let a be Real; :: thesis:
let Z be open Subset of REAL; :: thesis:
let f, f1 be PartFunc of REAL,REAL; :: thesis:
assume that
A1: Z c= dom ((2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1)))) and
A2: for x being Real st x in Z holds
( f . x = 1 & f1 . x = x * (log (number_e,a)) ) and
A3: a > 0 and
A4: a <> 1 ; :: thesis:
A5: for x being Real st x in Z holds
f . x = (0 * x) + 1 by A2;
A6: Z c= dom ((#R (3 / 2)) * (f + (exp_R * f1))) by A1, VALUED_1:def 5;
then for y being object st y in Z holds
y in dom (f + (exp_R * f1)) by FUNCT_1:11;
then A7: Z c= dom (f + (exp_R * f1)) by TARSKI:def 3;
then A8: Z c= (dom (exp_R * f1)) /\ (dom f) by VALUED_1:def 1;
then A9: Z c= dom (exp_R * f1) by XBOOLE_1:18;
A10: for x being Real st x in Z holds
f1 . x = x * (log (number_e,a)) by A2;
then A11: exp_R * f1 is_differentiable_on Z by A3, A9, Th11;
A12: Z c= dom f by A8, XBOOLE_1:18;
then A13: f is_differentiable_on Z by A5, FDIFF_1:23;
then A14: f + (exp_R * f1) is_differentiable_on Z by A7, A11, FDIFF_1:18;
A15: for x being Real st x in Z holds
(f + (exp_R * f1)) . x > 0

proof

let x be Real; :: thesis:
assume A16: x in Z ; :: thesis:
then (f + (exp_R * f1)) . x = (f . x) + ((exp_R * f1) . x) by A7, VALUED_1:def 1
.= (f . x) + (exp_R . (f1 . x)) by A9, A16, FUNCT_1:12
.= 1 + (exp_R . (f1 . x)) by A2, A16
.= 1 + (exp_R . (x * (log (number_e,a)))) by A2, A16 ;
hence (f + (exp_R * f1)) . x > 0 by SIN_COS:54, XREAL_1:34; :: thesis:

end;

now :: thesis:

let x be Real; :: thesis:
assume x in Z ; :: thesis:
then ( f + (exp_R * f1) is_differentiable_in x & (f + (exp_R * f1)) . x > 0 ) by A14, A15, FDIFF_1:9;
hence (#R (3 / 2)) * (f + (exp_R * f1)) is_differentiable_in x by TAYLOR_1:22; :: thesis:

end;

then A17: (#R (3 / 2)) * (f + (exp_R * f1)) is_differentiable_on Z by A6, FDIFF_1:9;
A18: log (number_e,a) <> 0

proof

A19: number_e <> 1 by TAYLOR_1:11;
assume log (number_e,a) = 0 ; :: thesis:
then log (number_e,a) = log (number_e,1) by SIN_COS2:13, TAYLOR_1:13;
then a = number_e to_power (log (number_e,1)) by A3, A19, POWER:def 3, TAYLOR_1:11
.= 1 by A19, POWER:def 3, TAYLOR_1:11 ;
hence contradiction by A4; :: thesis:

end;

for x being Real st x in Z holds
(((2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1)))) `| Z) . x = (a #R x) * ((1 + (a #R x)) #R (1 / 2))

proof

let x be Real; :: thesis:
A20: 3 * (log (number_e,a)) <> 0 by A18;
assume A21: x in Z ; :: thesis:
then A22: ((f + (exp_R * f1)) `| Z) . x = (diff (f,x)) + (diff ((exp_R * f1),x)) by A7, A13, A11, FDIFF_1:18
.= (diff (f,x)) + (((exp_R * f1) `| Z) . x) by A11, A21, FDIFF_1:def 7
.= ((f `| Z) . x) + (((exp_R * f1) `| Z) . x) by A13, A21, FDIFF_1:def 7
.= 0 + (((exp_R * f1) `| Z) . x) by A12, A5, A21, FDIFF_1:23
.= (a #R x) * (log (number_e,a)) by A3, A10, A9, A21, Th11 ;
A23: (f + (exp_R * f1)) . x = (f . x) + ((exp_R * f1) . x) by A7, A21, VALUED_1:def 1
.= (f . x) + (exp_R . (f1 . x)) by A9, A21, FUNCT_1:12
.= 1 + (exp_R . (f1 . x)) by A2, A21
.= 1 + (exp_R . (x * (log (number_e,a)))) by A2, A21
.= 1 + (a #R x) by A3, Th1 ;
( f + (exp_R * f1) is_differentiable_in x & (f + (exp_R * f1)) . x > 0 ) by A14, A15, A21, FDIFF_1:9;
then diff (((#R (3 / 2)) * (f + (exp_R * f1))),x) = ((3 / 2) * (((f + (exp_R * f1)) . x) #R ((3 / 2) - 1))) * (diff ((f + (exp_R * f1)),x)) by TAYLOR_1:22
.= ((3 / 2) * ((1 + (a #R x)) #R (1 / 2))) * ((a #R x) * (log (number_e,a))) by A14, A21, A23, A22, FDIFF_1:def 7
.= (((3 * (log (number_e,a))) / 2) * (a #R x)) * ((1 + (a #R x)) #R (1 / 2)) ;
then (((2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1)))) `| Z) . x = (2 / (3 * (log (number_e,a)))) * ((((3 * (log (number_e,a))) / 2) * (a #R x)) * ((1 + (a #R x)) #R (1 / 2))) by A1, A17, A21, FDIFF_1:20
.= (((2 / (3 * (log (number_e,a)))) * ((3 * (log (number_e,a))) / 2)) * (a #R x)) * ((1 + (a #R x)) #R (1 / 2))
.= (1 * (a #R x)) * ((1 + (a #R x)) #R (1 / 2)) by A20, XCMPLX_1:112
.= (a #R x) * ((1 + (a #R x)) #R (1 / 2)) ;
hence (((2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1)))) `| Z) . x = (a #R x) * ((1 + (a #R x)) #R (1 / 2)) ; :: thesis:

end;

hence ( (2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1))) is_differentiable_on Z & ( for x being Real st x in Z holds
(((2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1)))) `| Z) . x = (a #R x) * ((1 + (a #R x)) #R (1 / 2)) ) ) by A1, A17, FDIFF_1:20; :: thesis: