let a be Real; :: thesis: for Z being open Subset of REAL

for f being PartFunc of REAL,REAL st Z c= dom (2 (#) ((#R (1 / 2)) * f)) & ( for x being Real st x in Z holds

( f . x = a + x & f . x > 0 ) ) holds

( 2 (#) ((#R (1 / 2)) * f) is_differentiable_on Z & ( for x being Real st x in Z holds

((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = (a + x) #R (- (1 / 2)) ) )

let Z be open Subset of REAL; :: thesis: for f being PartFunc of REAL,REAL st Z c= dom (2 (#) ((#R (1 / 2)) * f)) & ( for x being Real st x in Z holds

( f . x = a + x & f . x > 0 ) ) holds

( 2 (#) ((#R (1 / 2)) * f) is_differentiable_on Z & ( for x being Real st x in Z holds

((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = (a + x) #R (- (1 / 2)) ) )

let f be PartFunc of REAL,REAL; :: thesis: ( Z c= dom (2 (#) ((#R (1 / 2)) * f)) & ( for x being Real st x in Z holds

( f . x = a + x & f . x > 0 ) ) implies ( 2 (#) ((#R (1 / 2)) * f) is_differentiable_on Z & ( for x being Real st x in Z holds

((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = (a + x) #R (- (1 / 2)) ) ) )

assume that

A1: Z c= dom (2 (#) ((#R (1 / 2)) * f)) and

A2: for x being Real st x in Z holds

( f . x = a + x & f . x > 0 ) ; :: thesis: ( 2 (#) ((#R (1 / 2)) * f) is_differentiable_on Z & ( for x being Real st x in Z holds

((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = (a + x) #R (- (1 / 2)) ) )

A3: Z c= dom ((#R (1 / 2)) * f) by A1, VALUED_1:def 5;

then for y being object st y in Z holds

y in dom f by FUNCT_1:11;

then A4: Z c= dom f by TARSKI:def 3;

A5: for x being Real st x in Z holds

f . x = (1 * x) + a by A2;

then A6: f is_differentiable_on Z by A4, FDIFF_1:23;

for x being Real st x in Z holds

((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = (a + x) #R (- (1 / 2))

((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = (a + x) #R (- (1 / 2)) ) ) by A1, A7, FDIFF_1:20; :: thesis: verum

for f being PartFunc of REAL,REAL st Z c= dom (2 (#) ((#R (1 / 2)) * f)) & ( for x being Real st x in Z holds

( f . x = a + x & f . x > 0 ) ) holds

( 2 (#) ((#R (1 / 2)) * f) is_differentiable_on Z & ( for x being Real st x in Z holds

((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = (a + x) #R (- (1 / 2)) ) )

let Z be open Subset of REAL; :: thesis: for f being PartFunc of REAL,REAL st Z c= dom (2 (#) ((#R (1 / 2)) * f)) & ( for x being Real st x in Z holds

( f . x = a + x & f . x > 0 ) ) holds

( 2 (#) ((#R (1 / 2)) * f) is_differentiable_on Z & ( for x being Real st x in Z holds

((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = (a + x) #R (- (1 / 2)) ) )

let f be PartFunc of REAL,REAL; :: thesis: ( Z c= dom (2 (#) ((#R (1 / 2)) * f)) & ( for x being Real st x in Z holds

( f . x = a + x & f . x > 0 ) ) implies ( 2 (#) ((#R (1 / 2)) * f) is_differentiable_on Z & ( for x being Real st x in Z holds

((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = (a + x) #R (- (1 / 2)) ) ) )

assume that

A1: Z c= dom (2 (#) ((#R (1 / 2)) * f)) and

A2: for x being Real st x in Z holds

( f . x = a + x & f . x > 0 ) ; :: thesis: ( 2 (#) ((#R (1 / 2)) * f) is_differentiable_on Z & ( for x being Real st x in Z holds

((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = (a + x) #R (- (1 / 2)) ) )

A3: Z c= dom ((#R (1 / 2)) * f) by A1, VALUED_1:def 5;

then for y being object st y in Z holds

y in dom f by FUNCT_1:11;

then A4: Z c= dom f by TARSKI:def 3;

A5: for x being Real st x in Z holds

f . x = (1 * x) + a by A2;

then A6: f is_differentiable_on Z by A4, FDIFF_1:23;

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

(#R (1 / 2)) * f is_differentiable_in x

then A7:
(#R (1 / 2)) * f is_differentiable_on Z
by A3, FDIFF_1:9;(#R (1 / 2)) * f is_differentiable_in x

let x be Real; :: thesis: ( x in Z implies (#R (1 / 2)) * f is_differentiable_in x )

assume x in Z ; :: thesis: (#R (1 / 2)) * f is_differentiable_in x

then ( f is_differentiable_in x & f . x > 0 ) by A2, A6, FDIFF_1:9;

hence (#R (1 / 2)) * f is_differentiable_in x by TAYLOR_1:22; :: thesis: verum

end;assume x in Z ; :: thesis: (#R (1 / 2)) * f is_differentiable_in x

then ( f is_differentiable_in x & f . x > 0 ) by A2, A6, FDIFF_1:9;

hence (#R (1 / 2)) * f is_differentiable_in x by TAYLOR_1:22; :: thesis: verum

for x being Real st x in Z holds

((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = (a + x) #R (- (1 / 2))

proof

hence
( 2 (#) ((#R (1 / 2)) * f) is_differentiable_on Z & ( for x being Real st x in Z holds
let x be Real; :: thesis: ( x in Z implies ((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = (a + x) #R (- (1 / 2)) )

assume A8: x in Z ; :: thesis: ((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = (a + x) #R (- (1 / 2))

then A9: f . x = a + x by A2;

A10: ( f is_differentiable_in x & f . x > 0 ) by A2, A6, A8, FDIFF_1:9;

((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = 2 * (diff (((#R (1 / 2)) * f),x)) by A1, A7, A8, FDIFF_1:20

.= 2 * (((1 / 2) * ((f . x) #R ((1 / 2) - 1))) * (diff (f,x))) by A10, TAYLOR_1:22

.= 2 * (((1 / 2) * ((f . x) #R ((1 / 2) - 1))) * ((f `| Z) . x)) by A6, A8, FDIFF_1:def 7

.= 2 * (((1 / 2) * ((a + x) #R ((1 / 2) - 1))) * 1) by A4, A5, A8, A9, FDIFF_1:23

.= (a + x) #R (- (1 / 2)) ;

hence ((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = (a + x) #R (- (1 / 2)) ; :: thesis: verum

end;assume A8: x in Z ; :: thesis: ((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = (a + x) #R (- (1 / 2))

then A9: f . x = a + x by A2;

A10: ( f is_differentiable_in x & f . x > 0 ) by A2, A6, A8, FDIFF_1:9;

((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = 2 * (diff (((#R (1 / 2)) * f),x)) by A1, A7, A8, FDIFF_1:20

.= 2 * (((1 / 2) * ((f . x) #R ((1 / 2) - 1))) * (diff (f,x))) by A10, TAYLOR_1:22

.= 2 * (((1 / 2) * ((f . x) #R ((1 / 2) - 1))) * ((f `| Z) . x)) by A6, A8, FDIFF_1:def 7

.= 2 * (((1 / 2) * ((a + x) #R ((1 / 2) - 1))) * 1) by A4, A5, A8, A9, FDIFF_1:23

.= (a + x) #R (- (1 / 2)) ;

hence ((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = (a + x) #R (- (1 / 2)) ; :: thesis: verum

((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = (a + x) #R (- (1 / 2)) ) ) by A1, A7, FDIFF_1:20; :: thesis: verum