let a, b be Real; :: thesis: 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 ; :: thesis: 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 ; :: thesis: ( 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 ) ; :: thesis: ( 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 set st y in Z holds
y in dom f by A1, FUNCT_1:21;
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_1:31;
A6: for x being Real st x in Z holds
arccos * f is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies arccos * f is_differentiable_in x )
assume A7: x in Z ; :: thesis: arccos * f is_differentiable_in x
then A8: f . x < 1 by A2;
( f is_differentiable_in x & f . x > - 1 ) by A2, A5, A7, FDIFF_1:16;
hence arccos * f is_differentiable_in x by A8, Th7; :: thesis: verum

end;

then A9: arccos * f is_differentiable_on Z by A1, FDIFF_1:16;
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; :: thesis: ( x in Z implies ((arccos * f) `| Z) . x = - (a / (sqrt (1 - (((a * x) + b) ^2 )))) )
assume A10: x in Z ; :: thesis: ((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_1:16;
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 8
.= - (a / (sqrt (1 - ((f . x) ^2 )))) by A4, A3, A10, FDIFF_1:31
.= - (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 8; :: thesis: 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_1:16; :: thesis: verum