let n be Element of NAT ; :: thesis:
let A be closed-interval Subset of REAL; :: thesis:
let f1, f be PartFunc of REAL,REAL; :: thesis:
let Z be open Subset of REAL; :: thesis:
assume A1: ( A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f1 . x = 1 ) & Z c= dom ((#Z n) * arccos) & Z = dom f & 1 < n & f = (n (#) ((#Z (n - 1)) * arccos)) / ((#R (1 / 2)) * (f1 - (#Z 2))) ) ; :: thesis:
then Z = (dom (n (#) ((#Z (n - 1)) * arccos))) /\ ((dom ((#R (1 / 2)) * (f1 - (#Z 2)))) \ (((#R (1 / 2)) * (f1 - (#Z 2))) " {0})) by RFUNCT_1:def 4;
then A3: ( Z c= dom (n (#) ((#Z (n - 1)) * arccos)) & Z c= (dom ((#R (1 / 2)) * (f1 - (#Z 2)))) \ (((#R (1 / 2)) * (f1 - (#Z 2))) " {0}) ) by XBOOLE_1:18;
then A4: Z c= dom ((#Z (n - 1)) * arccos) by VALUED_1:def 5;
A5: Z c= dom (((#R (1 / 2)) * (f1 - (#Z 2))) ^) by RFUNCT_1:def 8, A3;
dom (((#R (1 / 2)) * (f1 - (#Z 2))) ^) c= dom ((#R (1 / 2)) * (f1 - (#Z 2))) by RFUNCT_1:11;
then A6: Z c= dom ((#R (1 / 2)) * (f1 - (#Z 2))) by A5, XBOOLE_1:1;
for x being Real st x in Z holds
(#Z (n - 1)) * arccos is_differentiable_in x

proof

let x be Real; :: thesis:
assume x in Z ; :: thesis:
then A8: arccos is_differentiable_in x by A1, FDIFF_1:16, SIN_COS6:108;
consider m being Nat such that
A9: n = m + 1 by A1, NAT_1:6;
thus (#Z (n - 1)) * arccos is_differentiable_in x by A8, A9, TAYLOR_1:3; :: thesis:

end;

then (#Z (n - 1)) * arccos is_differentiable_on Z by A4, FDIFF_1:16;
then A11: n (#) ((#Z (n - 1)) * arccos) is_differentiable_on Z by A3, FDIFF_1:28;
set f2 = #Z 2;
for x being Real st x in Z holds
(f1 - (#Z 2)) . x > 0

proof

let x be Real; :: thesis:
assume A14: x in Z ; :: thesis:
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 A15: 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, A6;
then (f1 - (#Z 2)) . x = (f1 . x) - ((#Z 2) . x) by A14, 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, A14 ;
hence (f1 - (#Z 2)) . x > 0 by A15; :: thesis:

end;

then for x being Real st x in Z holds
( f1 . x = 1 & (f1 - (#Z 2)) . x > 0 ) by A1;
then A23: (#R (1 / 2)) * (f1 - (#Z 2)) is_differentiable_on Z by A6, FDIFF_7:22;
for x being Real st x in Z holds
((#R (1 / 2)) * (f1 - (#Z 2))) . x <> 0 by RFUNCT_1:13, A5;
then f is_differentiable_on Z by A1, A11, A23, FDIFF_2:21;
then f | Z is continuous by FDIFF_1:33;
then f | A is continuous by A1, FCONT_1:17;
then A27: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A28: (#Z n) * arccos is_differentiable_on Z by A1, FDIFF_7:11;
A29: Z c= dom (- ((#Z n) * arccos)) by A1, VALUED_1:8;
then A30: (- 1) (#) ((#Z n) * arccos) is_differentiable_on Z by A28, FDIFF_1:28, X;
B1: for x being Real st x in Z holds
f . x = (n * ((arccos . x) #Z (n - 1))) / (sqrt (1 - (x ^2)))

proof

let x be Real; :: thesis:
assume B2: x in Z ; :: thesis:
then B3: ( x in dom (f1 - (#Z 2)) & (f1 - (#Z 2)) . x in dom (#R (1 / 2)) ) by FUNCT_1:21, A6;
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;
((n (#) ((#Z (n - 1)) * arccos)) / ((#R (1 / 2)) * (f1 - (#Z 2)))) . x = ((n (#) ((#Z (n - 1)) * arccos)) . x) / (((#R (1 / 2)) * (f1 - (#Z 2))) . x) by RFUNCT_1:def 4, A1, B2
.= (n * (((#Z (n - 1)) * arccos) . x)) / (((#R (1 / 2)) * (f1 - (#Z 2))) . x) by VALUED_1:6
.= (n * ((#Z (n - 1)) . (arccos . x))) / (((#R (1 / 2)) * (f1 - (#Z 2))) . x) by FUNCT_1:22, A4, B2
.= (n * ((arccos . x) #Z (n - 1))) / (((#R (1 / 2)) * (f1 - (#Z 2))) . x) by TAYLOR_1:def 1
.= (n * ((arccos . x) #Z (n - 1))) / ((#R (1 / 2)) . ((f1 - (#Z 2)) . x)) by FUNCT_1:22, A6, B2
.= (n * ((arccos . x) #Z (n - 1))) / (((f1 - (#Z 2)) . x) #R (1 / 2)) by TAYLOR_1:def 4, B4
.= (n * ((arccos . x) #Z (n - 1))) / (((f1 . x) - ((#Z 2) . x)) #R (1 / 2)) by VALUED_1:13, B3
.= (n * ((arccos . x) #Z (n - 1))) / (((f1 . x) - (x #Z 2)) #R (1 / 2)) by TAYLOR_1:def 1
.= (n * ((arccos . x) #Z (n - 1))) / (((f1 . x) - (x ^2)) #R (1 / 2)) by FDIFF_7:1
.= (n * ((arccos . x) #Z (n - 1))) / ((1 - (x ^2)) #R (1 / 2)) by A1, B2
.= (n * ((arccos . x) #Z (n - 1))) / (sqrt (1 - (x ^2))) by FDIFF_7:2, B6 ;
hence f . x = (n * ((arccos . x) #Z (n - 1))) / (sqrt (1 - (x ^2))) by A1; :: thesis:

end;

A31: for x being Real st x in Z holds
((- ((#Z n) * arccos)) `| Z) . x = (n * ((arccos . x) #Z (n - 1))) / (sqrt (1 - (x ^2)))

proof

let x be Real; :: thesis:
assume A32: x in Z ; :: thesis:
then A33: ( - 1 < x & x < 1 ) by A1, XXREAL_1:4;
A34: arccos is_differentiable_in x by A1, A32, FDIFF_1:16, SIN_COS6:108;
A35: (#Z n) * arccos is_differentiable_in x by A28, A32, FDIFF_1:16;
((- ((#Z n) * arccos)) `| Z) . x = diff ((- ((#Z n) * arccos)),x) by A30, A32, FDIFF_1:def 8
.= (- 1) * (diff (((#Z n) * arccos),x)) by A35, FDIFF_1:23, X
.= (- 1) * ((n * ((arccos . x) #Z (n - 1))) * (diff (arccos,x))) by A34, TAYLOR_1:3
.= (- 1) * ((n * ((arccos . x) #Z (n - 1))) * (- (1 / (sqrt (1 - (x ^2)))))) by A33, SIN_COS6:108
.= (n * ((arccos . x) #Z (n - 1))) / (sqrt (1 - (x ^2))) ;
hence ((- ((#Z n) * arccos)) `| Z) . x = (n * ((arccos . x) #Z (n - 1))) / (sqrt (1 - (x ^2))) ; :: thesis:

end;

A36: for x being Real st x in dom ((- ((#Z n) * arccos)) `| Z) holds
((- ((#Z n) * arccos)) `| Z) . x = f . x

proof

let x be Real; :: thesis:
assume x in dom ((- ((#Z n) * arccos)) `| Z) ; :: thesis:
then A37: x in Z by A30, FDIFF_1:def 8;
then ((- ((#Z n) * arccos)) `| Z) . x = (n * ((arccos . x) #Z (n - 1))) / (sqrt (1 - (x ^2))) by A31
.= f . x by B1, A37 ;
hence ((- ((#Z n) * arccos)) `| Z) . x = f . x ; :: thesis:

end;

dom ((- ((#Z n) * arccos)) `| Z) = dom f by A1, A30, FDIFF_1:def 8;
then (- ((#Z n) * arccos)) `| Z = f by A36, PARTFUN1:34;
hence integral (f,A) = ((- ((#Z n) * arccos)) . (upper_bound A)) - ((- ((#Z n) * arccos)) . (lower_bound A)) by A1, A27, A28, A29, FDIFF_1:28, X, INTEGRA5:13; :: thesis: