( ) ( V11() real ext-real ) Real) holds
ex n being ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) st
for m being ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) holds
for x being ( ( real ) ( V11() real ext-real ) number ) st x : ( ( real ) ( V11() real ext-real ) number ) in ].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) holds
( abs ((sin : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) - ((Partial_Sums (Maclaurin (sin : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ,].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . m : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) < e : ( ( ) ( V11() real ext-real ) Real) & abs ((cos : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) - ((Partial_Sums (Maclaurin (cos : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ,].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . m : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) < e : ( ( ) ( V11() real ext-real ) Real) ) ;

theorem :: TAYLOR_2:25
for r, x being ( ( ) ( V11() real ext-real ) Real)
for m being ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) st 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V61() V74() V75() V76() V77() V78() V79() V80() V81() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) holds
( (Partial_Sums (Maclaurin (sin : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ,].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( V11() real ext-real ) Real) )) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . ((2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) * m : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) = (Partial_Sums (x : ( ( ) ( V11() real ext-real ) Real) P_sin) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . m : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) & (Partial_Sums (Maclaurin (cos : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ,].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( V11() real ext-real ) Real) )) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . ((2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) * m : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) = (Partial_Sums (x : ( ( ) ( V11() real ext-real ) Real) P_cos) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . m : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ;

theorem :: TAYLOR_2:26
for r, x being ( ( ) ( V11() real ext-real ) Real)
for m being ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) st 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V61() V74() V75() V76() V77() V78() V79() V80() V81() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) & m : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V61() V74() V75() V76() V77() V78() V79() V80() V81() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( (Partial_Sums (Maclaurin (sin : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ,].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( V11() real ext-real ) Real) )) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . (2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) * m : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) = (Partial_Sums (x : ( ( ) ( V11() real ext-real ) Real) P_sin) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . (m : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) & (Partial_Sums (Maclaurin (cos : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ,].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( V11() real ext-real ) Real) )) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . (2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) * m : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) = (Partial_Sums (x : ( ( ) ( V11() real ext-real ) Real) P_cos) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . m : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ;

theorem :: TAYLOR_2:27
for r, x being ( ( ) ( V11() real ext-real ) Real)
for m being ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) st 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V61() V74() V75() V76() V77() V78() V79() V80() V81() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) holds
(Partial_Sums (Maclaurin (cos : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ,].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( V11() real ext-real ) Real) )) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . (2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) * m : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) = (Partial_Sums (x : ( ( ) ( V11() real ext-real ) Real) P_cos) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . m : ( ( ) ( ordinal natural V11() real ext-real non negative V61() V74() V75() V76() V77() V78() V79() V80() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ;

theorem :: TAYLOR_2:28
for r, x being ( ( ) ( V11() real ext-real ) Real) st r : ( ( ) ( V11() real ext-real ) Real) > 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V61() V74() V75() V76() V77() V78() V79() V80() V81() ) Element of NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( Partial_Sums (Maclaurin (sin : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ,].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( V11() real ext-real ) Real) )) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Real_Sequence) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) is convergent & sin : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) = Sum (Maclaurin (sin : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ,].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( V11() real ext-real ) Real) )) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Real_Sequence) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) & Partial_Sums (Maclaurin (cos : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ,].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( V11() real ext-real ) Real) )) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Real_Sequence) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) is convergent & cos : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) = Sum (Maclaurin (cos : ( ( V21() V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Element of K6(K7(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ,REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ,].(- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ,r : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( open V75() V76() V77() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( V11() real ext-real ) Real) )) : ( ( V21() V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) -valued V21() total V30( NAT : ( ( ) ( V75() V76() V77() V78() V79() V80() V81() ) Element of K6(REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) V33() V34() V35() ) Real_Sequence) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V63() V75() V76() V77() V81() ) set ) ) ) ;