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_1:9;
exp_R is_differentiable_in arctan . x by SIN_COS:65;
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_1:9;
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:65;
A9: arctan is_differentiable_on Z by A2, Th81;
then arctan is_differentiable_in x by A7, FDIFF_1:9;
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 7
.= (diff (exp_R,(arctan . x))) * (1 / (1 + (x ^2))) by A2, A7, Th81
.= (exp_R . (arctan . x)) / (1 + (x ^2)) by SIN_COS:65 ;
hence ((exp_R * arctan) `| Z) . x = (exp_R . (arctan . x)) / (1 + (x ^2)) by A6, A7, FDIFF_1:def 7; :: 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_1:9; :: thesis: verum