let A be closed-interval Subset of REAL; :: thesis: for f1, f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
( arcsin . x > 0 & f1 . x = 1 ) ) & Z c= dom (ln * arcsin) & Z = dom f & f = (((#R (1 / 2)) * (f1 - (#Z 2))) (#) arcsin) ^ holds
integral (f,A) = ((ln * arcsin) . (upper_bound A)) - ((ln * arcsin) . (lower_bound A))
let f1, f be PartFunc of REAL,REAL; :: thesis: for Z being open Subset of REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
( arcsin . x > 0 & f1 . x = 1 ) ) & Z c= dom (ln * arcsin) & Z = dom f & f = (((#R (1 / 2)) * (f1 - (#Z 2))) (#) arcsin) ^ holds
integral (f,A) = ((ln * arcsin) . (upper_bound A)) - ((ln * arcsin) . (lower_bound A))
let Z be open Subset of REAL; :: thesis: ( A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
( arcsin . x > 0 & f1 . x = 1 ) ) & Z c= dom (ln * arcsin) & Z = dom f & f = (((#R (1 / 2)) * (f1 - (#Z 2))) (#) arcsin) ^ implies integral (f,A) = ((ln * arcsin) . (upper_bound A)) - ((ln * arcsin) . (lower_bound A)) )
assume A1: ( A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
( arcsin . x > 0 & f1 . x = 1 ) ) & Z c= dom (ln * arcsin) & Z = dom f & f = (((#R (1 / 2)) * (f1 - (#Z 2))) (#) arcsin) ^ ) ; :: thesis: integral (f,A) = ((ln * arcsin) . (upper_bound A)) - ((ln * arcsin) . (lower_bound A))
set g = ((#R (1 / 2)) * (f1 - (#Z 2))) (#) arcsin;
A3: Z c= dom (((#R (1 / 2)) * (f1 - (#Z 2))) (#) arcsin) by A1, RFUNCT_1:11;
dom (((#R (1 / 2)) * (f1 - (#Z 2))) (#) arcsin) = (dom ((#R (1 / 2)) * (f1 - (#Z 2)))) /\ (dom arcsin) by VALUED_1:def 4;
then A5: ( Z c= dom ((#R (1 / 2)) * (f1 - (#Z 2))) & Z c= dom arcsin ) by A3, XBOOLE_1:18;
A6: arcsin is_differentiable_on Z by A1, FDIFF_1:34, SIN_COS6:84;
set f2 = #Z 2;
for x being Real st x in Z holds
(f1 - (#Z 2)) . x > 0

proof

let x be Real; :: thesis: ( x in Z implies (f1 - (#Z 2)) . x > 0 )
assume A9: x in Z ; :: thesis: (f1 - (#Z 2)) . x > 0
then ( - 1 < x & x < 1 ) by A1, XXREAL_1:4;
then ( 0 < 1 + x & 0 < 1 - x ) by XREAL_1:150, XREAL_1:52;
then A10: 0 < (1 + x) * (1 - x) by XREAL_1:131;
for x being Real st x in Z holds
x in dom (f1 - (#Z 2)) by FUNCT_1:21, A5;
then (f1 - (#Z 2)) . x = (f1 . x) - ((#Z 2) . x) by A9, VALUED_1:13
.= (f1 . x) - (x #Z (1 + 1)) by TAYLOR_1:def 1
.= (f1 . x) - ((x #Z 1) * (x #Z 1)) by TAYLOR_1:1
.= (f1 . x) - (x * (x #Z 1)) by PREPOWER:45
.= (f1 . x) - (x * x) by PREPOWER:45
.= 1 - (x * x) by A1, A9 ;
hence (f1 - (#Z 2)) . x > 0 by A10; :: thesis: verum

end;

then for x being Real st x in Z holds
( f1 . x = 1 & (f1 - (#Z 2)) . x > 0 ) by A1;
then (#R (1 / 2)) * (f1 - (#Z 2)) is_differentiable_on Z by A5, FDIFF_7:22;
then A19: ((#R (1 / 2)) * (f1 - (#Z 2))) (#) arcsin is_differentiable_on Z by A3, A6, FDIFF_1:29;
for x being Real st x in Z holds
(((#R (1 / 2)) * (f1 - (#Z 2))) (#) arcsin) . x <> 0 by RFUNCT_1:13, A1;
then f is_differentiable_on Z by A1, A19, FDIFF_2:22;
then f | Z is continuous by FDIFF_1:33;
then f | A is continuous by A1, FCONT_1:17;
then A23: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A24: for x being Real st x in Z holds
arcsin . x > 0 by A1;
then A25: ln * arcsin is_differentiable_on Z by A1, FDIFF_7:8;
B1: for x being Real st x in Z holds
f . x = 1 / ((sqrt (1 - (x ^2))) * (arcsin . x))

proof

let x be Real; :: thesis: ( x in Z implies f . x = 1 / ((sqrt (1 - (x ^2))) * (arcsin . x)) )
assume B2: x in Z ; :: thesis: f . x = 1 / ((sqrt (1 - (x ^2))) * (arcsin . x))
then B3: ( x in dom (f1 - (#Z 2)) & (f1 - (#Z 2)) . x in dom (#R (1 / 2)) ) by FUNCT_1:21, A5;
then B4: (f1 - (#Z 2)) . x in right_open_halfline 0 by TAYLOR_1:def 4;
( - 1 < x & x < 1 ) by A1, XXREAL_1:4, B2;
then ( 0 < 1 + x & 0 < 1 - x ) by XREAL_1:150, XREAL_1:52;
then B6: 0 < (1 + x) * (1 - x) by XREAL_1:131;
((((#R (1 / 2)) * (f1 - (#Z 2))) (#) arcsin) ^) . x = 1 / ((((#R (1 / 2)) * (f1 - (#Z 2))) (#) arcsin) . x) by RFUNCT_1:def 8, A1, B2
.= 1 / ((((#R (1 / 2)) * (f1 - (#Z 2))) . x) * (arcsin . x)) by VALUED_1:5
.= 1 / (((#R (1 / 2)) . ((f1 - (#Z 2)) . x)) * (arcsin . x)) by FUNCT_1:22, A5, B2
.= 1 / ((((f1 - (#Z 2)) . x) #R (1 / 2)) * (arcsin . x)) by TAYLOR_1:def 4, B4
.= 1 / ((((f1 . x) - ((#Z 2) . x)) #R (1 / 2)) * (arcsin . x)) by VALUED_1:13, B3
.= 1 / ((((f1 . x) - (x #Z 2)) #R (1 / 2)) * (arcsin . x)) by TAYLOR_1:def 1
.= 1 / ((((f1 . x) - (x ^2)) #R (1 / 2)) * (arcsin . x)) by FDIFF_7:1
.= 1 / (((1 - (x ^2)) #R (1 / 2)) * (arcsin . x)) by A1, B2
.= 1 / ((sqrt (1 - (x ^2))) * (arcsin . x)) by FDIFF_7:2, B6 ;
hence f . x = 1 / ((sqrt (1 - (x ^2))) * (arcsin . x)) by A1; :: thesis: verum

end;

A26: for x being Real st x in dom ((ln * arcsin) `| Z) holds
((ln * arcsin) `| Z) . x = f . x

proof

let x be Real; :: thesis: ( x in dom ((ln * arcsin) `| Z) implies ((ln * arcsin) `| Z) . x = f . x )
assume x in dom ((ln * arcsin) `| Z) ; :: thesis: ((ln * arcsin) `| Z) . x = f . x
then A27: x in Z by A25, FDIFF_1:def 8;
then ((ln * arcsin) `| Z) . x = 1 / ((sqrt (1 - (x ^2))) * (arcsin . x)) by A1, A24, FDIFF_7:8
.= f . x by B1, A27 ;
hence ((ln * arcsin) `| Z) . x = f . x ; :: thesis: verum

end;

dom ((ln * arcsin) `| Z) = dom f by A1, A25, FDIFF_1:def 8;
then (ln * arcsin) `| Z = f by A26, PARTFUN1:34;
hence integral (f,A) = ((ln * arcsin) . (upper_bound A)) - ((ln * arcsin) . (lower_bound A)) by A1, A23, A25, INTEGRA5:13; :: thesis: verum