:: DIFF_1 semantic presentation

begin

definition
let f be ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ;
let h be ( ( real ) ( V11() real ext-real ) number ) ;
func Shift (f,h) -> ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) means :: DIFF_1:def 1
( dom it : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (f : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,(PFuncs (f : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) )) : ( ( ) ( non empty functional ) set ) :] : ( ( ) ( V13() ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty V57() V58() V59() V60() V61() V62() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) = (- h : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (f : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,(PFuncs (f : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) )) : ( ( ) ( non empty functional ) set ) :] : ( ( ) ( V13() ) set ) : ( ( ) ( ) set ) ) ) : ( ( V11() ) ( V11() ) set ) ++ (dom f : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( V57() V58() V59() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( V57() V58() V59() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in (- h : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (f : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,(PFuncs (f : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) )) : ( ( ) ( non empty functional ) set ) :] : ( ( ) ( V13() ) set ) : ( ( ) ( ) set ) ) ) : ( ( V11() ) ( V11() ) set ) ++ (dom f : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( V57() V58() V59() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( V57() V58() V59() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) holds
it : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (f : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,(PFuncs (f : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) )) : ( ( ) ( non empty functional ) set ) :] : ( ( ) ( V13() ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) = f : ( ( non empty ) ( non empty ) set ) . (x : ( ( ) ( V11() real ext-real ) Real) + h : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (f : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,(PFuncs (f : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) )) : ( ( ) ( non empty functional ) set ) :] : ( ( ) ( V13() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) );
end;

definition
let f be ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ;
let h be ( ( real ) ( V11() real ext-real ) number ) ;
:: original: Shift
redefine func Shift (f,h) -> ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) means :: DIFF_1:def 2
 for x being ( ( ) ( V11() real ext-real ) Real) holds it : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (f : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,(PFuncs (f : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) )) : ( ( ) ( non empty functional ) set ) :] : ( ( ) ( V13() ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) = f : ( ( non empty ) ( non empty ) set ) . (x : ( ( ) ( V11() real ext-real ) Real) + h : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (f : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,(PFuncs (f : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) )) : ( ( ) ( non empty functional ) set ) :] : ( ( ) ( V13() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ;
end;

definition
let f be ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ;
let h be ( ( real ) ( V11() real ext-real ) number ) ;
func fD (f,h) -> ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) equals :: DIFF_1:def 3
(Shift (f : ( ( non empty ) ( non empty ) set ) ,h : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (f : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,(PFuncs (f : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) )) : ( ( ) ( non empty functional ) set ) :] : ( ( ) ( V13() ) set ) : ( ( ) ( ) set ) ) )) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) - f : ( ( non empty ) ( non empty ) set ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) ;
end;

registration
let f be ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ;
let h be ( ( real ) ( V11() real ext-real ) number ) ;
cluster fD (f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) ,h : ( ( real ) ( V11() real ext-real ) set ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) -> Function-like quasi_total ;
end;

definition
let f be ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ;
let h be ( ( real ) ( V11() real ext-real ) number ) ;
func bD (f,h) -> ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) equals :: DIFF_1:def 4
f : ( ( ) ( ) set ) - (Shift (f : ( ( ) ( ) set ) ,(- h : ( ( ) ( ) set ) ) : ( ( V11() ) ( V11() ) set ) )) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) ;
end;

registration
let f be ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ;
let h be ( ( real ) ( V11() real ext-real ) number ) ;
cluster bD (f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) ,h : ( ( real ) ( V11() real ext-real ) set ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) -> Function-like quasi_total ;
end;

definition
let f be ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ;
let h be ( ( real ) ( V11() real ext-real ) number ) ;
func cD (f,h) -> ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) equals :: DIFF_1:def 5
(Shift (f : ( ( ) ( ) set ) ,(h : ( ( ) ( ) set ) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) )) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) - (Shift (f : ( ( ) ( ) set ) ,(- (h : ( ( ) ( ) set ) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) )) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) ;
end;

registration
let f be ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ;
let h be ( ( real ) ( V11() real ext-real ) number ) ;
cluster cD (f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) ,h : ( ( real ) ( V11() real ext-real ) set ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) -> Function-like quasi_total ;
end;

definition
let f be ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ;
let h be ( ( real ) ( V11() real ext-real ) number ) ;
func forward_difference (f,h) -> ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) means :: DIFF_1:def 6
( it : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (f : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,(PFuncs (f : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) )) : ( ( ) ( non empty functional ) set ) :] : ( ( ) ( V13() ) set ) : ( ( ) ( ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real V13() non-empty empty-yielding V17( RAT : ( ( ) ( non empty V32() V57() V58() V59() V60() V63() ) set ) ) V30() ext-real complex-valued ext-real-valued real-valued natural-valued V56() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) = f : ( ( ) ( ) set ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real ) Nat) holds it : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (f : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,(PFuncs (f : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) )) : ( ( ) ( non empty functional ) set ) :] : ( ( ) ( V13() ) set ) : ( ( ) ( ) set ) ) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real ) Nat) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) = fD ((it : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (f : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,(PFuncs (f : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) )) : ( ( ) ( non empty functional ) set ) :] : ( ( ) ( V13() ) set ) : ( ( ) ( ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real ) Nat) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ) );
end;

notation
let f be ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ;
let h be ( ( real ) ( V11() real ext-real ) number ) ;
synonym fdif (f,h) for forward_difference (f,h);
end;

theorem :: DIFF_1:1
for x, h being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) st x : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( V57() V58() V59() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) & x : ( ( ) ( V11() real ext-real ) Real) + h : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) in dom f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( V57() V58() V59() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) holds
(fD (f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) = (f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) . (x : ( ( ) ( V11() real ext-real ) Real) + h : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) - (f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ;

theorem :: DIFF_1:2
for h being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) )
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real ) Nat) holds (fdif (f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real ) Nat) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) is ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ;

theorem :: DIFF_1:3
for h, x being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) holds (fD (f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) = (f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (x : ( ( ) ( V11() real ext-real ) Real) + h : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) - (f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ;

theorem :: DIFF_1:4
for h, x being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) holds (bD (f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) = (f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) - (f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (x : ( ( ) ( V11() real ext-real ) Real) - h : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ;

theorem :: DIFF_1:5
for h, x being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) holds (cD (f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) = (f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (x : ( ( ) ( V11() real ext-real ) Real) + (h : ( ( ) ( V11() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) - (f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (x : ( ( ) ( V11() real ext-real ) Real) - (h : ( ( ) ( V11() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ;

theorem :: DIFF_1:6
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) )
for h being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) st f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) is constant holds
for x being ( ( ) ( V11() real ext-real ) Real) holds ((fdif (f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) = 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real V13() non-empty empty-yielding V17( RAT : ( ( ) ( non empty V32() V57() V58() V59() V60() V63() ) set ) ) V30() ext-real complex-valued ext-real-valued real-valued natural-valued V56() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ;

theorem :: DIFF_1:7
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) )
for r, h, x being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) holds ((fdif ((r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( Function-like ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) = r : ( ( ) ( V11() real ext-real ) Real) * (((fdif (f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ;

theorem :: DIFF_1:8
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) )
for h, x being ( ( ) ( V11() real ext-real ) Real)
for f1, f2 being ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) holds ((fdif ((f1 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) + f2 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( Function-like ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) = (((fdif (f1 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) + (((fdif (f2 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ;

theorem :: DIFF_1:9
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) )
for h, x being ( ( ) ( V11() real ext-real ) Real)
for f1, f2 being ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) holds ((fdif ((f1 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) - f2 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( Function-like ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) = (((fdif (f1 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) - (((fdif (f2 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ;

theorem :: DIFF_1:10
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) )
for r1, r2, h, x being ( ( ) ( V11() real ext-real ) Real)
for f1, f2 being ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) holds ((fdif (((r1 : ( ( ) ( V11() real ext-real ) Real) (#) f1 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( Function-like ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) + (r2 : ( ( ) ( V11() real ext-real ) Real) (#) f2 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( Function-like ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) = (r1 : ( ( ) ( V11() real ext-real ) Real) * (((fdif (f1 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) + (r2 : ( ( ) ( V11() real ext-real ) Real) * (((fdif (f2 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ;

theorem :: DIFF_1:11
for h, x being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) holds ((fdif (f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) = ((Shift (f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) - (f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ;

definition
let f be ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ;
let h be ( ( real ) ( V11() real ext-real ) number ) ;
func backward_difference (f,h) -> ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) means :: DIFF_1:def 7
( it : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (f : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,(PFuncs (f : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) )) : ( ( ) ( non empty functional ) set ) :] : ( ( ) ( V13() ) set ) : ( ( ) ( ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real V13() non-empty empty-yielding V17( RAT : ( ( ) ( non empty V32() V57() V58() V59() V60() V63() ) set ) ) V30() ext-real complex-valued ext-real-valued real-valued natural-valued V56() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) = f : ( ( ) ( ) set ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real ) Nat) holds it : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (f : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,(PFuncs (f : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) )) : ( ( ) ( non empty functional ) set ) :] : ( ( ) ( V13() ) set ) : ( ( ) ( ) set ) ) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real ) Nat) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) = bD ((it : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (f : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,(PFuncs (f : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) )) : ( ( ) ( non empty functional ) set ) :] : ( ( ) ( V13() ) set ) : ( ( ) ( ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real ) Nat) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ) );
end;

notation
let f be ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ;
let h be ( ( real ) ( V11() real ext-real ) number ) ;
synonym bdif (f,h) for backward_difference (f,h);
end;

theorem :: DIFF_1:12
for h being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) )
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real ) Nat) holds (bdif (f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real ) Nat) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) is ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ;

theorem :: DIFF_1:13
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) )
for h being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) st f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) is constant holds
for x being ( ( ) ( V11() real ext-real ) Real) holds ((bdif (f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) = 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real V13() non-empty empty-yielding V17( RAT : ( ( ) ( non empty V32() V57() V58() V59() V60() V63() ) set ) ) V30() ext-real complex-valued ext-real-valued real-valued natural-valued V56() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ;

theorem :: DIFF_1:14
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) )
for r, h, x being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) holds ((bdif ((r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( Function-like ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) = r : ( ( ) ( V11() real ext-real ) Real) * (((bdif (f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ;

theorem :: DIFF_1:15
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) )
for h, x being ( ( ) ( V11() real ext-real ) Real)
for f1, f2 being ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) holds ((bdif ((f1 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) + f2 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( Function-like ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) = (((bdif (f1 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) + (((bdif (f2 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ;

theorem :: DIFF_1:16
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) )
for h, x being ( ( ) ( V11() real ext-real ) Real)
for f1, f2 being ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) holds ((bdif ((f1 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) - f2 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( Function-like ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) = (((bdif (f1 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) - (((bdif (f2 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ;

theorem :: DIFF_1:17
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) )
for r1, r2, h, x being ( ( ) ( V11() real ext-real ) Real)
for f1, f2 being ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) holds ((bdif (((r1 : ( ( ) ( V11() real ext-real ) Real) (#) f1 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( Function-like ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) + (r2 : ( ( ) ( V11() real ext-real ) Real) (#) f2 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( Function-like ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) = (r1 : ( ( ) ( V11() real ext-real ) Real) * (((bdif (f1 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) + (r2 : ( ( ) ( V11() real ext-real ) Real) * (((bdif (f2 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ;

theorem :: DIFF_1:18
for h, x being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) holds ((bdif (f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) = (f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) - ((Shift (f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,(- h : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ;

definition
let f be ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ;
let h be ( ( real ) ( V11() real ext-real ) number ) ;
func central_difference (f,h) -> ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) means :: DIFF_1:def 8
( it : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (f : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,(PFuncs (f : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) )) : ( ( ) ( non empty functional ) set ) :] : ( ( ) ( V13() ) set ) : ( ( ) ( ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real V13() non-empty empty-yielding V17( RAT : ( ( ) ( non empty V32() V57() V58() V59() V60() V63() ) set ) ) V30() ext-real complex-valued ext-real-valued real-valued natural-valued V56() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) = f : ( ( ) ( ) set ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real ) Nat) holds it : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (f : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,(PFuncs (f : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) )) : ( ( ) ( non empty functional ) set ) :] : ( ( ) ( V13() ) set ) : ( ( ) ( ) set ) ) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real ) Nat) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) = cD ((it : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (f : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,(PFuncs (f : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) )) : ( ( ) ( non empty functional ) set ) :] : ( ( ) ( V13() ) set ) : ( ( ) ( ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real ) Nat) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ) );
end;

notation
let f be ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ;
let h be ( ( real ) ( V11() real ext-real ) number ) ;
synonym cdif (f,h) for central_difference (f,h);
end;

theorem :: DIFF_1:19
for h being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) )
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real ) Nat) holds (cdif (f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real ) Nat) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) is ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ;

theorem :: DIFF_1:20
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) )
for h being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) st f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) is constant holds
for x being ( ( ) ( V11() real ext-real ) Real) holds ((cdif (f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) = 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real V13() non-empty empty-yielding V17( RAT : ( ( ) ( non empty V32() V57() V58() V59() V60() V63() ) set ) ) V30() ext-real complex-valued ext-real-valued real-valued natural-valued V56() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ;

theorem :: DIFF_1:21
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) )
for r, h, x being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) holds ((cdif ((r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( Function-like ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) = r : ( ( ) ( V11() real ext-real ) Real) * (((cdif (f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ;

theorem :: DIFF_1:22
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) )
for h, x being ( ( ) ( V11() real ext-real ) Real)
for f1, f2 being ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) holds ((cdif ((f1 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) + f2 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( Function-like ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) = (((cdif (f1 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) + (((cdif (f2 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ;

theorem :: DIFF_1:23
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) )
for h, x being ( ( ) ( V11() real ext-real ) Real)
for f1, f2 being ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) holds ((cdif ((f1 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) - f2 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( Function-like ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) = (((cdif (f1 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) - (((cdif (f2 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ;

theorem :: DIFF_1:24
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) )
for r1, r2, h, x being ( ( ) ( V11() real ext-real ) Real)
for f1, f2 being ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) holds ((cdif (((r1 : ( ( ) ( V11() real ext-real ) Real) (#) f1 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( Function-like ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) + (r2 : ( ( ) ( V11() real ext-real ) Real) (#) f2 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( Function-like ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) = (r1 : ( ( ) ( V11() real ext-real ) Real) * (((cdif (f1 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) + (r2 : ( ( ) ( V11() real ext-real ) Real) * (((cdif (f2 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ;

theorem :: DIFF_1:25
for h, x being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) holds ((cdif (f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) = ((Shift (f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,(h : ( ( ) ( V11() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) - ((Shift (f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,(- (h : ( ( ) ( V11() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ;

theorem :: DIFF_1:26
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) )
for h, x being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) holds ((fdif (f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) = ((bdif (f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V11() real ext-real ) Real) + (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) * h : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ;

theorem :: DIFF_1:27
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) )
for h, x being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) holds ((fdif (f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) = ((cdif (f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V11() real ext-real ) Real) + (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) * h : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ;

theorem :: DIFF_1:28
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) )
for h, x being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) holds ((fdif (f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) = ((cdif (f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . ((x : ( ( ) ( V11() real ext-real ) Real) + (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) * h : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) + (h : ( ( ) ( V11() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ;

definition
let f be ( ( V13() Function-like real-valued ) ( V13() Function-like complex-valued ext-real-valued real-valued ) Function) ;
let x0, x1 be ( ( real ) ( V11() real ext-real ) number ) ;
func [!f,x0,x1!] -> ( ( ) ( V11() real ext-real ) Real) equals :: DIFF_1:def 9
((f : ( ( ) ( ) set ) . x0 : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) - (f : ( ( ) ( ) set ) . x1 : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (f : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,(PFuncs (f : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) )) : ( ( ) ( non empty functional ) set ) :] : ( ( ) ( V13() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) / (x0 : ( ( ) ( ) set ) - x1 : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (f : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,(PFuncs (f : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) )) : ( ( ) ( non empty functional ) set ) :] : ( ( ) ( V13() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ;
end;

definition
let f be ( ( V13() Function-like real-valued ) ( V13() Function-like complex-valued ext-real-valued real-valued ) Function) ;
let x0, x1, x2 be ( ( real ) ( V11() real ext-real ) number ) ;
func [!f,x0,x1,x2!] -> ( ( ) ( V11() real ext-real ) Real) equals :: DIFF_1:def 10
([!f : ( ( ) ( ) set ) ,x0 : ( ( ) ( ) set ) ,x1 : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (f : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,(PFuncs (f : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) )) : ( ( ) ( non empty functional ) set ) :] : ( ( ) ( V13() ) set ) : ( ( ) ( ) set ) ) !] : ( ( ) ( V11() real ext-real ) Real) - [!f : ( ( ) ( ) set ) ,x1 : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (f : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,(PFuncs (f : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) )) : ( ( ) ( non empty functional ) set ) :] : ( ( ) ( V13() ) set ) : ( ( ) ( ) set ) ) ,x2 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real ) set ) !] : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) / (x0 : ( ( ) ( ) set ) - x2 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ;
end;

definition
let f be ( ( V13() Function-like real-valued ) ( V13() Function-like complex-valued ext-real-valued real-valued ) Function) ;
let x0, x1, x2, x3 be ( ( real ) ( V11() real ext-real ) number ) ;
func [!f,x0,x1,x2,x3!] -> ( ( ) ( V11() real ext-real ) Real) equals :: DIFF_1:def 11
([!f : ( ( ) ( ) set ) ,x0 : ( ( ) ( ) set ) ,x1 : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (f : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,(PFuncs (f : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) )) : ( ( ) ( non empty functional ) set ) :] : ( ( ) ( V13() ) set ) : ( ( ) ( ) set ) ) ,x2 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real ) set ) !] : ( ( ) ( V11() real ext-real ) Real) - [!f : ( ( ) ( ) set ) ,x1 : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (f : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,(PFuncs (f : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) )) : ( ( ) ( non empty functional ) set ) :] : ( ( ) ( V13() ) set ) : ( ( ) ( ) set ) ) ,x2 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real ) set ) ,x3 : ( ( ) ( V13() V16(f : ( ( ) ( ) set ) ) V17(x0 : ( ( ) ( ) set ) ) ) Element of bool [:f : ( ( ) ( ) set ) ,x0 : ( ( ) ( ) set ) :] : ( ( ) ( V13() ) set ) : ( ( ) ( ) set ) ) !] : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) / (x0 : ( ( ) ( ) set ) - x3 : ( ( ) ( V13() V16(f : ( ( ) ( ) set ) ) V17(x0 : ( ( ) ( ) set ) ) ) Element of bool [:f : ( ( ) ( ) set ) ,x0 : ( ( ) ( ) set ) :] : ( ( ) ( V13() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ;
end;

theorem :: DIFF_1:29
for x0, x1 being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) holds [!f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) ,x1 : ( ( ) ( V11() real ext-real ) Real) !] : ( ( ) ( V11() real ext-real ) Real) = [!f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,x1 : ( ( ) ( V11() real ext-real ) Real) ,x0 : ( ( ) ( V11() real ext-real ) Real) !] : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: DIFF_1:30
for x0, x1 being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) st f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) is constant holds
[!f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) ,x1 : ( ( ) ( V11() real ext-real ) Real) !] : ( ( ) ( V11() real ext-real ) Real) = 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real V13() non-empty empty-yielding V17( RAT : ( ( ) ( non empty V32() V57() V58() V59() V60() V63() ) set ) ) V30() ext-real complex-valued ext-real-valued real-valued natural-valued V56() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ;

theorem :: DIFF_1:31
for r, x0, x1 being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) holds [!(r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( Function-like ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) ,x1 : ( ( ) ( V11() real ext-real ) Real) !] : ( ( ) ( V11() real ext-real ) Real) = r : ( ( ) ( V11() real ext-real ) Real) * [!f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) ,x1 : ( ( ) ( V11() real ext-real ) Real) !] : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ;

theorem :: DIFF_1:32
for x0, x1 being ( ( ) ( V11() real ext-real ) Real)
for f1, f2 being ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) holds [!(f1 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) + f2 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( Function-like ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) ,x1 : ( ( ) ( V11() real ext-real ) Real) !] : ( ( ) ( V11() real ext-real ) Real) = [!f1 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) ,x1 : ( ( ) ( V11() real ext-real ) Real) !] : ( ( ) ( V11() real ext-real ) Real) + [!f2 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) ,x1 : ( ( ) ( V11() real ext-real ) Real) !] : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ;

theorem :: DIFF_1:33
for r1, r2, x0, x1 being ( ( ) ( V11() real ext-real ) Real)
for f1, f2 being ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) holds [!((r1 : ( ( ) ( V11() real ext-real ) Real) (#) f1 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( Function-like ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) + (r2 : ( ( ) ( V11() real ext-real ) Real) (#) f2 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( Function-like ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) ,x1 : ( ( ) ( V11() real ext-real ) Real) !] : ( ( ) ( V11() real ext-real ) Real) = (r1 : ( ( ) ( V11() real ext-real ) Real) * [!f1 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) ,x1 : ( ( ) ( V11() real ext-real ) Real) !] : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) + (r2 : ( ( ) ( V11() real ext-real ) Real) * [!f2 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) ,x1 : ( ( ) ( V11() real ext-real ) Real) !] : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ;

theorem :: DIFF_1:34
for x0, x1, x2 being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) st x0 : ( ( ) ( V11() real ext-real ) Real) ,x1 : ( ( ) ( V11() real ext-real ) Real) ,x2 : ( ( ) ( V11() real ext-real ) Real) are_mutually_different holds
( [!f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) ,x1 : ( ( ) ( V11() real ext-real ) Real) ,x2 : ( ( ) ( V11() real ext-real ) Real) !] : ( ( ) ( V11() real ext-real ) Real) = [!f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,x1 : ( ( ) ( V11() real ext-real ) Real) ,x2 : ( ( ) ( V11() real ext-real ) Real) ,x0 : ( ( ) ( V11() real ext-real ) Real) !] : ( ( ) ( V11() real ext-real ) Real) & [!f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) ,x1 : ( ( ) ( V11() real ext-real ) Real) ,x2 : ( ( ) ( V11() real ext-real ) Real) !] : ( ( ) ( V11() real ext-real ) Real) = [!f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,x2 : ( ( ) ( V11() real ext-real ) Real) ,x1 : ( ( ) ( V11() real ext-real ) Real) ,x0 : ( ( ) ( V11() real ext-real ) Real) !] : ( ( ) ( V11() real ext-real ) Real) ) ;

theorem :: DIFF_1:35
for x0, x1, x2 being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) st x0 : ( ( ) ( V11() real ext-real ) Real) ,x1 : ( ( ) ( V11() real ext-real ) Real) ,x2 : ( ( ) ( V11() real ext-real ) Real) are_mutually_different holds
( [!f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) ,x1 : ( ( ) ( V11() real ext-real ) Real) ,x2 : ( ( ) ( V11() real ext-real ) Real) !] : ( ( ) ( V11() real ext-real ) Real) = [!f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,x2 : ( ( ) ( V11() real ext-real ) Real) ,x0 : ( ( ) ( V11() real ext-real ) Real) ,x1 : ( ( ) ( V11() real ext-real ) Real) !] : ( ( ) ( V11() real ext-real ) Real) & [!f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) ,x1 : ( ( ) ( V11() real ext-real ) Real) ,x2 : ( ( ) ( V11() real ext-real ) Real) !] : ( ( ) ( V11() real ext-real ) Real) = [!f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,x1 : ( ( ) ( V11() real ext-real ) Real) ,x0 : ( ( ) ( V11() real ext-real ) Real) ,x2 : ( ( ) ( V11() real ext-real ) Real) !] : ( ( ) ( V11() real ext-real ) Real) ) ;

theorem :: DIFF_1:36
for m, n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) )
for h, x being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) holds ((fdif (((fdif (f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) = ((fdif (f : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ;

definition
let S be ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ;
attr S is Sequence-yielding means :: DIFF_1:def 12
 for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) holds S : ( ( ) ( ) set ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) is ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ;
end;

registration
cluster non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total Sequence-yielding for ( ( ) ( ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) )) : ( ( ) ( non empty functional ) set ) :] : ( ( ) ( V13() ) set ) : ( ( ) ( ) set ) ) ;
end;

definition
mode Seq_Sequence is ( ( Function-like quasi_total Sequence-yielding ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total Sequence-yielding ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ;
end;

definition
let S be ( ( Function-like quasi_total Sequence-yielding ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total Sequence-yielding ) Seq_Sequence) ;
let n be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ;
:: original: .
redefine func S . n -> ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ;
end;

theorem :: DIFF_1:37
for h, x being ( ( ) ( V11() real ext-real ) Real)
for f1, f2 being ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) )
for S being ( ( Function-like quasi_total Sequence-yielding ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total Sequence-yielding ) Seq_Sequence) st ( for n, i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) st i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) holds
(S : ( ( Function-like quasi_total Sequence-yielding ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total Sequence-yielding ) Seq_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) . i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) = ((n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) choose i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) * (((fdif (f1 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) * (((fdif (f2 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) -' i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real non negative V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V11() real ext-real ) Real) + (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) * h : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) holds
( ((fdif ((f1 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) (#) f2 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( Function-like ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) = Sum ((S : ( ( Function-like quasi_total Sequence-yielding ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total Sequence-yielding ) Seq_Sequence) . 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) & ((fdif ((f1 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) (#) f2 : ( ( Function-like quasi_total ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( Function-like ) ( non empty V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) . 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) :] : ( ( ) ( V13() complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) = Sum ((S : ( ( Function-like quasi_total Sequence-yielding ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( PFuncs (REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( non empty functional ) set ) ) Function-like total quasi_total Sequence-yielding ) Seq_Sequence) . 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like quasi_total ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ,2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) ;

theorem :: DIFF_1:38
for x, h being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) st x : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( V57() V58() V59() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) & x : ( ( ) ( V11() real ext-real ) Real) - h : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) in dom f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( V57() V58() V59() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) holds
(bD (f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) = (f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) - (f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) . (x : ( ( ) ( V11() real ext-real ) Real) - h : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ;

theorem :: DIFF_1:39
for x, h being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) st x : ( ( ) ( V11() real ext-real ) Real) + (h : ( ( ) ( V11() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) in dom f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( V57() V58() V59() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) & x : ( ( ) ( V11() real ext-real ) Real) - (h : ( ( ) ( V11() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) in dom f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( V57() V58() V59() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) holds
(cD (f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,h : ( ( ) ( V11() real ext-real ) Real) )) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) = (f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) . (x : ( ( ) ( V11() real ext-real ) Real) + (h : ( ( ) ( V11() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) - (f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) . (x : ( ( ) ( V11() real ext-real ) Real) - (h : ( ( ) ( V11() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real V30() ext-real positive V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V57() V58() V59() V63() ) set ) ) ;