let Z be open Subset of REAL; ( 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 . ((tan . x) + (cot . x))) * ((1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2))) ) ) )
assume A1:
Z c= dom (exp_R * (tan + cot))
; ( 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 . ((tan . x) + (cot . x))) * ((1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2))) ) )
dom (exp_R * (tan + cot)) c= dom (tan + cot)
by RELAT_1:25;
then A2:
Z c= dom (tan + cot)
by A1, XBOOLE_1:1;
then A3:
tan + cot is_differentiable_on Z
by Th6;
A4:
for x being Real st x in Z holds
exp_R * (tan + cot) is_differentiable_in x
then A6:
exp_R * (tan + cot) is_differentiable_on Z
by A1, FDIFF_1:9;
for x being Real st x in Z holds
((exp_R * (tan + cot)) `| Z) . x = (exp_R . ((tan . x) + (cot . x))) * ((1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2)))
proof
let x be
Real;
( x in Z implies ((exp_R * (tan + cot)) `| Z) . x = (exp_R . ((tan . x) + (cot . x))) * ((1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2))) )
A7:
exp_R is_differentiable_in (tan + cot) . x
by SIN_COS:65;
assume A8:
x in Z
;
((exp_R * (tan + cot)) `| Z) . x = (exp_R . ((tan . x) + (cot . x))) * ((1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2)))
then
tan + cot is_differentiable_in x
by A3, FDIFF_1:9;
then diff (
(exp_R * (tan + cot)),
x) =
(diff (exp_R,((tan + cot) . x))) * (diff ((tan + cot),x))
by A7, FDIFF_2:13
.=
(exp_R . ((tan + cot) . x)) * (diff ((tan + cot),x))
by SIN_COS:65
.=
(exp_R . ((tan + cot) . x)) * (((tan + cot) `| Z) . x)
by A3, A8, FDIFF_1:def 7
.=
(exp_R . ((tan + cot) . x)) * ((1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2)))
by A2, A8, Th6
.=
(exp_R . ((tan . x) + (cot . x))) * ((1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2)))
by A2, A8, VALUED_1:def 1
;
hence
((exp_R * (tan + cot)) `| Z) . x = (exp_R . ((tan . x) + (cot . x))) * ((1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2)))
by A6, A8, FDIFF_1:def 7;
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 . ((tan . x) + (cot . x))) * ((1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2))) ) )
by A1, A4, FDIFF_1:9; verum