let Z be open Subset of REAL; :: thesis: ( Z c= dom (tan + cot) implies ( 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)) ) ) )
assume A1: Z c= dom (tan + cot) ; :: thesis: ( 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)) ) )
then A2: Z c= (dom tan) /\ (dom cot) by VALUED_1:def 1;
then A3: Z c= dom tan by XBOOLE_1:18;
for x being Real st x in Z holds
tan is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies tan is_differentiable_in x )
assume x in Z ; :: thesis: tan is_differentiable_in x
then cos . x <> 0 by A3, FDIFF_8:1;
hence tan is_differentiable_in x by FDIFF_7:46; :: thesis: verum

end;

then A4: tan is_differentiable_on Z by A3, FDIFF_1:9;
A5: Z c= dom cot by A2, 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 A5, FDIFF_8:2;
hence cot is_differentiable_in x by FDIFF_7:47; :: thesis: verum

end;

then A6: cot is_differentiable_on Z by A5, FDIFF_1:9;
for x being Real st x in Z holds
((tan + cot) `| Z) . x = (1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2))

proof

let x be Real; :: thesis: ( x in Z implies ((tan + cot) `| Z) . x = (1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2)) )
assume A7: x in Z ; :: thesis: ((tan + cot) `| Z) . x = (1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2))
then A8: sin . x <> 0 by A5, FDIFF_8:2;
A9: cos . x <> 0 by A3, A7, FDIFF_8:1;
((tan + cot) `| Z) . x = (diff (tan,x)) + (diff (cot,x)) by A1, A6, A4, A7, FDIFF_1:18
.= (1 / ((cos . x) ^2)) + (diff (cot,x)) by A9, FDIFF_7:46
.= (1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2)) by A8, FDIFF_7:47 ;
hence ((tan + cot) `| Z) . x = (1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2)) ; :: thesis: verum

end;

hence ( 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 A1, A6, A4, FDIFF_1:18; :: thesis: verum