let Z be open Subset of REAL ; :: thesis: ( Z c= dom (cos (#) (tan - cot )) implies ( cos (#) (tan - cot ) is_differentiable_on Z & ( for x being Real st x in Z holds
((cos (#) (tan - cot )) `| Z) . x = (- ((sin . x) * ((tan . x) - (cot . x)))) + ((cos . x) * ((1 / ((cos . x) ^2 )) + (1 / ((sin . x) ^2 )))) ) ) )
assume A1: Z c= dom (cos (#) (tan - cot )) ; :: thesis: ( cos (#) (tan - cot ) is_differentiable_on Z & ( for x being Real st x in Z holds
((cos (#) (tan - cot )) `| Z) . x = (- ((sin . x) * ((tan . x) - (cot . x)))) + ((cos . x) * ((1 / ((cos . x) ^2 )) + (1 / ((sin . x) ^2 )))) ) )
then A2: Z c= (dom (tan - cot )) /\ (dom cos ) by VALUED_1:def 4;
then A3: Z c= dom cos by XBOOLE_1:18;
A4: Z c= dom (tan - cot ) by A2, XBOOLE_1:18;
then A5: ( tan - cot is_differentiable_on Z & ( for x being Real st x in Z holds
((tan - cot ) `| Z) . x = (1 / ((cos . x) ^2 )) + (1 / ((sin . x) ^2 )) ) ) by Th5;
for x being Real st x in Z holds
cos is_differentiable_in x by SIN_COS:68;
then A6: cos is_differentiable_on Z by A3, FDIFF_1:16;
for x being Real st x in Z holds
((cos (#) (tan - cot )) `| Z) . x = (- ((sin . x) * ((tan . x) - (cot . x)))) + ((cos . x) * ((1 / ((cos . x) ^2 )) + (1 / ((sin . x) ^2 ))))

proof

let x be Real; :: thesis: ( x in Z implies ((cos (#) (tan - cot )) `| Z) . x = (- ((sin . x) * ((tan . x) - (cot . x)))) + ((cos . x) * ((1 / ((cos . x) ^2 )) + (1 / ((sin . x) ^2 )))) )
assume A7: x in Z ; :: thesis: ((cos (#) (tan - cot )) `| Z) . x = (- ((sin . x) * ((tan . x) - (cot . x)))) + ((cos . x) * ((1 / ((cos . x) ^2 )) + (1 / ((sin . x) ^2 ))))
then ((cos (#) (tan - cot )) `| Z) . x = (((tan - cot ) . x) * (diff cos ,x)) + ((cos . x) * (diff (tan - cot ),x)) by A1, A5, A6, FDIFF_1:29
.= (((tan . x) - (cot . x)) * (diff cos ,x)) + ((cos . x) * (diff (tan - cot ),x)) by A4, A7, VALUED_1:13
.= (((tan . x) - (cot . x)) * (- (sin . x))) + ((cos . x) * (diff (tan - cot ),x)) by SIN_COS:68
.= (((tan . x) - (cot . x)) * (- (sin . x))) + ((cos . x) * (((tan - cot ) `| Z) . x)) by A5, A7, FDIFF_1:def 8
.= (((tan . x) - (cot . x)) * (- (sin . x))) + ((cos . x) * ((1 / ((cos . x) ^2 )) + (1 / ((sin . x) ^2 )))) by A4, A7, Th5 ;
hence ((cos (#) (tan - cot )) `| Z) . x = (- ((sin . x) * ((tan . x) - (cot . x)))) + ((cos . x) * ((1 / ((cos . x) ^2 )) + (1 / ((sin . x) ^2 )))) ; :: thesis: verum

end;

hence ( cos (#) (tan - cot ) is_differentiable_on Z & ( for x being Real st x in Z holds
((cos (#) (tan - cot )) `| Z) . x = (- ((sin . x) * ((tan . x) - (cot . x)))) + ((cos . x) * ((1 / ((cos . x) ^2 )) + (1 / ((sin . x) ^2 )))) ) ) by A1, A5, A6, FDIFF_1:29; :: thesis: verum