let Z be open Subset of REAL ; :: thesis: ( Z c= dom (sec (#) arctan ) & Z c= ].(- 1),1.[ implies ( sec (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((sec (#) arctan ) `| Z) . x = (((sin . x) * (arctan . x)) / ((cos . x) ^2 )) + (1 / ((cos . x) * (1 + (x ^2 )))) ) ) )
assume that
A1: Z c= dom (sec (#) arctan ) and
A2: Z c= ].(- 1),1.[ ; :: thesis: ( sec (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((sec (#) arctan ) `| Z) . x = (((sin . x) * (arctan . x)) / ((cos . x) ^2 )) + (1 / ((cos . x) * (1 + (x ^2 )))) ) )
A3: arctan is_differentiable_on Z by A2, SIN_COS9:81;
Z c= (dom sec ) /\ (dom arctan ) by A1, VALUED_1:def 4;
then A4: Z c= dom sec by XBOOLE_1:18;
for x being Real st x in Z holds
sec is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies sec is_differentiable_in x )
assume x in Z ; :: thesis: sec is_differentiable_in x
then cos . x <> 0 by A4, RFUNCT_1:13;
hence sec is_differentiable_in x by FDIFF_9:1; :: thesis: verum

end;

then A5: sec is_differentiable_on Z by A4, FDIFF_1:16;
for x being Real st x in Z holds
((sec (#) arctan ) `| Z) . x = (((sin . x) * (arctan . x)) / ((cos . x) ^2 )) + (1 / ((cos . x) * (1 + (x ^2 ))))

proof

let x be Real; :: thesis: ( x in Z implies ((sec (#) arctan ) `| Z) . x = (((sin . x) * (arctan . x)) / ((cos . x) ^2 )) + (1 / ((cos . x) * (1 + (x ^2 )))) )
assume A6: x in Z ; :: thesis: ((sec (#) arctan ) `| Z) . x = (((sin . x) * (arctan . x)) / ((cos . x) ^2 )) + (1 / ((cos . x) * (1 + (x ^2 ))))
then A7: cos . x <> 0 by A4, RFUNCT_1:13;
((sec (#) arctan ) `| Z) . x = ((arctan . x) * (diff sec ,x)) + ((sec . x) * (diff arctan ,x)) by A1, A5, A3, A6, FDIFF_1:29
.= ((arctan . x) * ((sin . x) / ((cos . x) ^2 ))) + ((sec . x) * (diff arctan ,x)) by A7, FDIFF_9:1
.= (((sin . x) * (arctan . x)) / ((cos . x) ^2 )) + ((sec . x) * ((arctan `| Z) . x)) by A3, A6, FDIFF_1:def 8
.= (((sin . x) * (arctan . x)) / ((cos . x) ^2 )) + ((sec . x) * (1 / (1 + (x ^2 )))) by A2, A6, SIN_COS9:81
.= (((sin . x) * (arctan . x)) / ((cos . x) ^2 )) + ((1 / (cos . x)) * (1 / (1 + (x ^2 )))) by A4, A6, RFUNCT_1:def 8
.= (((sin . x) * (arctan . x)) / ((cos . x) ^2 )) + (1 / ((cos . x) * (1 + (x ^2 )))) by XCMPLX_1:103 ;
hence ((sec (#) arctan ) `| Z) . x = (((sin . x) * (arctan . x)) / ((cos . x) ^2 )) + (1 / ((cos . x) * (1 + (x ^2 )))) ; :: thesis: verum

end;

hence ( sec (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((sec (#) arctan ) `| Z) . x = (((sin . x) * (arctan . x)) / ((cos . x) ^2 )) + (1 / ((cos . x) * (1 + (x ^2 )))) ) ) by A1, A5, A3, FDIFF_1:29; :: thesis: verum