let Z be open Subset of REAL ; :: thesis: for f being PartFunc of REAL ,REAL st Z c= dom ((2 / 3) (#) ((#R (3 / 2)) * (f + exp_R ))) & ( for x being Real st x in Z holds
f . x = 1 ) holds
( (2 / 3) (#) ((#R (3 / 2)) * (f + exp_R )) is_differentiable_on Z & ( for x being Real st x in Z holds
(((2 / 3) (#) ((#R (3 / 2)) * (f + exp_R ))) `| Z) . x = (exp_R . x) * ((1 + (exp_R . x)) #R (1 / 2)) ) )
let f be PartFunc of REAL ,REAL ; :: thesis: ( Z c= dom ((2 / 3) (#) ((#R (3 / 2)) * (f + exp_R ))) & ( for x being Real st x in Z holds
f . x = 1 ) implies ( (2 / 3) (#) ((#R (3 / 2)) * (f + exp_R )) is_differentiable_on Z & ( for x being Real st x in Z holds
(((2 / 3) (#) ((#R (3 / 2)) * (f + exp_R ))) `| Z) . x = (exp_R . x) * ((1 + (exp_R . x)) #R (1 / 2)) ) ) )
assume that
A1: Z c= dom ((2 / 3) (#) ((#R (3 / 2)) * (f + exp_R ))) and
A2: for x being Real st x in Z holds
f . x = 1 ; :: thesis: ( (2 / 3) (#) ((#R (3 / 2)) * (f + exp_R )) is_differentiable_on Z & ( for x being Real st x in Z holds
(((2 / 3) (#) ((#R (3 / 2)) * (f + exp_R ))) `| Z) . x = (exp_R . x) * ((1 + (exp_R . x)) #R (1 / 2)) ) )
A3: for x being Real st x in Z holds
f . x = (0 * x) + 1 by A2;
A4: Z c= dom ((#R (3 / 2)) * (f + exp_R )) by A1, VALUED_1:def 5;
then for y being set st y in Z holds
y in dom (f + exp_R ) by FUNCT_1:21;
then A5: Z c= dom (f + exp_R ) by TARSKI:def 3;
then Z c= (dom exp_R ) /\ (dom f) by VALUED_1:def 1;
then A6: Z c= dom f by XBOOLE_1:18;
then A7: f is_differentiable_on Z by A3, FDIFF_1:31;
A8: exp_R is_differentiable_on Z by FDIFF_1:34, TAYLOR_1:16;
then A9: f + exp_R is_differentiable_on Z by A5, A7, FDIFF_1:26;
A10: for x being Real st x in Z holds
(f + exp_R ) . x > 0

proof

let x be Real; :: thesis: ( x in Z implies (f + exp_R ) . x > 0 )
assume A11: x in Z ; :: thesis: (f + exp_R ) . x > 0
then (f + exp_R ) . x = (f . x) + (exp_R . x) by A5, VALUED_1:def 1
.= 1 + (exp_R . x) by A2, A11 ;
hence (f + exp_R ) . x > 0 by SIN_COS:59, XREAL_1:36; :: thesis: verum

end;

now

let x be Real; :: thesis: ( x in Z implies (#R (3 / 2)) * (f + exp_R ) is_differentiable_in x )
assume x in Z ; :: thesis: (#R (3 / 2)) * (f + exp_R ) is_differentiable_in x
then ( f + exp_R is_differentiable_in x & (f + exp_R ) . x > 0 ) by A9, A10, FDIFF_1:16;
hence (#R (3 / 2)) * (f + exp_R ) is_differentiable_in x by TAYLOR_1:22; :: thesis: verum

end;

then A12: (#R (3 / 2)) * (f + exp_R ) is_differentiable_on Z by A4, FDIFF_1:16;
for x being Real st x in Z holds
(((2 / 3) (#) ((#R (3 / 2)) * (f + exp_R ))) `| Z) . x = (exp_R . x) * ((1 + (exp_R . x)) #R (1 / 2))

proof

let x be Real; :: thesis: ( x in Z implies (((2 / 3) (#) ((#R (3 / 2)) * (f + exp_R ))) `| Z) . x = (exp_R . x) * ((1 + (exp_R . x)) #R (1 / 2)) )
assume A13: x in Z ; :: thesis: (((2 / 3) (#) ((#R (3 / 2)) * (f + exp_R ))) `| Z) . x = (exp_R . x) * ((1 + (exp_R . x)) #R (1 / 2))
then A14: ((f + exp_R ) `| Z) . x = (diff f,x) + (diff exp_R ,x) by A5, A7, A8, FDIFF_1:26
.= (diff f,x) + (exp_R . x) by SIN_COS:70
.= ((f `| Z) . x) + (exp_R . x) by A7, A13, FDIFF_1:def 8
.= 0 + (exp_R . x) by A6, A3, A13, FDIFF_1:31
.= exp_R . x ;
A15: ( f + exp_R is_differentiable_in x & (f + exp_R ) . x > 0 ) by A9, A10, A13, FDIFF_1:16;
A16: (f + exp_R ) . x = (f . x) + (exp_R . x) by A5, A13, VALUED_1:def 1
.= 1 + (exp_R . x) by A2, A13 ;
(((2 / 3) (#) ((#R (3 / 2)) * (f + exp_R ))) `| Z) . x = (2 / 3) * (diff ((#R (3 / 2)) * (f + exp_R )),x) by A1, A12, A13, FDIFF_1:28
.= (2 / 3) * (((3 / 2) * (((f + exp_R ) . x) #R ((3 / 2) - 1))) * (diff (f + exp_R ),x)) by A15, TAYLOR_1:22
.= (2 / 3) * (((3 / 2) * (((f + exp_R ) . x) #R ((3 / 2) - 1))) * (((f + exp_R ) `| Z) . x)) by A9, A13, FDIFF_1:def 8
.= (exp_R . x) * ((1 + (exp_R . x)) #R (1 / 2)) by A16, A14 ;
hence (((2 / 3) (#) ((#R (3 / 2)) * (f + exp_R ))) `| Z) . x = (exp_R . x) * ((1 + (exp_R . x)) #R (1 / 2)) ; :: thesis: verum

end;

hence ( (2 / 3) (#) ((#R (3 / 2)) * (f + exp_R )) is_differentiable_on Z & ( for x being Real st x in Z holds
(((2 / 3) (#) ((#R (3 / 2)) * (f + exp_R ))) `| Z) . x = (exp_R . x) * ((1 + (exp_R . x)) #R (1 / 2)) ) ) by A1, A12, FDIFF_1:28; :: thesis: verum