let Z be open Subset of REAL ; :: thesis: ( Z c= dom (sin (#) (tan + cot )) implies ( sin (#) (tan + cot ) is_differentiable_on Z & ( for x being Real st x in Z holds
((sin (#) (tan + cot )) `| Z) . x = ((cos . x) * ((tan . x) + (cot . x))) + ((sin . x) * ((1 / ((cos . x) ^2 )) - (1 / ((sin . x) ^2 )))) ) ) )
assume A1: Z c= dom (sin (#) (tan + cot )) ; :: thesis: ( sin (#) (tan + cot ) is_differentiable_on Z & ( for x being Real st x in Z holds
((sin (#) (tan + cot )) `| Z) . x = ((cos . x) * ((tan . x) + (cot . x))) + ((sin . x) * ((1 / ((cos . x) ^2 )) - (1 / ((sin . x) ^2 )))) ) )
then A2: Z c= (dom (tan + cot )) /\ (dom sin ) by VALUED_1:def 4;
then A3: Z c= dom sin 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 Th6;
for x being Real st x in Z holds
sin is_differentiable_in x by SIN_COS:69;
then A6: sin is_differentiable_on Z by A3, FDIFF_1:16;
for x being Real st x in Z holds
((sin (#) (tan + cot )) `| Z) . x = ((cos . x) * ((tan . x) + (cot . x))) + ((sin . x) * ((1 / ((cos . x) ^2 )) - (1 / ((sin . x) ^2 ))))

proof

let x be Real; :: thesis: ( x in Z implies ((sin (#) (tan + cot )) `| Z) . x = ((cos . x) * ((tan . x) + (cot . x))) + ((sin . x) * ((1 / ((cos . x) ^2 )) - (1 / ((sin . x) ^2 )))) )
assume A7: x in Z ; :: thesis: ((sin (#) (tan + cot )) `| Z) . x = ((cos . x) * ((tan . x) + (cot . x))) + ((sin . x) * ((1 / ((cos . x) ^2 )) - (1 / ((sin . x) ^2 ))))
then ((sin (#) (tan + cot )) `| Z) . x = (((tan + cot ) . x) * (diff sin ,x)) + ((sin . x) * (diff (tan + cot ),x)) by A1, A5, A6, FDIFF_1:29
.= (((tan . x) + (cot . x)) * (diff sin ,x)) + ((sin . x) * (diff (tan + cot ),x)) by A4, A7, VALUED_1:def 1
.= (((tan . x) + (cot . x)) * (cos . x)) + ((sin . x) * (diff (tan + cot ),x)) by SIN_COS:69
.= (((tan . x) + (cot . x)) * (cos . x)) + ((sin . x) * (((tan + cot ) `| Z) . x)) by A5, A7, FDIFF_1:def 8
.= (((tan . x) + (cot . x)) * (cos . x)) + ((sin . x) * ((1 / ((cos . x) ^2 )) - (1 / ((sin . x) ^2 )))) by A4, A7, Th6 ;
hence ((sin (#) (tan + cot )) `| Z) . x = ((cos . x) * ((tan . x) + (cot . x))) + ((sin . x) * ((1 / ((cos . x) ^2 )) - (1 / ((sin . x) ^2 )))) ; :: thesis: verum

end;

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