let a be Real; for A being non empty closed_interval Subset of REAL
for f, g, 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
( g . x = 1 & f1 . x = x / a & f1 . x > - 1 & f1 . x < 1 ) ) & Z = dom f & f | A is continuous & f = (arcsin * f1) + ((id Z) / (a (#) ((#R (1 / 2)) * (g - (f1 ^2))))) holds
integral (f,A) = (((id Z) (#) (arcsin * f1)) . (upper_bound A)) - (((id Z) (#) (arcsin * f1)) . (lower_bound A))
let A be non empty closed_interval Subset of REAL; for f, g, 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
( g . x = 1 & f1 . x = x / a & f1 . x > - 1 & f1 . x < 1 ) ) & Z = dom f & f | A is continuous & f = (arcsin * f1) + ((id Z) / (a (#) ((#R (1 / 2)) * (g - (f1 ^2))))) holds
integral (f,A) = (((id Z) (#) (arcsin * f1)) . (upper_bound A)) - (((id Z) (#) (arcsin * f1)) . (lower_bound A))
let f, g, f1 be PartFunc of REAL,REAL; for Z being open Subset of REAL st A c= Z & ( for x being Real st x in Z holds
( g . x = 1 & f1 . x = x / a & f1 . x > - 1 & f1 . x < 1 ) ) & Z = dom f & f | A is continuous & f = (arcsin * f1) + ((id Z) / (a (#) ((#R (1 / 2)) * (g - (f1 ^2))))) holds
integral (f,A) = (((id Z) (#) (arcsin * f1)) . (upper_bound A)) - (((id Z) (#) (arcsin * f1)) . (lower_bound A))
let Z be open Subset of REAL; ( A c= Z & ( for x being Real st x in Z holds
( g . x = 1 & f1 . x = x / a & f1 . x > - 1 & f1 . x < 1 ) ) & Z = dom f & f | A is continuous & f = (arcsin * f1) + ((id Z) / (a (#) ((#R (1 / 2)) * (g - (f1 ^2))))) implies integral (f,A) = (((id Z) (#) (arcsin * f1)) . (upper_bound A)) - (((id Z) (#) (arcsin * f1)) . (lower_bound A)) )
assume A1:
( A c= Z & ( for x being Real st x in Z holds
( g . x = 1 & f1 . x = x / a & f1 . x > - 1 & f1 . x < 1 ) ) & Z = dom f & f | A is continuous & f = (arcsin * f1) + ((id Z) / (a (#) ((#R (1 / 2)) * (g - (f1 ^2))))) )
; integral (f,A) = (((id Z) (#) (arcsin * f1)) . (upper_bound A)) - (((id Z) (#) (arcsin * f1)) . (lower_bound A))
then
Z = (dom (arcsin * f1)) /\ (dom ((id Z) / (a (#) ((#R (1 / 2)) * (g - (f1 ^2))))))
by VALUED_1:def 1;
then A2:
( Z c= dom (arcsin * f1) & Z c= dom ((id Z) / (a (#) ((#R (1 / 2)) * (g - (f1 ^2))))) )
by XBOOLE_1:18;
Z c= (dom (id Z)) /\ (dom (arcsin * f1))
by A2, XBOOLE_1:19;
then A3:
Z c= dom ((id Z) (#) (arcsin * f1))
by VALUED_1:def 4;
Z c= (dom (id Z)) /\ ((dom (a (#) ((#R (1 / 2)) * (g - (f1 ^2))))) \ ((a (#) ((#R (1 / 2)) * (g - (f1 ^2)))) " {0}))
by A2, RFUNCT_1:def 1;
then
Z c= (dom (a (#) ((#R (1 / 2)) * (g - (f1 ^2))))) \ ((a (#) ((#R (1 / 2)) * (g - (f1 ^2)))) " {0})
by XBOOLE_1:18;
then A4:
Z c= dom ((a (#) ((#R (1 / 2)) * (g - (f1 ^2)))) ^)
by RFUNCT_1:def 2;
dom ((a (#) ((#R (1 / 2)) * (g - (f1 ^2)))) ^) c= dom (a (#) ((#R (1 / 2)) * (g - (f1 ^2))))
by RFUNCT_1:1;
then
Z c= dom (a (#) ((#R (1 / 2)) * (g - (f1 ^2))))
by A4;
then A5:
Z c= dom ((#R (1 / 2)) * (g - (f1 ^2)))
by VALUED_1:def 5;
A6:
( f is_integrable_on A & f | A is bounded )
by A1, INTEGRA5:10, INTEGRA5:11;
A7:
for x being Real st x in Z holds
( f1 . x = x / a & f1 . x > - 1 & f1 . x < 1 )
by A1;
then A8:
(id Z) (#) (arcsin * f1) is_differentiable_on Z
by A3, FDIFF_7:25;
A9:
for x being Real st x in Z holds
f . x = (arcsin . (x / a)) + (x / (a * (sqrt (1 - ((x / a) ^2)))))
proof
let x be
Real;
( x in Z implies f . x = (arcsin . (x / a)) + (x / (a * (sqrt (1 - ((x / a) ^2))))) )
assume A10:
x in Z
;
f . x = (arcsin . (x / a)) + (x / (a * (sqrt (1 - ((x / a) ^2)))))
then A11:
(
x in dom (g - (f1 ^2)) &
(g - (f1 ^2)) . x in dom (#R (1 / 2)) )
by A5, FUNCT_1:11;
then A12:
(g - (f1 ^2)) . x in right_open_halfline 0
by TAYLOR_1:def 4;
(
- 1
< f1 . x &
f1 . x < 1 )
by A1, A10;
then
(
0 < 1
+ (f1 . x) &
0 < 1
- (f1 . x) )
by XREAL_1:50, XREAL_1:148;
then A13:
0 < (1 + (f1 . x)) * (1 - (f1 . x))
by XREAL_1:129;
A14:
f1 . x = x / a
by A1, A10;
((arcsin * f1) + ((id Z) / (a (#) ((#R (1 / 2)) * (g - (f1 ^2)))))) . x =
((arcsin * f1) . x) + (((id Z) / (a (#) ((#R (1 / 2)) * (g - (f1 ^2))))) . x)
by A1, A10, VALUED_1:def 1
.=
(arcsin . (f1 . x)) + (((id Z) / (a (#) ((#R (1 / 2)) * (g - (f1 ^2))))) . x)
by A2, A10, FUNCT_1:12
.=
(arcsin . (x / a)) + (((id Z) / (a (#) ((#R (1 / 2)) * (g - (f1 ^2))))) . x)
by A1, A10
.=
(arcsin . (x / a)) + (((id Z) . x) / ((a (#) ((#R (1 / 2)) * (g - (f1 ^2)))) . x))
by A2, A10, RFUNCT_1:def 1
.=
(arcsin . (x / a)) + (x / ((a (#) ((#R (1 / 2)) * (g - (f1 ^2)))) . x))
by A10, FUNCT_1:18
.=
(arcsin . (x / a)) + (x / (a * (((#R (1 / 2)) * (g - (f1 ^2))) . x)))
by VALUED_1:6
.=
(arcsin . (x / a)) + (x / (a * ((#R (1 / 2)) . ((g - (f1 ^2)) . x))))
by A5, A10, FUNCT_1:12
.=
(arcsin . (x / a)) + (x / (a * (((g - (f1 ^2)) . x) #R (1 / 2))))
by A12, TAYLOR_1:def 4
.=
(arcsin . (x / a)) + (x / (a * (((g . x) - ((f1 ^2) . x)) #R (1 / 2))))
by A11, VALUED_1:13
.=
(arcsin . (x / a)) + (x / (a * (((g . x) - ((f1 . x) ^2)) #R (1 / 2))))
by VALUED_1:11
.=
(arcsin . (x / a)) + (x / (a * ((1 - ((f1 . x) ^2)) #R (1 / 2))))
by A1, A10
.=
(arcsin . (x / a)) + (x / (a * ((1 - ((x / a) ^2)) #R (1 / 2))))
by A1, A10
.=
(arcsin . (x / a)) + (x / (a * (sqrt (1 - ((x / a) ^2)))))
by A14, A13, FDIFF_7:2
;
hence
f . x = (arcsin . (x / a)) + (x / (a * (sqrt (1 - ((x / a) ^2)))))
by A1;
verum
end;
A15:
for x being Element of REAL st x in dom (((id Z) (#) (arcsin * f1)) `| Z) holds
(((id Z) (#) (arcsin * f1)) `| Z) . x = f . x
dom (((id Z) (#) (arcsin * f1)) `| Z) = dom f
by A1, A8, FDIFF_1:def 7;
then
((id Z) (#) (arcsin * f1)) `| Z = f
by A15, PARTFUN1:5;
hence
integral (f,A) = (((id Z) (#) (arcsin * f1)) . (upper_bound A)) - (((id Z) (#) (arcsin * f1)) . (lower_bound A))
by A1, A6, A8, INTEGRA5:13; verum