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

proof

let x be Real; :: thesis: ( x in Z implies tan is_differentiable_in x )
assume x in Z ; :: thesis: tan is_differentiable_in x
then cos . x <> 0 by A3, FDIFF_8:1;
hence tan is_differentiable_in x by FDIFF_7:46; :: thesis: verum

end;

then A4: tan is_differentiable_on Z by A3, FDIFF_1:16;
A5: arctan is_differentiable_on Z by A2, SIN_COS9:81;
for x being Real st x in Z holds
((tan (#) arctan ) `| Z) . x = ((arctan . x) / ((cos . x) ^2 )) + ((tan . x) / (1 + (x ^2 )))

proof

let x be Real; :: thesis: ( x in Z implies ((tan (#) arctan ) `| Z) . x = ((arctan . x) / ((cos . x) ^2 )) + ((tan . x) / (1 + (x ^2 ))) )
assume A6: x in Z ; :: thesis: ((tan (#) arctan ) `| Z) . x = ((arctan . x) / ((cos . x) ^2 )) + ((tan . x) / (1 + (x ^2 )))
then A7: cos . x <> 0 by A3, FDIFF_8:1;
((tan (#) arctan ) `| Z) . x = ((arctan . x) * (diff tan ,x)) + ((tan . x) * (diff arctan ,x)) by A1, A4, A5, A6, FDIFF_1:29
.= ((arctan . x) * (1 / ((cos . x) ^2 ))) + ((tan . x) * (diff arctan ,x)) by A7, FDIFF_7:46
.= ((arctan . x) / ((cos . x) ^2 )) + ((tan . x) * ((arctan `| Z) . x)) by A5, A6, FDIFF_1:def 8
.= ((arctan . x) / ((cos . x) ^2 )) + ((tan . x) * (1 / (1 + (x ^2 )))) by A2, A6, SIN_COS9:81
.= ((arctan . x) / ((cos . x) ^2 )) + ((tan . x) / (1 + (x ^2 ))) ;
hence ((tan (#) arctan ) `| Z) . x = ((arctan . x) / ((cos . x) ^2 )) + ((tan . x) / (1 + (x ^2 ))) ; :: thesis: verum

end;

hence ( tan (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((tan (#) arctan ) `| Z) . x = ((arctan . x) / ((cos . x) ^2 )) + ((tan . x) / (1 + (x ^2 ))) ) ) by A1, A4, A5, FDIFF_1:29; :: thesis: verum