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

f . x = 1 ) holds

( (#Z 2) * (exp_R - f) is_differentiable_on Z & ( for x being Real st x in Z holds

(((#Z 2) * (exp_R - f)) `| Z) . x = (2 * (exp_R . x)) * ((exp_R . x) - 1) ) )

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

f . x = 1 ) implies ( (#Z 2) * (exp_R - f) is_differentiable_on Z & ( for x being Real st x in Z holds

(((#Z 2) * (exp_R - f)) `| Z) . x = (2 * (exp_R . x)) * ((exp_R . x) - 1) ) ) )

assume that

A1: Z c= dom ((#Z 2) * (exp_R - f)) and

A2: for x being Real st x in Z holds

f . x = 1 ; :: thesis: ( (#Z 2) * (exp_R - f) is_differentiable_on Z & ( for x being Real st x in Z holds

(((#Z 2) * (exp_R - f)) `| Z) . x = (2 * (exp_R . x)) * ((exp_R . x) - 1) ) )

for y being object st y in Z holds

y in dom (exp_R - f) by A1, FUNCT_1:11;

then A3: Z c= dom (exp_R - f) by TARSKI:def 3;

then Z c= (dom exp_R) /\ (dom f) by VALUED_1:12;

then A4: Z c= dom f by XBOOLE_1:18;

A5: for x being Real st x in Z holds

f . x = (0 * x) + 1 by A2;

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

A7: for x being Real st x in Z holds

(#Z 2) * (exp_R - f) is_differentiable_in x

for x being Real st x in Z holds

(((#Z 2) * (exp_R - f)) `| Z) . x = (2 * (exp_R . x)) * ((exp_R . x) - 1)

(((#Z 2) * (exp_R - f)) `| Z) . x = (2 * (exp_R . x)) * ((exp_R . x) - 1) ) ) by A1, A7, FDIFF_1:9; :: thesis: verum

f . x = 1 ) holds

( (#Z 2) * (exp_R - f) is_differentiable_on Z & ( for x being Real st x in Z holds

(((#Z 2) * (exp_R - f)) `| Z) . x = (2 * (exp_R . x)) * ((exp_R . x) - 1) ) )

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

f . x = 1 ) implies ( (#Z 2) * (exp_R - f) is_differentiable_on Z & ( for x being Real st x in Z holds

(((#Z 2) * (exp_R - f)) `| Z) . x = (2 * (exp_R . x)) * ((exp_R . x) - 1) ) ) )

assume that

A1: Z c= dom ((#Z 2) * (exp_R - f)) and

A2: for x being Real st x in Z holds

f . x = 1 ; :: thesis: ( (#Z 2) * (exp_R - f) is_differentiable_on Z & ( for x being Real st x in Z holds

(((#Z 2) * (exp_R - f)) `| Z) . x = (2 * (exp_R . x)) * ((exp_R . x) - 1) ) )

for y being object st y in Z holds

y in dom (exp_R - f) by A1, FUNCT_1:11;

then A3: Z c= dom (exp_R - f) by TARSKI:def 3;

then Z c= (dom exp_R) /\ (dom f) by VALUED_1:12;

then A4: Z c= dom f by XBOOLE_1:18;

A5: for x being Real st x in Z holds

f . x = (0 * x) + 1 by A2;

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

A7: for x being Real st x in Z holds

(#Z 2) * (exp_R - f) is_differentiable_in x

proof

then A9:
(#Z 2) * (exp_R - f) is_differentiable_on Z
by A1, FDIFF_1:9;
let x be Real; :: thesis: ( x in Z implies (#Z 2) * (exp_R - f) is_differentiable_in x )

assume x in Z ; :: thesis: (#Z 2) * (exp_R - f) is_differentiable_in x

then A8: f is_differentiable_in x by A6, FDIFF_1:9;

exp_R is_differentiable_in x by SIN_COS:65;

then exp_R - f is_differentiable_in x by A8, FDIFF_1:14;

hence (#Z 2) * (exp_R - f) is_differentiable_in x by TAYLOR_1:3; :: thesis: verum

end;assume x in Z ; :: thesis: (#Z 2) * (exp_R - f) is_differentiable_in x

then A8: f is_differentiable_in x by A6, FDIFF_1:9;

exp_R is_differentiable_in x by SIN_COS:65;

then exp_R - f is_differentiable_in x by A8, FDIFF_1:14;

hence (#Z 2) * (exp_R - f) is_differentiable_in x by TAYLOR_1:3; :: thesis: verum

for x being Real st x in Z holds

(((#Z 2) * (exp_R - f)) `| Z) . x = (2 * (exp_R . x)) * ((exp_R . x) - 1)

proof

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

A10: exp_R is_differentiable_in x by SIN_COS:65;

assume A11: x in Z ; :: thesis: (((#Z 2) * (exp_R - f)) `| Z) . x = (2 * (exp_R . x)) * ((exp_R . x) - 1)

then A12: (exp_R - f) . x = (exp_R . x) - (f . x) by A3, VALUED_1:13

.= (exp_R . x) - 1 by A2, A11 ;

A13: f is_differentiable_in x by A6, A11, FDIFF_1:9;

then A14: diff ((exp_R - f),x) = (diff (exp_R,x)) - (diff (f,x)) by A10, FDIFF_1:14

.= (diff (exp_R,x)) - ((f `| Z) . x) by A6, A11, FDIFF_1:def 7

.= (exp_R . x) - ((f `| Z) . x) by SIN_COS:65

.= (exp_R . x) - 0 by A4, A5, A11, FDIFF_1:23

.= exp_R . x ;

A15: exp_R - f is_differentiable_in x by A13, A10, FDIFF_1:14;

(((#Z 2) * (exp_R - f)) `| Z) . x = diff (((#Z 2) * (exp_R - f)),x) by A9, A11, FDIFF_1:def 7

.= (2 * (((exp_R - f) . x) #Z (2 - 1))) * (diff ((exp_R - f),x)) by A15, TAYLOR_1:3

.= (2 * ((exp_R . x) - 1)) * (exp_R . x) by A14, A12, PREPOWER:35

.= (2 * (exp_R . x)) * ((exp_R . x) - 1) ;

hence (((#Z 2) * (exp_R - f)) `| Z) . x = (2 * (exp_R . x)) * ((exp_R . x) - 1) ; :: thesis: verum

end;A10: exp_R is_differentiable_in x by SIN_COS:65;

assume A11: x in Z ; :: thesis: (((#Z 2) * (exp_R - f)) `| Z) . x = (2 * (exp_R . x)) * ((exp_R . x) - 1)

then A12: (exp_R - f) . x = (exp_R . x) - (f . x) by A3, VALUED_1:13

.= (exp_R . x) - 1 by A2, A11 ;

A13: f is_differentiable_in x by A6, A11, FDIFF_1:9;

then A14: diff ((exp_R - f),x) = (diff (exp_R,x)) - (diff (f,x)) by A10, FDIFF_1:14

.= (diff (exp_R,x)) - ((f `| Z) . x) by A6, A11, FDIFF_1:def 7

.= (exp_R . x) - ((f `| Z) . x) by SIN_COS:65

.= (exp_R . x) - 0 by A4, A5, A11, FDIFF_1:23

.= exp_R . x ;

A15: exp_R - f is_differentiable_in x by A13, A10, FDIFF_1:14;

(((#Z 2) * (exp_R - f)) `| Z) . x = diff (((#Z 2) * (exp_R - f)),x) by A9, A11, FDIFF_1:def 7

.= (2 * (((exp_R - f) . x) #Z (2 - 1))) * (diff ((exp_R - f),x)) by A15, TAYLOR_1:3

.= (2 * ((exp_R . x) - 1)) * (exp_R . x) by A14, A12, PREPOWER:35

.= (2 * (exp_R . x)) * ((exp_R . x) - 1) ;

hence (((#Z 2) * (exp_R - f)) `| Z) . x = (2 * (exp_R . x)) * ((exp_R . x) - 1) ; :: thesis: verum

(((#Z 2) * (exp_R - f)) `| Z) . x = (2 * (exp_R . x)) * ((exp_R . x) - 1) ) ) by A1, A7, FDIFF_1:9; :: thesis: verum