let Z be open Subset of REAL ; :: thesis: ( Z c= dom (arctan * arctan ) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
( arctan . x > - 1 & arctan . x < 1 ) ) implies ( arctan * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * arctan ) `| Z) . x = 1 / ((1 + (x ^2 )) * (1 + ((arctan . x) ^2 ))) ) ) )
assume that
A1: Z c= dom (arctan * arctan ) and
A2: Z c= ].(- 1),1.[ and
A3: for x being Real st x in Z holds
( arctan . x > - 1 & arctan . x < 1 ) ; :: thesis: ( arctan * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * arctan ) `| Z) . x = 1 / ((1 + (x ^2 )) * (1 + ((arctan . x) ^2 ))) ) )
A4: for x being Real st x in Z holds
arctan * arctan is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies arctan * arctan is_differentiable_in x )
assume A5: x in Z ; :: thesis: arctan * arctan is_differentiable_in x
then A6: ( arctan . x > - 1 & arctan . x < 1 ) by A3;
arctan is_differentiable_on Z by A2, SIN_COS9:81;
then arctan is_differentiable_in x by A5, FDIFF_1:16;
hence arctan * arctan is_differentiable_in x by A6, SIN_COS9:85; :: thesis: verum

end;

then A7: arctan * arctan is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
((arctan * arctan ) `| Z) . x = 1 / ((1 + (x ^2 )) * (1 + ((arctan . x) ^2 )))

proof

let x be Real; :: thesis: ( x in Z implies ((arctan * arctan ) `| Z) . x = 1 / ((1 + (x ^2 )) * (1 + ((arctan . x) ^2 ))) )
assume A8: x in Z ; :: thesis: ((arctan * arctan ) `| Z) . x = 1 / ((1 + (x ^2 )) * (1 + ((arctan . x) ^2 )))
then A9: ( arctan . x > - 1 & arctan . x < 1 ) by A3;
A10: arctan is_differentiable_on Z by A2, SIN_COS9:81;
then A11: arctan is_differentiable_in x by A8, FDIFF_1:16;
((arctan * arctan ) `| Z) . x = diff (arctan * arctan ),x by A7, A8, FDIFF_1:def 8
.= (diff arctan ,x) / (1 + ((arctan . x) ^2 )) by A11, A9, SIN_COS9:85
.= ((arctan `| Z) . x) / (1 + ((arctan . x) ^2 )) by A8, A10, FDIFF_1:def 8
.= (1 / (1 + (x ^2 ))) / (1 + ((arctan . x) ^2 )) by A2, A8, SIN_COS9:81
.= 1 / ((1 + (x ^2 )) * (1 + ((arctan . x) ^2 ))) by XCMPLX_1:79 ;
hence ((arctan * arctan ) `| Z) . x = 1 / ((1 + (x ^2 )) * (1 + ((arctan . x) ^2 ))) ; :: thesis: verum

end;

hence ( arctan * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * arctan ) `| Z) . x = 1 / ((1 + (x ^2 )) * (1 + ((arctan . x) ^2 ))) ) ) by A1, A4, FDIFF_1:16; :: thesis: verum