let Z be open Subset of REAL ; :: thesis: ( Z c= dom (arctan * tan ) & ( for x being Real st x in Z holds
( tan . x > - 1 & tan . x < 1 ) ) implies ( arctan * tan is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * tan ) `| Z) . x = 1 ) ) )
assume that
A1: Z c= dom (arctan * tan ) and
A2: for x being Real st x in Z holds
( tan . x > - 1 & tan . x < 1 ) ; :: thesis: ( arctan * tan is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * tan ) `| Z) . x = 1 ) )
dom (arctan * tan ) c= dom tan by RELAT_1:44;
then A3: Z c= dom tan by A1, XBOOLE_1:1;
A4: for x being Real st x in Z holds
arctan * tan is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies arctan * tan is_differentiable_in x )
assume A5: x in Z ; :: thesis: arctan * tan is_differentiable_in x
then cos . x <> 0 by A3, FDIFF_8:1;
then A6: tan is_differentiable_in x by FDIFF_7:46;
( tan . x > - 1 & tan . x < 1 ) by A2, A5;
hence arctan * tan is_differentiable_in x by A6, SIN_COS9:85; :: thesis: verum

end;

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

proof

let x be Real; :: thesis: ( x in Z implies ((arctan * tan ) `| Z) . x = 1 )
assume A8: x in Z ; :: thesis: ((arctan * tan ) `| Z) . x = 1
then A9: ( tan . x > - 1 & tan . x < 1 ) by A2;
A10: tan . x = (sin . x) / (cos . x) by A3, A8, RFUNCT_1:def 4;
A11: cos . x <> 0 by A3, A8, FDIFF_8:1;
then A12: tan is_differentiable_in x by FDIFF_7:46;
A13: (cos . x) ^2 <> 0 by A11, SQUARE_1:74;
((arctan * tan ) `| Z) . x = diff (arctan * tan ),x by A7, A8, FDIFF_1:def 8
.= (diff tan ,x) / (1 + ((tan . x) ^2 )) by A12, A9, SIN_COS9:85
.= (1 / ((cos . x) ^2 )) / (1 + ((tan . x) ^2 )) by A11, FDIFF_7:46
.= 1 / (((cos . x) ^2 ) * (1 + (((sin . x) / (cos . x)) * ((sin . x) / (cos . x))))) by A10, XCMPLX_1:79
.= 1 / (((cos . x) ^2 ) * (1 + (((sin . x) ^2 ) / ((cos . x) ^2 )))) by XCMPLX_1:77
.= 1 / (((cos . x) ^2 ) + ((((cos . x) ^2 ) * ((sin . x) ^2 )) / ((cos . x) ^2 )))
.= 1 / (((cos . x) ^2 ) + ((sin . x) ^2 )) by A13, XCMPLX_1:90
.= 1 / 1 by SIN_COS:31
.= 1 ;
hence ((arctan * tan ) `| Z) . x = 1 ; :: thesis: verum

end;

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