let S be RealNormSpace; :: thesis: for f being PartFunc of S,S

for r being Real

for p being Point of S

for Z being Subset of S st Z is open & Z c= dom f & ( for x being Point of S st x in Z holds

f /. x = (r * x) + p ) holds

( f is_differentiable_on Z & ( for x being Point of S st x in Z holds

(f `| Z) /. x = r * (FuncUnit S) ) )

let f be PartFunc of S,S; :: thesis: for r being Real

for p being Point of S

for Z being Subset of S st Z is open & Z c= dom f & ( for x being Point of S st x in Z holds

f /. x = (r * x) + p ) holds

( f is_differentiable_on Z & ( for x being Point of S st x in Z holds

(f `| Z) /. x = r * (FuncUnit S) ) )

let r be Real; :: thesis: for p being Point of S

for Z being Subset of S st Z is open & Z c= dom f & ( for x being Point of S st x in Z holds

f /. x = (r * x) + p ) holds

( f is_differentiable_on Z & ( for x being Point of S st x in Z holds

(f `| Z) /. x = r * (FuncUnit S) ) )

let p be Point of S; :: thesis: for Z being Subset of S st Z is open & Z c= dom f & ( for x being Point of S st x in Z holds

f /. x = (r * x) + p ) holds

( f is_differentiable_on Z & ( for x being Point of S st x in Z holds

(f `| Z) /. x = r * (FuncUnit S) ) )

let Z be Subset of S; :: thesis: ( Z is open & Z c= dom f & ( for x being Point of S st x in Z holds

f /. x = (r * x) + p ) implies ( f is_differentiable_on Z & ( for x being Point of S st x in Z holds

(f `| Z) /. x = r * (FuncUnit S) ) ) )

assume A1: Z is open ; :: thesis: ( not Z c= dom f or ex x being Point of S st

( x in Z & not f /. x = (r * x) + p ) or ( f is_differentiable_on Z & ( for x being Point of S st x in Z holds

(f `| Z) /. x = r * (FuncUnit S) ) ) )

A2: R_NormSpace_of_BoundedLinearOperators (S,S) = NORMSTR(# (BoundedLinearOperators (S,S)),(Zero_ ((BoundedLinearOperators (S,S)),(R_VectorSpace_of_LinearOperators (S,S)))),(Add_ ((BoundedLinearOperators (S,S)),(R_VectorSpace_of_LinearOperators (S,S)))),(Mult_ ((BoundedLinearOperators (S,S)),(R_VectorSpace_of_LinearOperators (S,S)))),(BoundedLinearOperatorsNorm (S,S)) #) by LOPBAN_1:def 14;

then reconsider II = FuncUnit S as Point of (R_NormSpace_of_BoundedLinearOperators (S,S)) ;

set L = r * II;

reconsider L = r * II as Point of (R_NormSpace_of_BoundedLinearOperators (S,S)) ;

reconsider R = the carrier of S --> (0. S) as PartFunc of S,S ;

assume that

A3: Z c= dom f and

A4: for x being Point of S st x in Z holds

f /. x = (r * x) + p ; :: thesis: ( f is_differentiable_on Z & ( for x being Point of S st x in Z holds

(f `| Z) /. x = r * (FuncUnit S) ) )

A5: L = r * (FuncUnit S) by A2, LOPBAN_2:def 3;

A6: dom R = the carrier of S ;

(f `| Z) /. x = r * (FuncUnit S)

let x0 be Point of S; :: thesis: ( x0 in Z implies (f `| Z) /. x0 = r * (FuncUnit S) )

assume A19: x0 in Z ; :: thesis: (f `| Z) /. x0 = r * (FuncUnit S)

then consider N being Neighbourhood of x0 such that

A20: N c= Z by A1, Th2;

A21: for x being Point of S st x in N holds

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

A25: f is_differentiable_in x0 by A12, A19;

thus (f `| Z) /. x0 = diff (f,x0) by A18, A19, Def9

.= r * (FuncUnit S) by A5, A25, A24, A21, Def7 ; :: thesis: verum

for r being Real

for p being Point of S

for Z being Subset of S st Z is open & Z c= dom f & ( for x being Point of S st x in Z holds

f /. x = (r * x) + p ) holds

( f is_differentiable_on Z & ( for x being Point of S st x in Z holds

(f `| Z) /. x = r * (FuncUnit S) ) )

let f be PartFunc of S,S; :: thesis: for r being Real

for p being Point of S

for Z being Subset of S st Z is open & Z c= dom f & ( for x being Point of S st x in Z holds

f /. x = (r * x) + p ) holds

( f is_differentiable_on Z & ( for x being Point of S st x in Z holds

(f `| Z) /. x = r * (FuncUnit S) ) )

let r be Real; :: thesis: for p being Point of S

for Z being Subset of S st Z is open & Z c= dom f & ( for x being Point of S st x in Z holds

f /. x = (r * x) + p ) holds

( f is_differentiable_on Z & ( for x being Point of S st x in Z holds

(f `| Z) /. x = r * (FuncUnit S) ) )

let p be Point of S; :: thesis: for Z being Subset of S st Z is open & Z c= dom f & ( for x being Point of S st x in Z holds

f /. x = (r * x) + p ) holds

( f is_differentiable_on Z & ( for x being Point of S st x in Z holds

(f `| Z) /. x = r * (FuncUnit S) ) )

let Z be Subset of S; :: thesis: ( Z is open & Z c= dom f & ( for x being Point of S st x in Z holds

f /. x = (r * x) + p ) implies ( f is_differentiable_on Z & ( for x being Point of S st x in Z holds

(f `| Z) /. x = r * (FuncUnit S) ) ) )

assume A1: Z is open ; :: thesis: ( not Z c= dom f or ex x being Point of S st

( x in Z & not f /. x = (r * x) + p ) or ( f is_differentiable_on Z & ( for x being Point of S st x in Z holds

(f `| Z) /. x = r * (FuncUnit S) ) ) )

A2: R_NormSpace_of_BoundedLinearOperators (S,S) = NORMSTR(# (BoundedLinearOperators (S,S)),(Zero_ ((BoundedLinearOperators (S,S)),(R_VectorSpace_of_LinearOperators (S,S)))),(Add_ ((BoundedLinearOperators (S,S)),(R_VectorSpace_of_LinearOperators (S,S)))),(Mult_ ((BoundedLinearOperators (S,S)),(R_VectorSpace_of_LinearOperators (S,S)))),(BoundedLinearOperatorsNorm (S,S)) #) by LOPBAN_1:def 14;

then reconsider II = FuncUnit S as Point of (R_NormSpace_of_BoundedLinearOperators (S,S)) ;

set L = r * II;

reconsider L = r * II as Point of (R_NormSpace_of_BoundedLinearOperators (S,S)) ;

reconsider R = the carrier of S --> (0. S) as PartFunc of S,S ;

assume that

A3: Z c= dom f and

A4: for x being Point of S st x in Z holds

f /. x = (r * x) + p ; :: thesis: ( f is_differentiable_on Z & ( for x being Point of S st x in Z holds

(f `| Z) /. x = r * (FuncUnit S) ) )

A5: L = r * (FuncUnit S) by A2, LOPBAN_2:def 3;

A6: dom R = the carrier of S ;

now :: thesis: for h being 0. S -convergent sequence of S st h is non-zero holds

( (||.h.|| ") (#) (R /* h) is convergent & lim ((||.h.|| ") (#) (R /* h)) = 0. S )

then reconsider R = R as RestFunc of S,S by Def5;( (||.h.|| ") (#) (R /* h) is convergent & lim ((||.h.|| ") (#) (R /* h)) = 0. S )

let h be 0. S -convergent sequence of S; :: thesis: ( h is non-zero implies ( (||.h.|| ") (#) (R /* h) is convergent & lim ((||.h.|| ") (#) (R /* h)) = 0. S ) )

assume h is non-zero ; :: thesis: ( (||.h.|| ") (#) (R /* h) is convergent & lim ((||.h.|| ") (#) (R /* h)) = 0. S )

hence (||.h.|| ") (#) (R /* h) is convergent by Th18; :: thesis: lim ((||.h.|| ") (#) (R /* h)) = 0. S

((||.h.|| ") (#) (R /* h)) . 0 = 0. S by A7;

hence lim ((||.h.|| ") (#) (R /* h)) = 0. S by A11, Th18; :: thesis: verum

end;assume h is non-zero ; :: thesis: ( (||.h.|| ") (#) (R /* h) is convergent & lim ((||.h.|| ") (#) (R /* h)) = 0. S )

A7: now :: thesis: for n being Nat holds ((||.h.|| ") (#) (R /* h)) . n = 0. S

then A11:
(||.h.|| ") (#) (R /* h) is constant
by VALUED_0:def 18;let n be Nat; :: thesis: ((||.h.|| ") (#) (R /* h)) . n = 0. S

A8: R /. (h . n) = R . (h . n) by A6, PARTFUN1:def 6

.= 0. S ;

A9: rng h c= dom R ;

A10: n in NAT by ORDINAL1:def 12;

thus ((||.h.|| ") (#) (R /* h)) . n = ((||.h.|| ") . n) * ((R /* h) . n) by Def2

.= ((||.h.|| ") . n) * (R /. (h . n)) by A10, A9, FUNCT_2:109

.= 0. S by A8, RLVECT_1:10 ; :: thesis: verum

end;A8: R /. (h . n) = R . (h . n) by A6, PARTFUN1:def 6

.= 0. S ;

A9: rng h c= dom R ;

A10: n in NAT by ORDINAL1:def 12;

thus ((||.h.|| ") (#) (R /* h)) . n = ((||.h.|| ") . n) * ((R /* h) . n) by Def2

.= ((||.h.|| ") . n) * (R /. (h . n)) by A10, A9, FUNCT_2:109

.= 0. S by A8, RLVECT_1:10 ; :: thesis: verum

hence (||.h.|| ") (#) (R /* h) is convergent by Th18; :: thesis: lim ((||.h.|| ") (#) (R /* h)) = 0. S

((||.h.|| ") (#) (R /* h)) . 0 = 0. S by A7;

hence lim ((||.h.|| ") (#) (R /* h)) = 0. S by A11, Th18; :: thesis: verum

A12: now :: thesis: for x0 being Point of S st x0 in Z holds

f is_differentiable_in x0

hence A18:
f is_differentiable_on Z
by A1, A3, Th31; :: thesis: for x being Point of S st x in Z holds f is_differentiable_in x0

let x0 be Point of S; :: thesis: ( x0 in Z implies f is_differentiable_in x0 )

assume A13: x0 in Z ; :: thesis: f is_differentiable_in x0

then consider N being Neighbourhood of x0 such that

A14: N c= Z by A1, Th2;

A15: for x being Point of S st x in N holds

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

hence f is_differentiable_in x0 by A15; :: thesis: verum

end;assume A13: x0 in Z ; :: thesis: f is_differentiable_in x0

then consider N being Neighbourhood of x0 such that

A14: N c= Z by A1, Th2;

A15: for x being Point of S st x in N holds

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

proof

N c= dom f
by A3, A14;
let x be Point of S; :: thesis: ( x in N implies (f /. x) - (f /. x0) = (L . (x - x0)) + (R /. (x - x0)) )

A16: R /. (x - x0) = R . (x - x0) by A6, PARTFUN1:def 6

.= 0. S ;

x - x0 = (id the carrier of S) . (x - x0) ;

then A17: r * (x - x0) = r * ((FuncUnit S) . (x - x0)) by LOPBAN_2:def 5

.= L . (x - x0) by LOPBAN_1:36 ;

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

hence (f /. x) - (f /. x0) = ((r * x) + p) - (f /. x0) by A4, A14

.= ((r * x) + p) - ((r * x0) + p) by A4, A13

.= (r * x) + (p - ((r * x0) + p)) by RLVECT_1:def 3

.= (r * x) + ((p - (r * x0)) - p) by RLVECT_1:27

.= (r * x) + ((p + (- (r * x0))) - p)

.= (r * x) + ((- (r * x0)) + (p - p)) by RLVECT_1:def 3

.= (r * x) + ((- (r * x0)) + (0. S)) by RLVECT_1:15

.= (r * x) - (r * x0) by RLVECT_1:4

.= r * (x - x0) by RLVECT_1:34

.= (L . (x - x0)) + (R /. (x - x0)) by A16, A17, RLVECT_1:4 ;

:: thesis: verum

end;A16: R /. (x - x0) = R . (x - x0) by A6, PARTFUN1:def 6

.= 0. S ;

x - x0 = (id the carrier of S) . (x - x0) ;

then A17: r * (x - x0) = r * ((FuncUnit S) . (x - x0)) by LOPBAN_2:def 5

.= L . (x - x0) by LOPBAN_1:36 ;

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

hence (f /. x) - (f /. x0) = ((r * x) + p) - (f /. x0) by A4, A14

.= ((r * x) + p) - ((r * x0) + p) by A4, A13

.= (r * x) + (p - ((r * x0) + p)) by RLVECT_1:def 3

.= (r * x) + ((p - (r * x0)) - p) by RLVECT_1:27

.= (r * x) + ((p + (- (r * x0))) - p)

.= (r * x) + ((- (r * x0)) + (p - p)) by RLVECT_1:def 3

.= (r * x) + ((- (r * x0)) + (0. S)) by RLVECT_1:15

.= (r * x) - (r * x0) by RLVECT_1:4

.= r * (x - x0) by RLVECT_1:34

.= (L . (x - x0)) + (R /. (x - x0)) by A16, A17, RLVECT_1:4 ;

:: thesis: verum

hence f is_differentiable_in x0 by A15; :: thesis: verum

(f `| Z) /. x = r * (FuncUnit S)

let x0 be Point of S; :: thesis: ( x0 in Z implies (f `| Z) /. x0 = r * (FuncUnit S) )

assume A19: x0 in Z ; :: thesis: (f `| Z) /. x0 = r * (FuncUnit S)

then consider N being Neighbourhood of x0 such that

A20: N c= Z by A1, Th2;

A21: for x being Point of S st x in N holds

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

proof

A24:
N c= dom f
by A3, A20;
let x be Point of S; :: thesis: ( x in N implies (f /. x) - (f /. x0) = (L . (x - x0)) + (R /. (x - x0)) )

A22: R /. (x - x0) = R . (x - x0) by A6, PARTFUN1:def 6

.= 0. S ;

x - x0 = (id the carrier of S) . (x - x0) ;

then A23: r * (x - x0) = r * ((FuncUnit S) . (x - x0)) by LOPBAN_2:def 5

.= L . (x - x0) by LOPBAN_1:36 ;

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

hence (f /. x) - (f /. x0) = ((r * x) + p) - (f /. x0) by A4, A20

.= ((r * x) + p) - ((r * x0) + p) by A4, A19

.= (r * x) + (p - ((r * x0) + p)) by RLVECT_1:def 3

.= (r * x) + ((p - (r * x0)) - p) by RLVECT_1:27

.= (r * x) + ((p + (- (r * x0))) - p)

.= (r * x) + ((- (r * x0)) + (p - p)) by RLVECT_1:def 3

.= (r * x) + ((- (r * x0)) + (0. S)) by RLVECT_1:15

.= (r * x) - (r * x0) by RLVECT_1:4

.= r * (x - x0) by RLVECT_1:34

.= (L . (x - x0)) + (R /. (x - x0)) by A22, A23, RLVECT_1:4 ;

:: thesis: verum

end;A22: R /. (x - x0) = R . (x - x0) by A6, PARTFUN1:def 6

.= 0. S ;

x - x0 = (id the carrier of S) . (x - x0) ;

then A23: r * (x - x0) = r * ((FuncUnit S) . (x - x0)) by LOPBAN_2:def 5

.= L . (x - x0) by LOPBAN_1:36 ;

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

hence (f /. x) - (f /. x0) = ((r * x) + p) - (f /. x0) by A4, A20

.= ((r * x) + p) - ((r * x0) + p) by A4, A19

.= (r * x) + (p - ((r * x0) + p)) by RLVECT_1:def 3

.= (r * x) + ((p - (r * x0)) - p) by RLVECT_1:27

.= (r * x) + ((p + (- (r * x0))) - p)

.= (r * x) + ((- (r * x0)) + (p - p)) by RLVECT_1:def 3

.= (r * x) + ((- (r * x0)) + (0. S)) by RLVECT_1:15

.= (r * x) - (r * x0) by RLVECT_1:4

.= r * (x - x0) by RLVECT_1:34

.= (L . (x - x0)) + (R /. (x - x0)) by A22, A23, RLVECT_1:4 ;

:: thesis: verum

A25: f is_differentiable_in x0 by A12, A19;

thus (f `| Z) /. x0 = diff (f,x0) by A18, A19, Def9

.= r * (FuncUnit S) by A5, A25, A24, A21, Def7 ; :: thesis: verum