let r be Real; :: thesis: 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; :: thesis: 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; :: thesis: 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; :: thesis: ( 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 ) ; :: thesis: 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; :: thesis: ( x in Z implies f . x = (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2)))) )
assume A10: x in Z ; :: thesis: 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; :: thesis: 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

proof

let x be Element of REAL ; :: thesis: ( x in dom (((id Z) (#) (arctan * f1)) `| Z) implies (((id Z) (#) (arctan * f1)) `| Z) . x = f . x )
assume x in dom (((id Z) (#) (arctan * f1)) `| Z) ; :: thesis: (((id Z) (#) (arctan * f1)) `| Z) . x = f . x
then A12: x in Z by A8, FDIFF_1:def 7;
then (((id Z) (#) (arctan * f1)) `| Z) . x = (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2)))) by A3, A7, SIN_COS9:105
.= f . x by A9, A12 ;
hence (((id Z) (#) (arctan * f1)) `| Z) . x = f . x ; :: thesis: verum

end;

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; :: thesis: verum