let r be Real; :: thesis: for A being non empty closed_interval Subset of REAL
for f, g, f1, f2 being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z c= dom (ln * (f1 + f2)) & r <> 0 & ( for x being Real st x in Z holds
( g . x = x / r & g . x > - 1 & g . x < 1 & f1 . x = 1 ) ) & f2 = (#Z 2) * g & Z = dom f & f = arccot * g holds
integral (f,A) = ((((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2)))) . (upper_bound A)) - ((((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2)))) . (lower_bound A))
let A be non empty closed_interval Subset of REAL; :: thesis: for f, g, f1, f2 being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z c= dom (ln * (f1 + f2)) & r <> 0 & ( for x being Real st x in Z holds
( g . x = x / r & g . x > - 1 & g . x < 1 & f1 . x = 1 ) ) & f2 = (#Z 2) * g & Z = dom f & f = arccot * g holds
integral (f,A) = ((((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2)))) . (upper_bound A)) - ((((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2)))) . (lower_bound A))
let f, g, f1, f2 be PartFunc of REAL,REAL; :: thesis: for Z being open Subset of REAL st A c= Z & Z c= dom (ln * (f1 + f2)) & r <> 0 & ( for x being Real st x in Z holds
( g . x = x / r & g . x > - 1 & g . x < 1 & f1 . x = 1 ) ) & f2 = (#Z 2) * g & Z = dom f & f = arccot * g holds
integral (f,A) = ((((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2)))) . (upper_bound A)) - ((((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2)))) . (lower_bound A))
let Z be open Subset of REAL; :: thesis: ( A c= Z & Z c= dom (ln * (f1 + f2)) & r <> 0 & ( for x being Real st x in Z holds
( g . x = x / r & g . x > - 1 & g . x < 1 & f1 . x = 1 ) ) & f2 = (#Z 2) * g & Z = dom f & f = arccot * g implies integral (f,A) = ((((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2)))) . (upper_bound A)) - ((((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2)))) . (lower_bound A)) )
assume A1: ( A c= Z & Z c= dom (ln * (f1 + f2)) & r <> 0 & ( for x being Real st x in Z holds
( g . x = x / r & g . x > - 1 & g . x < 1 & f1 . x = 1 ) ) & f2 = (#Z 2) * g & Z = dom f & f = arccot * g ) ; :: thesis: integral (f,A) = ((((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2)))) . (upper_bound A)) - ((((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2)))) . (lower_bound A))
Z c= (dom (id Z)) /\ (dom f) by A1;
then A2: Z c= dom ((id Z) (#) (arccot * g)) by A1, VALUED_1:def 4;
Z c= dom ((r / 2) (#) (ln * (f1 + f2))) by A1, VALUED_1:def 5;
then Z c= (dom ((id Z) (#) (arccot * g))) /\ (dom ((r / 2) (#) (ln * (f1 + f2)))) by A2, XBOOLE_1:19;
then A3: Z c= dom (((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2)))) by VALUED_1:def 1;
A4: for x being Real st x in Z holds
( g . x = x / r & g . x > - 1 & g . x < 1 ) by A1;
for x being Real st x in Z holds
g . x = ((1 / r) * x) + 0

proof

let x be Real; :: thesis: ( x in Z implies g . x = ((1 / r) * x) + 0 )
assume x in Z ; :: thesis: g . x = ((1 / r) * x) + 0
then g . x = x / r by A1;
hence g . x = ((1 / r) * x) + 0 ; :: thesis: verum

end;

then for x being Real st x in Z holds
( g . x = ((1 / r) * x) + 0 & g . x > - 1 & g . x < 1 ) by A1;
then f is_differentiable_on Z by A1, SIN_COS9:88;
then f | Z is continuous by FDIFF_1:25;
then f | A is continuous by A1, FCONT_1:16;
then A5: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A6: ( ( for x being Real st x in Z holds
f1 . x = 1 ) & ( for x being Real st x in Z holds
g . x = x / r ) ) by A1;
then A7: ((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z by A1, A3, A4, SIN_COS9:110;
A8: for x being Real st x in Z holds
f . x = arccot . (x / r)

proof

let x be Real; :: thesis: ( x in Z implies f . x = arccot . (x / r) )
assume A9: x in Z ; :: thesis: f . x = arccot . (x / r)
then (arccot * g) . x = arccot . (g . x) by A1, FUNCT_1:12
.= arccot . (x / r) by A1, A9 ;
hence f . x = arccot . (x / r) by A1; :: thesis: verum

end;

A10: for x being Element of REAL st x in dom ((((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) holds
((((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = f . x

proof

let x be Element of REAL ; :: thesis: ( x in dom ((((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) implies ((((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = f . x )
assume x in dom ((((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) ; :: thesis: ((((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = f . x
then A11: x in Z by A7, FDIFF_1:def 7;
then ((((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . (x / r) by A1, A3, A4, A6, SIN_COS9:110
.= f . x by A8, A11 ;
hence ((((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = f . x ; :: thesis: verum

end;

dom ((((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) = dom f by A1, A7, FDIFF_1:def 7;
then (((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z = f by A10, PARTFUN1:5;
hence integral (f,A) = ((((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2)))) . (upper_bound A)) - ((((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2)))) . (lower_bound A)) by A1, A5, A7, INTEGRA5:13; :: thesis: verum