let Z be open Subset of REAL; :: thesis: ( not 0 in Z & Z c= dom (cot * ((id Z) ^)) implies ( cot * ((id Z) ^) is_differentiable_on Z & ( for x being Real st x in Z holds
((cot * ((id Z) ^)) `| Z) . x = 1 / ((x ^2) * ((sin . (1 / x)) ^2)) ) ) )
set f = id Z;
assume that
A1: not 0 in Z and
A2: Z c= dom (cot * ((id Z) ^)) ; :: thesis: ( cot * ((id Z) ^) is_differentiable_on Z & ( for x being Real st x in Z holds
((cot * ((id Z) ^)) `| Z) . x = 1 / ((x ^2) * ((sin . (1 / x)) ^2)) ) )
A3: (id Z) ^ is_differentiable_on Z by A1, FDIFF_5:4;
dom (cot * ((id Z) ^)) c= dom ((id Z) ^) by RELAT_1:25;
then A4: Z c= dom ((id Z) ^) by A2, XBOOLE_1:1;
A5: for x being Real st x in Z holds
sin . (((id Z) ^) . x) <> 0

proof

let x be Real; :: thesis: ( x in Z implies sin . (((id Z) ^) . x) <> 0 )
assume x in Z ; :: thesis: sin . (((id Z) ^) . x) <> 0
then ((id Z) ^) . x in dom (cos / sin) by A2, FUNCT_1:11;
hence sin . (((id Z) ^) . x) <> 0 by Th2; :: thesis: verum

end;

A6: for x being Real st x in Z holds
cot * ((id Z) ^) is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies cot * ((id Z) ^) is_differentiable_in x )
assume A7: x in Z ; :: thesis: cot * ((id Z) ^) is_differentiable_in x
then sin . (((id Z) ^) . x) <> 0 by A5;
then A8: cot is_differentiable_in ((id Z) ^) . x by FDIFF_7:47;
(id Z) ^ is_differentiable_in x by A3, A7, FDIFF_1:9;
hence cot * ((id Z) ^) is_differentiable_in x by A8, FDIFF_2:13; :: thesis: verum

end;

then A9: cot * ((id Z) ^) is_differentiable_on Z by A2, FDIFF_1:9;
for x being Real st x in Z holds
((cot * ((id Z) ^)) `| Z) . x = 1 / ((x ^2) * ((sin . (1 / x)) ^2))

proof

let x be Real; :: thesis: ( x in Z implies ((cot * ((id Z) ^)) `| Z) . x = 1 / ((x ^2) * ((sin . (1 / x)) ^2)) )
assume A10: x in Z ; :: thesis: ((cot * ((id Z) ^)) `| Z) . x = 1 / ((x ^2) * ((sin . (1 / x)) ^2))
then A11: (id Z) ^ is_differentiable_in x by A3, FDIFF_1:9;
A12: sin . (((id Z) ^) . x) <> 0 by A5, A10;
then cot is_differentiable_in ((id Z) ^) . x by FDIFF_7:47;
then diff ((cot * ((id Z) ^)),x) = (diff (cot,(((id Z) ^) . x))) * (diff (((id Z) ^),x)) by A11, FDIFF_2:13
.= (- (1 / ((sin . (((id Z) ^) . x)) ^2))) * (diff (((id Z) ^),x)) by A12, FDIFF_7:47
.= - ((diff (((id Z) ^),x)) / ((sin . (((id Z) ^) . x)) ^2))
.= - ((diff (((id Z) ^),x)) / ((sin . (((id Z) . x) ")) ^2)) by A4, A10, RFUNCT_1:def 2
.= - ((diff (((id Z) ^),x)) / ((sin . (1 * (x "))) ^2)) by A10, FUNCT_1:18
.= - (((((id Z) ^) `| Z) . x) / ((sin . (1 * (x "))) ^2)) by A3, A10, FDIFF_1:def 7
.= - ((- (1 / (x ^2))) / ((sin . (1 * (x "))) ^2)) by A1, A10, FDIFF_5:4
.= - (((- 1) / (x ^2)) / ((sin . (1 / x)) ^2))
.= - ((- 1) / ((x ^2) * ((sin . (1 / x)) ^2))) by XCMPLX_1:78
.= 1 / ((x ^2) * ((sin . (1 / x)) ^2)) ;
hence ((cot * ((id Z) ^)) `| Z) . x = 1 / ((x ^2) * ((sin . (1 / x)) ^2)) by A9, A10, FDIFF_1:def 7; :: thesis: verum

end;

hence ( cot * ((id Z) ^) is_differentiable_on Z & ( for x being Real st x in Z holds
((cot * ((id Z) ^)) `| Z) . x = 1 / ((x ^2) * ((sin . (1 / x)) ^2)) ) ) by A2, A6, FDIFF_1:9; :: thesis: verum