let A be non empty closed_interval Subset of REAL; :: thesis: for f, 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
( exp_R . x < 1 & f1 . x = 1 ) ) & Z c= dom (arctan * exp_R) & Z = dom f & f = exp_R / (f1 + (exp_R ^2)) holds
integral (f,A) = ((arctan * exp_R) . (upper_bound A)) - ((arctan * exp_R) . (lower_bound A))
let f, f1 be PartFunc of REAL,REAL; :: thesis: for Z being open Subset of REAL st A c= Z & ( for x being Real st x in Z holds
( exp_R . x < 1 & f1 . x = 1 ) ) & Z c= dom (arctan * exp_R) & Z = dom f & f = exp_R / (f1 + (exp_R ^2)) holds
integral (f,A) = ((arctan * exp_R) . (upper_bound A)) - ((arctan * exp_R) . (lower_bound A))
let Z be open Subset of REAL; :: thesis: ( A c= Z & ( for x being Real st x in Z holds
( exp_R . x < 1 & f1 . x = 1 ) ) & Z c= dom (arctan * exp_R) & Z = dom f & f = exp_R / (f1 + (exp_R ^2)) implies integral (f,A) = ((arctan * exp_R) . (upper_bound A)) - ((arctan * exp_R) . (lower_bound A)) )
assume A1: ( A c= Z & ( for x being Real st x in Z holds
( exp_R . x < 1 & f1 . x = 1 ) ) & Z c= dom (arctan * exp_R) & Z = dom f & f = exp_R / (f1 + (exp_R ^2)) ) ; :: thesis: integral (f,A) = ((arctan * exp_R) . (upper_bound A)) - ((arctan * exp_R) . (lower_bound A))
then Z c= (dom exp_R) /\ ((dom (f1 + (exp_R ^2))) \ ((f1 + (exp_R ^2)) " {0})) by RFUNCT_1:def 1;
then ( Z c= dom exp_R & Z c= (dom (f1 + (exp_R ^2))) \ ((f1 + (exp_R ^2)) " {0}) ) by XBOOLE_1:18;
then A2: Z c= dom ((f1 + (exp_R ^2)) ^) by RFUNCT_1:def 2;
dom ((f1 + (exp_R ^2)) ^) c= dom (f1 + (exp_R ^2)) by RFUNCT_1:1;
then A3: Z c= dom (f1 + (exp_R ^2)) by A2;
then A4: Z c= (dom f1) /\ (dom (exp_R ^2)) by VALUED_1:def 1;
then A5: ( Z c= dom f1 & Z c= dom (exp_R ^2) ) by XBOOLE_1:18;
A6: Z c= dom (exp_R (#) exp_R) by A4, XBOOLE_1:18;
A7: exp_R is_differentiable_on Z by FDIFF_1:26, TAYLOR_1:16;
then A8: exp_R (#) exp_R is_differentiable_on Z by A6, FDIFF_1:21;
for x being Real st x in Z holds
f1 . x = (0 * x) + 1 by A1;
then f1 is_differentiable_on Z by A5, FDIFF_1:23;
then A9: f1 + (exp_R ^2) is_differentiable_on Z by A3, A8, FDIFF_1:18;
for x being Real st x in Z holds
(f1 + (exp_R ^2)) . x <> 0

proof

let x be Real; :: thesis: ( x in Z implies (f1 + (exp_R ^2)) . x <> 0 )
assume x in Z ; :: thesis: (f1 + (exp_R ^2)) . x <> 0
then x in (dom exp_R) /\ ((dom (f1 + (exp_R ^2))) \ ((f1 + (exp_R ^2)) " {0})) by A1, RFUNCT_1:def 1;
then x in (dom (f1 + (exp_R ^2))) \ ((f1 + (exp_R ^2)) " {0}) by XBOOLE_0:def 4;
then x in dom ((f1 + (exp_R ^2)) ^) by RFUNCT_1:def 2;
hence (f1 + (exp_R ^2)) . x <> 0 by RFUNCT_1:3; :: thesis: verum

end;

then f is_differentiable_on Z by A1, A7, A9, FDIFF_2:21;
then f | Z is continuous by FDIFF_1:25;
then f | A is continuous by A1, FCONT_1:16;
then A10: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A11: for x being Real st x in Z holds
exp_R . x < 1 by A1;
then A12: arctan * exp_R is_differentiable_on Z by A1, SIN_COS9:115;
A13: for x being Real st x in Z holds
f . x = (exp_R . x) / (1 + ((exp_R . x) ^2))

proof

let x be Real; :: thesis: ( x in Z implies f . x = (exp_R . x) / (1 + ((exp_R . x) ^2)) )
assume A14: x in Z ; :: thesis: f . x = (exp_R . x) / (1 + ((exp_R . x) ^2))
then (exp_R / (f1 + (exp_R ^2))) . x = (exp_R . x) * (((f1 + (exp_R ^2)) . x) ") by A1, RFUNCT_1:def 1
.= (exp_R . x) * (((f1 . x) + ((exp_R ^2) . x)) ") by A14, A3, VALUED_1:def 1
.= (exp_R . x) * (((f1 . x) + ((exp_R . x) ^2)) ") by VALUED_1:11
.= (exp_R . x) / (1 + ((exp_R . x) ^2)) by A1, A14 ;
hence f . x = (exp_R . x) / (1 + ((exp_R . x) ^2)) by A1; :: thesis: verum

end;

A15: for x being Element of REAL st x in dom ((arctan * exp_R) `| Z) holds
((arctan * exp_R) `| Z) . x = f . x

proof

let x be Element of REAL ; :: thesis: ( x in dom ((arctan * exp_R) `| Z) implies ((arctan * exp_R) `| Z) . x = f . x )
assume x in dom ((arctan * exp_R) `| Z) ; :: thesis: ((arctan * exp_R) `| Z) . x = f . x
then A16: x in Z by A12, FDIFF_1:def 7;
then ((arctan * exp_R) `| Z) . x = (exp_R . x) / (1 + ((exp_R . x) ^2)) by A1, A11, SIN_COS9:115
.= f . x by A16, A13 ;
hence ((arctan * exp_R) `| Z) . x = f . x ; :: thesis: verum

end;

dom ((arctan * exp_R) `| Z) = dom f by A1, A12, FDIFF_1:def 7;
then (arctan * exp_R) `| Z = f by A15, PARTFUN1:5;
hence integral (f,A) = ((arctan * exp_R) . (upper_bound A)) - ((arctan * exp_R) . (lower_bound A)) by A1, A10, A12, INTEGRA5:13; :: thesis: verum