let r, p be Real; :: thesis: for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom f & ( for x being Real st x in Z holds
f . x = (r * x) + p ) holds
( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = r ) )
let Z be open Subset of REAL; :: thesis: for f being PartFunc of REAL,REAL st Z c= dom f & ( for x being Real st x in Z holds
f . x = (r * x) + p ) holds
( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = r ) )
let f be PartFunc of REAL,REAL; :: thesis: ( Z c= dom f & ( for x being Real st x in Z holds
f . x = (r * x) + p ) implies ( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = r ) ) )
reconsider cf = REAL --> (In (0,REAL)) as Function of REAL,REAL ;
set R = cf;
defpred S₁[ set ] means $1 in REAL ;

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

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

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;

then A6: (h ") (#) (cf /* h) is constant by VALUED_0:def 18;
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;

then reconsider R = cf as RestFunc by Def2;
assume that
A7: Z c= dom f and
A8: for x being Real st x in Z holds
f . x = (r * x) + p ; :: thesis: ( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = r ) )
deffunc H₁( Real) -> Element of REAL = In ((r * $1),REAL);
consider L being PartFunc of REAL,REAL such that
A9: ( ( for x being Element of REAL holds
( x in dom L iff S₁[x] ) ) & ( for x being Element of REAL st x in dom L holds
L . x = H₁(x) ) ) from SEQ_1:sch 3();
for r being Real holds
( r in dom L iff r in REAL ) by A9;
then dom L = REAL by Th1;
then A10: L is total by PARTFUN1:def 2;
A11: for x being Real holds L . x = r * x

proof

let x be Real; :: thesis: L . x = r * x
reconsider x = x as Element of REAL by XREAL_0:def 1;
L . x = H₁(x) by A9;
hence L . x = r * x ; :: thesis: verum

end;

then reconsider L = L as LinearFunc by A10, Def3;

A12: now :: thesis: for x0 being Real st x0 in Z holds
f is_differentiable_in x0

let x0 be Real; :: 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 RCOMP_1:18;
A15: for x being Real st x in N holds
(f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0))

proof

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) = ((r * x) + p) - (f . x0) by A8, A14
.= ((r * x) + p) - ((r * x0) + p) by A8, A13
.= (r * (x - x0)) + 0
.= (L . (xx - xx0)) + 0 by A11
.= (L . (x - x0)) + (R . (x - x0)) ;
:: thesis: verum

end;

N c= dom f by A7, A14;
hence f is_differentiable_in x0 by A15; :: thesis: verum

end;

hence A16: f is_differentiable_on Z by A7, Th9; :: thesis: for x being Real st x in Z holds
(f `| Z) . x = r
let x0 be Real; :: thesis: ( x0 in Z implies (f `| Z) . x0 = r )
assume A17: x0 in Z ; :: thesis: (f `| Z) . x0 = r
then consider N being Neighbourhood of x0 such that
A18: N c= Z by RCOMP_1:18;
A19: for x being Real st x in N holds
(f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0))

proof

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) = ((r * x) + p) - (f . x0) by A8, A18
.= ((r * x) + p) - ((r * x0) + p) by A8, A17
.= (r * (x - x0)) + 0
.= (L . (xx - xx0)) + 0 by A11
.= (L . (x - x0)) + (R . (x - x0)) ;
:: thesis: verum

end;

A20: N c= dom f by A7, A18;
A21: f is_differentiable_in x0 by A12, A17;
thus (f `| Z) . x0 = diff (f,x0) by A16, A17, Def7
.= L . 1 by A21, A20, A19, Def5
.= r * 1 by A11
.= r ; :: thesis: verum