let a, b be Real; for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom (arccos * f) & ( for x being Real st x in Z holds
( f . x = (a * x) + b & f . x > - 1 & f . x < 1 ) ) holds
( arccos * f is_differentiable_on Z & ( for x being Real st x in Z holds
((arccos * f) `| Z) . x = - (a / (sqrt (1 - (((a * x) + b) ^2)))) ) )
let Z be open Subset of REAL; for f being PartFunc of REAL,REAL st Z c= dom (arccos * f) & ( for x being Real st x in Z holds
( f . x = (a * x) + b & f . x > - 1 & f . x < 1 ) ) holds
( arccos * f is_differentiable_on Z & ( for x being Real st x in Z holds
((arccos * f) `| Z) . x = - (a / (sqrt (1 - (((a * x) + b) ^2)))) ) )
let f be PartFunc of REAL,REAL; ( Z c= dom (arccos * f) & ( for x being Real st x in Z holds
( f . x = (a * x) + b & f . x > - 1 & f . x < 1 ) ) implies ( arccos * f is_differentiable_on Z & ( for x being Real st x in Z holds
((arccos * f) `| Z) . x = - (a / (sqrt (1 - (((a * x) + b) ^2)))) ) ) )
assume that
A1:
Z c= dom (arccos * f)
and
A2:
for x being Real st x in Z holds
( f . x = (a * x) + b & f . x > - 1 & f . x < 1 )
; ( arccos * f is_differentiable_on Z & ( for x being Real st x in Z holds
((arccos * f) `| Z) . x = - (a / (sqrt (1 - (((a * x) + b) ^2)))) ) )
for y being object st y in Z holds
y in dom f
by A1, FUNCT_1:11;
then A3:
Z c= dom f
by TARSKI:def 3;
A4:
for x being Real st x in Z holds
f . x = (a * x) + b
by A2;
then A5:
f is_differentiable_on Z
by A3, FDIFF_1:23;
A6:
for x being Real st x in Z holds
arccos * f is_differentiable_in x
then A9:
arccos * f is_differentiable_on Z
by A1, FDIFF_1:9;
for x being Real st x in Z holds
((arccos * f) `| Z) . x = - (a / (sqrt (1 - (((a * x) + b) ^2))))
proof
let x be
Real;
( x in Z implies ((arccos * f) `| Z) . x = - (a / (sqrt (1 - (((a * x) + b) ^2)))) )
assume A10:
x in Z
;
((arccos * f) `| Z) . x = - (a / (sqrt (1 - (((a * x) + b) ^2))))
then A11:
f . x < 1
by A2;
(
f is_differentiable_in x &
f . x > - 1 )
by A2, A5, A10, FDIFF_1:9;
then diff (
(arccos * f),
x) =
- ((diff (f,x)) / (sqrt (1 - ((f . x) ^2))))
by A11, Th7
.=
- (((f `| Z) . x) / (sqrt (1 - ((f . x) ^2))))
by A5, A10, FDIFF_1:def 7
.=
- (a / (sqrt (1 - ((f . x) ^2))))
by A4, A3, A10, FDIFF_1:23
.=
- (a / (sqrt (1 - (((a * x) + b) ^2))))
by A2, A10
;
hence
((arccos * f) `| Z) . x = - (a / (sqrt (1 - (((a * x) + b) ^2))))
by A9, A10, FDIFF_1:def 7;
verum
end;
hence
( arccos * f is_differentiable_on Z & ( for x being Real st x in Z holds
((arccos * f) `| Z) . x = - (a / (sqrt (1 - (((a * x) + b) ^2)))) ) )
by A1, A6, FDIFF_1:9; verum