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))) ) )
A3: arctan is_differentiable_on Z by A2, SIN_COS9:81;
Z c= (dom tan) /\ (dom arctan) by A1, VALUED_1:def 4;
then A4: 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 A4, FDIFF_8:1;
hence tan is_differentiable_in x by FDIFF_7:46; :: thesis: verum

end;

then A5: tan is_differentiable_on Z by A4, FDIFF_1:9;
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 A4, FDIFF_8:1;
((tan (#) arctan) `| Z) . x = ((arctan . x) * (diff (tan,x))) + ((tan . x) * (diff (arctan,x))) by A1, A5, A3, A6, FDIFF_1:21
.= ((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 A3, A6, FDIFF_1:def 7
.= ((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, A5, A3, FDIFF_1:21; :: thesis: verum