let Z be open Subset of REAL; ( not 0 in Z & Z c= dom (((id Z) ^) (#) tan) implies ( ((id Z) ^) (#) tan is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) ^) (#) tan) `| Z) . x = (- (((sin . x) / (cos . x)) / (x ^2))) + ((1 / x) / ((cos . x) ^2)) ) ) )
set f = id Z;
assume that
A1:
not 0 in Z
and
A2:
Z c= dom (((id Z) ^) (#) tan)
; ( ((id Z) ^) (#) tan is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) ^) (#) tan) `| Z) . x = (- (((sin . x) / (cos . x)) / (x ^2))) + ((1 / x) / ((cos . x) ^2)) ) )
A3:
(id Z) ^ is_differentiable_on Z
by A1, FDIFF_5:4;
A4:
Z c= (dom ((id Z) ^)) /\ (dom tan)
by A2, VALUED_1:def 4;
then A5:
Z c= dom tan
by XBOOLE_1:18;
A6:
for x being Real st x in Z holds
( tan is_differentiable_in x & diff (tan,x) = 1 / ((cos . x) ^2) )
then
for x being Real st x in Z holds
tan is_differentiable_in x
;
then A7:
tan is_differentiable_on Z
by A5, FDIFF_1:9;
A8:
Z c= dom ((id Z) ^)
by A4, XBOOLE_1:18;
for x being Real st x in Z holds
((((id Z) ^) (#) tan) `| Z) . x = (- (((sin . x) / (cos . x)) / (x ^2))) + ((1 / x) / ((cos . x) ^2))
proof
let x be
Real;
( x in Z implies ((((id Z) ^) (#) tan) `| Z) . x = (- (((sin . x) / (cos . x)) / (x ^2))) + ((1 / x) / ((cos . x) ^2)) )
assume A9:
x in Z
;
((((id Z) ^) (#) tan) `| Z) . x = (- (((sin . x) / (cos . x)) / (x ^2))) + ((1 / x) / ((cos . x) ^2))
then ((((id Z) ^) (#) tan) `| Z) . x =
((tan . x) * (diff (((id Z) ^),x))) + ((((id Z) ^) . x) * (diff (tan,x)))
by A2, A3, A7, FDIFF_1:21
.=
((tan . x) * ((((id Z) ^) `| Z) . x)) + ((((id Z) ^) . x) * (diff (tan,x)))
by A3, A9, FDIFF_1:def 7
.=
((tan . x) * (- (1 / (x ^2)))) + ((((id Z) ^) . x) * (diff (tan,x)))
by A1, A9, FDIFF_5:4
.=
(- ((tan . x) * (1 / (x ^2)))) + ((((id Z) ^) . x) * (1 / ((cos . x) ^2)))
by A6, A9
.=
(- (((sin . x) / (cos . x)) * (1 / (x ^2)))) + ((((id Z) ^) . x) / ((cos . x) ^2))
by A5, A9, RFUNCT_1:def 1
.=
(- (((sin . x) / (cos . x)) / (x ^2))) + ((((id Z) . x) ") / ((cos . x) ^2))
by A8, A9, RFUNCT_1:def 2
.=
(- (((sin . x) / (cos . x)) / (x ^2))) + ((1 / x) / ((cos . x) ^2))
by A9, FUNCT_1:18
;
hence
((((id Z) ^) (#) tan) `| Z) . x = (- (((sin . x) / (cos . x)) / (x ^2))) + ((1 / x) / ((cos . x) ^2))
;
verum
end;
hence
( ((id Z) ^) (#) tan is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) ^) (#) tan) `| Z) . x = (- (((sin . x) / (cos . x)) / (x ^2))) + ((1 / x) / ((cos . x) ^2)) ) )
by A2, A3, A7, FDIFF_1:21; verum