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 = 1 / (((cos . (arctan . x)) ^2 ) * (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 = 1 / (((cos . (arctan . x)) ^2 ) * (1 + (x ^2 ))) ) )
AA: for x being Real st x in Z holds
tan * arctan is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies tan * arctan is_differentiable_in x )
assume A3: x in Z ; :: thesis: tan * arctan is_differentiable_in x
arctan is_differentiable_on Z by A2, SIN_COS9:81;
then A4: arctan is_differentiable_in x by A3, FDIFF_1:16;
arctan . x in dom tan by A1, A3, FUNCT_1:21;
then cos . (arctan . x) <> 0 by FDIFF_8:1;
then tan is_differentiable_in arctan . x by FDIFF_7:46;
hence tan * arctan is_differentiable_in x by A4, FDIFF_2:13; :: thesis: verum

end;

then A5: tan * arctan is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
((tan * arctan ) `| Z) . x = 1 / (((cos . (arctan . x)) ^2 ) * (1 + (x ^2 )))

proof

let x be Real; :: thesis: ( x in Z implies ((tan * arctan ) `| Z) . x = 1 / (((cos . (arctan . x)) ^2 ) * (1 + (x ^2 ))) )
assume A6: x in Z ; :: thesis: ((tan * arctan ) `| Z) . x = 1 / (((cos . (arctan . x)) ^2 ) * (1 + (x ^2 )))
A7: arctan is_differentiable_on Z by A2, SIN_COS9:81;
then A8: arctan is_differentiable_in x by A6, FDIFF_1:16;
arctan . x in dom tan by A1, A6, FUNCT_1:21;
then A9: cos . (arctan . x) <> 0 by FDIFF_8:1;
then A10: tan is_differentiable_in arctan . x by FDIFF_7:46;
((tan * arctan ) `| Z) . x = diff (tan * arctan ),x by A5, A6, FDIFF_1:def 8
.= (diff tan ,(arctan . x)) * (diff arctan ,x) by A8, A10, FDIFF_2:13
.= (1 / ((cos . (arctan . x)) ^2 )) * (diff arctan ,x) by A9, FDIFF_7:46
.= (1 / ((cos . (arctan . x)) ^2 )) * ((arctan `| Z) . x) by A6, A7, FDIFF_1:def 8
.= (1 / ((cos . (arctan . x)) ^2 )) * (1 / (1 + (x ^2 ))) by A2, A6, SIN_COS9:81
.= 1 / (((cos . (arctan . x)) ^2 ) * (1 + (x ^2 ))) by XCMPLX_1:103 ;
hence ((tan * arctan ) `| Z) . x = 1 / (((cos . (arctan . x)) ^2 ) * (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 = 1 / (((cos . (arctan . x)) ^2 ) * (1 + (x ^2 ))) ) ) by A1, AA, FDIFF_1:16; :: thesis: verum