let Z be open Subset of REAL ; :: thesis:
assume A1: Z c= ].(- 1),1.[ ; :: thesis:
then A2: arctan is_differentiable_on Z by Th81;
].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then ].(- 1),1.[ c= dom arctan by Th23, XBOOLE_1:1;
then Z c= dom arctan by A1, XBOOLE_1:1;
then Z c= (dom exp_R ) /\ (dom arctan ) by SIN_COS:51, XBOOLE_1:19;
then A3: Z c= dom (exp_R (#) arctan ) by VALUED_1:def 4;
for x being Real st x in Z holds
exp_R is_differentiable_in x by SIN_COS:70;
then A4: exp_R is_differentiable_on Z by FDIFF_1:16, SIN_COS:51;
for x being Real st x in Z holds
((exp_R (#) arctan ) `| Z) . x = ((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2 )))

proof

let x be Real; :: thesis:
assume A5: x in Z ; :: thesis:
then ((exp_R (#) arctan ) `| Z) . x = ((arctan . x) * (diff exp_R ,x)) + ((exp_R . x) * (diff arctan ,x)) by A3, A4, A2, FDIFF_1:29
.= ((arctan . x) * (exp_R . x)) + ((exp_R . x) * (diff arctan ,x)) by SIN_COS:70
.= ((exp_R . x) * (arctan . x)) + ((exp_R . x) * ((arctan `| Z) . x)) by A2, A5, FDIFF_1:def 8
.= ((exp_R . x) * (arctan . x)) + ((exp_R . x) * (1 / (1 + (x ^2 )))) by A1, A5, Th81
.= ((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2 ))) ;
hence ((exp_R (#) arctan ) `| Z) . x = ((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2 ))) ; :: thesis:

end;

hence ( exp_R (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) arctan ) `| Z) . x = ((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2 ))) ) ) by A3, A4, A2, FDIFF_1:29; :: thesis: