let Z be open Subset of REAL ; :: thesis: ( Z c= dom (cot (#) sec ) implies ( cot (#) sec is_differentiable_on Z & ( for x being Real st x in Z holds
((cot (#) sec ) `| Z) . x = (- ((1 / ((sin . x) ^2 )) / (cos . x))) + (((cot . x) * (sin . x)) / ((cos . x) ^2 )) ) ) )
assume A1: Z c= dom (cot (#) sec ) ; :: thesis: ( cot (#) sec is_differentiable_on Z & ( for x being Real st x in Z holds
((cot (#) sec ) `| Z) . x = (- ((1 / ((sin . x) ^2 )) / (cos . x))) + (((cot . x) * (sin . x)) / ((cos . x) ^2 )) ) )
then A2: Z c= (dom cot ) /\ (dom sec ) by VALUED_1:def 4;
then A3: Z c= dom cot by XBOOLE_1:18;
for x being Real st x in Z holds
cot is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies cot is_differentiable_in x )
assume x in Z ; :: thesis: cot is_differentiable_in x
then sin . x <> 0 by A3, FDIFF_8:2;
hence cot is_differentiable_in x by FDIFF_7:47; :: thesis: verum

end;

then A4: cot is_differentiable_on Z by A3, FDIFF_1:16;
A5: Z c= dom sec by A2, XBOOLE_1:18;
A6: for x being Real st x in Z holds
( sec is_differentiable_in x & diff sec ,x = (sin . x) / ((cos . x) ^2 ) )

proof

let x be Real; :: thesis: ( x in Z implies ( sec is_differentiable_in x & diff sec ,x = (sin . x) / ((cos . x) ^2 ) ) )
assume x in Z ; :: thesis: ( sec is_differentiable_in x & diff sec ,x = (sin . x) / ((cos . x) ^2 ) )
then cos . x <> 0 by A5, RFUNCT_1:13;
hence ( sec is_differentiable_in x & diff sec ,x = (sin . x) / ((cos . x) ^2 ) ) by Th1; :: thesis: verum

end;

then for x being Real st x in Z holds
sec is_differentiable_in x ;
then A7: sec is_differentiable_on Z by A5, FDIFF_1:16;
for x being Real st x in Z holds
((cot (#) sec ) `| Z) . x = (- ((1 / ((sin . x) ^2 )) / (cos . x))) + (((cot . x) * (sin . x)) / ((cos . x) ^2 ))

proof

let x be Real; :: thesis: ( x in Z implies ((cot (#) sec ) `| Z) . x = (- ((1 / ((sin . x) ^2 )) / (cos . x))) + (((cot . x) * (sin . x)) / ((cos . x) ^2 )) )
assume A8: x in Z ; :: thesis: ((cot (#) sec ) `| Z) . x = (- ((1 / ((sin . x) ^2 )) / (cos . x))) + (((cot . x) * (sin . x)) / ((cos . x) ^2 ))
then A9: sin . x <> 0 by A3, FDIFF_8:2;
((cot (#) sec ) `| Z) . x = ((sec . x) * (diff cot ,x)) + ((cot . x) * (diff sec ,x)) by A1, A4, A7, A8, FDIFF_1:29
.= ((sec . x) * (- (1 / ((sin . x) ^2 )))) + ((cot . x) * (diff sec ,x)) by A9, FDIFF_7:47
.= ((sec . x) * (- (1 / ((sin . x) ^2 )))) + ((cot . x) * ((sin . x) / ((cos . x) ^2 ))) by A6, A8
.= ((- (1 / ((sin . x) ^2 ))) / (cos . x)) + (((cot . x) * (sin . x)) / ((cos . x) ^2 )) by A5, A8, RFUNCT_1:def 8 ;
hence ((cot (#) sec ) `| Z) . x = (- ((1 / ((sin . x) ^2 )) / (cos . x))) + (((cot . x) * (sin . x)) / ((cos . x) ^2 )) ; :: thesis: verum

end;

hence ( cot (#) sec is_differentiable_on Z & ( for x being Real st x in Z holds
((cot (#) sec ) `| Z) . x = (- ((1 / ((sin . x) ^2 )) / (cos . x))) + (((cot . x) * (sin . x)) / ((cos . x) ^2 )) ) ) by A1, A4, A7, FDIFF_1:29; :: thesis: verum