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

proof

let x be Real; :: thesis: ( x in Z implies arccot * tan is_differentiable_in x )
assume A4: x in Z ; :: thesis: arccot * tan is_differentiable_in x
then cos . x <> 0 by A3, FDIFF_8:1;
then A5: tan is_differentiable_in x by FDIFF_7:46;
( tan . x > - 1 & tan . x < 1 ) by A2, A4;
hence arccot * tan is_differentiable_in x by A5, SIN_COS9:86; :: thesis: verum

end;

then A6: arccot * tan is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
((arccot * tan ) `| Z) . x = - 1

proof

let x be Real; :: thesis: ( x in Z implies ((arccot * tan ) `| Z) . x = - 1 )
assume A7: x in Z ; :: thesis: ((arccot * tan ) `| Z) . x = - 1
then A8: cos . x <> 0 by A3, FDIFF_8:1;
then A9: tan is_differentiable_in x by FDIFF_7:46;
A10: ( tan . x > - 1 & tan . x < 1 ) by A2, A7;
A13: tan . x = (sin . x) / (cos . x) by A3, A7, RFUNCT_1:def 4;
A14: (cos . x) ^2 <> 0 by A8, SQUARE_1:74;
((arccot * tan ) `| Z) . x = diff (arccot * tan ),x by A6, A7, FDIFF_1:def 8
.= - ((diff tan ,x) / (1 + ((tan . x) ^2 ))) by A9, A10, SIN_COS9:86
.= - ((1 / ((cos . x) ^2 )) / (1 + ((tan . x) ^2 ))) by A8, FDIFF_7:46
.= - (1 / (((cos . x) ^2 ) * (1 + (((sin . x) / (cos . x)) * ((sin . x) / (cos . x)))))) by A13, 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 A14, XCMPLX_1:90
.= - (1 / 1) by SIN_COS:31
.= - 1 ;
hence ((arccot * tan ) `| Z) . x = - 1 ; :: thesis: verum

end;

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