let A be closed-interval Subset of REAL; :: thesis: for f1, f 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 (arccot * exp_R) & Z = dom f & f = (- exp_R) / (f1 + (exp_R ^2)) holds
integral (f,A) = ((arccot * exp_R) . (upper_bound A)) - ((arccot * exp_R) . (lower_bound A))
let f1, f 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 (arccot * exp_R) & Z = dom f & f = (- exp_R) / (f1 + (exp_R ^2)) holds
integral (f,A) = ((arccot * exp_R) . (upper_bound A)) - ((arccot * 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 (arccot * exp_R) & Z = dom f & f = (- exp_R) / (f1 + (exp_R ^2)) implies integral (f,A) = ((arccot * exp_R) . (upper_bound A)) - ((arccot * 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 (arccot * exp_R) & Z = dom f & f = (- exp_R) / (f1 + (exp_R ^2)) ) ; :: thesis: integral (f,A) = ((arccot * exp_R) . (upper_bound A)) - ((arccot * 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 4;
then A4: ( Z c= dom (- exp_R) & Z c= (dom (f1 + (exp_R ^2))) \ ((f1 + (exp_R ^2)) " {0}) ) by XBOOLE_1:18;
then A5: Z c= dom ((f1 + (exp_R ^2)) ^) by RFUNCT_1:def 8;
dom ((f1 + (exp_R ^2)) ^) c= dom (f1 + (exp_R ^2)) by RFUNCT_1:11;
then A6: Z c= dom (f1 + (exp_R ^2)) by XBOOLE_1:1, A5;
then A7: Z c= (dom f1) /\ (dom (exp_R ^2)) by VALUED_1:def 1;
then A8: ( Z c= dom f1 & Z c= dom (exp_R ^2) ) by XBOOLE_1:18;
A9: Z c= dom (exp_R (#) exp_R) by XBOOLE_1:18, A7;
A10: exp_R is_differentiable_on Z by FDIFF_1:34, TAYLOR_1:16;
then A11: (- 1) (#) exp_R is_differentiable_on Z by A4, FDIFF_1:28, X;
A12: exp_R (#) exp_R is_differentiable_on Z by A9, A10, FDIFF_1:29;
for x being Real st x in Z holds
f1 . x = (0 * x) + 1 by A1;
then f1 is_differentiable_on Z by FDIFF_1:31, A8;
then A16: f1 + (exp_R ^2) is_differentiable_on Z by A6, A12, FDIFF_1:26;
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 RFUNCT_1:def 4, A1;
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 8;
hence (f1 + (exp_R ^2)) . x <> 0 by RFUNCT_1:13; :: thesis: verum

end;

then f is_differentiable_on Z by A1, A11, A16, FDIFF_2:21;
then f | Z is continuous by FDIFF_1:33;
then f | A is continuous by A1, FCONT_1:17;
then A21: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A22: for x being Real st x in Z holds
exp_R . x < 1 by A1;
then A23: arccot * exp_R is_differentiable_on Z by A1, SIN_COS9:116;
B1: 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 B2: 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 RFUNCT_1:def 4, A1
.= (- (exp_R . x)) * (((f1 + (exp_R ^2)) . x) ") by RFUNCT_1:74
.= (- (exp_R . x)) * (((f1 . x) + ((exp_R ^2) . x)) ") by VALUED_1:def 1, B2, A6
.= (- (exp_R . x)) * (((f1 . x) + ((exp_R . x) ^2)) ") by VALUED_1:11
.= (- (exp_R . x)) / (1 + ((exp_R . x) ^2)) by A1, B2
.= - ((exp_R . x) / (1 + ((exp_R . x) ^2))) ;
hence f . x = - ((exp_R . x) / (1 + ((exp_R . x) ^2))) by A1; :: thesis: verum

end;

A24: for x being Real st x in dom ((arccot * exp_R) `| Z) holds
((arccot * exp_R) `| Z) . x = f . x

proof

let x be Real; :: thesis: ( x in dom ((arccot * exp_R) `| Z) implies ((arccot * exp_R) `| Z) . x = f . x )
assume x in dom ((arccot * exp_R) `| Z) ; :: thesis: ((arccot * exp_R) `| Z) . x = f . x
then A25: x in Z by A23, FDIFF_1:def 8;
then ((arccot * exp_R) `| Z) . x = - ((exp_R . x) / (1 + ((exp_R . x) ^2))) by A1, A22, SIN_COS9:116
.= f . x by A25, B1 ;
hence ((arccot * exp_R) `| Z) . x = f . x ; :: thesis: verum

end;

dom ((arccot * exp_R) `| Z) = dom f by A1, A23, FDIFF_1:def 8;
then (arccot * exp_R) `| Z = f by A24, PARTFUN1:34;
hence integral (f,A) = ((arccot * exp_R) . (upper_bound A)) - ((arccot * exp_R) . (lower_bound A)) by A1, A21, A23, INTEGRA5:13; :: thesis: verum