let n be Element of NAT ; :: thesis: for A being closed-interval Subset of REAL
for f, f1 being PartFunc of REAL ,REAL
for Z being open Subset of REAL st A c= Z & f = n (#) (((#Z (n - 1)) * arctan ) / (f1 + (#Z 2))) & ( for x being Real st x in Z holds
f1 . x = 1 ) & Z c= ].(- 1),1.[ & Z c= dom ((#Z n) * arctan ) & Z = dom f & f | A is continuous holds
integral f,A = (((#Z n) * arctan ) . (upper_bound A)) - (((#Z n) * arctan ) . (lower_bound A))
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 & f = n (#) (((#Z (n - 1)) * arctan ) / (f1 + (#Z 2))) & ( for x being Real st x in Z holds
f1 . x = 1 ) & Z c= ].(- 1),1.[ & Z c= dom ((#Z n) * arctan ) & Z = dom f & f | A is continuous holds
integral f,A = (((#Z n) * arctan ) . (upper_bound A)) - (((#Z n) * arctan ) . (lower_bound A))
let f, f1 be PartFunc of REAL ,REAL ; :: thesis: for Z being open Subset of REAL st A c= Z & f = n (#) (((#Z (n - 1)) * arctan ) / (f1 + (#Z 2))) & ( for x being Real st x in Z holds
f1 . x = 1 ) & Z c= ].(- 1),1.[ & Z c= dom ((#Z n) * arctan ) & Z = dom f & f | A is continuous holds
integral f,A = (((#Z n) * arctan ) . (upper_bound A)) - (((#Z n) * arctan ) . (lower_bound A))
let Z be open Subset of REAL ; :: thesis: ( A c= Z & f = n (#) (((#Z (n - 1)) * arctan ) / (f1 + (#Z 2))) & ( for x being Real st x in Z holds
f1 . x = 1 ) & Z c= ].(- 1),1.[ & Z c= dom ((#Z n) * arctan ) & Z = dom f & f | A is continuous implies integral f,A = (((#Z n) * arctan ) . (upper_bound A)) - (((#Z n) * arctan ) . (lower_bound A)) )
assume A1: ( A c= Z & f = n (#) (((#Z (n - 1)) * arctan ) / (f1 + (#Z 2))) & ( for x being Real st x in Z holds
f1 . x = 1 ) & Z c= ].(- 1),1.[ & Z c= dom ((#Z n) * arctan ) & Z = dom f & f | A is continuous ) ; :: thesis: integral f,A = (((#Z n) * arctan ) . (upper_bound A)) - (((#Z n) * arctan ) . (lower_bound A))
then A2: ( f is_integrable_on A & f | A is bounded ) by INTEGRA5:10, INTEGRA5:11;
A3: (#Z n) * arctan is_differentiable_on Z by A1, SIN_COS9:91;
A4: Z = dom (((#Z (n - 1)) * arctan ) / (f1 + (#Z 2))) by A1, VALUED_1:def 5;
then Z c= (dom ((#Z (n - 1)) * arctan )) /\ ((dom (f1 + (#Z 2))) \ ((f1 + (#Z 2)) " {0 })) by RFUNCT_1:def 4;
then A5: ( Z c= dom ((#Z (n - 1)) * arctan ) & 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 8;
dom ((f1 + (#Z 2)) ^ ) c= dom (f1 + (#Z 2)) by RFUNCT_1:11;
then A7: Z c= dom (f1 + (#Z 2)) by XBOOLE_1:1, A6;
A8: for x being Real st x in Z holds
f . x = (n * ((arctan . x) #Z (n - 1))) / (1 + (x ^2 ))

proof

let x be Real; :: thesis: ( x in Z implies f . x = (n * ((arctan . x) #Z (n - 1))) / (1 + (x ^2 )) )
assume A9: x in Z ; :: thesis: f . x = (n * ((arctan . x) #Z (n - 1))) / (1 + (x ^2 ))
(n (#) (((#Z (n - 1)) * arctan ) / (f1 + (#Z 2)))) . x = n * ((((#Z (n - 1)) * arctan ) / (f1 + (#Z 2))) . x) by VALUED_1:6
.= n * ((((#Z (n - 1)) * arctan ) . x) * (((f1 + (#Z 2)) . x) " )) by RFUNCT_1:def 4, A4, A9
.= (n * (((#Z (n - 1)) * arctan ) . x)) / ((f1 + (#Z 2)) . x)
.= (n * ((#Z (n - 1)) . (arctan . x))) / ((f1 + (#Z 2)) . x) by FUNCT_1:22, A5, A9
.= (n * ((arctan . x) #Z (n - 1))) / ((f1 + (#Z 2)) . x) by TAYLOR_1:def 1
.= (n * ((arctan . x) #Z (n - 1))) / ((f1 . x) + ((#Z 2) . x)) by A7, VALUED_1:def 1, A9
.= (n * ((arctan . x) #Z (n - 1))) / (1 + ((#Z 2) . x)) by A1, A9
.= (n * ((arctan . x) #Z (n - 1))) / (1 + (x #Z 2)) by TAYLOR_1:def 1
.= (n * ((arctan . x) #Z (n - 1))) / (1 + (x ^2 )) by FDIFF_7:1 ;
hence f . x = (n * ((arctan . x) #Z (n - 1))) / (1 + (x ^2 )) by A1; :: thesis: verum

end;

A10: for x being Real st x in dom (((#Z n) * arctan ) `| Z) holds
(((#Z n) * arctan ) `| Z) . x = f . x

proof

let x be Real; :: thesis: ( x in dom (((#Z n) * arctan ) `| Z) implies (((#Z n) * arctan ) `| Z) . x = f . x )
assume x in dom (((#Z n) * arctan ) `| Z) ; :: thesis: (((#Z n) * arctan ) `| Z) . x = f . x
then A11: x in Z by A3, FDIFF_1:def 8;
then (((#Z n) * arctan ) `| Z) . x = (n * ((arctan . x) #Z (n - 1))) / (1 + (x ^2 )) by A1, SIN_COS9:91
.= f . x by A8, A11 ;
hence (((#Z n) * arctan ) `| Z) . x = f . x ; :: thesis: verum

end;

dom (((#Z n) * arctan ) `| Z) = dom f by A1, A3, FDIFF_1:def 8;
then ((#Z n) * arctan ) `| Z = f by A10, PARTFUN1:34;
hence integral f,A = (((#Z n) * arctan ) . (upper_bound A)) - (((#Z n) * arctan ) . (lower_bound A)) by A1, A2, SIN_COS9:91, INTEGRA5:13; :: thesis: verum