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_1:25;
then A3: Z c= dom tan by A1, XBOOLE_1:1;
A4: 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 A5: x in Z ; :: thesis: arccot * 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 arccot * tan is_differentiable_in x by A6, SIN_COS9:86; :: thesis: verum

end;

then A7: arccot * tan is_differentiable_on Z by A1, FDIFF_1:9;
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 A8: x in Z ; :: thesis: ((arccot * 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 1;
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_1:12;
((arccot * tan) `| Z) . x = diff ((arccot * tan),x) by A7, A8, FDIFF_1:def 7
.= - ((diff (tan,x)) / (1 + ((tan . x) ^2))) by A12, A9, SIN_COS9:86
.= - ((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_1:78
.= - (1 / (((cos . x) ^2) * (1 + (((sin . x) ^2) / ((cos . x) ^2))))) by XCMPLX_1:76
.= - (1 / (((cos . x) ^2) + ((((cos . x) ^2) * ((sin . x) ^2)) / ((cos . x) ^2))))
.= - (1 / (((cos . x) ^2) + ((sin . x) ^2))) by A13, XCMPLX_1:89
.= - (1 / 1) by SIN_COS:28
.= - 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, A4, FDIFF_1:9; :: thesis: verum