let a, b be Real; :: thesis: for A being non empty closed_interval Subset of REAL
for f, f1 being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & ( for x being Real st x in Z holds
( f . x = a / (sqrt (1 - (((a * x) + b) ^2))) & f1 . x = (a * x) + b & f1 . x > - 1 & f1 . x < 1 ) ) & Z c= dom (arccos * f1) & Z = dom f & f | A is continuous holds
integral (f,A) = ((- (arccos * f1)) . (upper_bound A)) - ((- (arccos * f1)) . (lower_bound A))
let A be non empty closed_interval Subset of REAL; :: thesis: for f, f1 being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & ( for x being Real st x in Z holds
( f . x = a / (sqrt (1 - (((a * x) + b) ^2))) & f1 . x = (a * x) + b & f1 . x > - 1 & f1 . x < 1 ) ) & Z c= dom (arccos * f1) & Z = dom f & f | A is continuous holds
integral (f,A) = ((- (arccos * f1)) . (upper_bound A)) - ((- (arccos * f1)) . (lower_bound A))
let f, f1 be PartFunc of REAL,REAL; :: thesis: for Z being open Subset of REAL st A c= Z & ( for x being Real st x in Z holds
( f . x = a / (sqrt (1 - (((a * x) + b) ^2))) & f1 . x = (a * x) + b & f1 . x > - 1 & f1 . x < 1 ) ) & Z c= dom (arccos * f1) & Z = dom f & f | A is continuous holds
integral (f,A) = ((- (arccos * f1)) . (upper_bound A)) - ((- (arccos * f1)) . (lower_bound A))
let Z be open Subset of REAL; :: thesis: ( A c= Z & ( for x being Real st x in Z holds
( f . x = a / (sqrt (1 - (((a * x) + b) ^2))) & f1 . x = (a * x) + b & f1 . x > - 1 & f1 . x < 1 ) ) & Z c= dom (arccos * f1) & Z = dom f & f | A is continuous implies integral (f,A) = ((- (arccos * f1)) . (upper_bound A)) - ((- (arccos * f1)) . (lower_bound A)) )
assume A1: ( A c= Z & ( for x being Real st x in Z holds
( f . x = a / (sqrt (1 - (((a * x) + b) ^2))) & f1 . x = (a * x) + b & f1 . x > - 1 & f1 . x < 1 ) ) & Z c= dom (arccos * f1) & Z = dom f & f | A is continuous ) ; :: thesis: integral (f,A) = ((- (arccos * f1)) . (upper_bound A)) - ((- (arccos * f1)) . (lower_bound A))
then A2: ( f is_integrable_on A & f | A is bounded ) by INTEGRA5:10, INTEGRA5:11;
A3: for x being Real st x in Z holds
f1 . x = (a * x) + b by A1;
A4: for x being Real st x in Z holds
( f1 . x = (a * x) + b & f1 . x > - 1 & f1 . x < 1 ) by A1;
A5: Z c= dom (- (arccos * f1)) by A1, VALUED_1:8;
A6: arccos * f1 is_differentiable_on Z by A1, A4, FDIFF_7:15;
then A7: (- 1) (#) (arccos * f1) is_differentiable_on Z by A5, FDIFF_1:20;
for y being object st y in Z holds
y in dom f1 by A1, FUNCT_1:11;
then A8: Z c= dom f1 ;
then A9: ( f1 is_differentiable_on Z & ( for x being Real st x in Z holds
(f1 `| Z) . x = a ) ) by A3, FDIFF_1:23;
A10: for x being Real st x in Z holds
((- (arccos * f1)) `| Z) . x = a / (sqrt (1 - (((a * x) + b) ^2)))

proof

let x be Real; :: thesis: ( x in Z implies ((- (arccos * f1)) `| Z) . x = a / (sqrt (1 - (((a * x) + b) ^2))) )
assume A11: x in Z ; :: thesis: ((- (arccos * f1)) `| Z) . x = a / (sqrt (1 - (((a * x) + b) ^2)))
then A12: f1 is_differentiable_in x by A9, FDIFF_1:9;
A13: ( f1 . x > - 1 & f1 . x < 1 ) by A1, A11;
A14: arccos * f1 is_differentiable_in x by A6, A11, FDIFF_1:9;
((- (arccos * f1)) `| Z) . x = diff ((- (arccos * f1)),x) by A7, A11, FDIFF_1:def 7
.= (- 1) * (diff ((arccos * f1),x)) by A14, FDIFF_1:15
.= (- 1) * (- ((diff (f1,x)) / (sqrt (1 - ((f1 . x) ^2))))) by A12, A13, FDIFF_7:7
.= (- 1) * (- (((f1 `| Z) . x) / (sqrt (1 - ((f1 . x) ^2))))) by A9, A11, FDIFF_1:def 7
.= (- 1) * (- (a / (sqrt (1 - ((f1 . x) ^2))))) by A3, A8, A11, FDIFF_1:23
.= a / (sqrt (1 - (((a * x) + b) ^2))) by A1, A11 ;
hence ((- (arccos * f1)) `| Z) . x = a / (sqrt (1 - (((a * x) + b) ^2))) ; :: thesis: verum

end;

A15: for x being Element of REAL st x in dom ((- (arccos * f1)) `| Z) holds
((- (arccos * f1)) `| Z) . x = f . x

proof

let x be Element of REAL ; :: thesis: ( x in dom ((- (arccos * f1)) `| Z) implies ((- (arccos * f1)) `| Z) . x = f . x )
assume x in dom ((- (arccos * f1)) `| Z) ; :: thesis: ((- (arccos * f1)) `| Z) . x = f . x
then A16: x in Z by A7, FDIFF_1:def 7;
then ((- (arccos * f1)) `| Z) . x = a / (sqrt (1 - (((a * x) + b) ^2))) by A10
.= f . x by A1, A16 ;
hence ((- (arccos * f1)) `| Z) . x = f . x ; :: thesis: verum

end;

dom ((- (arccos * f1)) `| Z) = dom f by A1, A7, FDIFF_1:def 7;
then (- (arccos * f1)) `| Z = f by A15, PARTFUN1:5;
hence integral (f,A) = ((- (arccos * f1)) . (upper_bound A)) - ((- (arccos * f1)) . (lower_bound A)) by A1, A2, A7, INTEGRA5:13; :: thesis: verum