let Z be open Subset of REAL ; :: thesis: ( Z c= dom (exp_R * arctan ) & Z c= ].(- 1),1.[ implies ( exp_R * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R * arctan ) `| Z) . x = (exp_R . (arctan . x)) / (1 + (x ^2 )) ) ) )
assume that
A1: Z c= dom (exp_R * arctan ) and
A2: Z c= ].(- 1),1.[ ; :: thesis: ( exp_R * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R * arctan ) `| Z) . x = (exp_R . (arctan . x)) / (1 + (x ^2 )) ) )
A3: for x being Real st x in Z holds
exp_R * arctan is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies exp_R * arctan is_differentiable_in x )
assume A4: x in Z ; :: thesis: exp_R * arctan is_differentiable_in x
arctan is_differentiable_on Z by A2, Th81;
then A5: arctan is_differentiable_in x by A4, FDIFF_1:16;
exp_R is_differentiable_in arctan . x by SIN_COS:70;
hence exp_R * arctan is_differentiable_in x by A5, FDIFF_2:13; :: thesis: verum

end;

then A6: exp_R * arctan is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
((exp_R * arctan ) `| Z) . x = (exp_R . (arctan . x)) / (1 + (x ^2 ))

proof

let x be Real; :: thesis: ( x in Z implies ((exp_R * arctan ) `| Z) . x = (exp_R . (arctan . x)) / (1 + (x ^2 )) )
assume A7: x in Z ; :: thesis: ((exp_R * arctan ) `| Z) . x = (exp_R . (arctan . x)) / (1 + (x ^2 ))
A8: exp_R is_differentiable_in arctan . x by SIN_COS:70;
A9: arctan is_differentiable_on Z by A2, Th81;
then arctan is_differentiable_in x by A7, FDIFF_1:16;
then diff (exp_R * arctan ),x = (diff exp_R ,(arctan . x)) * (diff arctan ,x) by A8, FDIFF_2:13
.= (diff exp_R ,(arctan . x)) * ((arctan `| Z) . x) by A7, A9, FDIFF_1:def 8
.= (diff exp_R ,(arctan . x)) * (1 / (1 + (x ^2 ))) by A2, A7, Th81
.= (exp_R . (arctan . x)) / (1 + (x ^2 )) by SIN_COS:70 ;
hence ((exp_R * arctan ) `| Z) . x = (exp_R . (arctan . x)) / (1 + (x ^2 )) by A6, A7, FDIFF_1:def 8; :: thesis: verum

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 . (arctan . x)) / (1 + (x ^2 )) ) ) by A1, A3, FDIFF_1:16; :: thesis: verum