let Z be open Subset of REAL ; :: thesis: ( Z c= dom (exp_R (#) (tan - cot )) implies ( exp_R (#) (tan - cot ) is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) (tan - cot )) `| Z) . x = ((exp_R . x) * ((tan . x) - (cot . x))) + ((exp_R . x) * ((1 / ((cos . x) ^2 )) + (1 / ((sin . x) ^2 )))) ) ) )
assume A1: Z c= dom (exp_R (#) (tan - cot )) ; :: thesis: ( exp_R (#) (tan - cot ) is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) (tan - cot )) `| Z) . x = ((exp_R . x) * ((tan . x) - (cot . x))) + ((exp_R . x) * ((1 / ((cos . x) ^2 )) + (1 / ((sin . x) ^2 )))) ) )
then Z c= (dom (tan - cot )) /\ (dom exp_R ) by VALUED_1:def 4;
then A2: Z c= dom (tan - cot ) by XBOOLE_1:18;
then A3: tan - cot is_differentiable_on Z by Th5;
A4: exp_R is_differentiable_on Z by FDIFF_1:34, TAYLOR_1:16;
for x being Real st x in Z holds
((exp_R (#) (tan - cot )) `| Z) . x = ((exp_R . x) * ((tan . x) - (cot . x))) + ((exp_R . x) * ((1 / ((cos . x) ^2 )) + (1 / ((sin . x) ^2 ))))

proof

let x be Real; :: thesis: ( x in Z implies ((exp_R (#) (tan - cot )) `| Z) . x = ((exp_R . x) * ((tan . x) - (cot . x))) + ((exp_R . x) * ((1 / ((cos . x) ^2 )) + (1 / ((sin . x) ^2 )))) )
assume A5: x in Z ; :: thesis: ((exp_R (#) (tan - cot )) `| Z) . x = ((exp_R . x) * ((tan . x) - (cot . x))) + ((exp_R . x) * ((1 / ((cos . x) ^2 )) + (1 / ((sin . x) ^2 ))))
then ((exp_R (#) (tan - cot )) `| Z) . x = (((tan - cot ) . x) * (diff exp_R ,x)) + ((exp_R . x) * (diff (tan - cot ),x)) by A1, A3, A4, FDIFF_1:29
.= (((tan . x) - (cot . x)) * (diff exp_R ,x)) + ((exp_R . x) * (diff (tan - cot ),x)) by A2, A5, VALUED_1:13
.= ((exp_R . x) * ((tan . x) - (cot . x))) + ((exp_R . x) * (diff (tan - cot ),x)) by TAYLOR_1:16
.= ((exp_R . x) * ((tan . x) - (cot . x))) + ((exp_R . x) * (((tan - cot ) `| Z) . x)) by A3, A5, FDIFF_1:def 8
.= ((exp_R . x) * ((tan . x) - (cot . x))) + ((exp_R . x) * ((1 / ((cos . x) ^2 )) + (1 / ((sin . x) ^2 )))) by A2, A5, Th5 ;
hence ((exp_R (#) (tan - cot )) `| Z) . x = ((exp_R . x) * ((tan . x) - (cot . x))) + ((exp_R . x) * ((1 / ((cos . x) ^2 )) + (1 / ((sin . x) ^2 )))) ; :: thesis: verum

end;

hence ( exp_R (#) (tan - cot ) is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) (tan - cot )) `| Z) . x = ((exp_R . x) * ((tan . x) - (cot . x))) + ((exp_R . x) * ((1 / ((cos . x) ^2 )) + (1 / ((sin . x) ^2 )))) ) ) by A1, A3, A4, FDIFF_1:29; :: thesis: verum