let r 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 & f = (arctan * f1) + ((id Z) / (r (#) (g + (f1 ^2)))) & ( for x being Real st x in Z holds
( g . x = 1 & f1 . x = x / r & f1 . x > - 1 & f1 . x < 1 ) ) & Z = dom f & f | A is continuous holds
integral (f,A) = (((id Z) (#) (arctan * f1)) . (upper_bound A)) - (((id Z) (#) (arctan * 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 & f = (arctan * f1) + ((id Z) / (r (#) (g + (f1 ^2)))) & ( for x being Real st x in Z holds
( g . x = 1 & f1 . x = x / r & f1 . x > - 1 & f1 . x < 1 ) ) & Z = dom f & f | A is continuous holds
integral (f,A) = (((id Z) (#) (arctan * f1)) . (upper_bound A)) - (((id Z) (#) (arctan * f1)) . (lower_bound A))
let f, g, f1 be PartFunc of REAL,REAL; for Z being open Subset of REAL st A c= Z & f = (arctan * f1) + ((id Z) / (r (#) (g + (f1 ^2)))) & ( for x being Real st x in Z holds
( g . x = 1 & f1 . x = x / r & f1 . x > - 1 & f1 . x < 1 ) ) & Z = dom f & f | A is continuous holds
integral (f,A) = (((id Z) (#) (arctan * f1)) . (upper_bound A)) - (((id Z) (#) (arctan * f1)) . (lower_bound A))
let Z be open Subset of REAL; ( A c= Z & f = (arctan * f1) + ((id Z) / (r (#) (g + (f1 ^2)))) & ( for x being Real st x in Z holds
( g . x = 1 & f1 . x = x / r & f1 . x > - 1 & f1 . x < 1 ) ) & Z = dom f & f | A is continuous implies integral (f,A) = (((id Z) (#) (arctan * f1)) . (upper_bound A)) - (((id Z) (#) (arctan * f1)) . (lower_bound A)) )
assume A1:
( A c= Z & f = (arctan * f1) + ((id Z) / (r (#) (g + (f1 ^2)))) & ( for x being Real st x in Z holds
( g . x = 1 & f1 . x = x / r & f1 . x > - 1 & f1 . x < 1 ) ) & Z = dom f & f | A is continuous )
; integral (f,A) = (((id Z) (#) (arctan * f1)) . (upper_bound A)) - (((id Z) (#) (arctan * f1)) . (lower_bound A))
then
Z = (dom (arctan * f1)) /\ (dom ((id Z) / (r (#) (g + (f1 ^2)))))
by VALUED_1:def 1;
then A2:
( Z c= dom (arctan * f1) & Z c= dom ((id Z) / (r (#) (g + (f1 ^2)))) )
by XBOOLE_1:18;
Z c= (dom (id Z)) /\ (dom (arctan * f1))
by A2, XBOOLE_1:19;
then A3:
Z c= dom ((id Z) (#) (arctan * f1))
by VALUED_1:def 4;
Z c= (dom (id Z)) /\ ((dom (r (#) (g + (f1 ^2)))) \ ((r (#) (g + (f1 ^2))) " {0}))
by A2, RFUNCT_1:def 1;
then
Z c= (dom (r (#) (g + (f1 ^2)))) \ ((r (#) (g + (f1 ^2))) " {0})
by XBOOLE_1:18;
then A4:
Z c= dom ((r (#) (g + (f1 ^2))) ^)
by RFUNCT_1:def 2;
dom ((r (#) (g + (f1 ^2))) ^) c= dom (r (#) (g + (f1 ^2)))
by RFUNCT_1:1;
then
Z c= dom (r (#) (g + (f1 ^2)))
by A4;
then A5:
Z c= dom (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 / r & f1 . x > - 1 & f1 . x < 1 )
by A1;
then A8:
(id Z) (#) (arctan * f1) is_differentiable_on Z
by A3, SIN_COS9:105;
A9:
for x being Real st x in Z holds
f . x = (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2))))
proof
let x be
Real;
( x in Z implies f . x = (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2)))) )
assume A10:
x in Z
;
f . x = (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2))))
then ((arctan * f1) + ((id Z) / (r (#) (g + (f1 ^2))))) . x =
((arctan * f1) . x) + (((id Z) / (r (#) (g + (f1 ^2)))) . x)
by A1, VALUED_1:def 1
.=
(arctan . (f1 . x)) + (((id Z) / (r (#) (g + (f1 ^2)))) . x)
by A2, A10, FUNCT_1:12
.=
(arctan . (f1 . x)) + (((id Z) . x) / ((r (#) (g + (f1 ^2))) . x))
by A2, A10, RFUNCT_1:def 1
.=
(arctan . (f1 . x)) + (x / ((r (#) (g + (f1 ^2))) . x))
by A10, FUNCT_1:18
.=
(arctan . (f1 . x)) + (x / (r * ((g + (f1 ^2)) . x)))
by VALUED_1:6
.=
(arctan . (f1 . x)) + (x / (r * ((g . x) + ((f1 ^2) . x))))
by A5, A10, VALUED_1:def 1
.=
(arctan . (f1 . x)) + (x / (r * ((g . x) + ((f1 . x) ^2))))
by VALUED_1:11
.=
(arctan . (x / r)) + (x / (r * ((g . x) + ((f1 . x) ^2))))
by A1, A10
.=
(arctan . (x / r)) + (x / (r * (1 + ((f1 . x) ^2))))
by A1, A10
.=
(arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2))))
by A1, A10
;
hence
f . x = (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2))))
by A1;
verum
end;
A11:
for x being Element of REAL st x in dom (((id Z) (#) (arctan * f1)) `| Z) holds
(((id Z) (#) (arctan * f1)) `| Z) . x = f . x
dom (((id Z) (#) (arctan * f1)) `| Z) = dom f
by A1, A8, FDIFF_1:def 7;
then
((id Z) (#) (arctan * f1)) `| Z = f
by A11, PARTFUN1:5;
hence
integral (f,A) = (((id Z) (#) (arctan * f1)) . (upper_bound A)) - (((id Z) (#) (arctan * f1)) . (lower_bound A))
by A1, A6, A8, INTEGRA5:13; verum