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

proof

let x be Real; :: thesis: ( x in Z implies arctan * exp_R is_differentiable_in x )
assume A4: x in Z ; :: thesis: arctan * exp_R is_differentiable_in x
(exp_R . x) + 0 > 0 + (- 1) by SIN_COS:59, XREAL_1:10;
then A5: ( exp_R . x > - 1 & exp_R . x < 1 ) by A2, A4;
exp_R is_differentiable_in x by SIN_COS:70;
hence arctan * exp_R is_differentiable_in x by A5, Th83; :: thesis: verum

end;

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

proof

let x be Real; :: thesis: ( x in Z implies ((arctan * exp_R ) `| Z) . x = (exp_R . x) / (1 + ((exp_R . x) ^2 )) )
assume A7: x in Z ; :: thesis: ((arctan * exp_R ) `| Z) . x = (exp_R . x) / (1 + ((exp_R . x) ^2 ))
(exp_R . x) + 0 > 0 + (- 1) by SIN_COS:59, XREAL_1:10;
then A8: ( exp_R . x > - 1 & exp_R . x < 1 ) by A2, A7;
A9: exp_R is_differentiable_in x by SIN_COS:70;
((arctan * exp_R ) `| Z) . x = diff (arctan * exp_R ),x by A6, A7, FDIFF_1:def 8
.= (diff exp_R ,x) / (1 + ((exp_R . x) ^2 )) by A8, A9, Th83
.= (exp_R . x) / (1 + ((exp_R . x) ^2 )) by SIN_COS:70 ;
hence ((arctan * exp_R ) `| Z) . x = (exp_R . x) / (1 + ((exp_R . x) ^2 )) ; :: thesis: verum

end;

hence ( arctan * exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * exp_R ) `| Z) . x = (exp_R . x) / (1 + ((exp_R . x) ^2 )) ) ) by A1, A3, FDIFF_1:16; :: thesis: verum