let A be 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 & Z c= ].(- 1),1.[ & f = (exp_R * arccot) / (f1 + (#Z 2)) & ( for x being Real st x in Z holds
f1 . x = 1 ) & Z = dom f & f | A is continuous holds
integral (f,A) = ((- (exp_R * arccot)) . (upper_bound A)) - ((- (exp_R * arccot)) . (lower_bound A))
let f, f1 be PartFunc of REAL,REAL; :: thesis: for Z being open Subset of REAL st A c= Z & Z c= ].(- 1),1.[ & f = (exp_R * arccot) / (f1 + (#Z 2)) & ( for x being Real st x in Z holds
f1 . x = 1 ) & Z = dom f & f | A is continuous holds
integral (f,A) = ((- (exp_R * arccot)) . (upper_bound A)) - ((- (exp_R * arccot)) . (lower_bound A))
let Z be open Subset of REAL; :: thesis: ( A c= Z & Z c= ].(- 1),1.[ & f = (exp_R * arccot) / (f1 + (#Z 2)) & ( for x being Real st x in Z holds
f1 . x = 1 ) & Z = dom f & f | A is continuous implies integral (f,A) = ((- (exp_R * arccot)) . (upper_bound A)) - ((- (exp_R * arccot)) . (lower_bound A)) )
assume A1: ( A c= Z & Z c= ].(- 1),1.[ & f = (exp_R * arccot) / (f1 + (#Z 2)) & ( for x being Real st x in Z holds
f1 . x = 1 ) & Z = dom f & f | A is continuous ) ; :: thesis: integral (f,A) = ((- (exp_R * arccot)) . (upper_bound A)) - ((- (exp_R * arccot)) . (lower_bound A))
then A2: ( f is_integrable_on A & f | A is bounded ) by INTEGRA5:10, INTEGRA5:11;
Z = (dom (exp_R * arccot)) /\ ((dom (f1 + (#Z 2))) \ ((f1 + (#Z 2)) " {0})) by RFUNCT_1:def 4, A1;
then A3: ( Z c= dom (exp_R * arccot) & Z c= (dom (f1 + (#Z 2))) \ ((f1 + (#Z 2)) " {0}) ) by XBOOLE_1:18;
then A4: Z c= dom ((f1 + (#Z 2)) ^) by RFUNCT_1:def 8;
dom ((f1 + (#Z 2)) ^) c= dom (f1 + (#Z 2)) by RFUNCT_1:11;
then A5: Z c= dom (f1 + (#Z 2)) by XBOOLE_1:1, A4;
A6: - (exp_R * arccot) is_differentiable_on Z by A1, A3, Th9;
B: for x being Real st x in Z holds
f . x = (exp_R . (arccot . x)) / (1 + (x ^2))

proof

let x be Real; :: thesis: ( x in Z implies f . x = (exp_R . (arccot . x)) / (1 + (x ^2)) )
assume A7: x in Z ; :: thesis: f . x = (exp_R . (arccot . x)) / (1 + (x ^2))
then ((exp_R * arccot) / (f1 + (#Z 2))) . x = ((exp_R * arccot) . x) / ((f1 + (#Z 2)) . x) by RFUNCT_1:def 4, A1
.= (exp_R . (arccot . x)) / ((f1 + (#Z 2)) . x) by FUNCT_1:22, A3, A7
.= (exp_R . (arccot . x)) / ((f1 . x) + ((#Z 2) . x)) by VALUED_1:def 1, A5, A7
.= (exp_R . (arccot . x)) / (1 + ((#Z 2) . x)) by A1, A7
.= (exp_R . (arccot . x)) / (1 + (x #Z 2)) by TAYLOR_1:def 1
.= (exp_R . (arccot . x)) / (1 + (x ^2)) by FDIFF_7:1 ;
hence f . x = (exp_R . (arccot . x)) / (1 + (x ^2)) by A1; :: thesis: verum

end;

A8: for x being Real st x in dom ((- (exp_R * arccot)) `| Z) holds
((- (exp_R * arccot)) `| Z) . x = f . x

proof

let x be Real; :: thesis: ( x in dom ((- (exp_R * arccot)) `| Z) implies ((- (exp_R * arccot)) `| Z) . x = f . x )
assume x in dom ((- (exp_R * arccot)) `| Z) ; :: thesis: ((- (exp_R * arccot)) `| Z) . x = f . x
then A9: x in Z by A6, FDIFF_1:def 8;
then ((- (exp_R * arccot)) `| Z) . x = (exp_R . (arccot . x)) / (1 + (x ^2)) by A1, A3, Th9
.= f . x by B, A9 ;
hence ((- (exp_R * arccot)) `| Z) . x = f . x ; :: thesis: verum

end;

dom ((- (exp_R * arccot)) `| Z) = dom f by A1, A6, FDIFF_1:def 8;
then (- (exp_R * arccot)) `| Z = f by A8, PARTFUN1:34;
hence integral (f,A) = ((- (exp_R * arccot)) . (upper_bound A)) - ((- (exp_R * arccot)) . (lower_bound A)) by A1, A2, A3, Th9, INTEGRA5:13; :: thesis: verum