let a, b be Real; for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom (f (#) cosec) & ( for x being Real st x in Z holds
f . x = (a * x) + b ) holds
( f (#) cosec is_differentiable_on Z & ( for x being Real st x in Z holds
((f (#) cosec) `| Z) . x = (a / (sin . x)) - ((((a * x) + b) * (cos . x)) / ((sin . x) ^2)) ) )
let Z be open Subset of REAL; for f being PartFunc of REAL,REAL st Z c= dom (f (#) cosec) & ( for x being Real st x in Z holds
f . x = (a * x) + b ) holds
( f (#) cosec is_differentiable_on Z & ( for x being Real st x in Z holds
((f (#) cosec) `| Z) . x = (a / (sin . x)) - ((((a * x) + b) * (cos . x)) / ((sin . x) ^2)) ) )
let f be PartFunc of REAL,REAL; ( Z c= dom (f (#) cosec) & ( for x being Real st x in Z holds
f . x = (a * x) + b ) implies ( f (#) cosec is_differentiable_on Z & ( for x being Real st x in Z holds
((f (#) cosec) `| Z) . x = (a / (sin . x)) - ((((a * x) + b) * (cos . x)) / ((sin . x) ^2)) ) ) )
assume that
A1:
Z c= dom (f (#) cosec)
and
A2:
for x being Real st x in Z holds
f . x = (a * x) + b
; ( f (#) cosec is_differentiable_on Z & ( for x being Real st x in Z holds
((f (#) cosec) `| Z) . x = (a / (sin . x)) - ((((a * x) + b) * (cos . x)) / ((sin . x) ^2)) ) )
A3:
Z c= (dom f) /\ (dom cosec)
by A1, VALUED_1:def 4;
then A4:
Z c= dom cosec
by XBOOLE_1:18;
A5:
Z c= dom f
by A3, XBOOLE_1:18;
then A6:
f is_differentiable_on Z
by A2, FDIFF_1:23;
A7:
for x being Real st x in Z holds
( cosec is_differentiable_in x & diff (cosec,x) = - ((cos . x) / ((sin . x) ^2)) )
then
for x being Real st x in Z holds
cosec is_differentiable_in x
;
then A8:
cosec is_differentiable_on Z
by A4, FDIFF_1:9;
for x being Real st x in Z holds
((f (#) cosec) `| Z) . x = (a / (sin . x)) - ((((a * x) + b) * (cos . x)) / ((sin . x) ^2))
proof
let x be
Real;
( x in Z implies ((f (#) cosec) `| Z) . x = (a / (sin . x)) - ((((a * x) + b) * (cos . x)) / ((sin . x) ^2)) )
assume A9:
x in Z
;
((f (#) cosec) `| Z) . x = (a / (sin . x)) - ((((a * x) + b) * (cos . x)) / ((sin . x) ^2))
then ((f (#) cosec) `| Z) . x =
((cosec . x) * (diff (f,x))) + ((f . x) * (diff (cosec,x)))
by A1, A6, A8, FDIFF_1:21
.=
((cosec . x) * ((f `| Z) . x)) + ((f . x) * (diff (cosec,x)))
by A6, A9, FDIFF_1:def 7
.=
((cosec . x) * a) + ((f . x) * (diff (cosec,x)))
by A2, A5, A9, FDIFF_1:23
.=
((cosec . x) * a) + (((a * x) + b) * (diff (cosec,x)))
by A2, A9
.=
((cosec . x) * a) + (((a * x) + b) * (- ((cos . x) / ((sin . x) ^2))))
by A7, A9
.=
(a / (sin . x)) - ((((a * x) + b) * (cos . x)) / ((sin . x) ^2))
by A4, A9, RFUNCT_1:def 2
;
hence
((f (#) cosec) `| Z) . x = (a / (sin . x)) - ((((a * x) + b) * (cos . x)) / ((sin . x) ^2))
;
verum
end;
hence
( f (#) cosec is_differentiable_on Z & ( for x being Real st x in Z holds
((f (#) cosec) `| Z) . x = (a / (sin . x)) - ((((a * x) + b) * (cos . x)) / ((sin . x) ^2)) ) )
by A1, A6, A8, FDIFF_1:21; verum