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