let Z be open Subset of REAL; :: thesis: for f being PartFunc of REAL,REAL st Z c= dom f & f | Z = id Z holds

( f is_differentiable_on Z & ( for x being Real st x in Z holds

(f `| Z) . x = 1 ) )

let f be PartFunc of REAL,REAL; :: thesis: ( Z c= dom f & f | Z = id Z implies ( f is_differentiable_on Z & ( for x being Real st x in Z holds

(f `| Z) . x = 1 ) ) )

reconsider j = 1 as Element of REAL by XREAL_0:def 1;

reconsider cf = REAL --> (In (0,REAL)) as Function of REAL,REAL ;

set R = cf;

reconsider L = id REAL as PartFunc of REAL,REAL ;

for p being Real holds L . p = 1 * p by XREAL_0:def 1, FUNCT_1:18;

then reconsider L = L as LinearFunc by Def3;

assume that

A7: Z c= dom f and

A8: f | Z = id Z ; :: thesis: ( f is_differentiable_on Z & ( for x being Real st x in Z holds

(f `| Z) . x = 1 ) )

(f `| Z) . x = 1

let x0 be Real; :: thesis: ( x0 in Z implies (f `| Z) . x0 = 1 )

assume A16: x0 in Z ; :: thesis: (f `| Z) . x0 = 1

then consider N1 being Neighbourhood of x0 such that

A17: N1 c= Z by RCOMP_1:18;

A18: f is_differentiable_in x0 by A11, A16;

then ex N being Neighbourhood of x0 st

( N c= dom f & ex L being LinearFunc ex R being RestFunc st

for x being Real st x in N holds

(f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0)) ) ;

then consider N being Neighbourhood of x0 such that

A19: N c= dom f ;

consider N2 being Neighbourhood of x0 such that

A20: N2 c= N1 and

A21: N2 c= N by RCOMP_1:17;

A22: N2 c= dom f by A19, A21;

A23: for x being Real st x in N2 holds

(f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0))

.= L . j by A18, A22, A23, Def5

.= 1 ; :: thesis: verum

( f is_differentiable_on Z & ( for x being Real st x in Z holds

(f `| Z) . x = 1 ) )

let f be PartFunc of REAL,REAL; :: thesis: ( Z c= dom f & f | Z = id Z implies ( f is_differentiable_on Z & ( for x being Real st x in Z holds

(f `| Z) . x = 1 ) ) )

reconsider j = 1 as Element of REAL by XREAL_0:def 1;

reconsider cf = REAL --> (In (0,REAL)) as Function of REAL,REAL ;

set R = cf;

now :: thesis: for h being non-zero 0 -convergent Real_Sequence holds

( (h ") (#) (cf /* h) is convergent & lim ((h ") (#) (cf /* h)) = 0 )

then reconsider R = cf as RestFunc by Def2;( (h ") (#) (cf /* h) is convergent & lim ((h ") (#) (cf /* h)) = 0 )

let h be non-zero 0 -convergent Real_Sequence; :: thesis: ( (h ") (#) (cf /* h) is convergent & lim ((h ") (#) (cf /* h)) = 0 )

hence (h ") (#) (cf /* h) is convergent ; :: thesis: lim ((h ") (#) (cf /* h)) = 0

((h ") (#) (cf /* h)) . 0 = 0 by A2;

hence lim ((h ") (#) (cf /* h)) = 0 by A6, SEQ_4:25; :: thesis: verum

end;A2: now :: thesis: for n being Nat holds ((h ") (#) (cf /* h)) . n = 0

then A6:
(h ") (#) (cf /* h) is V8()
by VALUED_0:def 18;let n be Nat; :: thesis: ((h ") (#) (cf /* h)) . n = 0

A3: rng h c= dom cf ;

A5: n in NAT by ORDINAL1:def 12;

thus ((h ") (#) (cf /* h)) . n = ((h ") . n) * ((cf /* h) . n) by SEQ_1:8

.= ((h ") . n) * (cf . (h . n)) by A5, A3, FUNCT_2:108

.= ((h ") . n) * 0

.= 0 ; :: thesis: verum

end;A3: rng h c= dom cf ;

A5: n in NAT by ORDINAL1:def 12;

thus ((h ") (#) (cf /* h)) . n = ((h ") . n) * ((cf /* h) . n) by SEQ_1:8

.= ((h ") . n) * (cf . (h . n)) by A5, A3, FUNCT_2:108

.= ((h ") . n) * 0

.= 0 ; :: thesis: verum

hence (h ") (#) (cf /* h) is convergent ; :: thesis: lim ((h ") (#) (cf /* h)) = 0

((h ") (#) (cf /* h)) . 0 = 0 by A2;

hence lim ((h ") (#) (cf /* h)) = 0 by A6, SEQ_4:25; :: thesis: verum

reconsider L = id REAL as PartFunc of REAL,REAL ;

for p being Real holds L . p = 1 * p by XREAL_0:def 1, FUNCT_1:18;

then reconsider L = L as LinearFunc by Def3;

assume that

A7: Z c= dom f and

A8: f | Z = id Z ; :: thesis: ( f is_differentiable_on Z & ( for x being Real st x in Z holds

(f `| Z) . x = 1 ) )

A9: now :: thesis: for x being Real st x in Z holds

f . x = x

f . x = x

let x be Real; :: thesis: ( x in Z implies f . x = x )

assume A10: x in Z ; :: thesis: f . x = x

then (f | Z) . x = x by A8, FUNCT_1:18;

hence f . x = x by A10, FUNCT_1:49; :: thesis: verum

end;assume A10: x in Z ; :: thesis: f . x = x

then (f | Z) . x = x by A8, FUNCT_1:18;

hence f . x = x by A10, FUNCT_1:49; :: thesis: verum

A11: now :: thesis: for x0 being Real st x0 in Z holds

f is_differentiable_in x0

hence A15:
f is_differentiable_on Z
by A7, Th9; :: thesis: for x being Real st x in Z holds f is_differentiable_in x0

let x0 be Real; :: thesis: ( x0 in Z implies f is_differentiable_in x0 )

assume A12: x0 in Z ; :: thesis: f is_differentiable_in x0

then consider N being Neighbourhood of x0 such that

A13: N c= Z by RCOMP_1:18;

A14: for x being Real st x in N holds

(f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0))

hence f is_differentiable_in x0 by A14; :: thesis: verum

end;assume A12: x0 in Z ; :: thesis: f is_differentiable_in x0

then consider N being Neighbourhood of x0 such that

A13: N c= Z by RCOMP_1:18;

A14: for x being Real st x in N holds

(f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0))

proof

N c= dom f
by A7, A13;
let x be Real; :: thesis: ( x in N implies (f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0)) )

reconsider xx = x, xx0 = x0 as Element of REAL by XREAL_0:def 1;

assume x in N ; :: thesis: (f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0))

hence (f . x) - (f . x0) = x - (f . x0) by A9, A13

.= x - x0 by A9, A12

.= (L . (xx - xx0)) + 0

.= (L . (x - x0)) + (R . (x - x0)) ;

:: thesis: verum

end;reconsider xx = x, xx0 = x0 as Element of REAL by XREAL_0:def 1;

assume x in N ; :: thesis: (f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0))

hence (f . x) - (f . x0) = x - (f . x0) by A9, A13

.= x - x0 by A9, A12

.= (L . (xx - xx0)) + 0

.= (L . (x - x0)) + (R . (x - x0)) ;

:: thesis: verum

hence f is_differentiable_in x0 by A14; :: thesis: verum

(f `| Z) . x = 1

let x0 be Real; :: thesis: ( x0 in Z implies (f `| Z) . x0 = 1 )

assume A16: x0 in Z ; :: thesis: (f `| Z) . x0 = 1

then consider N1 being Neighbourhood of x0 such that

A17: N1 c= Z by RCOMP_1:18;

A18: f is_differentiable_in x0 by A11, A16;

then ex N being Neighbourhood of x0 st

( N c= dom f & ex L being LinearFunc ex R being RestFunc st

for x being Real st x in N holds

(f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0)) ) ;

then consider N being Neighbourhood of x0 such that

A19: N c= dom f ;

consider N2 being Neighbourhood of x0 such that

A20: N2 c= N1 and

A21: N2 c= N by RCOMP_1:17;

A22: N2 c= dom f by A19, A21;

A23: for x being Real st x in N2 holds

(f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0))

proof

thus (f `| Z) . x0 =
diff (f,x0)
by A15, A16, Def7
let x be Real; :: thesis: ( x in N2 implies (f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0)) )

reconsider xx = x, xx0 = x0 as Element of REAL by XREAL_0:def 1;

assume x in N2 ; :: thesis: (f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0))

then x in N1 by A20;

hence (f . x) - (f . x0) = x - (f . x0) by A9, A17

.= x - x0 by A9, A16

.= (L . (xx - xx0)) + 0

.= (L . (x - x0)) + (R . (x - x0)) ;

:: thesis: verum

end;reconsider xx = x, xx0 = x0 as Element of REAL by XREAL_0:def 1;

assume x in N2 ; :: thesis: (f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0))

then x in N1 by A20;

hence (f . x) - (f . x0) = x - (f . x0) by A9, A17

.= x - x0 by A9, A16

.= (L . (xx - xx0)) + 0

.= (L . (x - x0)) + (R . (x - x0)) ;

:: thesis: verum

.= L . j by A18, A22, A23, Def5

.= 1 ; :: thesis: verum