let Z be open Subset of REAL ; :: thesis: ( not 0 in Z & Z c= dom (((id Z) ^ ) (#) sec ) implies ( ((id Z) ^ ) (#) sec is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) ^ ) (#) sec ) `| Z) . x = (- ((1 / (cos . x)) / (x ^2 ))) + (((sin . x) / x) / ((cos . x) ^2 )) ) ) )
set f = id Z;
assume A1:
( not 0 in Z & Z c= dom (((id Z) ^ ) (#) sec ) )
; :: thesis: ( ((id Z) ^ ) (#) sec is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) ^ ) (#) sec ) `| Z) . x = (- ((1 / (cos . x)) / (x ^2 ))) + (((sin . x) / x) / ((cos . x) ^2 )) ) )
A2:
for x being Real st x in Z holds
(id Z) . x = x
by FUNCT_1:35;
A3:
Z c= (dom ((id Z) ^ )) /\ (dom sec )
by A1, VALUED_1:def 4;
then A4:
Z c= dom ((id Z) ^ )
by XBOOLE_1:18;
A5:
Z c= dom sec
by A3, XBOOLE_1:18;
A8:
( (id Z) ^ is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) ^ ) `| Z) . x = - (1 / (x ^2 )) ) )
by FDIFF_5:4, A1;
A9:
for x being Real st x in Z holds
( sec is_differentiable_in x & diff sec ,x = (sin . x) / ((cos . x) ^2 ) )
then
for x being Real st x in Z holds
sec is_differentiable_in x
;
then A10:
sec is_differentiable_on Z
by A5, FDIFF_1:16;
for x being Real st x in Z holds
((((id Z) ^ ) (#) sec ) `| Z) . x = (- ((1 / (cos . x)) / (x ^2 ))) + (((sin . x) / x) / ((cos . x) ^2 ))
proof
let x be
Real;
:: thesis: ( x in Z implies ((((id Z) ^ ) (#) sec ) `| Z) . x = (- ((1 / (cos . x)) / (x ^2 ))) + (((sin . x) / x) / ((cos . x) ^2 )) )
assume A11:
x in Z
;
:: thesis: ((((id Z) ^ ) (#) sec ) `| Z) . x = (- ((1 / (cos . x)) / (x ^2 ))) + (((sin . x) / x) / ((cos . x) ^2 ))
then ((((id Z) ^ ) (#) sec ) `| Z) . x =
((sec . x) * (diff ((id Z) ^ ),x)) + ((((id Z) ^ ) . x) * (diff sec ,x))
by A1, A8, A10, FDIFF_1:29
.=
((sec . x) * ((((id Z) ^ ) `| Z) . x)) + ((((id Z) ^ ) . x) * (diff sec ,x))
by A8, A11, FDIFF_1:def 8
.=
((sec . x) * (- (1 / (x ^2 )))) + ((((id Z) ^ ) . x) * (diff sec ,x))
by A11, FDIFF_5:4, A1
.=
(- ((sec . x) * (1 / (x ^2 )))) + ((((id Z) ^ ) . x) * ((sin . x) / ((cos . x) ^2 )))
by A9, A11
.=
(- (((cos . x) " ) * (1 / (x ^2 )))) + ((((id Z) ^ ) . x) * ((sin . x) / ((cos . x) ^2 )))
by A5, A11, RFUNCT_1:def 8
.=
(- ((1 / (cos . x)) / (x ^2 ))) + ((((id Z) . x) " ) * ((sin . x) / ((cos . x) ^2 )))
by A4, A11, RFUNCT_1:def 8
.=
(- ((1 / (cos . x)) / (x ^2 ))) + ((1 / x) * ((sin . x) / ((cos . x) ^2 )))
by A2, A11
.=
(- ((1 / (cos . x)) / (x ^2 ))) + (((sin . x) / x) / ((cos . x) ^2 ))
;
hence
((((id Z) ^ ) (#) sec ) `| Z) . x = (- ((1 / (cos . x)) / (x ^2 ))) + (((sin . x) / x) / ((cos . x) ^2 ))
;
:: thesis: verum
end;
hence
( ((id Z) ^ ) (#) sec is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) ^ ) (#) sec ) `| Z) . x = (- ((1 / (cos . x)) / (x ^2 ))) + (((sin . x) / x) / ((cos . x) ^2 )) ) )
by A1, A8, A10, FDIFF_1:29; :: thesis: verum