let a, b be Real; for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom (sec * f) & ( for x being Real st x in Z holds
f . x = (a * x) + b ) holds
( sec * f is_differentiable_on Z & ( for x being Real st x in Z holds
((sec * f) `| Z) . x = (a * (sin . ((a * x) + b))) / ((cos . ((a * x) + b)) ^2) ) )
let Z be open Subset of REAL; for f being PartFunc of REAL,REAL st Z c= dom (sec * f) & ( for x being Real st x in Z holds
f . x = (a * x) + b ) holds
( sec * f is_differentiable_on Z & ( for x being Real st x in Z holds
((sec * f) `| Z) . x = (a * (sin . ((a * x) + b))) / ((cos . ((a * x) + b)) ^2) ) )
let f be PartFunc of REAL,REAL; ( Z c= dom (sec * f) & ( for x being Real st x in Z holds
f . x = (a * x) + b ) implies ( sec * f is_differentiable_on Z & ( for x being Real st x in Z holds
((sec * f) `| Z) . x = (a * (sin . ((a * x) + b))) / ((cos . ((a * x) + b)) ^2) ) ) )
assume that
A1:
Z c= dom (sec * f)
and
A2:
for x being Real st x in Z holds
f . x = (a * x) + b
; ( sec * f is_differentiable_on Z & ( for x being Real st x in Z holds
((sec * f) `| Z) . x = (a * (sin . ((a * x) + b))) / ((cos . ((a * x) + b)) ^2) ) )
dom (sec * f) c= dom f
by RELAT_1:25;
then A3:
Z c= dom f
by A1, XBOOLE_1:1;
then A4:
f is_differentiable_on Z
by A2, FDIFF_1:23;
A5:
for x being Real st x in Z holds
cos . (f . x) <> 0
A6:
for x being Real st x in Z holds
sec * f is_differentiable_in x
then A9:
sec * f is_differentiable_on Z
by A1, FDIFF_1:9;
for x being Real st x in Z holds
((sec * f) `| Z) . x = (a * (sin . ((a * x) + b))) / ((cos . ((a * x) + b)) ^2)
proof
let x be
Real;
( x in Z implies ((sec * f) `| Z) . x = (a * (sin . ((a * x) + b))) / ((cos . ((a * x) + b)) ^2) )
assume A10:
x in Z
;
((sec * f) `| Z) . x = (a * (sin . ((a * x) + b))) / ((cos . ((a * x) + b)) ^2)
then A11:
f is_differentiable_in x
by A4, FDIFF_1:9;
A12:
cos . (f . x) <> 0
by A5, A10;
then
sec is_differentiable_in f . x
by Th1;
then diff (
(sec * f),
x) =
(diff (sec,(f . x))) * (diff (f,x))
by A11, FDIFF_2:13
.=
((sin . (f . x)) / ((cos . (f . x)) ^2)) * (diff (f,x))
by A12, Th1
.=
(diff (f,x)) * ((sin . (f . x)) / ((cos . ((a * x) + b)) ^2))
by A2, A10
.=
((f `| Z) . x) * ((sin . (f . x)) / ((cos . ((a * x) + b)) ^2))
by A4, A10, FDIFF_1:def 7
.=
a * ((sin . (f . x)) / ((cos . ((a * x) + b)) ^2))
by A2, A3, A10, FDIFF_1:23
.=
a * ((sin . ((a * x) + b)) / ((cos . ((a * x) + b)) ^2))
by A2, A10
;
hence
((sec * f) `| Z) . x = (a * (sin . ((a * x) + b))) / ((cos . ((a * x) + b)) ^2)
by A9, A10, FDIFF_1:def 7;
verum
end;
hence
( sec * f is_differentiable_on Z & ( for x being Real st x in Z holds
((sec * f) `| Z) . x = (a * (sin . ((a * x) + b))) / ((cos . ((a * x) + b)) ^2) ) )
by A1, A6, FDIFF_1:9; verum