let n be Element of NAT ; :: thesis: for A being non empty closed_interval Subset of REAL
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
f1 . x = 1 ) & f = - (n (#) (((#Z (n - 1)) * arccot) / (f1 + (#Z 2)))) & Z c= ].(- 1),1.[ & Z c= dom ((#Z n) * arccot) & Z = dom f & f | A is continuous holds
integral (f,A) = (((#Z n) * arccot) . (upper_bound A)) - (((#Z n) * arccot) . (lower_bound A))
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
f1 . x = 1 ) & f = - (n (#) (((#Z (n - 1)) * arccot) / (f1 + (#Z 2)))) & Z c= ].(- 1),1.[ & Z c= dom ((#Z n) * arccot) & Z = dom f & f | A is continuous holds
integral (f,A) = (((#Z n) * arccot) . (upper_bound A)) - (((#Z n) * arccot) . (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
f1 . x = 1 ) & f = - (n (#) (((#Z (n - 1)) * arccot) / (f1 + (#Z 2)))) & Z c= ].(- 1),1.[ & Z c= dom ((#Z n) * arccot) & Z = dom f & f | A is continuous holds
integral (f,A) = (((#Z n) * arccot) . (upper_bound A)) - (((#Z n) * arccot) . (lower_bound A))
let Z be open Subset of REAL; :: thesis: ( A c= Z & ( for x being Real st x in Z holds
f1 . x = 1 ) & f = - (n (#) (((#Z (n - 1)) * arccot) / (f1 + (#Z 2)))) & Z c= ].(- 1),1.[ & Z c= dom ((#Z n) * arccot) & Z = dom f & f | A is continuous implies integral (f,A) = (((#Z n) * arccot) . (upper_bound A)) - (((#Z n) * arccot) . (lower_bound A)) )
assume A1: ( A c= Z & ( for x being Real st x in Z holds
f1 . x = 1 ) & f = - (n (#) (((#Z (n - 1)) * arccot) / (f1 + (#Z 2)))) & Z c= ].(- 1),1.[ & Z c= dom ((#Z n) * arccot) & Z = dom f & f | A is continuous ) ; :: thesis: integral (f,A) = (((#Z n) * arccot) . (upper_bound A)) - (((#Z n) * arccot) . (lower_bound A))
then A2: ( f is_integrable_on A & f | A is bounded ) by INTEGRA5:10, INTEGRA5:11;
A3: (#Z n) * arccot is_differentiable_on Z by A1, SIN_COS9:92;
Z = dom (n (#) (((#Z (n - 1)) * arccot) / (f1 + (#Z 2)))) by A1, VALUED_1:8;
then A4: Z = dom (((#Z (n - 1)) * arccot) / (f1 + (#Z 2))) by VALUED_1:def 5;
then Z c= (dom ((#Z (n - 1)) * arccot)) /\ ((dom (f1 + (#Z 2))) \ ((f1 + (#Z 2)) " {0})) by RFUNCT_1:def 1;
then A5: ( Z c= dom ((#Z (n - 1)) * arccot) & Z c= (dom (f1 + (#Z 2))) \ ((f1 + (#Z 2)) " {0}) ) by XBOOLE_1:18;
then A6: Z c= dom ((f1 + (#Z 2)) ^) by RFUNCT_1:def 2;
dom ((f1 + (#Z 2)) ^) c= dom (f1 + (#Z 2)) by RFUNCT_1:1;
then A7: Z c= dom (f1 + (#Z 2)) by A6;
A8: for x being Real st x in Z holds
f . x = - ((n * ((arccot . x) #Z (n - 1))) / (1 + (x ^2)))

proof

let x be Real; :: thesis: ( x in Z implies f . x = - ((n * ((arccot . x) #Z (n - 1))) / (1 + (x ^2))) )
assume A9: x in Z ; :: thesis: f . x = - ((n * ((arccot . x) #Z (n - 1))) / (1 + (x ^2)))
(- (n (#) (((#Z (n - 1)) * arccot) / (f1 + (#Z 2))))) . x = - ((n (#) (((#Z (n - 1)) * arccot) / (f1 + (#Z 2)))) . x) by VALUED_1:8
.= - (n * ((((#Z (n - 1)) * arccot) / (f1 + (#Z 2))) . x)) by VALUED_1:6
.= - (n * ((((#Z (n - 1)) * arccot) . x) * (((f1 + (#Z 2)) . x) "))) by A4, A9, RFUNCT_1:def 1
.= - ((n * (((#Z (n - 1)) * arccot) . x)) / ((f1 + (#Z 2)) . x))
.= - ((n * ((#Z (n - 1)) . (arccot . x))) / ((f1 + (#Z 2)) . x)) by A5, A9, FUNCT_1:12
.= - ((n * ((arccot . x) #Z (n - 1))) / ((f1 + (#Z 2)) . x)) by TAYLOR_1:def 1
.= - ((n * ((arccot . x) #Z (n - 1))) / ((f1 . x) + ((#Z 2) . x))) by A7, A9, VALUED_1:def 1
.= - ((n * ((arccot . x) #Z (n - 1))) / (1 + ((#Z 2) . x))) by A1, A9
.= - ((n * ((arccot . x) #Z (n - 1))) / (1 + (x #Z 2))) by TAYLOR_1:def 1
.= - ((n * ((arccot . x) #Z (n - 1))) / (1 + (x ^2))) by FDIFF_7:1 ;
hence f . x = - ((n * ((arccot . x) #Z (n - 1))) / (1 + (x ^2))) by A1; :: thesis: verum

end;

A10: for x being Element of REAL st x in dom (((#Z n) * arccot) `| Z) holds
(((#Z n) * arccot) `| Z) . x = f . x

proof

let x be Element of REAL ; :: thesis: ( x in dom (((#Z n) * arccot) `| Z) implies (((#Z n) * arccot) `| Z) . x = f . x )
assume x in dom (((#Z n) * arccot) `| Z) ; :: thesis: (((#Z n) * arccot) `| Z) . x = f . x
then A11: x in Z by A3, FDIFF_1:def 7;
then (((#Z n) * arccot) `| Z) . x = - ((n * ((arccot . x) #Z (n - 1))) / (1 + (x ^2))) by A1, SIN_COS9:92
.= f . x by A11, A8 ;
hence (((#Z n) * arccot) `| Z) . x = f . x ; :: thesis: verum

end;

dom (((#Z n) * arccot) `| Z) = dom f by A1, A3, FDIFF_1:def 7;
then ((#Z n) * arccot) `| Z = f by A10, PARTFUN1:5;
hence integral (f,A) = (((#Z n) * arccot) . (upper_bound A)) - (((#Z n) * arccot) . (lower_bound A)) by A1, A2, INTEGRA5:13, SIN_COS9:92; :: thesis: verum