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
f1 . x = x * (log number_e ,a) by A2;
A6: Z c= dom ((#R (3 / 2)) * (f + (exp_R * f1))) by A1, VALUED_1:def 5;
then for y being set st y in Z holds
y in dom (f + (exp_R * f1)) by FUNCT_1:21;
then A7: Z c= dom (f + (exp_R * f1)) by TARSKI:def 3;
then Z c= (dom (exp_R * f1)) /\ (dom f) by VALUED_1:def 1;
then A8: ( Z c= dom f & Z c= dom (exp_R * f1) ) by XBOOLE_1:18;
A9: for x being Real st x in Z holds
f . x = (0 * x) + 1 by A2;
then A10: ( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = 0 ) ) by A8, FDIFF_1:31;
A11: ( exp_R * f1 is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R * f1) `| Z) . x = (a #R x) * (log number_e ,a) ) ) by A3, A5, A8, Th11;
then A12: f + (exp_R * f1) is_differentiable_on Z by A7, A10, FDIFF_1:26;
A13: for x being Real st x in Z holds
(f + (exp_R * f1)) . x > 0

proof

let x be Real; :: thesis:
assume A14: 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 A8, A14, FUNCT_1:22
.= 1 + (exp_R . (f1 . x)) by A2, A14
.= 1 + (exp_R . (x * (log number_e ,a))) by A2, A14 ;
hence (f + (exp_R * f1)) . x > 0 by SIN_COS:59, XREAL_1:36; :: thesis:

end;

now

let x be Real; :: thesis:
assume A15: x in Z ; :: thesis:
then A16: f + (exp_R * f1) is_differentiable_in x by A12, FDIFF_1:16;
(f + (exp_R * f1)) . x > 0 by A13, A15;
hence (#R (3 / 2)) * (f + (exp_R * f1)) is_differentiable_in x by A16, TAYLOR_1:22; :: thesis:

end;

then A17: (#R (3 / 2)) * (f + (exp_R * f1)) is_differentiable_on Z by A6, FDIFF_1:16;
A18: log number_e ,a <> 0

proof

assume log number_e ,a = 0 ; :: thesis:
then A19: log number_e ,a = log number_e ,1 by SIN_COS2:13, TAYLOR_1:13;
A20: ( number_e > 0 & number_e <> 1 ) by TAYLOR_1:11;
then a = number_e to_power (log number_e ,1) by A3, A19, POWER:def 3
.= 1 by A20, POWER:def 3 ;
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:
assume A21: x in Z ; :: thesis:
then A22: f + (exp_R * f1) is_differentiable_in x by A12, FDIFF_1:16;
A23: (f + (exp_R * f1)) . x > 0 by A13, A21;
A24: (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 A8, A21, FUNCT_1:22
.= 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 ;
A25: ((f + (exp_R * f1)) `| Z) . x = (diff f,x) + (diff (exp_R * f1),x) by A7, A10, A11, A21, FDIFF_1:26
.= (diff f,x) + (((exp_R * f1) `| Z) . x) by A11, A21, FDIFF_1:def 8
.= ((f `| Z) . x) + (((exp_R * f1) `| Z) . x) by A10, A21, FDIFF_1:def 8
.= 0 + (((exp_R * f1) `| Z) . x) by A8, A9, A21, FDIFF_1:31
.= (a #R x) * (log number_e ,a) by A3, A5, A8, A21, Th11 ;
A26: 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 A22, A23, TAYLOR_1:22
.= ((3 / 2) * ((1 + (a #R x)) #R (1 / 2))) * ((a #R x) * (log number_e ,a)) by A12, A21, A24, A25, FDIFF_1:def 8
.= (((3 * (log number_e ,a)) / 2) * (a #R x)) * ((1 + (a #R x)) #R (1 / 2)) ;
A27: 3 * (log number_e ,a) <> 0 by A18;
(((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, A26, FDIFF_1:28
.= (((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 A27, XCMPLX_1:113
.= (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_1:28; :: thesis: