let r be Real; :: thesis: for A being closed-interval Subset of REAL
for f, f1, g being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & f = (arccot * f1) - ((id Z) / (r (#) (g + (f1 ^2)))) & ( for x being Real st x in Z holds
( g . x = 1 & f1 . x = x / r & f1 . x > - 1 & f1 . x < 1 ) ) & Z = dom f & f | A is continuous holds
integral (f,A) = (((id Z) (#) (arccot * f1)) . (upper_bound A)) - (((id Z) (#) (arccot * f1)) . (lower_bound A))
let A be closed-interval Subset of REAL; :: thesis: for f, f1, g being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & f = (arccot * f1) - ((id Z) / (r (#) (g + (f1 ^2)))) & ( for x being Real st x in Z holds
( g . x = 1 & f1 . x = x / r & f1 . x > - 1 & f1 . x < 1 ) ) & Z = dom f & f | A is continuous holds
integral (f,A) = (((id Z) (#) (arccot * f1)) . (upper_bound A)) - (((id Z) (#) (arccot * f1)) . (lower_bound A))
let f, f1, g be PartFunc of REAL,REAL; :: thesis: for Z being open Subset of REAL st A c= Z & f = (arccot * f1) - ((id Z) / (r (#) (g + (f1 ^2)))) & ( for x being Real st x in Z holds
( g . x = 1 & f1 . x = x / r & f1 . x > - 1 & f1 . x < 1 ) ) & Z = dom f & f | A is continuous holds
integral (f,A) = (((id Z) (#) (arccot * f1)) . (upper_bound A)) - (((id Z) (#) (arccot * f1)) . (lower_bound A))
let Z be open Subset of REAL; :: thesis: ( A c= Z & f = (arccot * f1) - ((id Z) / (r (#) (g + (f1 ^2)))) & ( for x being Real st x in Z holds
( g . x = 1 & f1 . x = x / r & f1 . x > - 1 & f1 . x < 1 ) ) & Z = dom f & f | A is continuous implies integral (f,A) = (((id Z) (#) (arccot * f1)) . (upper_bound A)) - (((id Z) (#) (arccot * f1)) . (lower_bound A)) )
assume A1: ( A c= Z & f = (arccot * f1) - ((id Z) / (r (#) (g + (f1 ^2)))) & ( for x being Real st x in Z holds
( g . x = 1 & f1 . x = x / r & f1 . x > - 1 & f1 . x < 1 ) ) & Z = dom f & f | A is continuous ) ; :: thesis: integral (f,A) = (((id Z) (#) (arccot * f1)) . (upper_bound A)) - (((id Z) (#) (arccot * f1)) . (lower_bound A))
then Z = (dom (arccot * f1)) /\ (dom ((id Z) / (r (#) (g + (f1 ^2))))) by VALUED_1:12;
then A3: ( Z c= dom (arccot * f1) & Z c= dom ((id Z) / (r (#) (g + (f1 ^2)))) ) by XBOOLE_1:18;
Z = dom (id Z) by RELAT_1:71;
then Z c= (dom (id Z)) /\ (dom (arccot * f1)) by A3, XBOOLE_1:19;
then A5: Z c= dom ((id Z) (#) (arccot * f1)) by VALUED_1:def 4;
Z c= (dom (id Z)) /\ ((dom (r (#) (g + (f1 ^2)))) \ ((r (#) (g + (f1 ^2))) " {0})) by RFUNCT_1:def 4, A3;
then Z c= (dom (r (#) (g + (f1 ^2)))) \ ((r (#) (g + (f1 ^2))) " {0}) by XBOOLE_1:18;
then A8: Z c= dom ((r (#) (g + (f1 ^2))) ^) by RFUNCT_1:def 8;
dom ((r (#) (g + (f1 ^2))) ^) c= dom (r (#) (g + (f1 ^2))) by RFUNCT_1:11;
then Z c= dom (r (#) (g + (f1 ^2))) by XBOOLE_1:1, A8;
then A10: Z c= dom (g + (f1 ^2)) by VALUED_1:def 5;
A11: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A12: for x being Real st x in Z holds
( f1 . x = x / r & f1 . x > - 1 & f1 . x < 1 ) by A1;
then A13: (id Z) (#) (arccot * f1) is_differentiable_on Z by A5, SIN_COS9:106;
A14: for x being Real st x in Z holds
f . x = (arccot . (x / r)) - (x / (r * (1 + ((x / r) ^2))))

proof

let x be Real; :: thesis: ( x in Z implies f . x = (arccot . (x / r)) - (x / (r * (1 + ((x / r) ^2)))) )
assume A15: x in Z ; :: thesis: f . x = (arccot . (x / r)) - (x / (r * (1 + ((x / r) ^2))))
then ((arccot * f1) - ((id Z) / (r (#) (g + (f1 ^2))))) . x = ((arccot * f1) . x) - (((id Z) / (r (#) (g + (f1 ^2)))) . x) by VALUED_1:13, A1
.= (arccot . (f1 . x)) - (((id Z) / (r (#) (g + (f1 ^2)))) . x) by FUNCT_1:22, A3, A15
.= (arccot . (f1 . x)) - (((id Z) . x) / ((r (#) (g + (f1 ^2))) . x)) by RFUNCT_1:def 4, A3, A15
.= (arccot . (f1 . x)) - (x / ((r (#) (g + (f1 ^2))) . x)) by FUNCT_1:35, A15
.= (arccot . (f1 . x)) - (x / (r * ((g + (f1 ^2)) . x))) by VALUED_1:6
.= (arccot . (f1 . x)) - (x / (r * ((g . x) + ((f1 ^2) . x)))) by VALUED_1:def 1, A10, A15
.= (arccot . (f1 . x)) - (x / (r * ((g . x) + ((f1 . x) ^2)))) by VALUED_1:11
.= (arccot . (x / r)) - (x / (r * ((g . x) + ((f1 . x) ^2)))) by A1, A15
.= (arccot . (x / r)) - (x / (r * (1 + ((f1 . x) ^2)))) by A1, A15
.= (arccot . (x / r)) - (x / (r * (1 + ((x / r) ^2)))) by A1, A15 ;
hence f . x = (arccot . (x / r)) - (x / (r * (1 + ((x / r) ^2)))) by A1; :: thesis: verum

end;

A16: for x being Real st x in dom (((id Z) (#) (arccot * f1)) `| Z) holds
(((id Z) (#) (arccot * f1)) `| Z) . x = f . x

proof

let x be Real; :: thesis: ( x in dom (((id Z) (#) (arccot * f1)) `| Z) implies (((id Z) (#) (arccot * f1)) `| Z) . x = f . x )
assume x in dom (((id Z) (#) (arccot * f1)) `| Z) ; :: thesis: (((id Z) (#) (arccot * f1)) `| Z) . x = f . x
then A17: x in Z by A13, FDIFF_1:def 8;
then (((id Z) (#) (arccot * f1)) `| Z) . x = (arccot . (x / r)) - (x / (r * (1 + ((x / r) ^2)))) by A5, A12, SIN_COS9:106
.= f . x by A14, A17 ;
hence (((id Z) (#) (arccot * f1)) `| Z) . x = f . x ; :: thesis: verum

end;

dom (((id Z) (#) (arccot * f1)) `| Z) = dom f by A1, A13, FDIFF_1:def 8;
then ((id Z) (#) (arccot * f1)) `| Z = f by A16, PARTFUN1:34;
hence integral (f,A) = (((id Z) (#) (arccot * f1)) . (upper_bound A)) - (((id Z) (#) (arccot * f1)) . (lower_bound A)) by A1, A11, A13, INTEGRA5:13; :: thesis: verum