:: INTEGR12 semantic presentation

begin

              Lm1:  - 1 is Real
by XREAL_0:def_1;

theorem Th1: :: INTEGR12:1
for f1, f2 being PartFunc of REAL,REAL
for Z being open Subset of REAL st Z c= dom ((f1 + f2) ^) & ( for x being Real st x in Z holds
f1 . x = 1 ) & f2 = #Z 2 holds
( (f1 + f2) ^ is_differentiable_on Z & ( for x being Real st x in Z holds
(((f1 + f2) ^) `| Z) . x = - ((2 * x) / ((1 + (x |^ 2)) ^2)) ) )
proof
let f1, f2 be PartFunc of REAL,REAL; ::_thesis: for Z being open Subset of REAL st Z c= dom ((f1 + f2) ^) & ( for x being Real st x in Z holds
f1 . x = 1 ) & f2 = #Z 2 holds
( (f1 + f2) ^ is_differentiable_on Z & ( for x being Real st x in Z holds
(((f1 + f2) ^) `| Z) . x = - ((2 * x) / ((1 + (x |^ 2)) ^2)) ) )
let Z be open Subset of REAL; ::_thesis: ( Z c= dom ((f1 + f2) ^) & ( for x being Real st x in Z holds
f1 . x = 1 ) & f2 = #Z 2 implies ( (f1 + f2) ^ is_differentiable_on Z & ( for x being Real st x in Z holds
(((f1 + f2) ^) `| Z) . x = - ((2 * x) / ((1 + (x |^ 2)) ^2)) ) ) )
assume A1: ( Z c= dom ((f1 + f2) ^) & ( for x being Real st x in Z holds
f1 . x = 1 ) & f2 = #Z 2 ) ; ::_thesis: ( (f1 + f2) ^ is_differentiable_on Z & ( for x being Real st x in Z holds
(((f1 + f2) ^) `| Z) . x = - ((2 * x) / ((1 + (x |^ 2)) ^2)) ) )
 dom ((f1 + f2) ^) c= dom (f1 + f2) by RFUNCT_1:1;
then A2: Z c= dom (f1 + f2) by A1, XBOOLE_1:1;
then A3: ( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = 2 * x ) ) by A1, SIN_COS9:101;
A4: for x being Real st x in Z holds
(f1 + f2) . x <> 0 by A1, RFUNCT_1:3;
then A5: (f1 + f2) ^ is_differentiable_on Z by A3, FDIFF_2:22;
 for x being Real st x in Z holds
(((f1 + f2) ^) `| Z) . x = - ((2 * x) / ((1 + (x |^ 2)) ^2))
proof
let x be Real; ::_thesis: ( x in Z implies (((f1 + f2) ^) `| Z) . x = - ((2 * x) / ((1 + (x |^ 2)) ^2)) )
assume A6: x in Z ; ::_thesis: (((f1 + f2) ^) `| Z) . x = - ((2 * x) / ((1 + (x |^ 2)) ^2))
then A7: (f1 + f2) . x <> 0 by A1, RFUNCT_1:3;
A8: f1 + f2 is_differentiable_in x by A3, A6, FDIFF_1:9;
A9: f2 . x = x #Z 2 by A1, TAYLOR_1:def_1
.= x |^ 2 by PREPOWER:36 ;
A10: (f1 + f2) . x = (f1 . x) + (f2 . x) by A2, A6, VALUED_1:def_1
.= 1 + (x |^ 2) by A1, A6, A9 ;
(((f1 + f2) ^) `| Z) . x = diff (((f1 + f2) ^),x) by A5, A6, FDIFF_1:def_7
.= - ((diff ((f1 + f2),x)) / (((f1 + f2) . x) ^2)) by A7, A8, FDIFF_2:15
.= - ((((f1 + f2) `| Z) . x) / (((f1 + f2) . x) ^2)) by A3, A6, FDIFF_1:def_7
.= - ((2 * x) / ((1 + (x |^ 2)) ^2)) by A1, A2, A6, A10, SIN_COS9:101 ;
hence  (((f1 + f2) ^) `| Z) . x = - ((2 * x) / ((1 + (x |^ 2)) ^2)) ; ::_thesis: verum
end;
hence  ( (f1 + f2) ^ is_differentiable_on Z & ( for x being Real st x in Z holds
(((f1 + f2) ^) `| Z) . x = - ((2 * x) / ((1 + (x |^ 2)) ^2)) ) ) by A3, A4, FDIFF_2:22; ::_thesis: verum
end;

theorem :: INTEGR12:2
for A being non empty closed_interval Subset of REAL
for f, g1, g2, f2 being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & f = ((g1 + g2) ^) / f2 & f2 = arccot & Z c= ].(- 1),1.[ & g2 = #Z 2 & ( for x being Real st x in Z holds
( g1 . x = 1 & f2 . x > 0 ) ) & Z = dom f holds
integral (f,A) = ((- (ln * arccot)) . (upper_bound A)) - ((- (ln * arccot)) . (lower_bound A))
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for f, g1, g2, f2 being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & f = ((g1 + g2) ^) / f2 & f2 = arccot & Z c= ].(- 1),1.[ & g2 = #Z 2 & ( for x being Real st x in Z holds
( g1 . x = 1 & f2 . x > 0 ) ) & Z = dom f holds
integral (f,A) = ((- (ln * arccot)) . (upper_bound A)) - ((- (ln * arccot)) . (lower_bound A))
let f, g1, g2, f2 be PartFunc of REAL,REAL; ::_thesis: for Z being open Subset of REAL st A c= Z & f = ((g1 + g2) ^) / f2 & f2 = arccot & Z c= ].(- 1),1.[ & g2 = #Z 2 & ( for x being Real st x in Z holds
( g1 . x = 1 & f2 . x > 0 ) ) & Z = dom f holds
integral (f,A) = ((- (ln * arccot)) . (upper_bound A)) - ((- (ln * arccot)) . (lower_bound A))
let Z be open Subset of REAL; ::_thesis: ( A c= Z & f = ((g1 + g2) ^) / f2 & f2 = arccot & Z c= ].(- 1),1.[ & g2 = #Z 2 & ( for x being Real st x in Z holds
( g1 . x = 1 & f2 . x > 0 ) ) & Z = dom f implies integral (f,A) = ((- (ln * arccot)) . (upper_bound A)) - ((- (ln * arccot)) . (lower_bound A)) )
assume A1: ( A c= Z & f = ((g1 + g2) ^) / f2 & f2 = arccot & Z c= ].(- 1),1.[ & g2 = #Z 2 & ( for x being Real st x in Z holds
( g1 . x = 1 & f2 . x > 0 ) ) & Z = dom f ) ; ::_thesis: integral (f,A) = ((- (ln * arccot)) . (upper_bound A)) - ((- (ln * arccot)) . (lower_bound A))
then  Z = (dom ((g1 + g2) ^)) /\ ((dom f2) \ (f2 " {0})) by RFUNCT_1:def_1;
then A2: ( Z c= dom ((g1 + g2) ^) & Z c= (dom f2) \ (f2 " {0}) ) by XBOOLE_1:18;
 dom ((g1 + g2) ^) c= dom (g1 + g2) by RFUNCT_1:1;
then A3: Z c= dom (g1 + g2) by A2, XBOOLE_1:1;
 for x being Real st x in Z holds
g1 . x = 1 by A1;
then A4: (g1 + g2) ^ is_differentiable_on Z by A1, A2, Th1;
A5: f2 is_differentiable_on Z by A1, SIN_COS9:82;
 for x being Real st x in Z holds
f2 . x <> 0 by A1;
then  f is_differentiable_on Z by A1, A4, A5, FDIFF_2:21;
then  f | Z is continuous by FDIFF_1:25;
then A6: f | A is continuous by A1, FCONT_1:16;
A7: Z c= dom (f2 ^) by A2, RFUNCT_1:def_2;
 dom (f2 ^) c= dom f2 by RFUNCT_1:1;
then A8: Z c= dom f2 by A7, XBOOLE_1:1;
A9: for x being Real st x in Z holds
f2 . x > 0 by A1;
 rng (f2 | Z) c= right_open_halfline 0
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (f2 | Z) or x in right_open_halfline 0 )
assume  x in rng (f2 | Z) ; ::_thesis: x in right_open_halfline 0
then consider y being set  such that
A10: ( y in dom (f2 | Z) & x = (f2 | Z) . y ) by FUNCT_1:def_3;
 y in Z by A10;
then  f2 . y > 0 by A1;
then  (f2 | Z) . y > 0 by A10, FUNCT_1:47;
hence  x in right_open_halfline 0 by A10, XXREAL_1:235; ::_thesis: verum
end;
then  f2 .: Z c= dom ln by RELAT_1:115, TAYLOR_1:18;
then A11: Z c= dom (ln * arccot) by A1, A8, FUNCT_1:101;
A12: ( f is_integrable_on A & f | A is bounded ) by A1, A6, INTEGRA5:10, INTEGRA5:11;
A13: ln * arccot is_differentiable_on Z by A1, A11, A9, SIN_COS9:90;
 Z c= dom (- (ln * arccot)) by A11, VALUED_1:8;
then A14: - (ln * arccot) is_differentiable_on Z by A13, Lm1, FDIFF_1:20;
A15: for x being Real st x in Z holds
((- (ln * arccot)) `| Z) . x = 1 / ((1 + (x ^2)) * (arccot . x))
proof
let x be Real; ::_thesis: ( x in Z implies ((- (ln * arccot)) `| Z) . x = 1 / ((1 + (x ^2)) * (arccot . x)) )
assume A16: x in Z ; ::_thesis: ((- (ln * arccot)) `| Z) . x = 1 / ((1 + (x ^2)) * (arccot . x))
then A17: ( - 1 < x & x < 1 ) by A1, XXREAL_1:4;
 arccot is_differentiable_on Z by A1, SIN_COS9:82;
then A18: arccot is_differentiable_in x by A16, FDIFF_1:9;
A19: arccot . x > 0 by A1, A16;
A20: ln * arccot is_differentiable_in x by A13, A16, FDIFF_1:9;
((- (ln * arccot)) `| Z) . x = diff ((- (ln * arccot)),x) by A14, A16, FDIFF_1:def_7
.= (- 1) * (diff ((ln * arccot),x)) by A20, Lm1, FDIFF_1:15
.= (- 1) * ((diff (arccot,x)) / (arccot . x)) by A18, A19, TAYLOR_1:20
.= (- 1) * ((- (1 / (1 + (x ^2)))) / (arccot . x)) by A17, SIN_COS9:76
.= (1 / (1 + (x ^2))) / (arccot . x)
.= 1 / ((1 + (x ^2)) * (arccot . x)) by XCMPLX_1:78 ;
hence  ((- (ln * arccot)) `| Z) . x = 1 / ((1 + (x ^2)) * (arccot . x)) ; ::_thesis: verum
end;
A21: for x being Real st x in Z holds
f . x = 1 / ((1 + (x ^2)) * (arccot . x))
proof
let x be Real; ::_thesis: ( x in Z implies f . x = 1 / ((1 + (x ^2)) * (arccot . x)) )
assume A22: x in Z ; ::_thesis: f . x = 1 / ((1 + (x ^2)) * (arccot . x))
then (((g1 + g2) ^) / f2) . x = (((g1 + g2) ^) . x) / (f2 . x) by A1, RFUNCT_1:def_1
.= (((g1 + g2) . x) ") / (f2 . x) by A2, A22, RFUNCT_1:def_2
.= (((g1 . x) + (g2 . x)) ") / (f2 . x) by A3, A22, VALUED_1:def_1
.= 1 / (((g1 . x) + (g2 . x)) * (f2 . x)) by XCMPLX_1:221
.= 1 / ((1 + ((#Z 2) . x)) * (f2 . x)) by A1, A22
.= 1 / ((1 + (x #Z 2)) * (f2 . x)) by TAYLOR_1:def_1
.= 1 / ((1 + (x ^2)) * (arccot . x)) by A1, FDIFF_7:1 ;
hence  f . x = 1 / ((1 + (x ^2)) * (arccot . x)) by A1; ::_thesis: verum
end;
A23: for x being Real st x in dom ((- (ln * arccot)) `| Z) holds
((- (ln * arccot)) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom ((- (ln * arccot)) `| Z) implies ((- (ln * arccot)) `| Z) . x = f . x )
assume  x in dom ((- (ln * arccot)) `| Z) ; ::_thesis: ((- (ln * arccot)) `| Z) . x = f . x
then A24: x in Z by A14, FDIFF_1:def_7;
then ((- (ln * arccot)) `| Z) . x = 1 / ((1 + (x ^2)) * (arccot . x)) by A15
.= f . x by A21, A24 ;
hence  ((- (ln * arccot)) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom ((- (ln * arccot)) `| Z) = dom f by A1, A14, FDIFF_1:def_7;
then  (- (ln * arccot)) `| Z = f by A23, PARTFUN1:5;
hence  integral (f,A) = ((- (ln * arccot)) . (upper_bound A)) - ((- (ln * arccot)) . (lower_bound A)) by A1, A12, A14, INTEGRA5:13; ::_thesis: verum
end;

theorem :: INTEGR12:3
for A being non empty closed_interval Subset of REAL
for f1, f 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
( exp_R . x < 1 & f1 . x = 1 ) ) & Z c= dom (arctan * exp_R) & Z = dom f & f = exp_R / (f1 + (exp_R ^2)) holds
integral (f,A) = ((arctan * exp_R) . (upper_bound A)) - ((arctan * exp_R) . (lower_bound A))
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for f1, f 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
( exp_R . x < 1 & f1 . x = 1 ) ) & Z c= dom (arctan * exp_R) & Z = dom f & f = exp_R / (f1 + (exp_R ^2)) holds
integral (f,A) = ((arctan * exp_R) . (upper_bound A)) - ((arctan * exp_R) . (lower_bound A))
let f1, f 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
( exp_R . x < 1 & f1 . x = 1 ) ) & Z c= dom (arctan * exp_R) & Z = dom f & f = exp_R / (f1 + (exp_R ^2)) holds
integral (f,A) = ((arctan * exp_R) . (upper_bound A)) - ((arctan * exp_R) . (lower_bound A))
let Z be open Subset of REAL; ::_thesis: ( A c= Z & ( for x being Real st x in Z holds
( exp_R . x < 1 & f1 . x = 1 ) ) & Z c= dom (arctan * exp_R) & Z = dom f & f = exp_R / (f1 + (exp_R ^2)) implies integral (f,A) = ((arctan * exp_R) . (upper_bound A)) - ((arctan * exp_R) . (lower_bound A)) )
assume A1: ( A c= Z & ( for x being Real st x in Z holds
( exp_R . x < 1 & f1 . x = 1 ) ) & Z c= dom (arctan * exp_R) & Z = dom f & f = exp_R / (f1 + (exp_R ^2)) ) ; ::_thesis: integral (f,A) = ((arctan * exp_R) . (upper_bound A)) - ((arctan * exp_R) . (lower_bound A))
then  Z c= (dom exp_R) /\ ((dom (f1 + (exp_R ^2))) \ ((f1 + (exp_R ^2)) " {0})) by RFUNCT_1:def_1;
then  ( Z c= dom exp_R & Z c= (dom (f1 + (exp_R ^2))) \ ((f1 + (exp_R ^2)) " {0}) ) by XBOOLE_1:18;
then A2: Z c= dom ((f1 + (exp_R ^2)) ^) by RFUNCT_1:def_2;
 dom ((f1 + (exp_R ^2)) ^) c= dom (f1 + (exp_R ^2)) by RFUNCT_1:1;
then A3: Z c= dom (f1 + (exp_R ^2)) by A2, XBOOLE_1:1;
then A4: Z c= (dom f1) /\ (dom (exp_R ^2)) by VALUED_1:def_1;
then A5: ( Z c= dom f1 & Z c= dom (exp_R ^2) ) by XBOOLE_1:18;
A6: Z c= dom (exp_R (#) exp_R) by A4, XBOOLE_1:18;
A7: exp_R is_differentiable_on Z by FDIFF_1:26, TAYLOR_1:16;
then A8: exp_R (#) exp_R is_differentiable_on Z by A6, FDIFF_1:21;
 for x being Real st x in Z holds
f1 . x = (0 * x) + 1 by A1;
then  f1 is_differentiable_on Z by A5, FDIFF_1:23;
then A9: f1 + (exp_R ^2) is_differentiable_on Z by A3, A8, FDIFF_1:18;
 for x being Real st x in Z holds
(f1 + (exp_R ^2)) . x <> 0
proof
let x be Real; ::_thesis: ( x in Z implies (f1 + (exp_R ^2)) . x <> 0 )
assume  x in Z ; ::_thesis: (f1 + (exp_R ^2)) . x <> 0
then  x in (dom exp_R) /\ ((dom (f1 + (exp_R ^2))) \ ((f1 + (exp_R ^2)) " {0})) by A1, RFUNCT_1:def_1;
then  x in (dom (f1 + (exp_R ^2))) \ ((f1 + (exp_R ^2)) " {0}) by XBOOLE_0:def_4;
then  x in dom ((f1 + (exp_R ^2)) ^) by RFUNCT_1:def_2;
hence  (f1 + (exp_R ^2)) . x <> 0 by RFUNCT_1:3; ::_thesis: verum
end;
then  f is_differentiable_on Z by A1, A7, A9, FDIFF_2:21;
then  f | Z is continuous by FDIFF_1:25;
then  f | A is continuous by A1, FCONT_1:16;
then A10: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A11: for x being Real st x in Z holds
exp_R . x < 1 by A1;
then A12: arctan * exp_R is_differentiable_on Z by A1, SIN_COS9:115;
A13: for x being Real st x in Z holds
f . x = (exp_R . x) / (1 + ((exp_R . x) ^2))
proof
let x be Real; ::_thesis: ( x in Z implies f . x = (exp_R . x) / (1 + ((exp_R . x) ^2)) )
assume A14: x in Z ; ::_thesis: f . x = (exp_R . x) / (1 + ((exp_R . x) ^2))
then (exp_R / (f1 + (exp_R ^2))) . x = (exp_R . x) * (((f1 + (exp_R ^2)) . x) ") by A1, RFUNCT_1:def_1
.= (exp_R . x) * (((f1 . x) + ((exp_R ^2) . x)) ") by A14, A3, VALUED_1:def_1
.= (exp_R . x) * (((f1 . x) + ((exp_R . x) ^2)) ") by VALUED_1:11
.= (exp_R . x) / (1 + ((exp_R . x) ^2)) by A1, A14 ;
hence  f . x = (exp_R . x) / (1 + ((exp_R . x) ^2)) by A1; ::_thesis: verum
end;
A15: for x being Real st x in dom ((arctan * exp_R) `| Z) holds
((arctan * exp_R) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom ((arctan * exp_R) `| Z) implies ((arctan * exp_R) `| Z) . x = f . x )
assume  x in dom ((arctan * exp_R) `| Z) ; ::_thesis: ((arctan * exp_R) `| Z) . x = f . x
then A16: x in Z by A12, FDIFF_1:def_7;
then ((arctan * exp_R) `| Z) . x = (exp_R . x) / (1 + ((exp_R . x) ^2)) by A1, A11, SIN_COS9:115
.= f . x by A16, A13 ;
hence  ((arctan * exp_R) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom ((arctan * exp_R) `| Z) = dom f by A1, A12, FDIFF_1:def_7;
then  (arctan * exp_R) `| Z = f by A15, PARTFUN1:5;
hence  integral (f,A) = ((arctan * exp_R) . (upper_bound A)) - ((arctan * exp_R) . (lower_bound A)) by A1, A10, A12, INTEGRA5:13; ::_thesis: verum
end;

theorem :: INTEGR12:4
for A being non empty closed_interval Subset of REAL
for f1, f 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
( exp_R . x < 1 & f1 . x = 1 ) ) & Z c= dom (arccot * exp_R) & Z = dom f & f = (- exp_R) / (f1 + (exp_R ^2)) holds
integral (f,A) = ((arccot * exp_R) . (upper_bound A)) - ((arccot * exp_R) . (lower_bound A))
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for f1, f 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
( exp_R . x < 1 & f1 . x = 1 ) ) & Z c= dom (arccot * exp_R) & Z = dom f & f = (- exp_R) / (f1 + (exp_R ^2)) holds
integral (f,A) = ((arccot * exp_R) . (upper_bound A)) - ((arccot * exp_R) . (lower_bound A))
let f1, f 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
( exp_R . x < 1 & f1 . x = 1 ) ) & Z c= dom (arccot * exp_R) & Z = dom f & f = (- exp_R) / (f1 + (exp_R ^2)) holds
integral (f,A) = ((arccot * exp_R) . (upper_bound A)) - ((arccot * exp_R) . (lower_bound A))
let Z be open Subset of REAL; ::_thesis: ( A c= Z & ( for x being Real st x in Z holds
( exp_R . x < 1 & f1 . x = 1 ) ) & Z c= dom (arccot * exp_R) & Z = dom f & f = (- exp_R) / (f1 + (exp_R ^2)) implies integral (f,A) = ((arccot * exp_R) . (upper_bound A)) - ((arccot * exp_R) . (lower_bound A)) )
assume A1: ( A c= Z & ( for x being Real st x in Z holds
( exp_R . x < 1 & f1 . x = 1 ) ) & Z c= dom (arccot * exp_R) & Z = dom f & f = (- exp_R) / (f1 + (exp_R ^2)) ) ; ::_thesis: integral (f,A) = ((arccot * exp_R) . (upper_bound A)) - ((arccot * exp_R) . (lower_bound A))
then  Z c= (dom (- exp_R)) /\ ((dom (f1 + (exp_R ^2))) \ ((f1 + (exp_R ^2)) " {0})) by RFUNCT_1:def_1;
then A2: ( Z c= dom (- exp_R) & Z c= (dom (f1 + (exp_R ^2))) \ ((f1 + (exp_R ^2)) " {0}) ) by XBOOLE_1:18;
then A3: Z c= dom ((f1 + (exp_R ^2)) ^) by RFUNCT_1:def_2;
 dom ((f1 + (exp_R ^2)) ^) c= dom (f1 + (exp_R ^2)) by RFUNCT_1:1;
then A4: Z c= dom (f1 + (exp_R ^2)) by A3, XBOOLE_1:1;
then A5: Z c= (dom f1) /\ (dom (exp_R ^2)) by VALUED_1:def_1;
then A6: ( Z c= dom f1 & Z c= dom (exp_R ^2) ) by XBOOLE_1:18;
A7: Z c= dom (exp_R (#) exp_R) by A5, XBOOLE_1:18;
A8: exp_R is_differentiable_on Z by FDIFF_1:26, TAYLOR_1:16;
then A9: (- 1) (#) exp_R is_differentiable_on Z by A2, Lm1, FDIFF_1:20;
A10: exp_R (#) exp_R is_differentiable_on Z by A7, A8, FDIFF_1:21;
 for x being Real st x in Z holds
f1 . x = (0 * x) + 1 by A1;
then  f1 is_differentiable_on Z by A6, FDIFF_1:23;
then A11: f1 + (exp_R ^2) is_differentiable_on Z by A4, A10, FDIFF_1:18;
 for x being Real st x in Z holds
(f1 + (exp_R ^2)) . x <> 0
proof
let x be Real; ::_thesis: ( x in Z implies (f1 + (exp_R ^2)) . x <> 0 )
assume  x in Z ; ::_thesis: (f1 + (exp_R ^2)) . x <> 0
then  x in (dom (- exp_R)) /\ ((dom (f1 + (exp_R ^2))) \ ((f1 + (exp_R ^2)) " {0})) by A1, RFUNCT_1:def_1;
then  x in (dom (f1 + (exp_R ^2))) \ ((f1 + (exp_R ^2)) " {0}) by XBOOLE_0:def_4;
then  x in dom ((f1 + (exp_R ^2)) ^) by RFUNCT_1:def_2;
hence  (f1 + (exp_R ^2)) . x <> 0 by RFUNCT_1:3; ::_thesis: verum
end;
then  f is_differentiable_on Z by A1, A9, A11, FDIFF_2:21;
then  f | Z is continuous by FDIFF_1:25;
then  f | A is continuous by A1, FCONT_1:16;
then A12: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A13: for x being Real st x in Z holds
exp_R . x < 1 by A1;
then A14: arccot * exp_R is_differentiable_on Z by A1, SIN_COS9:116;
A15: for x being Real st x in Z holds
f . x = - ((exp_R . x) / (1 + ((exp_R . x) ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies f . x = - ((exp_R . x) / (1 + ((exp_R . x) ^2))) )
assume A16: x in Z ; ::_thesis: f . x = - ((exp_R . x) / (1 + ((exp_R . x) ^2)))
then ((- exp_R) / (f1 + (exp_R ^2))) . x = ((- exp_R) . x) * (((f1 + (exp_R ^2)) . x) ") by A1, RFUNCT_1:def_1
.= (- (exp_R . x)) * (((f1 + (exp_R ^2)) . x) ") by RFUNCT_1:58
.= (- (exp_R . x)) * (((f1 . x) + ((exp_R ^2) . x)) ") by A16, A4, VALUED_1:def_1
.= (- (exp_R . x)) * (((f1 . x) + ((exp_R . x) ^2)) ") by VALUED_1:11
.= (- (exp_R . x)) / (1 + ((exp_R . x) ^2)) by A1, A16
.= - ((exp_R . x) / (1 + ((exp_R . x) ^2))) ;
hence  f . x = - ((exp_R . x) / (1 + ((exp_R . x) ^2))) by A1; ::_thesis: verum
end;
A17: for x being Real st x in dom ((arccot * exp_R) `| Z) holds
((arccot * exp_R) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom ((arccot * exp_R) `| Z) implies ((arccot * exp_R) `| Z) . x = f . x )
assume  x in dom ((arccot * exp_R) `| Z) ; ::_thesis: ((arccot * exp_R) `| Z) . x = f . x
then A18: x in Z by A14, FDIFF_1:def_7;
then ((arccot * exp_R) `| Z) . x = - ((exp_R . x) / (1 + ((exp_R . x) ^2))) by A1, A13, SIN_COS9:116
.= f . x by A18, A15 ;
hence  ((arccot * exp_R) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom ((arccot * exp_R) `| Z) = dom f by A1, A14, FDIFF_1:def_7;
then  (arccot * exp_R) `| Z = f by A17, PARTFUN1:5;
hence  integral (f,A) = ((arccot * exp_R) . (upper_bound A)) - ((arccot * exp_R) . (lower_bound A)) by A1, A12, A14, INTEGRA5:13; ::_thesis: verum
end;

theorem :: INTEGR12:5
for A being non empty closed_interval Subset of REAL
for f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z = dom f & f = (exp_R (#) (sin / cos)) + (exp_R / (cos ^2)) holds
integral (f,A) = ((exp_R (#) tan) . (upper_bound A)) - ((exp_R (#) tan) . (lower_bound A))
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z = dom f & f = (exp_R (#) (sin / cos)) + (exp_R / (cos ^2)) holds
integral (f,A) = ((exp_R (#) tan) . (upper_bound A)) - ((exp_R (#) tan) . (lower_bound A))
let f be PartFunc of REAL,REAL; ::_thesis: for Z being open Subset of REAL st A c= Z & Z = dom f & f = (exp_R (#) (sin / cos)) + (exp_R / (cos ^2)) holds
integral (f,A) = ((exp_R (#) tan) . (upper_bound A)) - ((exp_R (#) tan) . (lower_bound A))
let Z be open Subset of REAL; ::_thesis: ( A c= Z & Z = dom f & f = (exp_R (#) (sin / cos)) + (exp_R / (cos ^2)) implies integral (f,A) = ((exp_R (#) tan) . (upper_bound A)) - ((exp_R (#) tan) . (lower_bound A)) )
assume A1: ( A c= Z & Z = dom f & f = (exp_R (#) (sin / cos)) + (exp_R / (cos ^2)) ) ; ::_thesis: integral (f,A) = ((exp_R (#) tan) . (upper_bound A)) - ((exp_R (#) tan) . (lower_bound A))
then  Z = (dom (exp_R (#) (sin / cos))) /\ (dom (exp_R / (cos ^2))) by VALUED_1:def_1;
then A2: ( Z c= dom (exp_R (#) (sin / cos)) & Z c= dom (exp_R / (cos ^2)) ) by XBOOLE_1:18;
A3: dom (exp_R (#) (sin / cos)) c= (dom exp_R) /\ (dom (sin / cos)) by VALUED_1:def_4;
 dom (exp_R / (cos ^2)) c= (dom exp_R) /\ ((dom (cos ^2)) \ ((cos ^2) " {0})) by RFUNCT_1:def_1;
then  ( dom (exp_R (#) (sin / cos)) c= dom exp_R & dom (exp_R (#) (sin / cos)) c= dom (sin / cos) & dom (exp_R / (cos ^2)) c= (dom (cos ^2)) \ ((cos ^2) " {0}) ) by A3, XBOOLE_1:18;
then A4: ( Z c= dom exp_R & Z c= dom (sin / cos) & Z c= (dom (cos ^2)) \ ((cos ^2) " {0}) ) by A2, XBOOLE_1:1;
A5: exp_R is_differentiable_on Z by FDIFF_1:26, TAYLOR_1:16;
 for x being Real st x in Z holds
sin / cos is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies sin / cos is_differentiable_in x )
assume  x in Z ; ::_thesis: sin / cos is_differentiable_in x
then  cos . x <> 0 by A4, FDIFF_8:1;
hence  sin / cos is_differentiable_in x by FDIFF_7:46; ::_thesis: verum
end;
then  sin / cos is_differentiable_on Z by A4, FDIFF_1:9;
then A6: exp_R (#) (sin / cos) is_differentiable_on Z by A2, A5, FDIFF_1:21;
 cos is_differentiable_on Z by FDIFF_1:26, SIN_COS:67;
then A7: cos ^2 is_differentiable_on Z by FDIFF_2:20;
 for x being Real st x in Z holds
(cos ^2) . x <> 0
proof
let x be Real; ::_thesis: ( x in Z implies (cos ^2) . x <> 0 )
assume  x in Z ; ::_thesis: (cos ^2) . x <> 0
then  x in dom (exp_R / (cos ^2)) by A2;
then  x in (dom exp_R) /\ ((dom (cos ^2)) \ ((cos ^2) " {0})) by RFUNCT_1:def_1;
then  x in (dom (cos ^2)) \ ((cos ^2) " {0}) by XBOOLE_0:def_4;
then  x in dom ((cos ^2) ^) by RFUNCT_1:def_2;
hence  (cos ^2) . x <> 0 by RFUNCT_1:3; ::_thesis: verum
end;
then  exp_R / (cos ^2) is_differentiable_on Z by A5, A7, FDIFF_2:21;
then  f | Z is continuous by A1, A6, FDIFF_1:18, FDIFF_1:25;
then  f | A is continuous by A1, FCONT_1:16;
then A8: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A9: exp_R (#) tan is_differentiable_on Z by A2, FDIFF_8:30;
A10: for x being Real st x in Z holds
f . x = (((exp_R . x) * (sin . x)) / (cos . x)) + ((exp_R . x) / ((cos . x) ^2))
proof
let x be Real; ::_thesis: ( x in Z implies f . x = (((exp_R . x) * (sin . x)) / (cos . x)) + ((exp_R . x) / ((cos . x) ^2)) )
assume A11: x in Z ; ::_thesis: f . x = (((exp_R . x) * (sin . x)) / (cos . x)) + ((exp_R . x) / ((cos . x) ^2))
then ((exp_R (#) (sin / cos)) + (exp_R / (cos ^2))) . x = ((exp_R (#) (sin / cos)) . x) + ((exp_R / (cos ^2)) . x) by A1, VALUED_1:def_1
.= ((exp_R . x) * ((sin / cos) . x)) + ((exp_R / (cos ^2)) . x) by VALUED_1:5
.= ((exp_R . x) * ((sin . x) * ((cos . x) "))) + ((exp_R / (cos ^2)) . x) by A4, A11, RFUNCT_1:def_1
.= (((exp_R . x) * (sin . x)) / (cos . x)) + ((exp_R . x) / ((cos ^2) . x)) by A2, A11, RFUNCT_1:def_1
.= (((exp_R . x) * (sin . x)) / (cos . x)) + ((exp_R . x) / ((cos . x) ^2)) by VALUED_1:11 ;
hence  f . x = (((exp_R . x) * (sin . x)) / (cos . x)) + ((exp_R . x) / ((cos . x) ^2)) by A1; ::_thesis: verum
end;
A12: for x being Real st x in dom ((exp_R (#) tan) `| Z) holds
((exp_R (#) tan) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom ((exp_R (#) tan) `| Z) implies ((exp_R (#) tan) `| Z) . x = f . x )
assume  x in dom ((exp_R (#) tan) `| Z) ; ::_thesis: ((exp_R (#) tan) `| Z) . x = f . x
then A13: x in Z by A9, FDIFF_1:def_7;
then ((exp_R (#) tan) `| Z) . x = (((exp_R . x) * (sin . x)) / (cos . x)) + ((exp_R . x) / ((cos . x) ^2)) by A2, FDIFF_8:30
.= f . x by A13, A10 ;
hence  ((exp_R (#) tan) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom ((exp_R (#) tan) `| Z) = dom f by A1, A9, FDIFF_1:def_7;
then  (exp_R (#) tan) `| Z = f by A12, PARTFUN1:5;
hence  integral (f,A) = ((exp_R (#) tan) . (upper_bound A)) - ((exp_R (#) tan) . (lower_bound A)) by A1, A8, A2, FDIFF_8:30, INTEGRA5:13; ::_thesis: verum
end;

theorem :: INTEGR12:6
for A being non empty closed_interval Subset of REAL
for f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z = dom f & f = (exp_R (#) (cos / sin)) - (exp_R / (sin ^2)) holds
integral (f,A) = ((exp_R (#) cot) . (upper_bound A)) - ((exp_R (#) cot) . (lower_bound A))
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z = dom f & f = (exp_R (#) (cos / sin)) - (exp_R / (sin ^2)) holds
integral (f,A) = ((exp_R (#) cot) . (upper_bound A)) - ((exp_R (#) cot) . (lower_bound A))
let f be PartFunc of REAL,REAL; ::_thesis: for Z being open Subset of REAL st A c= Z & Z = dom f & f = (exp_R (#) (cos / sin)) - (exp_R / (sin ^2)) holds
integral (f,A) = ((exp_R (#) cot) . (upper_bound A)) - ((exp_R (#) cot) . (lower_bound A))
let Z be open Subset of REAL; ::_thesis: ( A c= Z & Z = dom f & f = (exp_R (#) (cos / sin)) - (exp_R / (sin ^2)) implies integral (f,A) = ((exp_R (#) cot) . (upper_bound A)) - ((exp_R (#) cot) . (lower_bound A)) )
assume A1: ( A c= Z & Z = dom f & f = (exp_R (#) (cos / sin)) - (exp_R / (sin ^2)) ) ; ::_thesis: integral (f,A) = ((exp_R (#) cot) . (upper_bound A)) - ((exp_R (#) cot) . (lower_bound A))
then  Z = (dom (exp_R (#) (cos / sin))) /\ (dom (exp_R / (sin ^2))) by VALUED_1:12;
then A2: ( Z c= dom (exp_R (#) (cos / sin)) & Z c= dom (exp_R / (sin ^2)) ) by XBOOLE_1:18;
A3: dom (exp_R (#) (cos / sin)) c= (dom exp_R) /\ (dom (cos / sin)) by VALUED_1:def_4;
 dom (exp_R / (sin ^2)) c= (dom exp_R) /\ ((dom (sin ^2)) \ ((sin ^2) " {0})) by RFUNCT_1:def_1;
then  ( dom (exp_R (#) (cos / sin)) c= dom exp_R & dom (exp_R (#) (cos / sin)) c= dom (cos / sin) & dom (exp_R / (sin ^2)) c= (dom (sin ^2)) \ ((sin ^2) " {0}) ) by A3, XBOOLE_1:18;
then A4: ( Z c= dom exp_R & Z c= dom (cos / sin) & Z c= (dom (sin ^2)) \ ((sin ^2) " {0}) ) by A2, XBOOLE_1:1;
A5: exp_R is_differentiable_on Z by FDIFF_1:26, TAYLOR_1:16;
 for x being Real st x in Z holds
cos / sin is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies cos / sin is_differentiable_in x )
assume  x in Z ; ::_thesis: cos / sin is_differentiable_in x
then  sin . x <> 0 by A4, FDIFF_8:2;
hence  cos / sin is_differentiable_in x by FDIFF_7:47; ::_thesis: verum
end;
then  cos / sin is_differentiable_on Z by A4, FDIFF_1:9;
then A6: exp_R (#) (cos / sin) is_differentiable_on Z by A2, A5, FDIFF_1:21;
 sin is_differentiable_on Z by FDIFF_1:26, SIN_COS:68;
then A7: sin ^2 is_differentiable_on Z by FDIFF_2:20;
 for x being Real st x in Z holds
(sin ^2) . x <> 0
proof
let x be Real; ::_thesis: ( x in Z implies (sin ^2) . x <> 0 )
assume  x in Z ; ::_thesis: (sin ^2) . x <> 0
then  x in dom (exp_R / (sin ^2)) by A2;
then  x in (dom exp_R) /\ ((dom (sin ^2)) \ ((sin ^2) " {0})) by RFUNCT_1:def_1;
then  x in (dom (sin ^2)) \ ((sin ^2) " {0}) by XBOOLE_0:def_4;
then  x in dom ((sin ^2) ^) by RFUNCT_1:def_2;
hence  (sin ^2) . x <> 0 by RFUNCT_1:3; ::_thesis: verum
end;
then  exp_R / (sin ^2) is_differentiable_on Z by A5, A7, FDIFF_2:21;
then  f | Z is continuous by A1, A6, FDIFF_1:19, FDIFF_1:25;
then  f | A is continuous by A1, FCONT_1:16;
then A8: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A9: exp_R (#) cot is_differentiable_on Z by A2, FDIFF_8:31;
A10: for x being Real st x in Z holds
f . x = (((exp_R . x) * (cos . x)) / (sin . x)) - ((exp_R . x) / ((sin . x) ^2))
proof
let x be Real; ::_thesis: ( x in Z implies f . x = (((exp_R . x) * (cos . x)) / (sin . x)) - ((exp_R . x) / ((sin . x) ^2)) )
assume A11: x in Z ; ::_thesis: f . x = (((exp_R . x) * (cos . x)) / (sin . x)) - ((exp_R . x) / ((sin . x) ^2))
then ((exp_R (#) (cos / sin)) - (exp_R / (sin ^2))) . x = ((exp_R (#) (cos / sin)) . x) - ((exp_R / (sin ^2)) . x) by A1, VALUED_1:13
.= ((exp_R . x) * ((cos / sin) . x)) - ((exp_R / (sin ^2)) . x) by VALUED_1:5
.= ((exp_R . x) * ((cos . x) * ((sin . x) "))) - ((exp_R / (sin ^2)) . x) by A4, A11, RFUNCT_1:def_1
.= (((exp_R . x) * (cos . x)) / (sin . x)) - ((exp_R . x) / ((sin ^2) . x)) by A2, A11, RFUNCT_1:def_1
.= (((exp_R . x) * (cos . x)) / (sin . x)) - ((exp_R . x) / ((sin . x) ^2)) by VALUED_1:11 ;
hence  f . x = (((exp_R . x) * (cos . x)) / (sin . x)) - ((exp_R . x) / ((sin . x) ^2)) by A1; ::_thesis: verum
end;
A12: for x being Real st x in dom ((exp_R (#) cot) `| Z) holds
((exp_R (#) cot) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom ((exp_R (#) cot) `| Z) implies ((exp_R (#) cot) `| Z) . x = f . x )
assume  x in dom ((exp_R (#) cot) `| Z) ; ::_thesis: ((exp_R (#) cot) `| Z) . x = f . x
then A13: x in Z by A9, FDIFF_1:def_7;
then ((exp_R (#) cot) `| Z) . x = (((exp_R . x) * (cos . x)) / (sin . x)) - ((exp_R . x) / ((sin . x) ^2)) by A2, FDIFF_8:31
.= f . x by A13, A10 ;
hence  ((exp_R (#) cot) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom ((exp_R (#) cot) `| Z) = dom f by A1, A9, FDIFF_1:def_7;
then  (exp_R (#) cot) `| Z = f by A12, PARTFUN1:5;
hence  integral (f,A) = ((exp_R (#) cot) . (upper_bound A)) - ((exp_R (#) cot) . (lower_bound A)) by A1, A8, A2, FDIFF_8:31, INTEGRA5:13; ::_thesis: verum
end;

theorem :: INTEGR12:7
for A being non empty closed_interval Subset of REAL
for f1, f 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 ) & Z c= ].(- 1),1.[ & Z = dom f & f = (exp_R (#) arctan) + (exp_R / (f1 + (#Z 2))) holds
integral (f,A) = ((exp_R (#) arctan) . (upper_bound A)) - ((exp_R (#) arctan) . (lower_bound A))
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for f1, f 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 ) & Z c= ].(- 1),1.[ & Z = dom f & f = (exp_R (#) arctan) + (exp_R / (f1 + (#Z 2))) holds
integral (f,A) = ((exp_R (#) arctan) . (upper_bound A)) - ((exp_R (#) arctan) . (lower_bound A))
let f1, f 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 ) & Z c= ].(- 1),1.[ & Z = dom f & f = (exp_R (#) arctan) + (exp_R / (f1 + (#Z 2))) holds
integral (f,A) = ((exp_R (#) arctan) . (upper_bound A)) - ((exp_R (#) arctan) . (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 ) & Z c= ].(- 1),1.[ & Z = dom f & f = (exp_R (#) arctan) + (exp_R / (f1 + (#Z 2))) implies integral (f,A) = ((exp_R (#) arctan) . (upper_bound A)) - ((exp_R (#) arctan) . (lower_bound A)) )
assume A1: ( A c= Z & ( for x being Real st x in Z holds
f1 . x = 1 ) & Z c= ].(- 1),1.[ & Z = dom f & f = (exp_R (#) arctan) + (exp_R / (f1 + (#Z 2))) ) ; ::_thesis: integral (f,A) = ((exp_R (#) arctan) . (upper_bound A)) - ((exp_R (#) arctan) . (lower_bound A))
then A2: Z = (dom (exp_R (#) arctan)) /\ (dom (exp_R / (f1 + (#Z 2)))) by VALUED_1:def_1;
A3: exp_R is_differentiable_on Z by FDIFF_1:26, TAYLOR_1:16;
A4: exp_R (#) arctan is_differentiable_on Z by A1, SIN_COS9:123;
A5: Z c= dom (exp_R / (f1 + (#Z 2))) by A2, XBOOLE_1:18;
then A6: Z c= dom (exp_R (#) ((f1 + (#Z 2)) ^)) by RFUNCT_1:31;
then  Z c= (dom exp_R) /\ (dom ((f1 + (#Z 2)) ^)) by VALUED_1:def_4;
then A7: Z c= dom ((f1 + (#Z 2)) ^) by XBOOLE_1:18;
 dom ((f1 + (#Z 2)) ^) c= dom (f1 + (#Z 2)) by RFUNCT_1:1;
then A8: Z c= dom (f1 + (#Z 2)) by A7, XBOOLE_1:1;
 (f1 + (#Z 2)) ^ is_differentiable_on Z by A1, A7, Th1;
then  exp_R (#) ((f1 + (#Z 2)) ^) is_differentiable_on Z by A3, A6, FDIFF_1:21;
then  exp_R / (f1 + (#Z 2)) is_differentiable_on Z by RFUNCT_1:31;
then  f | Z is continuous by A1, A4, FDIFF_1:18, FDIFF_1:25;
then  f | A is continuous by A1, FCONT_1:16;
then A9: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A10: for x being Real st x in Z holds
f . x = ((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies f . x = ((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2))) )
assume A11: x in Z ; ::_thesis: f . x = ((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2)))
then ((exp_R (#) arctan) + (exp_R / (f1 + (#Z 2)))) . x = ((exp_R (#) arctan) . x) + ((exp_R / (f1 + (#Z 2))) . x) by A1, VALUED_1:def_1
.= ((exp_R . x) * (arctan . x)) + ((exp_R / (f1 + (#Z 2))) . x) by VALUED_1:5
.= ((exp_R . x) * (arctan . x)) + ((exp_R . x) / ((f1 + (#Z 2)) . x)) by A5, A11, RFUNCT_1:def_1
.= ((exp_R . x) * (arctan . x)) + ((exp_R . x) / ((f1 . x) + ((#Z 2) . x))) by A8, A11, VALUED_1:def_1
.= ((exp_R . x) * (arctan . x)) + ((exp_R . x) / ((f1 . x) + (x #Z 2))) by TAYLOR_1:def_1
.= ((exp_R . x) * (arctan . x)) + ((exp_R . x) / ((f1 . x) + (x ^2))) by FDIFF_7:1
.= ((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2))) by A1, A11 ;
hence  f . x = ((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2))) by A1; ::_thesis: verum
end;
A12: for x being Real st x in dom ((exp_R (#) arctan) `| Z) holds
((exp_R (#) arctan) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom ((exp_R (#) arctan) `| Z) implies ((exp_R (#) arctan) `| Z) . x = f . x )
assume  x in dom ((exp_R (#) arctan) `| Z) ; ::_thesis: ((exp_R (#) arctan) `| Z) . x = f . x
then A13: x in Z by A4, FDIFF_1:def_7;
then ((exp_R (#) arctan) `| Z) . x = ((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2))) by A1, SIN_COS9:123
.= f . x by A13, A10 ;
hence  ((exp_R (#) arctan) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom ((exp_R (#) arctan) `| Z) = dom f by A1, A4, FDIFF_1:def_7;
then  (exp_R (#) arctan) `| Z = f by A12, PARTFUN1:5;
hence  integral (f,A) = ((exp_R (#) arctan) . (upper_bound A)) - ((exp_R (#) arctan) . (lower_bound A)) by A1, A9, INTEGRA5:13, SIN_COS9:123; ::_thesis: verum
end;

theorem :: INTEGR12:8
for A being non empty closed_interval Subset of REAL
for f1, f 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 ) & Z c= ].(- 1),1.[ & Z = dom f & f = (exp_R (#) arccot) - (exp_R / (f1 + (#Z 2))) holds
integral (f,A) = ((exp_R (#) arccot) . (upper_bound A)) - ((exp_R (#) arccot) . (lower_bound A))
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for f1, f 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 ) & Z c= ].(- 1),1.[ & Z = dom f & f = (exp_R (#) arccot) - (exp_R / (f1 + (#Z 2))) holds
integral (f,A) = ((exp_R (#) arccot) . (upper_bound A)) - ((exp_R (#) arccot) . (lower_bound A))
let f1, f 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 ) & Z c= ].(- 1),1.[ & Z = dom f & f = (exp_R (#) arccot) - (exp_R / (f1 + (#Z 2))) holds
integral (f,A) = ((exp_R (#) arccot) . (upper_bound A)) - ((exp_R (#) 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 ) & Z c= ].(- 1),1.[ & Z = dom f & f = (exp_R (#) arccot) - (exp_R / (f1 + (#Z 2))) implies integral (f,A) = ((exp_R (#) arccot) . (upper_bound A)) - ((exp_R (#) arccot) . (lower_bound A)) )
assume A1: ( A c= Z & ( for x being Real st x in Z holds
f1 . x = 1 ) & Z c= ].(- 1),1.[ & Z = dom f & f = (exp_R (#) arccot) - (exp_R / (f1 + (#Z 2))) ) ; ::_thesis: integral (f,A) = ((exp_R (#) arccot) . (upper_bound A)) - ((exp_R (#) arccot) . (lower_bound A))
then A2: Z = (dom (exp_R (#) arccot)) /\ (dom (exp_R / (f1 + (#Z 2)))) by VALUED_1:12;
A3: exp_R is_differentiable_on Z by FDIFF_1:26, TAYLOR_1:16;
A4: exp_R (#) arccot is_differentiable_on Z by A1, SIN_COS9:124;
A5: Z c= dom (exp_R / (f1 + (#Z 2))) by A2, XBOOLE_1:18;
then A6: Z c= dom (exp_R (#) ((f1 + (#Z 2)) ^)) by RFUNCT_1:31;
then  Z c= (dom exp_R) /\ (dom ((f1 + (#Z 2)) ^)) by VALUED_1:def_4;
then A7: Z c= dom ((f1 + (#Z 2)) ^) by XBOOLE_1:18;
 dom ((f1 + (#Z 2)) ^) c= dom (f1 + (#Z 2)) by RFUNCT_1:1;
then A8: Z c= dom (f1 + (#Z 2)) by A7, XBOOLE_1:1;
 (f1 + (#Z 2)) ^ is_differentiable_on Z by A1, A7, Th1;
then  exp_R (#) ((f1 + (#Z 2)) ^) is_differentiable_on Z by A3, A6, FDIFF_1:21;
then  exp_R / (f1 + (#Z 2)) is_differentiable_on Z by RFUNCT_1:31;
then  f | Z is continuous by A1, A4, FDIFF_1:19, FDIFF_1:25;
then  f | A is continuous by A1, FCONT_1:16;
then A9: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A10: for x being Real st x in Z holds
f . x = ((exp_R . x) * (arccot . x)) - ((exp_R . x) / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies f . x = ((exp_R . x) * (arccot . x)) - ((exp_R . x) / (1 + (x ^2))) )
assume A11: x in Z ; ::_thesis: f . x = ((exp_R . x) * (arccot . x)) - ((exp_R . x) / (1 + (x ^2)))
then ((exp_R (#) arccot) - (exp_R / (f1 + (#Z 2)))) . x = ((exp_R (#) arccot) . x) - ((exp_R / (f1 + (#Z 2))) . x) by A1, VALUED_1:13
.= ((exp_R . x) * (arccot . x)) - ((exp_R / (f1 + (#Z 2))) . x) by VALUED_1:5
.= ((exp_R . x) * (arccot . x)) - ((exp_R . x) / ((f1 + (#Z 2)) . x)) by A5, A11, RFUNCT_1:def_1
.= ((exp_R . x) * (arccot . x)) - ((exp_R . x) / ((f1 . x) + ((#Z 2) . x))) by A8, A11, VALUED_1:def_1
.= ((exp_R . x) * (arccot . x)) - ((exp_R . x) / ((f1 . x) + (x #Z 2))) by TAYLOR_1:def_1
.= ((exp_R . x) * (arccot . x)) - ((exp_R . x) / ((f1 . x) + (x ^2))) by FDIFF_7:1
.= ((exp_R . x) * (arccot . x)) - ((exp_R . x) / (1 + (x ^2))) by A1, A11 ;
hence  f . x = ((exp_R . x) * (arccot . x)) - ((exp_R . x) / (1 + (x ^2))) by A1; ::_thesis: verum
end;
A12: for x being Real st x in dom ((exp_R (#) arccot) `| Z) holds
((exp_R (#) arccot) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom ((exp_R (#) arccot) `| Z) implies ((exp_R (#) arccot) `| Z) . x = f . x )
assume  x in dom ((exp_R (#) arccot) `| Z) ; ::_thesis: ((exp_R (#) arccot) `| Z) . x = f . x
then A13: x in Z by A4, FDIFF_1:def_7;
then ((exp_R (#) arccot) `| Z) . x = ((exp_R . x) * (arccot . x)) - ((exp_R . x) / (1 + (x ^2))) by A1, SIN_COS9:124
.= f . x by A13, A10 ;
hence  ((exp_R (#) arccot) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom ((exp_R (#) arccot) `| Z) = dom f by A1, A4, FDIFF_1:def_7;
then  (exp_R (#) arccot) `| Z = f by A12, PARTFUN1:5;
hence  integral (f,A) = ((exp_R (#) arccot) . (upper_bound A)) - ((exp_R (#) arccot) . (lower_bound A)) by A1, A9, INTEGRA5:13, SIN_COS9:124; ::_thesis: verum
end;

theorem :: INTEGR12:9
for A being non empty closed_interval Subset of REAL
for f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z = dom f & f = (exp_R * sin) (#) cos holds
integral (f,A) = ((exp_R * sin) . (upper_bound A)) - ((exp_R * sin) . (lower_bound A))
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z = dom f & f = (exp_R * sin) (#) cos holds
integral (f,A) = ((exp_R * sin) . (upper_bound A)) - ((exp_R * sin) . (lower_bound A))
let f be PartFunc of REAL,REAL; ::_thesis: for Z being open Subset of REAL st A c= Z & Z = dom f & f = (exp_R * sin) (#) cos holds
integral (f,A) = ((exp_R * sin) . (upper_bound A)) - ((exp_R * sin) . (lower_bound A))
let Z be open Subset of REAL; ::_thesis: ( A c= Z & Z = dom f & f = (exp_R * sin) (#) cos implies integral (f,A) = ((exp_R * sin) . (upper_bound A)) - ((exp_R * sin) . (lower_bound A)) )
assume A1: ( A c= Z & Z = dom f & f = (exp_R * sin) (#) cos ) ; ::_thesis: integral (f,A) = ((exp_R * sin) . (upper_bound A)) - ((exp_R * sin) . (lower_bound A))
then  Z = (dom (exp_R * sin)) /\ (dom cos) by VALUED_1:def_4;
then A2: Z c= dom (exp_R * sin) by XBOOLE_1:18;
then A3: exp_R * sin is_differentiable_on Z by FDIFF_7:37;
 cos is_differentiable_on Z by FDIFF_1:26, SIN_COS:67;
then  f | Z is continuous by A1, A3, FDIFF_1:21, FDIFF_1:25;
then  f | A is continuous by A1, FCONT_1:16;
then A4: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A5: for x being Real st x in Z holds
f . x = (exp_R . (sin . x)) * (cos . x)
proof
let x be Real; ::_thesis: ( x in Z implies f . x = (exp_R . (sin . x)) * (cos . x) )
assume A6: x in Z ; ::_thesis: f . x = (exp_R . (sin . x)) * (cos . x)
then ((exp_R * sin) (#) cos) . x = ((exp_R * sin) . x) * (cos . x) by A1, VALUED_1:def_4
.= (exp_R . (sin . x)) * (cos . x) by A2, A6, FUNCT_1:12 ;
hence  f . x = (exp_R . (sin . x)) * (cos . x) by A1; ::_thesis: verum
end;
A7: for x being Real st x in dom ((exp_R * sin) `| Z) holds
((exp_R * sin) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom ((exp_R * sin) `| Z) implies ((exp_R * sin) `| Z) . x = f . x )
assume  x in dom ((exp_R * sin) `| Z) ; ::_thesis: ((exp_R * sin) `| Z) . x = f . x
then A8: x in Z by A3, FDIFF_1:def_7;
then ((exp_R * sin) `| Z) . x = (exp_R . (sin . x)) * (cos . x) by A2, FDIFF_7:37
.= f . x by A5, A8 ;
hence  ((exp_R * sin) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom ((exp_R * sin) `| Z) = dom f by A1, A3, FDIFF_1:def_7;
then  (exp_R * sin) `| Z = f by A7, PARTFUN1:5;
hence  integral (f,A) = ((exp_R * sin) . (upper_bound A)) - ((exp_R * sin) . (lower_bound A)) by A1, A2, A4, FDIFF_7:37, INTEGRA5:13; ::_thesis: verum
end;

theorem :: INTEGR12:10
for A being non empty closed_interval Subset of REAL
for f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z = dom f & f = (exp_R * cos) (#) sin holds
integral (f,A) = ((- (exp_R * cos)) . (upper_bound A)) - ((- (exp_R * cos)) . (lower_bound A))
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z = dom f & f = (exp_R * cos) (#) sin holds
integral (f,A) = ((- (exp_R * cos)) . (upper_bound A)) - ((- (exp_R * cos)) . (lower_bound A))
let f be PartFunc of REAL,REAL; ::_thesis: for Z being open Subset of REAL st A c= Z & Z = dom f & f = (exp_R * cos) (#) sin holds
integral (f,A) = ((- (exp_R * cos)) . (upper_bound A)) - ((- (exp_R * cos)) . (lower_bound A))
let Z be open Subset of REAL; ::_thesis: ( A c= Z & Z = dom f & f = (exp_R * cos) (#) sin implies integral (f,A) = ((- (exp_R * cos)) . (upper_bound A)) - ((- (exp_R * cos)) . (lower_bound A)) )
assume A1: ( A c= Z & Z = dom f & f = (exp_R * cos) (#) sin ) ; ::_thesis: integral (f,A) = ((- (exp_R * cos)) . (upper_bound A)) - ((- (exp_R * cos)) . (lower_bound A))
then  Z = (dom (exp_R * cos)) /\ (dom sin) by VALUED_1:def_4;
then A2: Z c= dom (exp_R * cos) by XBOOLE_1:18;
then A3: exp_R * cos is_differentiable_on Z by FDIFF_7:36;
 sin is_differentiable_on Z by FDIFF_1:26, SIN_COS:68;
then  f | Z is continuous by A1, A3, FDIFF_1:21, FDIFF_1:25;
then  f | A is continuous by A1, FCONT_1:16;
then A4: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A5: Z c= dom (- (exp_R * cos)) by A2, VALUED_1:8;
then A6: (- 1) (#) (exp_R * cos) is_differentiable_on Z by A3, Lm1, FDIFF_1:20;
A7: for x being Real st x in Z holds
((- (exp_R * cos)) `| Z) . x = (exp_R . (cos . x)) * (sin . x)
proof
let x be Real; ::_thesis: ( x in Z implies ((- (exp_R * cos)) `| Z) . x = (exp_R . (cos . x)) * (sin . x) )
assume A8: x in Z ; ::_thesis: ((- (exp_R * cos)) `| Z) . x = (exp_R . (cos . x)) * (sin . x)
A9: cos is_differentiable_in x by SIN_COS:63;
A10: exp_R is_differentiable_in cos . x by SIN_COS:65;
A11: exp_R * cos is_differentiable_in x by A3, A8, FDIFF_1:9;
((- (exp_R * cos)) `| Z) . x = diff ((- (exp_R * cos)),x) by A6, A8, FDIFF_1:def_7
.= (- 1) * (diff ((exp_R * cos),x)) by A11, Lm1, FDIFF_1:15
.= (- 1) * ((diff (exp_R,(cos . x))) * (diff (cos,x))) by A9, A10, FDIFF_2:13
.= (- 1) * ((diff (exp_R,(cos . x))) * (- (sin . x))) by SIN_COS:63
.= (- 1) * ((exp_R . (cos . x)) * (- (sin . x))) by SIN_COS:65
.= (exp_R . (cos . x)) * (sin . x) ;
hence  ((- (exp_R * cos)) `| Z) . x = (exp_R . (cos . x)) * (sin . x) ; ::_thesis: verum
end;
A12: for x being Real st x in Z holds
f . x = (exp_R . (cos . x)) * (sin . x)
proof
let x be Real; ::_thesis: ( x in Z implies f . x = (exp_R . (cos . x)) * (sin . x) )
assume A13: x in Z ; ::_thesis: f . x = (exp_R . (cos . x)) * (sin . x)
then ((exp_R * cos) (#) sin) . x = ((exp_R * cos) . x) * (sin . x) by A1, VALUED_1:def_4
.= (exp_R . (cos . x)) * (sin . x) by A2, A13, FUNCT_1:12 ;
hence  f . x = (exp_R . (cos . x)) * (sin . x) by A1; ::_thesis: verum
end;
A14: for x being Real st x in dom ((- (exp_R * cos)) `| Z) holds
((- (exp_R * cos)) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom ((- (exp_R * cos)) `| Z) implies ((- (exp_R * cos)) `| Z) . x = f . x )
assume  x in dom ((- (exp_R * cos)) `| Z) ; ::_thesis: ((- (exp_R * cos)) `| Z) . x = f . x
then A15: x in Z by A6, FDIFF_1:def_7;
then ((- (exp_R * cos)) `| Z) . x = (exp_R . (cos . x)) * (sin . x) by A7
.= f . x by A15, A12 ;
hence  ((- (exp_R * cos)) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom ((- (exp_R * cos)) `| Z) = dom f by A1, A6, FDIFF_1:def_7;
then  (- (exp_R * cos)) `| Z = f by A14, PARTFUN1:5;
hence  integral (f,A) = ((- (exp_R * cos)) . (upper_bound A)) - ((- (exp_R * cos)) . (lower_bound A)) by A1, A4, A3, A5, Lm1, FDIFF_1:20, INTEGRA5:13; ::_thesis: verum
end;

theorem :: INTEGR12:11
for A being non empty closed_interval Subset of REAL
for f 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
x > 0 ) & Z = dom f & f = (cos * ln) (#) ((id Z) ^) holds
integral (f,A) = ((sin * ln) . (upper_bound A)) - ((sin * ln) . (lower_bound A))
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for f 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
x > 0 ) & Z = dom f & f = (cos * ln) (#) ((id Z) ^) holds
integral (f,A) = ((sin * ln) . (upper_bound A)) - ((sin * ln) . (lower_bound A))
let f 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
x > 0 ) & Z = dom f & f = (cos * ln) (#) ((id Z) ^) holds
integral (f,A) = ((sin * ln) . (upper_bound A)) - ((sin * ln) . (lower_bound A))
let Z be open Subset of REAL; ::_thesis: ( A c= Z & ( for x being Real st x in Z holds
x > 0 ) & Z = dom f & f = (cos * ln) (#) ((id Z) ^) implies integral (f,A) = ((sin * ln) . (upper_bound A)) - ((sin * ln) . (lower_bound A)) )
assume A1: ( A c= Z & ( for x being Real st x in Z holds
x > 0 ) & Z = dom f & f = (cos * ln) (#) ((id Z) ^) ) ; ::_thesis: integral (f,A) = ((sin * ln) . (upper_bound A)) - ((sin * ln) . (lower_bound A))
then A2: Z = dom ((cos * ln) / (id Z)) by RFUNCT_1:31;
 Z = (dom (cos * ln)) /\ (dom ((id Z) ^)) by A1, VALUED_1:def_4;
then A3: Z c= dom (cos * ln) by XBOOLE_1:18;
 for y being set st y in Z holds
y in dom (sin * ln)
proof
let y be set ; ::_thesis: ( y in Z implies y in dom (sin * ln) )
assume  y in Z ; ::_thesis: y in dom (sin * ln)
then  ( y in dom ln & ln . y in dom sin ) by A3, FUNCT_1:11, SIN_COS:24;
hence  y in dom (sin * ln) by FUNCT_1:11; ::_thesis: verum
end;
then A4: Z c= dom (sin * ln) by TARSKI:def_3;
A5: cos * ln is_differentiable_on Z by A3, A1, FDIFF_7:33;
 not 0 in Z by A1;
then  (id Z) ^ is_differentiable_on Z by FDIFF_5:4;
then  f | Z is continuous by A1, A5, FDIFF_1:21, FDIFF_1:25;
then  f | A is continuous by A1, FCONT_1:16;
then A6: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A7: sin * ln is_differentiable_on Z by A4, A1, FDIFF_7:32;
A8: for x being Real st x in Z holds
f . x = (cos . (ln . x)) / x
proof
let x be Real; ::_thesis: ( x in Z implies f . x = (cos . (ln . x)) / x )
assume A9: x in Z ; ::_thesis: f . x = (cos . (ln . x)) / x
((cos * ln) (#) ((id Z) ^)) . x = ((cos * ln) / (id Z)) . x by RFUNCT_1:31
.= ((cos * ln) . x) * (((id Z) . x) ") by A2, A9, RFUNCT_1:def_1
.= ((cos * ln) . x) / x by A9, FUNCT_1:18
.= (cos . (ln . x)) / x by A3, A9, FUNCT_1:12 ;
hence  f . x = (cos . (ln . x)) / x by A1; ::_thesis: verum
end;
A10: for x being Real st x in dom ((sin * ln) `| Z) holds
((sin * ln) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom ((sin * ln) `| Z) implies ((sin * ln) `| Z) . x = f . x )
assume  x in dom ((sin * ln) `| Z) ; ::_thesis: ((sin * ln) `| Z) . x = f . x
then A11: x in Z by A7, FDIFF_1:def_7;
then ((sin * ln) `| Z) . x = (cos . (ln . x)) / x by A4, A1, FDIFF_7:32
.= f . x by A11, A8 ;
hence  ((sin * ln) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom ((sin * ln) `| Z) = dom f by A1, A7, FDIFF_1:def_7;
then  (sin * ln) `| Z = f by A10, PARTFUN1:5;
hence  integral (f,A) = ((sin * ln) . (upper_bound A)) - ((sin * ln) . (lower_bound A)) by A6, A4, A1, FDIFF_7:32, INTEGRA5:13; ::_thesis: verum
end;

Lm2:  right_open_halfline 0 =  {  g where g is Real : 0 < g  }
by XXREAL_1:230;

theorem :: INTEGR12:12
for A being non empty closed_interval Subset of REAL
for f 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
x > 0 ) & Z = dom f & f = (sin * ln) (#) ((id Z) ^) holds
integral (f,A) = ((- (cos * ln)) . (upper_bound A)) - ((- (cos * ln)) . (lower_bound A))
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for f 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
x > 0 ) & Z = dom f & f = (sin * ln) (#) ((id Z) ^) holds
integral (f,A) = ((- (cos * ln)) . (upper_bound A)) - ((- (cos * ln)) . (lower_bound A))
let f 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
x > 0 ) & Z = dom f & f = (sin * ln) (#) ((id Z) ^) holds
integral (f,A) = ((- (cos * ln)) . (upper_bound A)) - ((- (cos * ln)) . (lower_bound A))
let Z be open Subset of REAL; ::_thesis: ( A c= Z & ( for x being Real st x in Z holds
x > 0 ) & Z = dom f & f = (sin * ln) (#) ((id Z) ^) implies integral (f,A) = ((- (cos * ln)) . (upper_bound A)) - ((- (cos * ln)) . (lower_bound A)) )
assume A1: ( A c= Z & ( for x being Real st x in Z holds
x > 0 ) & Z = dom f & f = (sin * ln) (#) ((id Z) ^) ) ; ::_thesis: integral (f,A) = ((- (cos * ln)) . (upper_bound A)) - ((- (cos * ln)) . (lower_bound A))
then A2: Z = dom ((sin * ln) / (id Z)) by RFUNCT_1:31;
 Z = (dom (sin * ln)) /\ (dom ((id Z) ^)) by A1, VALUED_1:def_4;
then A3: Z c= dom (sin * ln) by XBOOLE_1:18;
 for y being set st y in Z holds
y in dom (cos * ln)
proof
let y be set ; ::_thesis: ( y in Z implies y in dom (cos * ln) )
assume  y in Z ; ::_thesis: y in dom (cos * ln)
then  ( y in dom ln & ln . y in dom cos ) by A3, FUNCT_1:11, SIN_COS:24;
hence  y in dom (cos * ln) by FUNCT_1:11; ::_thesis: verum
end;
then A4: Z c= dom (cos * ln) by TARSKI:def_3;
A5: sin * ln is_differentiable_on Z by A3, A1, FDIFF_7:32;
 not 0 in Z by A1;
then  (id Z) ^ is_differentiable_on Z by FDIFF_5:4;
then  f | Z is continuous by A1, A5, FDIFF_1:21, FDIFF_1:25;
then  f | A is continuous by A1, FCONT_1:16;
then A6: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A7: cos * ln is_differentiable_on Z by A4, A1, FDIFF_7:33;
A8: Z c= dom (- (cos * ln)) by A4, VALUED_1:8;
then A9: (- 1) (#) (cos * ln) is_differentiable_on Z by A7, Lm1, FDIFF_1:20;
A10: for x being Real st x in Z holds
((- (cos * ln)) `| Z) . x = (sin . (ln . x)) / x
proof
let x be Real; ::_thesis: ( x in Z implies ((- (cos * ln)) `| Z) . x = (sin . (ln . x)) / x )
assume A11: x in Z ; ::_thesis: ((- (cos * ln)) `| Z) . x = (sin . (ln . x)) / x
then  x > 0 by A1;
then A12: x in right_open_halfline 0 by Lm2;
A13: ln is_differentiable_in x by A11, A1, TAYLOR_1:18;
A14: cos is_differentiable_in ln . x by SIN_COS:63;
A15: cos * ln is_differentiable_in x by A7, A11, FDIFF_1:9;
((- (cos * ln)) `| Z) . x = diff ((- (cos * ln)),x) by A9, A11, FDIFF_1:def_7
.= (- 1) * (diff ((cos * ln),x)) by A15, Lm1, FDIFF_1:15
.= (- 1) * ((diff (cos,(ln . x))) * (diff (ln,x))) by A13, A14, FDIFF_2:13
.= (- 1) * ((- (sin . (ln . x))) * (diff (ln,x))) by SIN_COS:63
.= (- 1) * ((- (sin . (ln . x))) * (1 / x)) by A12, TAYLOR_1:18
.= (sin . (ln . x)) / x ;
hence  ((- (cos * ln)) `| Z) . x = (sin . (ln . x)) / x ; ::_thesis: verum
end;
A16: for x being Real st x in Z holds
f . x = (sin . (ln . x)) / x
proof
let x be Real; ::_thesis: ( x in Z implies f . x = (sin . (ln . x)) / x )
assume A17: x in Z ; ::_thesis: f . x = (sin . (ln . x)) / x
((sin * ln) (#) ((id Z) ^)) . x = ((sin * ln) / (id Z)) . x by RFUNCT_1:31
.= ((sin * ln) . x) * (((id Z) . x) ") by A2, A17, RFUNCT_1:def_1
.= ((sin * ln) . x) / x by A17, FUNCT_1:18
.= (sin . (ln . x)) / x by A3, A17, FUNCT_1:12 ;
hence  f . x = (sin . (ln . x)) / x by A1; ::_thesis: verum
end;
A18: for x being Real st x in dom ((- (cos * ln)) `| Z) holds
((- (cos * ln)) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom ((- (cos * ln)) `| Z) implies ((- (cos * ln)) `| Z) . x = f . x )
assume  x in dom ((- (cos * ln)) `| Z) ; ::_thesis: ((- (cos * ln)) `| Z) . x = f . x
then A19: x in Z by A9, FDIFF_1:def_7;
then ((- (cos * ln)) `| Z) . x = (sin . (ln . x)) / x by A10
.= f . x by A19, A16 ;
hence  ((- (cos * ln)) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom ((- (cos * ln)) `| Z) = dom f by A1, A9, FDIFF_1:def_7;
then  (- (cos * ln)) `| Z = f by A18, PARTFUN1:5;
hence  integral (f,A) = ((- (cos * ln)) . (upper_bound A)) - ((- (cos * ln)) . (lower_bound A)) by A1, A6, A7, A8, Lm1, FDIFF_1:20, INTEGRA5:13; ::_thesis: verum
end;

theorem :: INTEGR12:13
for A being non empty closed_interval Subset of REAL
for f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z = dom f & f = exp_R (#) (cos * exp_R) holds
integral (f,A) = ((sin * exp_R) . (upper_bound A)) - ((sin * exp_R) . (lower_bound A))
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z = dom f & f = exp_R (#) (cos * exp_R) holds
integral (f,A) = ((sin * exp_R) . (upper_bound A)) - ((sin * exp_R) . (lower_bound A))
let f be PartFunc of REAL,REAL; ::_thesis: for Z being open Subset of REAL st A c= Z & Z = dom f & f = exp_R (#) (cos * exp_R) holds
integral (f,A) = ((sin * exp_R) . (upper_bound A)) - ((sin * exp_R) . (lower_bound A))
let Z be open Subset of REAL; ::_thesis: ( A c= Z & Z = dom f & f = exp_R (#) (cos * exp_R) implies integral (f,A) = ((sin * exp_R) . (upper_bound A)) - ((sin * exp_R) . (lower_bound A)) )
assume A1: ( A c= Z & Z = dom f & f = exp_R (#) (cos * exp_R) ) ; ::_thesis: integral (f,A) = ((sin * exp_R) . (upper_bound A)) - ((sin * exp_R) . (lower_bound A))
then  Z = (dom exp_R) /\ (dom (cos * exp_R)) by VALUED_1:def_4;
then A2: ( Z c= dom exp_R & Z c= dom (cos * exp_R) ) by XBOOLE_1:18;
 for y being set st y in Z holds
y in dom (sin * exp_R)
proof
let y be set ; ::_thesis: ( y in Z implies y in dom (sin * exp_R) )
assume  y in Z ; ::_thesis: y in dom (sin * exp_R)
then  ( y in dom exp_R & exp_R . y in dom sin ) by A2, SIN_COS:24;
hence  y in dom (sin * exp_R) by FUNCT_1:11; ::_thesis: verum
end;
then A3: Z c= dom (sin * exp_R) by TARSKI:def_3;
A4: cos * exp_R is_differentiable_on Z by A2, FDIFF_7:35;
 exp_R is_differentiable_on Z by FDIFF_1:26, TAYLOR_1:16;
then  f | Z is continuous by A1, A4, FDIFF_1:21, 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: sin * exp_R is_differentiable_on Z by A3, FDIFF_7:34;
A7: for x being Real st x in Z holds
f . x = (exp_R . x) * (cos . (exp_R . x))
proof
let x be Real; ::_thesis: ( x in Z implies f . x = (exp_R . x) * (cos . (exp_R . x)) )
assume A8: x in Z ; ::_thesis: f . x = (exp_R . x) * (cos . (exp_R . x))
then (exp_R (#) (cos * exp_R)) . x = (exp_R . x) * ((cos * exp_R) . x) by A1, VALUED_1:def_4
.= (exp_R . x) * (cos . (exp_R . x)) by A2, A8, FUNCT_1:12 ;
hence  f . x = (exp_R . x) * (cos . (exp_R . x)) by A1; ::_thesis: verum
end;
A9: for x being Real st x in dom ((sin * exp_R) `| Z) holds
((sin * exp_R) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom ((sin * exp_R) `| Z) implies ((sin * exp_R) `| Z) . x = f . x )
assume  x in dom ((sin * exp_R) `| Z) ; ::_thesis: ((sin * exp_R) `| Z) . x = f . x
then A10: x in Z by A6, FDIFF_1:def_7;
then ((sin * exp_R) `| Z) . x = (exp_R . x) * (cos . (exp_R . x)) by A3, FDIFF_7:34
.= f . x by A10, A7 ;
hence  ((sin * exp_R) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom ((sin * exp_R) `| Z) = dom f by A1, A6, FDIFF_1:def_7;
then  (sin * exp_R) `| Z = f by A9, PARTFUN1:5;
hence  integral (f,A) = ((sin * exp_R) . (upper_bound A)) - ((sin * exp_R) . (lower_bound A)) by A1, A5, A3, FDIFF_7:34, INTEGRA5:13; ::_thesis: verum
end;

theorem :: INTEGR12:14
for A being non empty closed_interval Subset of REAL
for f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z = dom f & f = exp_R (#) (sin * exp_R) holds
integral (f,A) = ((- (cos * exp_R)) . (upper_bound A)) - ((- (cos * exp_R)) . (lower_bound A))
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z = dom f & f = exp_R (#) (sin * exp_R) holds
integral (f,A) = ((- (cos * exp_R)) . (upper_bound A)) - ((- (cos * exp_R)) . (lower_bound A))
let f be PartFunc of REAL,REAL; ::_thesis: for Z being open Subset of REAL st A c= Z & Z = dom f & f = exp_R (#) (sin * exp_R) holds
integral (f,A) = ((- (cos * exp_R)) . (upper_bound A)) - ((- (cos * exp_R)) . (lower_bound A))
let Z be open Subset of REAL; ::_thesis: ( A c= Z & Z = dom f & f = exp_R (#) (sin * exp_R) implies integral (f,A) = ((- (cos * exp_R)) . (upper_bound A)) - ((- (cos * exp_R)) . (lower_bound A)) )
assume A1: ( A c= Z & Z = dom f & f = exp_R (#) (sin * exp_R) ) ; ::_thesis: integral (f,A) = ((- (cos * exp_R)) . (upper_bound A)) - ((- (cos * exp_R)) . (lower_bound A))
then  Z = (dom exp_R) /\ (dom (sin * exp_R)) by VALUED_1:def_4;
then A2: ( Z c= dom exp_R & Z c= dom (sin * exp_R) ) by XBOOLE_1:18;
 for y being set st y in Z holds
y in dom (cos * exp_R)
proof
let y be set ; ::_thesis: ( y in Z implies y in dom (cos * exp_R) )
assume  y in Z ; ::_thesis: y in dom (cos * exp_R)
then  ( y in dom exp_R & exp_R . y in dom cos ) by A2, SIN_COS:24;
hence  y in dom (cos * exp_R) by FUNCT_1:11; ::_thesis: verum
end;
then A3: Z c= dom (cos * exp_R) by TARSKI:def_3;
A4: sin * exp_R is_differentiable_on Z by A2, FDIFF_7:34;
 exp_R is_differentiable_on Z by FDIFF_1:26, TAYLOR_1:16;
then  f | Z is continuous by A1, A4, FDIFF_1:21, 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: cos * exp_R is_differentiable_on Z by A3, FDIFF_7:35;
A7: Z c= dom (- (cos * exp_R)) by A3, VALUED_1:8;
then A8: (- 1) (#) (cos * exp_R) is_differentiable_on Z by A6, Lm1, FDIFF_1:20;
A9: for x being Real st x in Z holds
((- (cos * exp_R)) `| Z) . x = (exp_R . x) * (sin . (exp_R . x))
proof
let x be Real; ::_thesis: ( x in Z implies ((- (cos * exp_R)) `| Z) . x = (exp_R . x) * (sin . (exp_R . x)) )
assume A10: x in Z ; ::_thesis: ((- (cos * exp_R)) `| Z) . x = (exp_R . x) * (sin . (exp_R . x))
A11: exp_R is_differentiable_in x by SIN_COS:65;
A12: cos is_differentiable_in exp_R . x by SIN_COS:63;
A13: cos * exp_R is_differentiable_in x by A6, A10, FDIFF_1:9;
((- (cos * exp_R)) `| Z) . x = diff ((- (cos * exp_R)),x) by A8, A10, FDIFF_1:def_7
.= (- 1) * (diff ((cos * exp_R),x)) by A13, Lm1, FDIFF_1:15
.= (- 1) * ((diff (cos,(exp_R . x))) * (diff (exp_R,x))) by A11, A12, FDIFF_2:13
.= (- 1) * ((- (sin . (exp_R . x))) * (diff (exp_R,x))) by SIN_COS:63
.= (- 1) * ((- (sin . (exp_R . x))) * (exp_R . x)) by SIN_COS:65
.= (exp_R . x) * (sin . (exp_R . x)) ;
hence  ((- (cos * exp_R)) `| Z) . x = (exp_R . x) * (sin . (exp_R . x)) ; ::_thesis: verum
end;
A14: for x being Real st x in Z holds
f . x = (exp_R . x) * (sin . (exp_R . x))
proof
let x be Real; ::_thesis: ( x in Z implies f . x = (exp_R . x) * (sin . (exp_R . x)) )
assume A15: x in Z ; ::_thesis: f . x = (exp_R . x) * (sin . (exp_R . x))
then (exp_R (#) (sin * exp_R)) . x = (exp_R . x) * ((sin * exp_R) . x) by A1, VALUED_1:def_4
.= (exp_R . x) * (sin . (exp_R . x)) by A2, A15, FUNCT_1:12 ;
hence  f . x = (exp_R . x) * (sin . (exp_R . x)) by A1; ::_thesis: verum
end;
A16: for x being Real st x in dom ((- (cos * exp_R)) `| Z) holds
((- (cos * exp_R)) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom ((- (cos * exp_R)) `| Z) implies ((- (cos * exp_R)) `| Z) . x = f . x )
assume  x in dom ((- (cos * exp_R)) `| Z) ; ::_thesis: ((- (cos * exp_R)) `| Z) . x = f . x
then A17: x in Z by A8, FDIFF_1:def_7;
then ((- (cos * exp_R)) `| Z) . x = (exp_R . x) * (sin . (exp_R . x)) by A9
.= f . x by A17, A14 ;
hence  ((- (cos * exp_R)) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom ((- (cos * exp_R)) `| Z) = dom f by A1, A8, FDIFF_1:def_7;
then  (- (cos * exp_R)) `| Z = f by A16, PARTFUN1:5;
hence  integral (f,A) = ((- (cos * exp_R)) . (upper_bound A)) - ((- (cos * exp_R)) . (lower_bound A)) by A1, A5, A6, A7, Lm1, FDIFF_1:20, INTEGRA5:13; ::_thesis: verum
end;

theorem :: INTEGR12:15
for r being Real
for A being non empty closed_interval Subset of REAL
for f1, f2, g, f 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 = arctan * g holds
integral (f,A) = ((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) . (upper_bound A)) - ((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) . (lower_bound A))
proof
let r be Real; ::_thesis: for A being non empty closed_interval Subset of REAL
for f1, f2, g, f 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 = arctan * g holds
integral (f,A) = ((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) . (upper_bound A)) - ((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) . (lower_bound A))
let A be non empty closed_interval Subset of REAL; ::_thesis: for f1, f2, g, f 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 = arctan * g holds
integral (f,A) = ((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) . (upper_bound A)) - ((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) . (lower_bound A))
let f1, f2, g, f 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 = arctan * g holds
integral (f,A) = ((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) . (upper_bound A)) - ((((id Z) (#) (arctan * 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 = arctan * g implies integral (f,A) = ((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) . (upper_bound A)) - ((((id Z) (#) (arctan * 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 = arctan * g ) ; ::_thesis: integral (f,A) = ((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) . (upper_bound A)) - ((((id Z) (#) (arctan * 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) (#) (arctan * 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) (#) (arctan * g))) /\ (dom ((r / 2) (#) (ln * (f1 + f2)))) by A2, XBOOLE_1:19;
then A3: Z c= dom (((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) by VALUED_1:12;
 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_COS9:87;
then  f | Z is continuous by FDIFF_1:25;
then  f | A is continuous by A1, FCONT_1:16;
then A4: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A5: ( ( for x being Real st x in Z holds
( g . x = x / r & g . x > - 1 & g . x < 1 ) ) & ( 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 A6: ((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z by A1, A3, SIN_COS9:109;
A7: for x being Real st x in Z holds
f . x = arctan . (x / r)
proof
let x be Real; ::_thesis: ( x in Z implies f . x = arctan . (x / r) )
assume A8: x in Z ; ::_thesis: f . x = arctan . (x / r)
then (arctan * g) . x = arctan . (g . x) by A1, FUNCT_1:12
.= arctan . (x / r) by A1, A8 ;
hence  f . x = arctan . (x / r) by A1; ::_thesis: verum
end;
A9: for x being Real st x in dom ((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) holds
((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom ((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) implies ((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = f . x )
assume  x in dom ((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) ; ::_thesis: ((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = f . x
then A10: x in Z by A6, FDIFF_1:def_7;
then ((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . (x / r) by A1, A3, A5, SIN_COS9:109
.= f . x by A7, A10 ;
hence  ((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom ((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) = dom f by A1, A6, FDIFF_1:def_7;
then  (((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z = f by A9, PARTFUN1:5;
hence  integral (f,A) = ((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) . (upper_bound A)) - ((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) . (lower_bound A)) by A1, A4, A6, INTEGRA5:13; ::_thesis: verum
end;

theorem :: INTEGR12:16
for r being Real
for A being non empty closed_interval Subset of REAL
for f1, f2, g, f 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))
proof
let r be Real; ::_thesis: for A being non empty closed_interval Subset of REAL
for f1, f2, g, f 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 f1, f2, g, f 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 f1, f2, g, f 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 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 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
end;

theorem :: INTEGR12:17
for r being Real
for A being non empty 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 = (arctan * 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) (#) (arctan * f1)) . (upper_bound A)) - (((id Z) (#) (arctan * f1)) . (lower_bound A))
proof
let r be Real; ::_thesis: for A being non empty 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 = (arctan * 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) (#) (arctan * f1)) . (upper_bound A)) - (((id Z) (#) (arctan * f1)) . (lower_bound A))
let A be non empty 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 = (arctan * 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) (#) (arctan * f1)) . (upper_bound A)) - (((id Z) (#) (arctan * 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 = (arctan * 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) (#) (arctan * f1)) . (upper_bound A)) - (((id Z) (#) (arctan * f1)) . (lower_bound A))
let Z be open Subset of REAL; ::_thesis: ( A c= Z & f = (arctan * 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) (#) (arctan * f1)) . (upper_bound A)) - (((id Z) (#) (arctan * f1)) . (lower_bound A)) )
assume A1: ( A c= Z & f = (arctan * 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) (#) (arctan * f1)) . (upper_bound A)) - (((id Z) (#) (arctan * f1)) . (lower_bound A))
then  Z = (dom (arctan * f1)) /\ (dom ((id Z) / (r (#) (g + (f1 ^2))))) by VALUED_1:def_1;
then A2: ( Z c= dom (arctan * f1) & Z c= dom ((id Z) / (r (#) (g + (f1 ^2)))) ) by XBOOLE_1:18;
 Z c= (dom (id Z)) /\ (dom (arctan * f1)) by A2, XBOOLE_1:19;
then A3: Z c= dom ((id Z) (#) (arctan * f1)) by VALUED_1:def_4;
 Z c= (dom (id Z)) /\ ((dom (r (#) (g + (f1 ^2)))) \ ((r (#) (g + (f1 ^2))) " {0})) by A2, RFUNCT_1:def_1;
then  Z c= (dom (r (#) (g + (f1 ^2)))) \ ((r (#) (g + (f1 ^2))) " {0}) by XBOOLE_1:18;
then A4: Z c= dom ((r (#) (g + (f1 ^2))) ^) by RFUNCT_1:def_2;
 dom ((r (#) (g + (f1 ^2))) ^) c= dom (r (#) (g + (f1 ^2))) by RFUNCT_1:1;
then  Z c= dom (r (#) (g + (f1 ^2))) by A4, XBOOLE_1:1;
then A5: Z c= dom (g + (f1 ^2)) by VALUED_1:def_5;
A6: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A7: for x being Real st x in Z holds
( f1 . x = x / r & f1 . x > - 1 & f1 . x < 1 ) by A1;
then A8: (id Z) (#) (arctan * f1) is_differentiable_on Z by A3, SIN_COS9:105;
A9: for x being Real st x in Z holds
f . x = (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2))))
proof
let x be Real; ::_thesis: ( x in Z implies f . x = (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2)))) )
assume A10: x in Z ; ::_thesis: f . x = (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2))))
then ((arctan * f1) + ((id Z) / (r (#) (g + (f1 ^2))))) . x = ((arctan * f1) . x) + (((id Z) / (r (#) (g + (f1 ^2)))) . x) by A1, VALUED_1:def_1
.= (arctan . (f1 . x)) + (((id Z) / (r (#) (g + (f1 ^2)))) . x) by A2, A10, FUNCT_1:12
.= (arctan . (f1 . x)) + (((id Z) . x) / ((r (#) (g + (f1 ^2))) . x)) by A2, A10, RFUNCT_1:def_1
.= (arctan . (f1 . x)) + (x / ((r (#) (g + (f1 ^2))) . x)) by A10, FUNCT_1:18
.= (arctan . (f1 . x)) + (x / (r * ((g + (f1 ^2)) . x))) by VALUED_1:6
.= (arctan . (f1 . x)) + (x / (r * ((g . x) + ((f1 ^2) . x)))) by A5, A10, VALUED_1:def_1
.= (arctan . (f1 . x)) + (x / (r * ((g . x) + ((f1 . x) ^2)))) by VALUED_1:11
.= (arctan . (x / r)) + (x / (r * ((g . x) + ((f1 . x) ^2)))) by A1, A10
.= (arctan . (x / r)) + (x / (r * (1 + ((f1 . x) ^2)))) by A1, A10
.= (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2)))) by A1, A10 ;
hence  f . x = (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2)))) by A1; ::_thesis: verum
end;
A11: for x being Real st x in dom (((id Z) (#) (arctan * f1)) `| Z) holds
(((id Z) (#) (arctan * f1)) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom (((id Z) (#) (arctan * f1)) `| Z) implies (((id Z) (#) (arctan * f1)) `| Z) . x = f . x )
assume  x in dom (((id Z) (#) (arctan * f1)) `| Z) ; ::_thesis: (((id Z) (#) (arctan * f1)) `| Z) . x = f . x
then A12: x in Z by A8, FDIFF_1:def_7;
then (((id Z) (#) (arctan * f1)) `| Z) . x = (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2)))) by A3, A7, SIN_COS9:105
.= f . x by A9, A12 ;
hence  (((id Z) (#) (arctan * f1)) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom (((id Z) (#) (arctan * f1)) `| Z) = dom f by A1, A8, FDIFF_1:def_7;
then  ((id Z) (#) (arctan * f1)) `| Z = f by A11, PARTFUN1:5;
hence  integral (f,A) = (((id Z) (#) (arctan * f1)) . (upper_bound A)) - (((id Z) (#) (arctan * f1)) . (lower_bound A)) by A1, A6, A8, INTEGRA5:13; ::_thesis: verum
end;

theorem :: INTEGR12:18
for r being Real
for A being non empty 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))
proof
let r be Real; ::_thesis: for A being non empty 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 non empty 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 A2: ( Z c= dom (arccot * f1) & Z c= dom ((id Z) / (r (#) (g + (f1 ^2)))) ) by XBOOLE_1:18;
 Z c= (dom (id Z)) /\ (dom (arccot * f1)) by A2, XBOOLE_1:19;
then A3: 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 A2, RFUNCT_1:def_1;
then  Z c= (dom (r (#) (g + (f1 ^2)))) \ ((r (#) (g + (f1 ^2))) " {0}) by XBOOLE_1:18;
then A4: Z c= dom ((r (#) (g + (f1 ^2))) ^) by RFUNCT_1:def_2;
 dom ((r (#) (g + (f1 ^2))) ^) c= dom (r (#) (g + (f1 ^2))) by RFUNCT_1:1;
then  Z c= dom (r (#) (g + (f1 ^2))) by A4, XBOOLE_1:1;
then A5: Z c= dom (g + (f1 ^2)) by VALUED_1:def_5;
A6: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A7: for x being Real st x in Z holds
( f1 . x = x / r & f1 . x > - 1 & f1 . x < 1 ) by A1;
then A8: (id Z) (#) (arccot * f1) is_differentiable_on Z by A3, SIN_COS9:106;
A9: 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 A10: 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 A1, VALUED_1:13
.= (arccot . (f1 . x)) - (((id Z) / (r (#) (g + (f1 ^2)))) . x) by A2, A10, FUNCT_1:12
.= (arccot . (f1 . x)) - (((id Z) . x) / ((r (#) (g + (f1 ^2))) . x)) by A2, A10, RFUNCT_1:def_1
.= (arccot . (f1 . x)) - (x / ((r (#) (g + (f1 ^2))) . x)) by A10, FUNCT_1:18
.= (arccot . (f1 . x)) - (x / (r * ((g + (f1 ^2)) . x))) by VALUED_1:6
.= (arccot . (f1 . x)) - (x / (r * ((g . x) + ((f1 ^2) . x)))) by A5, A10, VALUED_1:def_1
.= (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, A10
.= (arccot . (x / r)) - (x / (r * (1 + ((f1 . x) ^2)))) by A1, A10
.= (arccot . (x / r)) - (x / (r * (1 + ((x / r) ^2)))) by A1, A10 ;
hence  f . x = (arccot . (x / r)) - (x / (r * (1 + ((x / r) ^2)))) by A1; ::_thesis: verum
end;
A11: 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 A12: x in Z by A8, FDIFF_1:def_7;
then (((id Z) (#) (arccot * f1)) `| Z) . x = (arccot . (x / r)) - (x / (r * (1 + ((x / r) ^2)))) by A3, A7, SIN_COS9:106
.= f . x by A9, A12 ;
hence  (((id Z) (#) (arccot * f1)) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom (((id Z) (#) (arccot * f1)) `| Z) = dom f by A1, A8, FDIFF_1:def_7;
then  ((id Z) (#) (arccot * f1)) `| Z = f by A11, PARTFUN1:5;
hence  integral (f,A) = (((id Z) (#) (arccot * f1)) . (upper_bound A)) - (((id Z) (#) (arccot * f1)) . (lower_bound A)) by A1, A6, A8, INTEGRA5:13; ::_thesis: verum
end;

theorem :: INTEGR12:19
for n being Element of NAT
for A being non empty closed_interval Subset of REAL
for f1, f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f1 . x = 1 ) & Z = dom f & Z c= dom ((#Z n) * arcsin) & 1 < n & f = (n (#) ((#Z (n - 1)) * arcsin)) / ((#R (1 / 2)) * (f1 - (#Z 2))) holds
integral (f,A) = (((#Z n) * arcsin) . (upper_bound A)) - (((#Z n) * arcsin) . (lower_bound A))
proof
let n be Element of NAT ; ::_thesis: for A being non empty closed_interval Subset of REAL
for f1, f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f1 . x = 1 ) & Z = dom f & Z c= dom ((#Z n) * arcsin) & 1 < n & f = (n (#) ((#Z (n - 1)) * arcsin)) / ((#R (1 / 2)) * (f1 - (#Z 2))) holds
integral (f,A) = (((#Z n) * arcsin) . (upper_bound A)) - (((#Z n) * arcsin) . (lower_bound A))
let A be non empty closed_interval Subset of REAL; ::_thesis: for f1, f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f1 . x = 1 ) & Z = dom f & Z c= dom ((#Z n) * arcsin) & 1 < n & f = (n (#) ((#Z (n - 1)) * arcsin)) / ((#R (1 / 2)) * (f1 - (#Z 2))) holds
integral (f,A) = (((#Z n) * arcsin) . (upper_bound A)) - (((#Z n) * arcsin) . (lower_bound A))
let f1, f be PartFunc of REAL,REAL; ::_thesis: for Z being open Subset of REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f1 . x = 1 ) & Z = dom f & Z c= dom ((#Z n) * arcsin) & 1 < n & f = (n (#) ((#Z (n - 1)) * arcsin)) / ((#R (1 / 2)) * (f1 - (#Z 2))) holds
integral (f,A) = (((#Z n) * arcsin) . (upper_bound A)) - (((#Z n) * arcsin) . (lower_bound A))
let Z be open Subset of REAL; ::_thesis: ( A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f1 . x = 1 ) & Z = dom f & Z c= dom ((#Z n) * arcsin) & 1 < n & f = (n (#) ((#Z (n - 1)) * arcsin)) / ((#R (1 / 2)) * (f1 - (#Z 2))) implies integral (f,A) = (((#Z n) * arcsin) . (upper_bound A)) - (((#Z n) * arcsin) . (lower_bound A)) )
assume A1: ( A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f1 . x = 1 ) & Z = dom f & Z c= dom ((#Z n) * arcsin) & 1 < n & f = (n (#) ((#Z (n - 1)) * arcsin)) / ((#R (1 / 2)) * (f1 - (#Z 2))) ) ; ::_thesis: integral (f,A) = (((#Z n) * arcsin) . (upper_bound A)) - (((#Z n) * arcsin) . (lower_bound A))
then  Z = (dom (n (#) ((#Z (n - 1)) * arcsin))) /\ ((dom ((#R (1 / 2)) * (f1 - (#Z 2)))) \ (((#R (1 / 2)) * (f1 - (#Z 2))) " {0})) by RFUNCT_1:def_1;
then A2: ( Z c= dom (n (#) ((#Z (n - 1)) * arcsin)) & Z c= (dom ((#R (1 / 2)) * (f1 - (#Z 2)))) \ (((#R (1 / 2)) * (f1 - (#Z 2))) " {0}) ) by XBOOLE_1:18;
then A3: Z c= dom ((#Z (n - 1)) * arcsin) by VALUED_1:def_5;
A4: Z c= dom (((#R (1 / 2)) * (f1 - (#Z 2))) ^) by A2, RFUNCT_1:def_2;
 dom (((#R (1 / 2)) * (f1 - (#Z 2))) ^) c= dom ((#R (1 / 2)) * (f1 - (#Z 2))) by RFUNCT_1:1;
then A5: Z c= dom ((#R (1 / 2)) * (f1 - (#Z 2))) by A4, XBOOLE_1:1;
 for x being Real st x in Z holds
(#Z (n - 1)) * arcsin is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies (#Z (n - 1)) * arcsin is_differentiable_in x )
assume  x in Z ; ::_thesis: (#Z (n - 1)) * arcsin is_differentiable_in x
then A6: arcsin is_differentiable_in x by A1, FDIFF_1:9, SIN_COS6:83;
consider m being Nat such that
A7: n = m + 1 by A1, NAT_1:6;
thus  (#Z (n - 1)) * arcsin is_differentiable_in x by A6, A7, TAYLOR_1:3; ::_thesis: verum
end;
then  (#Z (n - 1)) * arcsin is_differentiable_on Z by A3, FDIFF_1:9;
then A8: n (#) ((#Z (n - 1)) * arcsin) is_differentiable_on Z by A2, FDIFF_1:20;
set f2 = #Z 2;
 for x being Real st x in Z holds
(f1 - (#Z 2)) . x > 0
proof
let x be Real; ::_thesis: ( x in Z implies (f1 - (#Z 2)) . x > 0 )
assume A9: x in Z ; ::_thesis: (f1 - (#Z 2)) . x > 0
then  ( - 1 < x & x < 1 ) by A1, XXREAL_1:4;
then  ( 0 < 1 + x & 0 < 1 - x ) by XREAL_1:50, XREAL_1:148;
then A10: 0 < (1 + x) * (1 - x) by XREAL_1:129;
 for x being Real st x in Z holds
x in dom (f1 - (#Z 2)) by A5, FUNCT_1:11;
then (f1 - (#Z 2)) . x = (f1 . x) - ((#Z 2) . x) by A9, VALUED_1:13
.= (f1 . x) - (x #Z (1 + 1)) by TAYLOR_1:def_1
.= (f1 . x) - ((x #Z 1) * (x #Z 1)) by TAYLOR_1:1
.= (f1 . x) - (x * (x #Z 1)) by PREPOWER:35
.= (f1 . x) - (x * x) by PREPOWER:35
.= 1 - (x * x) by A1, A9 ;
hence  (f1 - (#Z 2)) . x > 0 by A10; ::_thesis: verum
end;
then  for x being Real st x in Z holds
( f1 . x = 1 & (f1 - (#Z 2)) . x > 0 ) by A1;
then A11: (#R (1 / 2)) * (f1 - (#Z 2)) is_differentiable_on Z by A5, FDIFF_7:22;
 for x being Real st x in Z holds
((#R (1 / 2)) * (f1 - (#Z 2))) . x <> 0 by A4, RFUNCT_1:3;
then  f is_differentiable_on Z by A1, A8, A11, FDIFF_2:21;
then  f | Z is continuous by FDIFF_1:25;
then  f | A is continuous by A1, FCONT_1:16;
then A12: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A13: (#Z n) * arcsin is_differentiable_on Z by A1, FDIFF_7:10;
A14: for x being Real st x in Z holds
f . x = (n * ((arcsin . x) #Z (n - 1))) / (sqrt (1 - (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies f . x = (n * ((arcsin . x) #Z (n - 1))) / (sqrt (1 - (x ^2))) )
assume A15: x in Z ; ::_thesis: f . x = (n * ((arcsin . x) #Z (n - 1))) / (sqrt (1 - (x ^2)))
then A16: ( x in dom (f1 - (#Z 2)) & (f1 - (#Z 2)) . x in dom (#R (1 / 2)) ) by A5, FUNCT_1:11;
then A17: (f1 - (#Z 2)) . x in right_open_halfline 0 by TAYLOR_1:def_4;
 ( - 1 < x & x < 1 ) by A1, A15, XXREAL_1:4;
then  ( 0 < 1 + x & 0 < 1 - x ) by XREAL_1:50, XREAL_1:148;
then A18: 0 < (1 + x) * (1 - x) by XREAL_1:129;
((n (#) ((#Z (n - 1)) * arcsin)) / ((#R (1 / 2)) * (f1 - (#Z 2)))) . x = ((n (#) ((#Z (n - 1)) * arcsin)) . x) / (((#R (1 / 2)) * (f1 - (#Z 2))) . x) by A1, A15, RFUNCT_1:def_1
.= (n * (((#Z (n - 1)) * arcsin) . x)) / (((#R (1 / 2)) * (f1 - (#Z 2))) . x) by VALUED_1:6
.= (n * ((#Z (n - 1)) . (arcsin . x))) / (((#R (1 / 2)) * (f1 - (#Z 2))) . x) by A3, A15, FUNCT_1:12
.= (n * ((arcsin . x) #Z (n - 1))) / (((#R (1 / 2)) * (f1 - (#Z 2))) . x) by TAYLOR_1:def_1
.= (n * ((arcsin . x) #Z (n - 1))) / ((#R (1 / 2)) . ((f1 - (#Z 2)) . x)) by A5, A15, FUNCT_1:12
.= (n * ((arcsin . x) #Z (n - 1))) / (((f1 - (#Z 2)) . x) #R (1 / 2)) by A17, TAYLOR_1:def_4
.= (n * ((arcsin . x) #Z (n - 1))) / (((f1 . x) - ((#Z 2) . x)) #R (1 / 2)) by A16, VALUED_1:13
.= (n * ((arcsin . x) #Z (n - 1))) / (((f1 . x) - (x #Z 2)) #R (1 / 2)) by TAYLOR_1:def_1
.= (n * ((arcsin . x) #Z (n - 1))) / (((f1 . x) - (x ^2)) #R (1 / 2)) by FDIFF_7:1
.= (n * ((arcsin . x) #Z (n - 1))) / ((1 - (x ^2)) #R (1 / 2)) by A1, A15
.= (n * ((arcsin . x) #Z (n - 1))) / (sqrt (1 - (x ^2))) by A18, FDIFF_7:2 ;
hence  f . x = (n * ((arcsin . x) #Z (n - 1))) / (sqrt (1 - (x ^2))) by A1; ::_thesis: verum
end;
A19: for x being Real st x in dom (((#Z n) * arcsin) `| Z) holds
(((#Z n) * arcsin) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom (((#Z n) * arcsin) `| Z) implies (((#Z n) * arcsin) `| Z) . x = f . x )
assume  x in dom (((#Z n) * arcsin) `| Z) ; ::_thesis: (((#Z n) * arcsin) `| Z) . x = f . x
then A20: x in Z by A13, FDIFF_1:def_7;
then (((#Z n) * arcsin) `| Z) . x = (n * ((arcsin . x) #Z (n - 1))) / (sqrt (1 - (x ^2))) by A1, FDIFF_7:10
.= f . x by A14, A20 ;
hence  (((#Z n) * arcsin) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom (((#Z n) * arcsin) `| Z) = dom f by A1, A13, FDIFF_1:def_7;
then  ((#Z n) * arcsin) `| Z = f by A19, PARTFUN1:5;
hence  integral (f,A) = (((#Z n) * arcsin) . (upper_bound A)) - (((#Z n) * arcsin) . (lower_bound A)) by A1, A12, FDIFF_7:10, INTEGRA5:13; ::_thesis: verum
end;

theorem :: INTEGR12:20
for n being Element of NAT
for A being non empty closed_interval Subset of REAL
for f1, f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f1 . x = 1 ) & Z c= dom ((#Z n) * arccos) & Z = dom f & 1 < n & f = (n (#) ((#Z (n - 1)) * arccos)) / ((#R (1 / 2)) * (f1 - (#Z 2))) holds
integral (f,A) = ((- ((#Z n) * arccos)) . (upper_bound A)) - ((- ((#Z n) * arccos)) . (lower_bound A))
proof
let n be Element of NAT ; ::_thesis: for A being non empty closed_interval Subset of REAL
for f1, f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f1 . x = 1 ) & Z c= dom ((#Z n) * arccos) & Z = dom f & 1 < n & f = (n (#) ((#Z (n - 1)) * arccos)) / ((#R (1 / 2)) * (f1 - (#Z 2))) holds
integral (f,A) = ((- ((#Z n) * arccos)) . (upper_bound A)) - ((- ((#Z n) * arccos)) . (lower_bound A))
let A be non empty closed_interval Subset of REAL; ::_thesis: for f1, f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f1 . x = 1 ) & Z c= dom ((#Z n) * arccos) & Z = dom f & 1 < n & f = (n (#) ((#Z (n - 1)) * arccos)) / ((#R (1 / 2)) * (f1 - (#Z 2))) holds
integral (f,A) = ((- ((#Z n) * arccos)) . (upper_bound A)) - ((- ((#Z n) * arccos)) . (lower_bound A))
let f1, f be PartFunc of REAL,REAL; ::_thesis: for Z being open Subset of REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f1 . x = 1 ) & Z c= dom ((#Z n) * arccos) & Z = dom f & 1 < n & f = (n (#) ((#Z (n - 1)) * arccos)) / ((#R (1 / 2)) * (f1 - (#Z 2))) holds
integral (f,A) = ((- ((#Z n) * arccos)) . (upper_bound A)) - ((- ((#Z n) * arccos)) . (lower_bound A))
let Z be open Subset of REAL; ::_thesis: ( A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f1 . x = 1 ) & Z c= dom ((#Z n) * arccos) & Z = dom f & 1 < n & f = (n (#) ((#Z (n - 1)) * arccos)) / ((#R (1 / 2)) * (f1 - (#Z 2))) implies integral (f,A) = ((- ((#Z n) * arccos)) . (upper_bound A)) - ((- ((#Z n) * arccos)) . (lower_bound A)) )
assume A1: ( A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f1 . x = 1 ) & Z c= dom ((#Z n) * arccos) & Z = dom f & 1 < n & f = (n (#) ((#Z (n - 1)) * arccos)) / ((#R (1 / 2)) * (f1 - (#Z 2))) ) ; ::_thesis: integral (f,A) = ((- ((#Z n) * arccos)) . (upper_bound A)) - ((- ((#Z n) * arccos)) . (lower_bound A))
then  Z = (dom (n (#) ((#Z (n - 1)) * arccos))) /\ ((dom ((#R (1 / 2)) * (f1 - (#Z 2)))) \ (((#R (1 / 2)) * (f1 - (#Z 2))) " {0})) by RFUNCT_1:def_1;
then A2: ( Z c= dom (n (#) ((#Z (n - 1)) * arccos)) & Z c= (dom ((#R (1 / 2)) * (f1 - (#Z 2)))) \ (((#R (1 / 2)) * (f1 - (#Z 2))) " {0}) ) by XBOOLE_1:18;
then A3: Z c= dom ((#Z (n - 1)) * arccos) by VALUED_1:def_5;
A4: Z c= dom (((#R (1 / 2)) * (f1 - (#Z 2))) ^) by A2, RFUNCT_1:def_2;
 dom (((#R (1 / 2)) * (f1 - (#Z 2))) ^) c= dom ((#R (1 / 2)) * (f1 - (#Z 2))) by RFUNCT_1:1;
then A5: Z c= dom ((#R (1 / 2)) * (f1 - (#Z 2))) by A4, XBOOLE_1:1;
 for x being Real st x in Z holds
(#Z (n - 1)) * arccos is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies (#Z (n - 1)) * arccos is_differentiable_in x )
assume  x in Z ; ::_thesis: (#Z (n - 1)) * arccos is_differentiable_in x
then A6: arccos is_differentiable_in x by A1, FDIFF_1:9, SIN_COS6:106;
consider m being Nat such that
A7: n = m + 1 by A1, NAT_1:6;
thus  (#Z (n - 1)) * arccos is_differentiable_in x by A6, A7, TAYLOR_1:3; ::_thesis: verum
end;
then  (#Z (n - 1)) * arccos is_differentiable_on Z by A3, FDIFF_1:9;
then A8: n (#) ((#Z (n - 1)) * arccos) is_differentiable_on Z by A2, FDIFF_1:20;
set f2 = #Z 2;
 for x being Real st x in Z holds
(f1 - (#Z 2)) . x > 0
proof
let x be Real; ::_thesis: ( x in Z implies (f1 - (#Z 2)) . x > 0 )
assume A9: x in Z ; ::_thesis: (f1 - (#Z 2)) . x > 0
then  ( - 1 < x & x < 1 ) by A1, XXREAL_1:4;
then  ( 0 < 1 + x & 0 < 1 - x ) by XREAL_1:50, XREAL_1:148;
then A10: 0 < (1 + x) * (1 - x) by XREAL_1:129;
 for x being Real st x in Z holds
x in dom (f1 - (#Z 2)) by A5, FUNCT_1:11;
then (f1 - (#Z 2)) . x = (f1 . x) - ((#Z 2) . x) by A9, VALUED_1:13
.= (f1 . x) - (x #Z (1 + 1)) by TAYLOR_1:def_1
.= (f1 . x) - ((x #Z 1) * (x #Z 1)) by TAYLOR_1:1
.= (f1 . x) - (x * (x #Z 1)) by PREPOWER:35
.= (f1 . x) - (x * x) by PREPOWER:35
.= 1 - (x * x) by A1, A9 ;
hence  (f1 - (#Z 2)) . x > 0 by A10; ::_thesis: verum
end;
then  for x being Real st x in Z holds
( f1 . x = 1 & (f1 - (#Z 2)) . x > 0 ) by A1;
then A11: (#R (1 / 2)) * (f1 - (#Z 2)) is_differentiable_on Z by A5, FDIFF_7:22;
 for x being Real st x in Z holds
((#R (1 / 2)) * (f1 - (#Z 2))) . x <> 0 by A4, RFUNCT_1:3;
then  f is_differentiable_on Z by A1, A8, A11, FDIFF_2:21;
then  f | Z is continuous by FDIFF_1:25;
then  f | A is continuous by A1, FCONT_1:16;
then A12: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A13: (#Z n) * arccos is_differentiable_on Z by A1, FDIFF_7:11;
A14: Z c= dom (- ((#Z n) * arccos)) by A1, VALUED_1:8;
then A15: (- 1) (#) ((#Z n) * arccos) is_differentiable_on Z by A13, Lm1, FDIFF_1:20;
A16: for x being Real st x in Z holds
f . x = (n * ((arccos . x) #Z (n - 1))) / (sqrt (1 - (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies f . x = (n * ((arccos . x) #Z (n - 1))) / (sqrt (1 - (x ^2))) )
assume A17: x in Z ; ::_thesis: f . x = (n * ((arccos . x) #Z (n - 1))) / (sqrt (1 - (x ^2)))
then A18: ( x in dom (f1 - (#Z 2)) & (f1 - (#Z 2)) . x in dom (#R (1 / 2)) ) by A5, FUNCT_1:11;
then A19: (f1 - (#Z 2)) . x in right_open_halfline 0 by TAYLOR_1:def_4;
 ( - 1 < x & x < 1 ) by A1, A17, XXREAL_1:4;
then  ( 0 < 1 + x & 0 < 1 - x ) by XREAL_1:50, XREAL_1:148;
then A20: 0 < (1 + x) * (1 - x) by XREAL_1:129;
((n (#) ((#Z (n - 1)) * arccos)) / ((#R (1 / 2)) * (f1 - (#Z 2)))) . x = ((n (#) ((#Z (n - 1)) * arccos)) . x) / (((#R (1 / 2)) * (f1 - (#Z 2))) . x) by A1, A17, RFUNCT_1:def_1
.= (n * (((#Z (n - 1)) * arccos) . x)) / (((#R (1 / 2)) * (f1 - (#Z 2))) . x) by VALUED_1:6
.= (n * ((#Z (n - 1)) . (arccos . x))) / (((#R (1 / 2)) * (f1 - (#Z 2))) . x) by A3, A17, FUNCT_1:12
.= (n * ((arccos . x) #Z (n - 1))) / (((#R (1 / 2)) * (f1 - (#Z 2))) . x) by TAYLOR_1:def_1
.= (n * ((arccos . x) #Z (n - 1))) / ((#R (1 / 2)) . ((f1 - (#Z 2)) . x)) by A5, A17, FUNCT_1:12
.= (n * ((arccos . x) #Z (n - 1))) / (((f1 - (#Z 2)) . x) #R (1 / 2)) by A19, TAYLOR_1:def_4
.= (n * ((arccos . x) #Z (n - 1))) / (((f1 . x) - ((#Z 2) . x)) #R (1 / 2)) by A18, VALUED_1:13
.= (n * ((arccos . x) #Z (n - 1))) / (((f1 . x) - (x #Z 2)) #R (1 / 2)) by TAYLOR_1:def_1
.= (n * ((arccos . x) #Z (n - 1))) / (((f1 . x) - (x ^2)) #R (1 / 2)) by FDIFF_7:1
.= (n * ((arccos . x) #Z (n - 1))) / ((1 - (x ^2)) #R (1 / 2)) by A1, A17
.= (n * ((arccos . x) #Z (n - 1))) / (sqrt (1 - (x ^2))) by A20, FDIFF_7:2 ;
hence  f . x = (n * ((arccos . x) #Z (n - 1))) / (sqrt (1 - (x ^2))) by A1; ::_thesis: verum
end;
A21: for x being Real st x in Z holds
((- ((#Z n) * arccos)) `| Z) . x = (n * ((arccos . x) #Z (n - 1))) / (sqrt (1 - (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((- ((#Z n) * arccos)) `| Z) . x = (n * ((arccos . x) #Z (n - 1))) / (sqrt (1 - (x ^2))) )
assume A22: x in Z ; ::_thesis: ((- ((#Z n) * arccos)) `| Z) . x = (n * ((arccos . x) #Z (n - 1))) / (sqrt (1 - (x ^2)))
then A23: ( - 1 < x & x < 1 ) by A1, XXREAL_1:4;
A24: arccos is_differentiable_in x by A1, A22, FDIFF_1:9, SIN_COS6:106;
A25: (#Z n) * arccos is_differentiable_in x by A13, A22, FDIFF_1:9;
((- ((#Z n) * arccos)) `| Z) . x = diff ((- ((#Z n) * arccos)),x) by A15, A22, FDIFF_1:def_7
.= (- 1) * (diff (((#Z n) * arccos),x)) by A25, Lm1, FDIFF_1:15
.= (- 1) * ((n * ((arccos . x) #Z (n - 1))) * (diff (arccos,x))) by A24, TAYLOR_1:3
.= (- 1) * ((n * ((arccos . x) #Z (n - 1))) * (- (1 / (sqrt (1 - (x ^2)))))) by A23, SIN_COS6:106
.= (n * ((arccos . x) #Z (n - 1))) / (sqrt (1 - (x ^2))) ;
hence  ((- ((#Z n) * arccos)) `| Z) . x = (n * ((arccos . x) #Z (n - 1))) / (sqrt (1 - (x ^2))) ; ::_thesis: verum
end;
A26: for x being Real st x in dom ((- ((#Z n) * arccos)) `| Z) holds
((- ((#Z n) * arccos)) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom ((- ((#Z n) * arccos)) `| Z) implies ((- ((#Z n) * arccos)) `| Z) . x = f . x )
assume  x in dom ((- ((#Z n) * arccos)) `| Z) ; ::_thesis: ((- ((#Z n) * arccos)) `| Z) . x = f . x
then A27: x in Z by A15, FDIFF_1:def_7;
then ((- ((#Z n) * arccos)) `| Z) . x = (n * ((arccos . x) #Z (n - 1))) / (sqrt (1 - (x ^2))) by A21
.= f . x by A16, A27 ;
hence  ((- ((#Z n) * arccos)) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom ((- ((#Z n) * arccos)) `| Z) = dom f by A1, A15, FDIFF_1:def_7;
then  (- ((#Z n) * arccos)) `| Z = f by A26, PARTFUN1:5;
hence  integral (f,A) = ((- ((#Z n) * arccos)) . (upper_bound A)) - ((- ((#Z n) * arccos)) . (lower_bound A)) by A1, A12, A13, A14, Lm1, FDIFF_1:20, INTEGRA5:13; ::_thesis: verum
end;

theorem :: INTEGR12:21
for A being non empty closed_interval Subset of REAL
for f1, f 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 ) & Z c= ].(- 1),1.[ & Z = dom f & f = arcsin + ((id Z) / ((#R (1 / 2)) * (f1 - (#Z 2)))) holds
integral (f,A) = (((id Z) (#) arcsin) . (upper_bound A)) - (((id Z) (#) arcsin) . (lower_bound A))
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for f1, f 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 ) & Z c= ].(- 1),1.[ & Z = dom f & f = arcsin + ((id Z) / ((#R (1 / 2)) * (f1 - (#Z 2)))) holds
integral (f,A) = (((id Z) (#) arcsin) . (upper_bound A)) - (((id Z) (#) arcsin) . (lower_bound A))
let f1, f 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 ) & Z c= ].(- 1),1.[ & Z = dom f & f = arcsin + ((id Z) / ((#R (1 / 2)) * (f1 - (#Z 2)))) holds
integral (f,A) = (((id Z) (#) arcsin) . (upper_bound A)) - (((id Z) (#) arcsin) . (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 ) & Z c= ].(- 1),1.[ & Z = dom f & f = arcsin + ((id Z) / ((#R (1 / 2)) * (f1 - (#Z 2)))) implies integral (f,A) = (((id Z) (#) arcsin) . (upper_bound A)) - (((id Z) (#) arcsin) . (lower_bound A)) )
assume A1: ( A c= Z & ( for x being Real st x in Z holds
f1 . x = 1 ) & Z c= ].(- 1),1.[ & Z = dom f & f = arcsin + ((id Z) / ((#R (1 / 2)) * (f1 - (#Z 2)))) ) ; ::_thesis: integral (f,A) = (((id Z) (#) arcsin) . (upper_bound A)) - (((id Z) (#) arcsin) . (lower_bound A))
then  Z = (dom arcsin) /\ (dom ((id Z) / ((#R (1 / 2)) * (f1 - (#Z 2))))) by VALUED_1:def_1;
then A2: ( Z c= dom arcsin & Z c= dom ((id Z) / ((#R (1 / 2)) * (f1 - (#Z 2)))) ) by XBOOLE_1:18;
A3: Z = dom (id Z) ;
 Z c= (dom (id Z)) /\ (dom arcsin) by A2, XBOOLE_1:19;
then A4: Z c= dom ((id Z) (#) arcsin) by VALUED_1:def_4;
A5: arcsin is_differentiable_on Z by A1, FDIFF_1:26, SIN_COS6:83;
 for x being Real st x in Z holds
(id Z) . x = (1 * x) + 0 by FUNCT_1:18;
then A6: id Z is_differentiable_on Z by A3, FDIFF_1:23;
 Z c= (dom (id Z)) /\ ((dom ((#R (1 / 2)) * (f1 - (#Z 2)))) \ (((#R (1 / 2)) * (f1 - (#Z 2))) " {0})) by A2, RFUNCT_1:def_1;
then  Z c= (dom ((#R (1 / 2)) * (f1 - (#Z 2)))) \ (((#R (1 / 2)) * (f1 - (#Z 2))) " {0}) by XBOOLE_1:18;
then A7: Z c= dom (((#R (1 / 2)) * (f1 - (#Z 2))) ^) by RFUNCT_1:def_2;
 dom (((#R (1 / 2)) * (f1 - (#Z 2))) ^) c= dom ((#R (1 / 2)) * (f1 - (#Z 2))) by RFUNCT_1:1;
then A8: Z c= dom ((#R (1 / 2)) * (f1 - (#Z 2))) by A7, XBOOLE_1:1;
set f2 = #Z 2;
 for x being Real st x in Z holds
(f1 - (#Z 2)) . x > 0
proof
let x be Real; ::_thesis: ( x in Z implies (f1 - (#Z 2)) . x > 0 )
assume A9: x in Z ; ::_thesis: (f1 - (#Z 2)) . x > 0
then  ( - 1 < x & x < 1 ) by A1, XXREAL_1:4;
then  ( 0 < 1 + x & 0 < 1 - x ) by XREAL_1:50, XREAL_1:148;
then A10: 0 < (1 + x) * (1 - x) by XREAL_1:129;
 for x being Real st x in Z holds
x in dom (f1 - (#Z 2)) by A8, FUNCT_1:11;
then (f1 - (#Z 2)) . x = (f1 . x) - ((#Z 2) . x) by A9, VALUED_1:13
.= (f1 . x) - (x #Z (1 + 1)) by TAYLOR_1:def_1
.= (f1 . x) - ((x #Z 1) * (x #Z 1)) by TAYLOR_1:1
.= (f1 . x) - (x * (x #Z 1)) by PREPOWER:35
.= (f1 . x) - (x * x) by PREPOWER:35
.= 1 - (x * x) by A1, A9 ;
hence  (f1 - (#Z 2)) . x > 0 by A10; ::_thesis: verum
end;
then  for x being Real st x in Z holds
( f1 . x = 1 & (f1 - (#Z 2)) . x > 0 ) by A1;
then A11: (#R (1 / 2)) * (f1 - (#Z 2)) is_differentiable_on Z by A8, FDIFF_7:22;
 for x being Real st x in Z holds
((#R (1 / 2)) * (f1 - (#Z 2))) . x <> 0 by A7, RFUNCT_1:3;
then  (id Z) / ((#R (1 / 2)) * (f1 - (#Z 2))) is_differentiable_on Z by A6, A11, FDIFF_2:21;
then  f | Z is continuous by A1, A5, FDIFF_1:18, FDIFF_1:25;
then  f | A is continuous by A1, FCONT_1:16;
then A12: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A13: (id Z) (#) arcsin is_differentiable_on Z by A1, A4, FDIFF_7:16;
A14: for x being Real st x in Z holds
f . x = (arcsin . x) + (x / (sqrt (1 - (x ^2))))
proof
let x be Real; ::_thesis: ( x in Z implies f . x = (arcsin . x) + (x / (sqrt (1 - (x ^2)))) )
assume A15: x in Z ; ::_thesis: f . x = (arcsin . x) + (x / (sqrt (1 - (x ^2))))
then A16: ( x in dom (f1 - (#Z 2)) & (f1 - (#Z 2)) . x in dom (#R (1 / 2)) ) by A8, FUNCT_1:11;
then A17: (f1 - (#Z 2)) . x in right_open_halfline 0 by TAYLOR_1:def_4;
 ( - 1 < x & x < 1 ) by A1, A15, XXREAL_1:4;
then  ( 0 < 1 + x & 0 < 1 - x ) by XREAL_1:50, XREAL_1:148;
then A18: 0 < (1 + x) * (1 - x) by XREAL_1:129;
(arcsin + ((id Z) / ((#R (1 / 2)) * (f1 - (#Z 2))))) . x = (arcsin . x) + (((id Z) / ((#R (1 / 2)) * (f1 - (#Z 2)))) . x) by A1, A15, VALUED_1:def_1
.= (arcsin . x) + (((id Z) . x) / (((#R (1 / 2)) * (f1 - (#Z 2))) . x)) by A2, A15, RFUNCT_1:def_1
.= (arcsin . x) + (x / (((#R (1 / 2)) * (f1 - (#Z 2))) . x)) by A15, FUNCT_1:18
.= (arcsin . x) + (x / ((#R (1 / 2)) . ((f1 - (#Z 2)) . x))) by A8, A15, FUNCT_1:12
.= (arcsin . x) + (x / (((f1 - (#Z 2)) . x) #R (1 / 2))) by A17, TAYLOR_1:def_4
.= (arcsin . x) + (x / (((f1 . x) - ((#Z 2) . x)) #R (1 / 2))) by A16, VALUED_1:13
.= (arcsin . x) + (x / (((f1 . x) - (x #Z 2)) #R (1 / 2))) by TAYLOR_1:def_1
.= (arcsin . x) + (x / (((f1 . x) - (x ^2)) #R (1 / 2))) by FDIFF_7:1
.= (arcsin . x) + (x / ((1 - (x ^2)) #R (1 / 2))) by A1, A15
.= (arcsin . x) + (x / (sqrt (1 - (x ^2)))) by A18, FDIFF_7:2 ;
hence  f . x = (arcsin . x) + (x / (sqrt (1 - (x ^2)))) by A1; ::_thesis: verum
end;
A19: for x being Real st x in dom (((id Z) (#) arcsin) `| Z) holds
(((id Z) (#) arcsin) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom (((id Z) (#) arcsin) `| Z) implies (((id Z) (#) arcsin) `| Z) . x = f . x )
assume  x in dom (((id Z) (#) arcsin) `| Z) ; ::_thesis: (((id Z) (#) arcsin) `| Z) . x = f . x
then A20: x in Z by A13, FDIFF_1:def_7;
then (((id Z) (#) arcsin) `| Z) . x = (arcsin . x) + (x / (sqrt (1 - (x ^2)))) by A1, A4, FDIFF_7:16
.= f . x by A14, A20 ;
hence  (((id Z) (#) arcsin) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom (((id Z) (#) arcsin) `| Z) = dom f by A1, A13, FDIFF_1:def_7;
then  ((id Z) (#) arcsin) `| Z = f by A19, PARTFUN1:5;
hence  integral (f,A) = (((id Z) (#) arcsin) . (upper_bound A)) - (((id Z) (#) arcsin) . (lower_bound A)) by A1, A12, A4, FDIFF_7:16, INTEGRA5:13; ::_thesis: verum
end;

theorem :: INTEGR12:22
for A being non empty closed_interval Subset of REAL
for f1, f 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 ) & Z c= ].(- 1),1.[ & Z = dom f & f = arccos - ((id Z) / ((#R (1 / 2)) * (f1 - (#Z 2)))) holds
integral (f,A) = (((id Z) (#) arccos) . (upper_bound A)) - (((id Z) (#) arccos) . (lower_bound A))
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for f1, f 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 ) & Z c= ].(- 1),1.[ & Z = dom f & f = arccos - ((id Z) / ((#R (1 / 2)) * (f1 - (#Z 2)))) holds
integral (f,A) = (((id Z) (#) arccos) . (upper_bound A)) - (((id Z) (#) arccos) . (lower_bound A))
let f1, f 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 ) & Z c= ].(- 1),1.[ & Z = dom f & f = arccos - ((id Z) / ((#R (1 / 2)) * (f1 - (#Z 2)))) holds
integral (f,A) = (((id Z) (#) arccos) . (upper_bound A)) - (((id Z) (#) arccos) . (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 ) & Z c= ].(- 1),1.[ & Z = dom f & f = arccos - ((id Z) / ((#R (1 / 2)) * (f1 - (#Z 2)))) implies integral (f,A) = (((id Z) (#) arccos) . (upper_bound A)) - (((id Z) (#) arccos) . (lower_bound A)) )
assume A1: ( A c= Z & ( for x being Real st x in Z holds
f1 . x = 1 ) & Z c= ].(- 1),1.[ & Z = dom f & f = arccos - ((id Z) / ((#R (1 / 2)) * (f1 - (#Z 2)))) ) ; ::_thesis: integral (f,A) = (((id Z) (#) arccos) . (upper_bound A)) - (((id Z) (#) arccos) . (lower_bound A))
then  Z = (dom arccos) /\ (dom ((id Z) / ((#R (1 / 2)) * (f1 - (#Z 2))))) by VALUED_1:12;
then A2: ( Z c= dom arccos & Z c= dom ((id Z) / ((#R (1 / 2)) * (f1 - (#Z 2)))) ) by XBOOLE_1:18;
A3: Z = dom (id Z) ;
 Z c= (dom (id Z)) /\ (dom arccos) by A2, XBOOLE_1:19;
then A4: Z c= dom ((id Z) (#) arccos) by VALUED_1:def_4;
A5: arccos is_differentiable_on Z by A1, FDIFF_1:26, SIN_COS6:106;
 for x being Real st x in Z holds
(id Z) . x = (1 * x) + 0 by FUNCT_1:18;
then A6: id Z is_differentiable_on Z by A3, FDIFF_1:23;
 Z c= (dom (id Z)) /\ ((dom ((#R (1 / 2)) * (f1 - (#Z 2)))) \ (((#R (1 / 2)) * (f1 - (#Z 2))) " {0})) by A2, RFUNCT_1:def_1;
then  Z c= (dom ((#R (1 / 2)) * (f1 - (#Z 2)))) \ (((#R (1 / 2)) * (f1 - (#Z 2))) " {0}) by XBOOLE_1:18;
then A7: Z c= dom (((#R (1 / 2)) * (f1 - (#Z 2))) ^) by RFUNCT_1:def_2;
 dom (((#R (1 / 2)) * (f1 - (#Z 2))) ^) c= dom ((#R (1 / 2)) * (f1 - (#Z 2))) by RFUNCT_1:1;
then A8: Z c= dom ((#R (1 / 2)) * (f1 - (#Z 2))) by A7, XBOOLE_1:1;
set f2 = #Z 2;
 for x being Real st x in Z holds
(f1 - (#Z 2)) . x > 0
proof
let x be Real; ::_thesis: ( x in Z implies (f1 - (#Z 2)) . x > 0 )
assume A9: x in Z ; ::_thesis: (f1 - (#Z 2)) . x > 0
then  ( - 1 < x & x < 1 ) by A1, XXREAL_1:4;
then  ( 0 < 1 + x & 0 < 1 - x ) by XREAL_1:50, XREAL_1:148;
then A10: 0 < (1 + x) * (1 - x) by XREAL_1:129;
 for x being Real st x in Z holds
x in dom (f1 - (#Z 2)) by A8, FUNCT_1:11;
then (f1 - (#Z 2)) . x = (f1 . x) - ((#Z 2) . x) by A9, VALUED_1:13
.= (f1 . x) - (x #Z (1 + 1)) by TAYLOR_1:def_1
.= (f1 . x) - ((x #Z 1) * (x #Z 1)) by TAYLOR_1:1
.= (f1 . x) - (x * (x #Z 1)) by PREPOWER:35
.= (f1 . x) - (x * x) by PREPOWER:35
.= 1 - (x * x) by A1, A9 ;
hence  (f1 - (#Z 2)) . x > 0 by A10; ::_thesis: verum
end;
then  for x being Real st x in Z holds
( f1 . x = 1 & (f1 - (#Z 2)) . x > 0 ) by A1;
then A11: (#R (1 / 2)) * (f1 - (#Z 2)) is_differentiable_on Z by A8, FDIFF_7:22;
 for x being Real st x in Z holds
((#R (1 / 2)) * (f1 - (#Z 2))) . x <> 0 by A7, RFUNCT_1:3;
then  (id Z) / ((#R (1 / 2)) * (f1 - (#Z 2))) is_differentiable_on Z by A6, A11, FDIFF_2:21;
then  f | Z is continuous by A1, A5, FDIFF_1:19, FDIFF_1:25;
then  f | A is continuous by A1, FCONT_1:16;
then A12: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A13: (id Z) (#) arccos is_differentiable_on Z by A1, A4, FDIFF_7:17;
A14: for x being Real st x in Z holds
f . x = (arccos . x) - (x / (sqrt (1 - (x ^2))))
proof
let x be Real; ::_thesis: ( x in Z implies f . x = (arccos . x) - (x / (sqrt (1 - (x ^2)))) )
assume A15: x in Z ; ::_thesis: f . x = (arccos . x) - (x / (sqrt (1 - (x ^2))))
then A16: ( x in dom (f1 - (#Z 2)) & (f1 - (#Z 2)) . x in dom (#R (1 / 2)) ) by A8, FUNCT_1:11;
then A17: (f1 - (#Z 2)) . x in right_open_halfline 0 by TAYLOR_1:def_4;
 ( - 1 < x & x < 1 ) by A1, A15, XXREAL_1:4;
then  ( 0 < 1 + x & 0 < 1 - x ) by XREAL_1:50, XREAL_1:148;
then A18: 0 < (1 + x) * (1 - x) by XREAL_1:129;
(arccos - ((id Z) / ((#R (1 / 2)) * (f1 - (#Z 2))))) . x = (arccos . x) - (((id Z) / ((#R (1 / 2)) * (f1 - (#Z 2)))) . x) by A1, A15, VALUED_1:13
.= (arccos . x) - (((id Z) . x) / (((#R (1 / 2)) * (f1 - (#Z 2))) . x)) by A2, A15, RFUNCT_1:def_1
.= (arccos . x) - (x / (((#R (1 / 2)) * (f1 - (#Z 2))) . x)) by A15, FUNCT_1:18
.= (arccos . x) - (x / ((#R (1 / 2)) . ((f1 - (#Z 2)) . x))) by A8, A15, FUNCT_1:12
.= (arccos . x) - (x / (((f1 - (#Z 2)) . x) #R (1 / 2))) by A17, TAYLOR_1:def_4
.= (arccos . x) - (x / (((f1 . x) - ((#Z 2) . x)) #R (1 / 2))) by A16, VALUED_1:13
.= (arccos . x) - (x / (((f1 . x) - (x #Z 2)) #R (1 / 2))) by TAYLOR_1:def_1
.= (arccos . x) - (x / (((f1 . x) - (x ^2)) #R (1 / 2))) by FDIFF_7:1
.= (arccos . x) - (x / ((1 - (x ^2)) #R (1 / 2))) by A1, A15
.= (arccos . x) - (x / (sqrt (1 - (x ^2)))) by A18, FDIFF_7:2 ;
hence  f . x = (arccos . x) - (x / (sqrt (1 - (x ^2)))) by A1; ::_thesis: verum
end;
A19: for x being Real st x in dom (((id Z) (#) arccos) `| Z) holds
(((id Z) (#) arccos) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom (((id Z) (#) arccos) `| Z) implies (((id Z) (#) arccos) `| Z) . x = f . x )
assume  x in dom (((id Z) (#) arccos) `| Z) ; ::_thesis: (((id Z) (#) arccos) `| Z) . x = f . x
then A20: x in Z by A13, FDIFF_1:def_7;
then (((id Z) (#) arccos) `| Z) . x = (arccos . x) - (x / (sqrt (1 - (x ^2)))) by A1, A4, FDIFF_7:17
.= f . x by A14, A20 ;
hence  (((id Z) (#) arccos) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom (((id Z) (#) arccos) `| Z) = dom f by A1, A13, FDIFF_1:def_7;
then  ((id Z) (#) arccos) `| Z = f by A19, PARTFUN1:5;
hence  integral (f,A) = (((id Z) (#) arccos) . (upper_bound A)) - (((id Z) (#) arccos) . (lower_bound A)) by A1, A12, A4, FDIFF_7:17, INTEGRA5:13; ::_thesis: verum
end;

theorem :: INTEGR12:23
for a, b being Real
for A being non empty closed_interval Subset of REAL
for f1, f2, f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
( f1 . x = (a * x) + b & f2 . x = 1 ) ) & Z = dom f & f = (a (#) arcsin) + (f1 / ((#R (1 / 2)) * (f2 - (#Z 2)))) holds
integral (f,A) = ((f1 (#) arcsin) . (upper_bound A)) - ((f1 (#) arcsin) . (lower_bound A))
proof
let a, b be Real; ::_thesis: for A being non empty closed_interval Subset of REAL
for f1, f2, f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
( f1 . x = (a * x) + b & f2 . x = 1 ) ) & Z = dom f & f = (a (#) arcsin) + (f1 / ((#R (1 / 2)) * (f2 - (#Z 2)))) holds
integral (f,A) = ((f1 (#) arcsin) . (upper_bound A)) - ((f1 (#) arcsin) . (lower_bound A))
let A be non empty closed_interval Subset of REAL; ::_thesis: for f1, f2, f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
( f1 . x = (a * x) + b & f2 . x = 1 ) ) & Z = dom f & f = (a (#) arcsin) + (f1 / ((#R (1 / 2)) * (f2 - (#Z 2)))) holds
integral (f,A) = ((f1 (#) arcsin) . (upper_bound A)) - ((f1 (#) arcsin) . (lower_bound A))
let f1, f2, f be PartFunc of REAL,REAL; ::_thesis: for Z being open Subset of REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
( f1 . x = (a * x) + b & f2 . x = 1 ) ) & Z = dom f & f = (a (#) arcsin) + (f1 / ((#R (1 / 2)) * (f2 - (#Z 2)))) holds
integral (f,A) = ((f1 (#) arcsin) . (upper_bound A)) - ((f1 (#) arcsin) . (lower_bound A))
let Z be open Subset of REAL; ::_thesis: ( A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
( f1 . x = (a * x) + b & f2 . x = 1 ) ) & Z = dom f & f = (a (#) arcsin) + (f1 / ((#R (1 / 2)) * (f2 - (#Z 2)))) implies integral (f,A) = ((f1 (#) arcsin) . (upper_bound A)) - ((f1 (#) arcsin) . (lower_bound A)) )
assume A1: ( A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
( f1 . x = (a * x) + b & f2 . x = 1 ) ) & Z = dom f & f = (a (#) arcsin) + (f1 / ((#R (1 / 2)) * (f2 - (#Z 2)))) ) ; ::_thesis: integral (f,A) = ((f1 (#) arcsin) . (upper_bound A)) - ((f1 (#) arcsin) . (lower_bound A))
then  Z = (dom (a (#) arcsin)) /\ (dom (f1 / ((#R (1 / 2)) * (f2 - (#Z 2))))) by VALUED_1:def_1;
then A2: ( Z c= dom (a (#) arcsin) & Z c= dom (f1 / ((#R (1 / 2)) * (f2 - (#Z 2)))) ) by XBOOLE_1:18;
then A3: Z c= dom arcsin by VALUED_1:def_5;
 Z c= (dom f1) /\ ((dom ((#R (1 / 2)) * (f2 - (#Z 2)))) \ (((#R (1 / 2)) * (f2 - (#Z 2))) " {0})) by A2, RFUNCT_1:def_1;
then A4: ( Z c= dom f1 & Z c= (dom ((#R (1 / 2)) * (f2 - (#Z 2)))) \ (((#R (1 / 2)) * (f2 - (#Z 2))) " {0}) ) by XBOOLE_1:18;
then  Z c= (dom f1) /\ (dom arcsin) by A3, XBOOLE_1:19;
then A5: Z c= dom (f1 (#) arcsin) by VALUED_1:def_4;
A6: Z c= dom (((#R (1 / 2)) * (f2 - (#Z 2))) ^) by A4, RFUNCT_1:def_2;
 dom (((#R (1 / 2)) * (f2 - (#Z 2))) ^) c= dom ((#R (1 / 2)) * (f2 - (#Z 2))) by RFUNCT_1:1;
then A7: Z c= dom ((#R (1 / 2)) * (f2 - (#Z 2))) by A6, XBOOLE_1:1;
A8: arcsin is_differentiable_on Z by A1, FDIFF_1:26, SIN_COS6:83;
then A9: a (#) arcsin is_differentiable_on Z by A2, FDIFF_1:20;
A10: for x being Real st x in Z holds
f1 . x = (a * x) + b by A1;
then A11: f1 is_differentiable_on Z by A4, FDIFF_1:23;
set f3 = #Z 2;
 for x being Real st x in Z holds
(f2 - (#Z 2)) . x > 0
proof
let x be Real; ::_thesis: ( x in Z implies (f2 - (#Z 2)) . x > 0 )
assume A12: x in Z ; ::_thesis: (f2 - (#Z 2)) . x > 0
then  ( - 1 < x & x < 1 ) by A1, XXREAL_1:4;
then  ( 0 < 1 + x & 0 < 1 - x ) by XREAL_1:50, XREAL_1:148;
then A13: 0 < (1 + x) * (1 - x) by XREAL_1:129;
 for x being Real st x in Z holds
x in dom (f2 - (#Z 2)) by A7, FUNCT_1:11;
then (f2 - (#Z 2)) . x = (f2 . x) - ((#Z 2) . x) by A12, VALUED_1:13
.= (f2 . x) - (x #Z (1 + 1)) by TAYLOR_1:def_1
.= (f2 . x) - ((x #Z 1) * (x #Z 1)) by TAYLOR_1:1
.= (f2 . x) - (x * (x #Z 1)) by PREPOWER:35
.= (f2 . x) - (x * x) by PREPOWER:35
.= 1 - (x * x) by A1, A12 ;
hence  (f2 - (#Z 2)) . x > 0 by A13; ::_thesis: verum
end;
then  for x being Real st x in Z holds
( f2 . x = 1 & (f2 - (#Z 2)) . x > 0 ) by A1;
then A14: (#R (1 / 2)) * (f2 - (#Z 2)) is_differentiable_on Z by A7, FDIFF_7:22;
 for x being Real st x in Z holds
((#R (1 / 2)) * (f2 - (#Z 2))) . x <> 0 by A6, RFUNCT_1:3;
then  f1 / ((#R (1 / 2)) * (f2 - (#Z 2))) is_differentiable_on Z by A11, A14, FDIFF_2:21;
then  f is_differentiable_on Z by A1, A9, FDIFF_1:18;
then  f | Z is continuous by FDIFF_1:25;
then  f | A is continuous by A1, FCONT_1:16;
then A15: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A16: f1 (#) arcsin is_differentiable_on Z by A5, A8, A11, FDIFF_1:21;
A17: for x being Real st x in Z holds
f . x = (a * (arcsin . x)) + (((a * x) + b) / (sqrt (1 - (x ^2))))
proof
let x be Real; ::_thesis: ( x in Z implies f . x = (a * (arcsin . x)) + (((a * x) + b) / (sqrt (1 - (x ^2)))) )
assume A18: x in Z ; ::_thesis: f . x = (a * (arcsin . x)) + (((a * x) + b) / (sqrt (1 - (x ^2))))
then A19: ( x in dom (f2 - (#Z 2)) & (f2 - (#Z 2)) . x in dom (#R (1 / 2)) ) by A7, FUNCT_1:11;
then A20: (f2 - (#Z 2)) . x in right_open_halfline 0 by TAYLOR_1:def_4;
 ( - 1 < x & x < 1 ) by A1, A18, XXREAL_1:4;
then  ( 0 < 1 + x & 0 < 1 - x ) by XREAL_1:50, XREAL_1:148;
then A21: 0 < (1 + x) * (1 - x) by XREAL_1:129;
((a (#) arcsin) + (f1 / ((#R (1 / 2)) * (f2 - (#Z 2))))) . x = ((a (#) arcsin) . x) + ((f1 / ((#R (1 / 2)) * (f2 - (#Z 2)))) . x) by A1, A18, VALUED_1:def_1
.= (a * (arcsin . x)) + ((f1 / ((#R (1 / 2)) * (f2 - (#Z 2)))) . x) by VALUED_1:6
.= (a * (arcsin . x)) + ((f1 . x) / (((#R (1 / 2)) * (f2 - (#Z 2))) . x)) by A2, A18, RFUNCT_1:def_1
.= (a * (arcsin . x)) + (((a * x) + b) / (((#R (1 / 2)) * (f2 - (#Z 2))) . x)) by A1, A18
.= (a * (arcsin . x)) + (((a * x) + b) / ((#R (1 / 2)) . ((f2 - (#Z 2)) . x))) by A7, A18, FUNCT_1:12
.= (a * (arcsin . x)) + (((a * x) + b) / (((f2 - (#Z 2)) . x) #R (1 / 2))) by A20, TAYLOR_1:def_4
.= (a * (arcsin . x)) + (((a * x) + b) / (((f2 . x) - ((#Z 2) . x)) #R (1 / 2))) by A19, VALUED_1:13
.= (a * (arcsin . x)) + (((a * x) + b) / (((f2 . x) - (x #Z 2)) #R (1 / 2))) by TAYLOR_1:def_1
.= (a * (arcsin . x)) + (((a * x) + b) / (((f2 . x) - (x ^2)) #R (1 / 2))) by FDIFF_7:1
.= (a * (arcsin . x)) + (((a * x) + b) / ((1 - (x ^2)) #R (1 / 2))) by A1, A18
.= (a * (arcsin . x)) + (((a * x) + b) / (sqrt (1 - (x ^2)))) by A21, FDIFF_7:2 ;
hence  f . x = (a * (arcsin . x)) + (((a * x) + b) / (sqrt (1 - (x ^2)))) by A1; ::_thesis: verum
end;
A22: for x being Real st x in dom ((f1 (#) arcsin) `| Z) holds
((f1 (#) arcsin) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom ((f1 (#) arcsin) `| Z) implies ((f1 (#) arcsin) `| Z) . x = f . x )
assume  x in dom ((f1 (#) arcsin) `| Z) ; ::_thesis: ((f1 (#) arcsin) `| Z) . x = f . x
then A23: x in Z by A16, FDIFF_1:def_7;
then ((f1 (#) arcsin) `| Z) . x = (a * (arcsin . x)) + (((a * x) + b) / (sqrt (1 - (x ^2)))) by A1, A10, A5, FDIFF_7:18
.= f . x by A17, A23 ;
hence  ((f1 (#) arcsin) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom ((f1 (#) arcsin) `| Z) = dom f by A1, A16, FDIFF_1:def_7;
then  (f1 (#) arcsin) `| Z = f by A22, PARTFUN1:5;
hence  integral (f,A) = ((f1 (#) arcsin) . (upper_bound A)) - ((f1 (#) arcsin) . (lower_bound A)) by A1, A15, A16, INTEGRA5:13; ::_thesis: verum
end;

theorem :: INTEGR12:24
for a, b being Real
for A being non empty closed_interval Subset of REAL
for f1, f2, f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
( f1 . x = (a * x) + b & f2 . x = 1 ) ) & Z = dom f & f = (a (#) arccos) - (f1 / ((#R (1 / 2)) * (f2 - (#Z 2)))) holds
integral (f,A) = ((f1 (#) arccos) . (upper_bound A)) - ((f1 (#) arccos) . (lower_bound A))
proof
let a, b be Real; ::_thesis: for A being non empty closed_interval Subset of REAL
for f1, f2, f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
( f1 . x = (a * x) + b & f2 . x = 1 ) ) & Z = dom f & f = (a (#) arccos) - (f1 / ((#R (1 / 2)) * (f2 - (#Z 2)))) holds
integral (f,A) = ((f1 (#) arccos) . (upper_bound A)) - ((f1 (#) arccos) . (lower_bound A))
let A be non empty closed_interval Subset of REAL; ::_thesis: for f1, f2, f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
( f1 . x = (a * x) + b & f2 . x = 1 ) ) & Z = dom f & f = (a (#) arccos) - (f1 / ((#R (1 / 2)) * (f2 - (#Z 2)))) holds
integral (f,A) = ((f1 (#) arccos) . (upper_bound A)) - ((f1 (#) arccos) . (lower_bound A))
let f1, f2, f be PartFunc of REAL,REAL; ::_thesis: for Z being open Subset of REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
( f1 . x = (a * x) + b & f2 . x = 1 ) ) & Z = dom f & f = (a (#) arccos) - (f1 / ((#R (1 / 2)) * (f2 - (#Z 2)))) holds
integral (f,A) = ((f1 (#) arccos) . (upper_bound A)) - ((f1 (#) arccos) . (lower_bound A))
let Z be open Subset of REAL; ::_thesis: ( A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
( f1 . x = (a * x) + b & f2 . x = 1 ) ) & Z = dom f & f = (a (#) arccos) - (f1 / ((#R (1 / 2)) * (f2 - (#Z 2)))) implies integral (f,A) = ((f1 (#) arccos) . (upper_bound A)) - ((f1 (#) arccos) . (lower_bound A)) )
assume A1: ( A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
( f1 . x = (a * x) + b & f2 . x = 1 ) ) & Z = dom f & f = (a (#) arccos) - (f1 / ((#R (1 / 2)) * (f2 - (#Z 2)))) ) ; ::_thesis: integral (f,A) = ((f1 (#) arccos) . (upper_bound A)) - ((f1 (#) arccos) . (lower_bound A))
then  Z = (dom (a (#) arccos)) /\ (dom (f1 / ((#R (1 / 2)) * (f2 - (#Z 2))))) by VALUED_1:12;
then A2: ( Z c= dom (a (#) arccos) & Z c= dom (f1 / ((#R (1 / 2)) * (f2 - (#Z 2)))) ) by XBOOLE_1:18;
then A3: Z c= dom arccos by VALUED_1:def_5;
 Z c= (dom f1) /\ ((dom ((#R (1 / 2)) * (f2 - (#Z 2)))) \ (((#R (1 / 2)) * (f2 - (#Z 2))) " {0})) by A2, RFUNCT_1:def_1;
then A4: ( Z c= dom f1 & Z c= (dom ((#R (1 / 2)) * (f2 - (#Z 2)))) \ (((#R (1 / 2)) * (f2 - (#Z 2))) " {0}) ) by XBOOLE_1:18;
then  Z c= (dom f1) /\ (dom arccos) by A3, XBOOLE_1:19;
then A5: Z c= dom (f1 (#) arccos) by VALUED_1:def_4;
A6: Z c= dom (((#R (1 / 2)) * (f2 - (#Z 2))) ^) by A4, RFUNCT_1:def_2;
 dom (((#R (1 / 2)) * (f2 - (#Z 2))) ^) c= dom ((#R (1 / 2)) * (f2 - (#Z 2))) by RFUNCT_1:1;
then A7: Z c= dom ((#R (1 / 2)) * (f2 - (#Z 2))) by A6, XBOOLE_1:1;
A8: arccos is_differentiable_on Z by A1, FDIFF_1:26, SIN_COS6:106;
then A9: a (#) arccos is_differentiable_on Z by A2, FDIFF_1:20;
A10: for x being Real st x in Z holds
f1 . x = (a * x) + b by A1;
then A11: f1 is_differentiable_on Z by A4, FDIFF_1:23;
set f3 = #Z 2;
 for x being Real st x in Z holds
(f2 - (#Z 2)) . x > 0
proof
let x be Real; ::_thesis: ( x in Z implies (f2 - (#Z 2)) . x > 0 )
assume A12: x in Z ; ::_thesis: (f2 - (#Z 2)) . x > 0
then  ( - 1 < x & x < 1 ) by A1, XXREAL_1:4;
then  ( 0 < 1 + x & 0 < 1 - x ) by XREAL_1:50, XREAL_1:148;
then A13: 0 < (1 + x) * (1 - x) by XREAL_1:129;
 for x being Real st x in Z holds
x in dom (f2 - (#Z 2)) by A7, FUNCT_1:11;
then (f2 - (#Z 2)) . x = (f2 . x) - ((#Z 2) . x) by A12, VALUED_1:13
.= (f2 . x) - (x #Z (1 + 1)) by TAYLOR_1:def_1
.= (f2 . x) - ((x #Z 1) * (x #Z 1)) by TAYLOR_1:1
.= (f2 . x) - (x * (x #Z 1)) by PREPOWER:35
.= (f2 . x) - (x * x) by PREPOWER:35
.= 1 - (x * x) by A1, A12 ;
hence  (f2 - (#Z 2)) . x > 0 by A13; ::_thesis: verum
end;
then  for x being Real st x in Z holds
( f2 . x = 1 & (f2 - (#Z 2)) . x > 0 ) by A1;
then A14: (#R (1 / 2)) * (f2 - (#Z 2)) is_differentiable_on Z by A7, FDIFF_7:22;
 for x being Real st x in Z holds
((#R (1 / 2)) * (f2 - (#Z 2))) . x <> 0 by A6, RFUNCT_1:3;
then  f1 / ((#R (1 / 2)) * (f2 - (#Z 2))) is_differentiable_on Z by A11, A14, FDIFF_2:21;
then  f is_differentiable_on Z by A1, A9, FDIFF_1:19;
then  f | Z is continuous by FDIFF_1:25;
then  f | A is continuous by A1, FCONT_1:16;
then A15: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A16: f1 (#) arccos is_differentiable_on Z by A5, A8, A11, FDIFF_1:21;
A17: for x being Real st x in Z holds
f . x = (a * (arccos . x)) - (((a * x) + b) / (sqrt (1 - (x ^2))))
proof
let x be Real; ::_thesis: ( x in Z implies f . x = (a * (arccos . x)) - (((a * x) + b) / (sqrt (1 - (x ^2)))) )
assume A18: x in Z ; ::_thesis: f . x = (a * (arccos . x)) - (((a * x) + b) / (sqrt (1 - (x ^2))))
then A19: ( x in dom (f2 - (#Z 2)) & (f2 - (#Z 2)) . x in dom (#R (1 / 2)) ) by A7, FUNCT_1:11;
then A20: (f2 - (#Z 2)) . x in right_open_halfline 0 by TAYLOR_1:def_4;
 ( - 1 < x & x < 1 ) by A1, A18, XXREAL_1:4;
then  ( 0 < 1 + x & 0 < 1 - x ) by XREAL_1:50, XREAL_1:148;
then A21: 0 < (1 + x) * (1 - x) by XREAL_1:129;
((a (#) arccos) - (f1 / ((#R (1 / 2)) * (f2 - (#Z 2))))) . x = ((a (#) arccos) . x) - ((f1 / ((#R (1 / 2)) * (f2 - (#Z 2)))) . x) by A1, A18, VALUED_1:13
.= (a * (arccos . x)) - ((f1 / ((#R (1 / 2)) * (f2 - (#Z 2)))) . x) by VALUED_1:6
.= (a * (arccos . x)) - ((f1 . x) / (((#R (1 / 2)) * (f2 - (#Z 2))) . x)) by A2, A18, RFUNCT_1:def_1
.= (a * (arccos . x)) - (((a * x) + b) / (((#R (1 / 2)) * (f2 - (#Z 2))) . x)) by A1, A18
.= (a * (arccos . x)) - (((a * x) + b) / ((#R (1 / 2)) . ((f2 - (#Z 2)) . x))) by A7, A18, FUNCT_1:12
.= (a * (arccos . x)) - (((a * x) + b) / (((f2 - (#Z 2)) . x) #R (1 / 2))) by A20, TAYLOR_1:def_4
.= (a * (arccos . x)) - (((a * x) + b) / (((f2 . x) - ((#Z 2) . x)) #R (1 / 2))) by A19, VALUED_1:13
.= (a * (arccos . x)) - (((a * x) + b) / (((f2 . x) - (x #Z 2)) #R (1 / 2))) by TAYLOR_1:def_1
.= (a * (arccos . x)) - (((a * x) + b) / (((f2 . x) - (x ^2)) #R (1 / 2))) by FDIFF_7:1
.= (a * (arccos . x)) - (((a * x) + b) / ((1 - (x ^2)) #R (1 / 2))) by A1, A18
.= (a * (arccos . x)) - (((a * x) + b) / (sqrt (1 - (x ^2)))) by A21, FDIFF_7:2 ;
hence  f . x = (a * (arccos . x)) - (((a * x) + b) / (sqrt (1 - (x ^2)))) by A1; ::_thesis: verum
end;
A22: for x being Real st x in dom ((f1 (#) arccos) `| Z) holds
((f1 (#) arccos) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom ((f1 (#) arccos) `| Z) implies ((f1 (#) arccos) `| Z) . x = f . x )
assume  x in dom ((f1 (#) arccos) `| Z) ; ::_thesis: ((f1 (#) arccos) `| Z) . x = f . x
then A23: x in Z by A16, FDIFF_1:def_7;
then ((f1 (#) arccos) `| Z) . x = (a * (arccos . x)) - (((a * x) + b) / (sqrt (1 - (x ^2)))) by A1, A10, A5, FDIFF_7:19
.= f . x by A17, A23 ;
hence  ((f1 (#) arccos) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom ((f1 (#) arccos) `| Z) = dom f by A1, A16, FDIFF_1:def_7;
then  (f1 (#) arccos) `| Z = f by A22, PARTFUN1:5;
hence  integral (f,A) = ((f1 (#) arccos) . (upper_bound A)) - ((f1 (#) arccos) . (lower_bound A)) by A1, A15, A16, INTEGRA5:13; ::_thesis: verum
end;

theorem :: INTEGR12:25
for a being Real
for A being non empty closed_interval Subset of REAL
for g, f1, f 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
( g . x = 1 & f1 . x = x / a & f1 . x > - 1 & f1 . x < 1 ) ) & Z = dom f & f | A is continuous & f = (arcsin * f1) + ((id Z) / (a (#) ((#R (1 / 2)) * (g - (f1 ^2))))) holds
integral (f,A) = (((id Z) (#) (arcsin * f1)) . (upper_bound A)) - (((id Z) (#) (arcsin * f1)) . (lower_bound A))
proof
let a be Real; ::_thesis: for A being non empty closed_interval Subset of REAL
for g, f1, f 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
( g . x = 1 & f1 . x = x / a & f1 . x > - 1 & f1 . x < 1 ) ) & Z = dom f & f | A is continuous & f = (arcsin * f1) + ((id Z) / (a (#) ((#R (1 / 2)) * (g - (f1 ^2))))) holds
integral (f,A) = (((id Z) (#) (arcsin * f1)) . (upper_bound A)) - (((id Z) (#) (arcsin * f1)) . (lower_bound A))
let A be non empty closed_interval Subset of REAL; ::_thesis: for g, f1, f 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
( g . x = 1 & f1 . x = x / a & f1 . x > - 1 & f1 . x < 1 ) ) & Z = dom f & f | A is continuous & f = (arcsin * f1) + ((id Z) / (a (#) ((#R (1 / 2)) * (g - (f1 ^2))))) holds
integral (f,A) = (((id Z) (#) (arcsin * f1)) . (upper_bound A)) - (((id Z) (#) (arcsin * f1)) . (lower_bound A))
let g, f1, f 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
( g . x = 1 & f1 . x = x / a & f1 . x > - 1 & f1 . x < 1 ) ) & Z = dom f & f | A is continuous & f = (arcsin * f1) + ((id Z) / (a (#) ((#R (1 / 2)) * (g - (f1 ^2))))) holds
integral (f,A) = (((id Z) (#) (arcsin * f1)) . (upper_bound A)) - (((id Z) (#) (arcsin * f1)) . (lower_bound A))
let Z be open Subset of REAL; ::_thesis: ( A c= Z & ( for x being Real st x in Z holds
( g . x = 1 & f1 . x = x / a & f1 . x > - 1 & f1 . x < 1 ) ) & Z = dom f & f | A is continuous & f = (arcsin * f1) + ((id Z) / (a (#) ((#R (1 / 2)) * (g - (f1 ^2))))) implies integral (f,A) = (((id Z) (#) (arcsin * f1)) . (upper_bound A)) - (((id Z) (#) (arcsin * f1)) . (lower_bound A)) )
assume A1: ( A c= Z & ( for x being Real st x in Z holds
( g . x = 1 & f1 . x = x / a & f1 . x > - 1 & f1 . x < 1 ) ) & Z = dom f & f | A is continuous & f = (arcsin * f1) + ((id Z) / (a (#) ((#R (1 / 2)) * (g - (f1 ^2))))) ) ; ::_thesis: integral (f,A) = (((id Z) (#) (arcsin * f1)) . (upper_bound A)) - (((id Z) (#) (arcsin * f1)) . (lower_bound A))
then  Z = (dom (arcsin * f1)) /\ (dom ((id Z) / (a (#) ((#R (1 / 2)) * (g - (f1 ^2)))))) by VALUED_1:def_1;
then A2: ( Z c= dom (arcsin * f1) & Z c= dom ((id Z) / (a (#) ((#R (1 / 2)) * (g - (f1 ^2))))) ) by XBOOLE_1:18;
 Z c= (dom (id Z)) /\ (dom (arcsin * f1)) by A2, XBOOLE_1:19;
then A3: Z c= dom ((id Z) (#) (arcsin * f1)) by VALUED_1:def_4;
 Z c= (dom (id Z)) /\ ((dom (a (#) ((#R (1 / 2)) * (g - (f1 ^2))))) \ ((a (#) ((#R (1 / 2)) * (g - (f1 ^2)))) " {0})) by A2, RFUNCT_1:def_1;
then  Z c= (dom (a (#) ((#R (1 / 2)) * (g - (f1 ^2))))) \ ((a (#) ((#R (1 / 2)) * (g - (f1 ^2)))) " {0}) by XBOOLE_1:18;
then A4: Z c= dom ((a (#) ((#R (1 / 2)) * (g - (f1 ^2)))) ^) by RFUNCT_1:def_2;
 dom ((a (#) ((#R (1 / 2)) * (g - (f1 ^2)))) ^) c= dom (a (#) ((#R (1 / 2)) * (g - (f1 ^2)))) by RFUNCT_1:1;
then  Z c= dom (a (#) ((#R (1 / 2)) * (g - (f1 ^2)))) by A4, XBOOLE_1:1;
then A5: Z c= dom ((#R (1 / 2)) * (g - (f1 ^2))) by VALUED_1:def_5;
A6: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A7: for x being Real st x in Z holds
( f1 . x = x / a & f1 . x > - 1 & f1 . x < 1 ) by A1;
then A8: (id Z) (#) (arcsin * f1) is_differentiable_on Z by A3, FDIFF_7:25;
A9: for x being Real st x in Z holds
f . x = (arcsin . (x / a)) + (x / (a * (sqrt (1 - ((x / a) ^2)))))
proof
let x be Real; ::_thesis: ( x in Z implies f . x = (arcsin . (x / a)) + (x / (a * (sqrt (1 - ((x / a) ^2))))) )
assume A10: x in Z ; ::_thesis: f . x = (arcsin . (x / a)) + (x / (a * (sqrt (1 - ((x / a) ^2)))))
then A11: ( x in dom (g - (f1 ^2)) & (g - (f1 ^2)) . x in dom (#R (1 / 2)) ) by A5, FUNCT_1:11;
then A12: (g - (f1 ^2)) . x in right_open_halfline 0 by TAYLOR_1:def_4;
 ( - 1 < f1 . x & f1 . x < 1 ) by A1, A10;
then  ( 0 < 1 + (f1 . x) & 0 < 1 - (f1 . x) ) by XREAL_1:50, XREAL_1:148;
then A13: 0 < (1 + (f1 . x)) * (1 - (f1 . x)) by XREAL_1:129;
A14: f1 . x = x / a by A1, A10;
((arcsin * f1) + ((id Z) / (a (#) ((#R (1 / 2)) * (g - (f1 ^2)))))) . x = ((arcsin * f1) . x) + (((id Z) / (a (#) ((#R (1 / 2)) * (g - (f1 ^2))))) . x) by A1, A10, VALUED_1:def_1
.= (arcsin . (f1 . x)) + (((id Z) / (a (#) ((#R (1 / 2)) * (g - (f1 ^2))))) . x) by A2, A10, FUNCT_1:12
.= (arcsin . (x / a)) + (((id Z) / (a (#) ((#R (1 / 2)) * (g - (f1 ^2))))) . x) by A1, A10
.= (arcsin . (x / a)) + (((id Z) . x) / ((a (#) ((#R (1 / 2)) * (g - (f1 ^2)))) . x)) by A2, A10, RFUNCT_1:def_1
.= (arcsin . (x / a)) + (x / ((a (#) ((#R (1 / 2)) * (g - (f1 ^2)))) . x)) by A10, FUNCT_1:18
.= (arcsin . (x / a)) + (x / (a * (((#R (1 / 2)) * (g - (f1 ^2))) . x))) by VALUED_1:6
.= (arcsin . (x / a)) + (x / (a * ((#R (1 / 2)) . ((g - (f1 ^2)) . x)))) by A5, A10, FUNCT_1:12
.= (arcsin . (x / a)) + (x / (a * (((g - (f1 ^2)) . x) #R (1 / 2)))) by A12, TAYLOR_1:def_4
.= (arcsin . (x / a)) + (x / (a * (((g . x) - ((f1 ^2) . x)) #R (1 / 2)))) by A11, VALUED_1:13
.= (arcsin . (x / a)) + (x / (a * (((g . x) - ((f1 . x) ^2)) #R (1 / 2)))) by VALUED_1:11
.= (arcsin . (x / a)) + (x / (a * ((1 - ((f1 . x) ^2)) #R (1 / 2)))) by A1, A10
.= (arcsin . (x / a)) + (x / (a * ((1 - ((x / a) ^2)) #R (1 / 2)))) by A1, A10
.= (arcsin . (x / a)) + (x / (a * (sqrt (1 - ((x / a) ^2))))) by A14, A13, FDIFF_7:2 ;
hence  f . x = (arcsin . (x / a)) + (x / (a * (sqrt (1 - ((x / a) ^2))))) by A1; ::_thesis: verum
end;
A15: for x being Real st x in dom (((id Z) (#) (arcsin * f1)) `| Z) holds
(((id Z) (#) (arcsin * f1)) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom (((id Z) (#) (arcsin * f1)) `| Z) implies (((id Z) (#) (arcsin * f1)) `| Z) . x = f . x )
assume  x in dom (((id Z) (#) (arcsin * f1)) `| Z) ; ::_thesis: (((id Z) (#) (arcsin * f1)) `| Z) . x = f . x
then A16: x in Z by A8, FDIFF_1:def_7;
then (((id Z) (#) (arcsin * f1)) `| Z) . x = (arcsin . (x / a)) + (x / (a * (sqrt (1 - ((x / a) ^2))))) by A3, A7, FDIFF_7:25
.= f . x by A9, A16 ;
hence  (((id Z) (#) (arcsin * f1)) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom (((id Z) (#) (arcsin * f1)) `| Z) = dom f by A1, A8, FDIFF_1:def_7;
then  ((id Z) (#) (arcsin * f1)) `| Z = f by A15, PARTFUN1:5;
hence  integral (f,A) = (((id Z) (#) (arcsin * f1)) . (upper_bound A)) - (((id Z) (#) (arcsin * f1)) . (lower_bound A)) by A1, A6, A8, INTEGRA5:13; ::_thesis: verum
end;

theorem :: INTEGR12:26
for a being Real
for A being non empty closed_interval Subset of REAL
for g, f1, f 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
( g . x = 1 & f1 . x = x / a & f1 . x > - 1 & f1 . x < 1 ) ) & Z = dom f & f | A is continuous & f = (arccos * f1) - ((id Z) / (a (#) ((#R (1 / 2)) * (g - (f1 ^2))))) holds
integral (f,A) = (((id Z) (#) (arccos * f1)) . (upper_bound A)) - (((id Z) (#) (arccos * f1)) . (lower_bound A))
proof
let a be Real; ::_thesis: for A being non empty closed_interval Subset of REAL
for g, f1, f 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
( g . x = 1 & f1 . x = x / a & f1 . x > - 1 & f1 . x < 1 ) ) & Z = dom f & f | A is continuous & f = (arccos * f1) - ((id Z) / (a (#) ((#R (1 / 2)) * (g - (f1 ^2))))) holds
integral (f,A) = (((id Z) (#) (arccos * f1)) . (upper_bound A)) - (((id Z) (#) (arccos * f1)) . (lower_bound A))
let A be non empty closed_interval Subset of REAL; ::_thesis: for g, f1, f 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
( g . x = 1 & f1 . x = x / a & f1 . x > - 1 & f1 . x < 1 ) ) & Z = dom f & f | A is continuous & f = (arccos * f1) - ((id Z) / (a (#) ((#R (1 / 2)) * (g - (f1 ^2))))) holds
integral (f,A) = (((id Z) (#) (arccos * f1)) . (upper_bound A)) - (((id Z) (#) (arccos * f1)) . (lower_bound A))
let g, f1, f 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
( g . x = 1 & f1 . x = x / a & f1 . x > - 1 & f1 . x < 1 ) ) & Z = dom f & f | A is continuous & f = (arccos * f1) - ((id Z) / (a (#) ((#R (1 / 2)) * (g - (f1 ^2))))) holds
integral (f,A) = (((id Z) (#) (arccos * f1)) . (upper_bound A)) - (((id Z) (#) (arccos * f1)) . (lower_bound A))
let Z be open Subset of REAL; ::_thesis: ( A c= Z & ( for x being Real st x in Z holds
( g . x = 1 & f1 . x = x / a & f1 . x > - 1 & f1 . x < 1 ) ) & Z = dom f & f | A is continuous & f = (arccos * f1) - ((id Z) / (a (#) ((#R (1 / 2)) * (g - (f1 ^2))))) implies integral (f,A) = (((id Z) (#) (arccos * f1)) . (upper_bound A)) - (((id Z) (#) (arccos * f1)) . (lower_bound A)) )
assume A1: ( A c= Z & ( for x being Real st x in Z holds
( g . x = 1 & f1 . x = x / a & f1 . x > - 1 & f1 . x < 1 ) ) & Z = dom f & f | A is continuous & f = (arccos * f1) - ((id Z) / (a (#) ((#R (1 / 2)) * (g - (f1 ^2))))) ) ; ::_thesis: integral (f,A) = (((id Z) (#) (arccos * f1)) . (upper_bound A)) - (((id Z) (#) (arccos * f1)) . (lower_bound A))
then  Z = (dom (arccos * f1)) /\ (dom ((id Z) / (a (#) ((#R (1 / 2)) * (g - (f1 ^2)))))) by VALUED_1:12;
then A2: ( Z c= dom (arccos * f1) & Z c= dom ((id Z) / (a (#) ((#R (1 / 2)) * (g - (f1 ^2))))) ) by XBOOLE_1:18;
 Z c= (dom (id Z)) /\ (dom (arccos * f1)) by A2, XBOOLE_1:19;
then A3: Z c= dom ((id Z) (#) (arccos * f1)) by VALUED_1:def_4;
 Z c= (dom (id Z)) /\ ((dom (a (#) ((#R (1 / 2)) * (g - (f1 ^2))))) \ ((a (#) ((#R (1 / 2)) * (g - (f1 ^2)))) " {0})) by A2, RFUNCT_1:def_1;
then  Z c= (dom (a (#) ((#R (1 / 2)) * (g - (f1 ^2))))) \ ((a (#) ((#R (1 / 2)) * (g - (f1 ^2)))) " {0}) by XBOOLE_1:18;
then A4: Z c= dom ((a (#) ((#R (1 / 2)) * (g - (f1 ^2)))) ^) by RFUNCT_1:def_2;
 dom ((a (#) ((#R (1 / 2)) * (g - (f1 ^2)))) ^) c= dom (a (#) ((#R (1 / 2)) * (g - (f1 ^2)))) by RFUNCT_1:1;
then  Z c= dom (a (#) ((#R (1 / 2)) * (g - (f1 ^2)))) by A4, XBOOLE_1:1;
then A5: Z c= dom ((#R (1 / 2)) * (g - (f1 ^2))) by VALUED_1:def_5;
A6: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A7: for x being Real st x in Z holds
( f1 . x = x / a & f1 . x > - 1 & f1 . x < 1 ) by A1;
then A8: (id Z) (#) (arccos * f1) is_differentiable_on Z by A3, FDIFF_7:26;
A9: for x being Real st x in Z holds
f . x = (arccos . (x / a)) - (x / (a * (sqrt (1 - ((x / a) ^2)))))
proof
let x be Real; ::_thesis: ( x in Z implies f . x = (arccos . (x / a)) - (x / (a * (sqrt (1 - ((x / a) ^2))))) )
assume A10: x in Z ; ::_thesis: f . x = (arccos . (x / a)) - (x / (a * (sqrt (1 - ((x / a) ^2)))))
then A11: ( x in dom (g - (f1 ^2)) & (g - (f1 ^2)) . x in dom (#R (1 / 2)) ) by A5, FUNCT_1:11;
then A12: (g - (f1 ^2)) . x in right_open_halfline 0 by TAYLOR_1:def_4;
 ( - 1 < f1 . x & f1 . x < 1 ) by A1, A10;
then  ( 0 < 1 + (f1 . x) & 0 < 1 - (f1 . x) ) by XREAL_1:50, XREAL_1:148;
then A13: 0 < (1 + (f1 . x)) * (1 - (f1 . x)) by XREAL_1:129;
A14: f1 . x = x / a by A1, A10;
((arccos * f1) - ((id Z) / (a (#) ((#R (1 / 2)) * (g - (f1 ^2)))))) . x = ((arccos * f1) . x) - (((id Z) / (a (#) ((#R (1 / 2)) * (g - (f1 ^2))))) . x) by A1, A10, VALUED_1:13
.= (arccos . (f1 . x)) - (((id Z) / (a (#) ((#R (1 / 2)) * (g - (f1 ^2))))) . x) by A2, A10, FUNCT_1:12
.= (arccos . (x / a)) - (((id Z) / (a (#) ((#R (1 / 2)) * (g - (f1 ^2))))) . x) by A1, A10
.= (arccos . (x / a)) - (((id Z) . x) / ((a (#) ((#R (1 / 2)) * (g - (f1 ^2)))) . x)) by A2, A10, RFUNCT_1:def_1
.= (arccos . (x / a)) - (x / ((a (#) ((#R (1 / 2)) * (g - (f1 ^2)))) . x)) by A10, FUNCT_1:18
.= (arccos . (x / a)) - (x / (a * (((#R (1 / 2)) * (g - (f1 ^2))) . x))) by VALUED_1:6
.= (arccos . (x / a)) - (x / (a * ((#R (1 / 2)) . ((g - (f1 ^2)) . x)))) by A5, A10, FUNCT_1:12
.= (arccos . (x / a)) - (x / (a * (((g - (f1 ^2)) . x) #R (1 / 2)))) by A12, TAYLOR_1:def_4
.= (arccos . (x / a)) - (x / (a * (((g . x) - ((f1 ^2) . x)) #R (1 / 2)))) by A11, VALUED_1:13
.= (arccos . (x / a)) - (x / (a * (((g . x) - ((f1 . x) ^2)) #R (1 / 2)))) by VALUED_1:11
.= (arccos . (x / a)) - (x / (a * ((1 - ((f1 . x) ^2)) #R (1 / 2)))) by A1, A10
.= (arccos . (x / a)) - (x / (a * ((1 - ((x / a) ^2)) #R (1 / 2)))) by A1, A10
.= (arccos . (x / a)) - (x / (a * (sqrt (1 - ((x / a) ^2))))) by A14, A13, FDIFF_7:2 ;
hence  f . x = (arccos . (x / a)) - (x / (a * (sqrt (1 - ((x / a) ^2))))) by A1; ::_thesis: verum
end;
A15: for x being Real st x in dom (((id Z) (#) (arccos * f1)) `| Z) holds
(((id Z) (#) (arccos * f1)) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom (((id Z) (#) (arccos * f1)) `| Z) implies (((id Z) (#) (arccos * f1)) `| Z) . x = f . x )
assume  x in dom (((id Z) (#) (arccos * f1)) `| Z) ; ::_thesis: (((id Z) (#) (arccos * f1)) `| Z) . x = f . x
then A16: x in Z by A8, FDIFF_1:def_7;
then (((id Z) (#) (arccos * f1)) `| Z) . x = (arccos . (x / a)) - (x / (a * (sqrt (1 - ((x / a) ^2))))) by A3, A7, FDIFF_7:26
.= f . x by A9, A16 ;
hence  (((id Z) (#) (arccos * f1)) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom (((id Z) (#) (arccos * f1)) `| Z) = dom f by A1, A8, FDIFF_1:def_7;
then  ((id Z) (#) (arccos * f1)) `| Z = f by A15, PARTFUN1:5;
hence  integral (f,A) = (((id Z) (#) (arccos * f1)) . (upper_bound A)) - (((id Z) (#) (arccos * f1)) . (lower_bound A)) by A1, A6, A8, INTEGRA5:13; ::_thesis: verum
end;

theorem :: INTEGR12:27
for n being Element of NAT
for A being non empty closed_interval Subset of REAL
for f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & f = (n (#) ((#Z (n - 1)) * sin)) / ((#Z (n + 1)) * cos) & 1 <= n & Z c= dom ((#Z n) * tan) & Z = dom f holds
integral (f,A) = (((#Z n) * tan) . (upper_bound A)) - (((#Z n) * tan) . (lower_bound A))
proof
let n be Element of NAT ; ::_thesis: for A being non empty closed_interval Subset of REAL
for f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & f = (n (#) ((#Z (n - 1)) * sin)) / ((#Z (n + 1)) * cos) & 1 <= n & Z c= dom ((#Z n) * tan) & Z = dom f holds
integral (f,A) = (((#Z n) * tan) . (upper_bound A)) - (((#Z n) * tan) . (lower_bound A))
let A be non empty closed_interval Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & f = (n (#) ((#Z (n - 1)) * sin)) / ((#Z (n + 1)) * cos) & 1 <= n & Z c= dom ((#Z n) * tan) & Z = dom f holds
integral (f,A) = (((#Z n) * tan) . (upper_bound A)) - (((#Z n) * tan) . (lower_bound A))
let f be PartFunc of REAL,REAL; ::_thesis: for Z being open Subset of REAL st A c= Z & f = (n (#) ((#Z (n - 1)) * sin)) / ((#Z (n + 1)) * cos) & 1 <= n & Z c= dom ((#Z n) * tan) & Z = dom f holds
integral (f,A) = (((#Z n) * tan) . (upper_bound A)) - (((#Z n) * tan) . (lower_bound A))
let Z be open Subset of REAL; ::_thesis: ( A c= Z & f = (n (#) ((#Z (n - 1)) * sin)) / ((#Z (n + 1)) * cos) & 1 <= n & Z c= dom ((#Z n) * tan) & Z = dom f implies integral (f,A) = (((#Z n) * tan) . (upper_bound A)) - (((#Z n) * tan) . (lower_bound A)) )
assume A1: ( A c= Z & f = (n (#) ((#Z (n - 1)) * sin)) / ((#Z (n + 1)) * cos) & 1 <= n & Z c= dom ((#Z n) * tan) & Z = dom f ) ; ::_thesis: integral (f,A) = (((#Z n) * tan) . (upper_bound A)) - (((#Z n) * tan) . (lower_bound A))
then  Z = (dom (n (#) ((#Z (n - 1)) * sin))) /\ ((dom ((#Z (n + 1)) * cos)) \ (((#Z (n + 1)) * cos) " {0})) by RFUNCT_1:def_1;
then A2: ( Z c= dom (n (#) ((#Z (n - 1)) * sin)) & Z c= (dom ((#Z (n + 1)) * cos)) \ (((#Z (n + 1)) * cos) " {0}) ) by XBOOLE_1:18;
then A3: Z c= dom (((#Z (n + 1)) * cos) ^) by RFUNCT_1:def_2;
 dom (((#Z (n + 1)) * cos) ^) c= dom ((#Z (n + 1)) * cos) by RFUNCT_1:1;
then A4: Z c= dom ((#Z (n + 1)) * cos) by A3, XBOOLE_1:1;
A5: for x being Real st x in Z holds
((#Z (n + 1)) * cos) . x <> 0
proof
let x be Real; ::_thesis: ( x in Z implies ((#Z (n + 1)) * cos) . x <> 0 )
assume  x in Z ; ::_thesis: ((#Z (n + 1)) * cos) . x <> 0
then  x in (dom (n (#) ((#Z (n - 1)) * sin))) /\ ((dom ((#Z (n + 1)) * cos)) \ (((#Z (n + 1)) * cos) " {0})) by A1, RFUNCT_1:def_1;
then  x in (dom ((#Z (n + 1)) * cos)) \ (((#Z (n + 1)) * cos) " {0}) by XBOOLE_0:def_4;
then  x in dom (((#Z (n + 1)) * cos) ^) by RFUNCT_1:def_2;
hence  ((#Z (n + 1)) * cos) . x <> 0 by RFUNCT_1:3; ::_thesis: verum
end;
A6: Z c= dom ((#Z (n - 1)) * sin) by A2, VALUED_1:def_5;
A7: for x being Real holds (#Z (n - 1)) * sin is_differentiable_in x
proof
let x be Real; ::_thesis: (#Z (n - 1)) * sin is_differentiable_in x
consider m being Nat such that
A8: n = m + 1 by A1, NAT_1:6;
 sin is_differentiable_in x by SIN_COS:64;
hence  (#Z (n - 1)) * sin is_differentiable_in x by A8, TAYLOR_1:3; ::_thesis: verum
end;
 (#Z (n - 1)) * sin is_differentiable_on Z
proof
 for x being Real st x in Z holds
(#Z (n - 1)) * sin is_differentiable_in x by A7;
hence  (#Z (n - 1)) * sin is_differentiable_on Z by A6, FDIFF_1:9; ::_thesis: verum
end;
then A9: n (#) ((#Z (n - 1)) * sin) is_differentiable_on Z by A2, FDIFF_1:20;
A10: for x being Real holds (#Z (n + 1)) * cos is_differentiable_in x
proof
let x be Real; ::_thesis: (#Z (n + 1)) * cos is_differentiable_in x
 cos is_differentiable_in x by SIN_COS:63;
hence  (#Z (n + 1)) * cos is_differentiable_in x by TAYLOR_1:3; ::_thesis: verum
end;
 (#Z (n + 1)) * cos is_differentiable_on Z
proof
 for x being Real st x in Z holds
(#Z (n + 1)) * cos is_differentiable_in x by A10;
hence  (#Z (n + 1)) * cos is_differentiable_on Z by A4, FDIFF_1:9; ::_thesis: verum
end;
then  f | Z is continuous by A1, A5, A9, FDIFF_1:25, FDIFF_2:21;
then  f | A is continuous by A1, FCONT_1:16;
then A11: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A12: (#Z n) * tan is_differentiable_on Z by A1, FDIFF_8:20;
A13: for x being Real st x in Z holds
f . x = (n * ((sin . x) #Z (n - 1))) / ((cos . x) #Z (n + 1))
proof
let x be Real; ::_thesis: ( x in Z implies f . x = (n * ((sin . x) #Z (n - 1))) / ((cos . x) #Z (n + 1)) )
assume A14: x in Z ; ::_thesis: f . x = (n * ((sin . x) #Z (n - 1))) / ((cos . x) #Z (n + 1))
then ((n (#) ((#Z (n - 1)) * sin)) / ((#Z (n + 1)) * cos)) . x = ((n (#) ((#Z (n - 1)) * sin)) . x) / (((#Z (n + 1)) * cos) . x) by A1, RFUNCT_1:def_1
.= (n * (((#Z (n - 1)) * sin) . x)) / (((#Z (n + 1)) * cos) . x) by VALUED_1:6
.= (n * ((#Z (n - 1)) . (sin . x))) / (((#Z (n + 1)) * cos) . x) by A6, A14, FUNCT_1:12
.= (n * ((sin . x) #Z (n - 1))) / (((#Z (n + 1)) * cos) . x) by TAYLOR_1:def_1
.= (n * ((sin . x) #Z (n - 1))) / ((#Z (n + 1)) . (cos . x)) by A4, A14, FUNCT_1:12
.= (n * ((sin . x) #Z (n - 1))) / ((cos . x) #Z (n + 1)) by TAYLOR_1:def_1 ;
hence  f . x = (n * ((sin . x) #Z (n - 1))) / ((cos . x) #Z (n + 1)) by A1; ::_thesis: verum
end;
A15: for x being Real st x in dom (((#Z n) * tan) `| Z) holds
(((#Z n) * tan) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom (((#Z n) * tan) `| Z) implies (((#Z n) * tan) `| Z) . x = f . x )
assume  x in dom (((#Z n) * tan) `| Z) ; ::_thesis: (((#Z n) * tan) `| Z) . x = f . x
then A16: x in Z by A12, FDIFF_1:def_7;
then (((#Z n) * tan) `| Z) . x = (n * ((sin . x) #Z (n - 1))) / ((cos . x) #Z (n + 1)) by A1, FDIFF_8:20
.= f . x by A13, A16 ;
hence  (((#Z n) * tan) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom (((#Z n) * tan) `| Z) = dom f by A1, A12, FDIFF_1:def_7;
then  ((#Z n) * tan) `| Z = f by A15, PARTFUN1:5;
hence  integral (f,A) = (((#Z n) * tan) . (upper_bound A)) - (((#Z n) * tan) . (lower_bound A)) by A1, A11, FDIFF_8:20, INTEGRA5:13; ::_thesis: verum
end;

theorem :: INTEGR12:28
for n being Element of NAT
for A being non empty closed_interval Subset of REAL
for f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & f = (n (#) ((#Z (n - 1)) * cos)) / ((#Z (n + 1)) * sin) & 1 <= n & Z c= dom ((#Z n) * cot) & Z = dom f holds
integral (f,A) = ((- ((#Z n) * cot)) . (upper_bound A)) - ((- ((#Z n) * cot)) . (lower_bound A))
proof
let n be Element of NAT ; ::_thesis: for A being non empty closed_interval Subset of REAL
for f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & f = (n (#) ((#Z (n - 1)) * cos)) / ((#Z (n + 1)) * sin) & 1 <= n & Z c= dom ((#Z n) * cot) & Z = dom f holds
integral (f,A) = ((- ((#Z n) * cot)) . (upper_bound A)) - ((- ((#Z n) * cot)) . (lower_bound A))
let A be non empty closed_interval Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & f = (n (#) ((#Z (n - 1)) * cos)) / ((#Z (n + 1)) * sin) & 1 <= n & Z c= dom ((#Z n) * cot) & Z = dom f holds
integral (f,A) = ((- ((#Z n) * cot)) . (upper_bound A)) - ((- ((#Z n) * cot)) . (lower_bound A))
let f be PartFunc of REAL,REAL; ::_thesis: for Z being open Subset of REAL st A c= Z & f = (n (#) ((#Z (n - 1)) * cos)) / ((#Z (n + 1)) * sin) & 1 <= n & Z c= dom ((#Z n) * cot) & Z = dom f holds
integral (f,A) = ((- ((#Z n) * cot)) . (upper_bound A)) - ((- ((#Z n) * cot)) . (lower_bound A))
let Z be open Subset of REAL; ::_thesis: ( A c= Z & f = (n (#) ((#Z (n - 1)) * cos)) / ((#Z (n + 1)) * sin) & 1 <= n & Z c= dom ((#Z n) * cot) & Z = dom f implies integral (f,A) = ((- ((#Z n) * cot)) . (upper_bound A)) - ((- ((#Z n) * cot)) . (lower_bound A)) )
assume A1: ( A c= Z & f = (n (#) ((#Z (n - 1)) * cos)) / ((#Z (n + 1)) * sin) & 1 <= n & Z c= dom ((#Z n) * cot) & Z = dom f ) ; ::_thesis: integral (f,A) = ((- ((#Z n) * cot)) . (upper_bound A)) - ((- ((#Z n) * cot)) . (lower_bound A))
then  Z = (dom (n (#) ((#Z (n - 1)) * cos))) /\ ((dom ((#Z (n + 1)) * sin)) \ (((#Z (n + 1)) * sin) " {0})) by RFUNCT_1:def_1;
then A2: ( Z c= dom (n (#) ((#Z (n - 1)) * cos)) & Z c= (dom ((#Z (n + 1)) * sin)) \ (((#Z (n + 1)) * sin) " {0}) ) by XBOOLE_1:18;
then A3: Z c= dom (((#Z (n + 1)) * sin) ^) by RFUNCT_1:def_2;
 dom (((#Z (n + 1)) * sin) ^) c= dom ((#Z (n + 1)) * sin) by RFUNCT_1:1;
then A4: Z c= dom ((#Z (n + 1)) * sin) by A3, XBOOLE_1:1;
A5: for x being Real st x in Z holds
((#Z (n + 1)) * sin) . x <> 0
proof
let x be Real; ::_thesis: ( x in Z implies ((#Z (n + 1)) * sin) . x <> 0 )
assume  x in Z ; ::_thesis: ((#Z (n + 1)) * sin) . x <> 0
then  x in (dom (n (#) ((#Z (n - 1)) * cos))) /\ ((dom ((#Z (n + 1)) * sin)) \ (((#Z (n + 1)) * sin) " {0})) by A1, RFUNCT_1:def_1;
then  x in (dom ((#Z (n + 1)) * sin)) \ (((#Z (n + 1)) * sin) " {0}) by XBOOLE_0:def_4;
then  x in dom (((#Z (n + 1)) * sin) ^) by RFUNCT_1:def_2;
hence  ((#Z (n + 1)) * sin) . x <> 0 by RFUNCT_1:3; ::_thesis: verum
end;
A6: Z c= dom ((#Z (n - 1)) * cos) by A2, VALUED_1:def_5;
A7: for x being Real holds (#Z (n - 1)) * cos is_differentiable_in x
proof
let x be Real; ::_thesis: (#Z (n - 1)) * cos is_differentiable_in x
consider m being Nat such that
A8: n = m + 1 by A1, NAT_1:6;
 cos is_differentiable_in x by SIN_COS:63;
hence  (#Z (n - 1)) * cos is_differentiable_in x by A8, TAYLOR_1:3; ::_thesis: verum
end;
 (#Z (n - 1)) * cos is_differentiable_on Z
proof
 for x being Real st x in Z holds
(#Z (n - 1)) * cos is_differentiable_in x by A7;
hence  (#Z (n - 1)) * cos is_differentiable_on Z by A6, FDIFF_1:9; ::_thesis: verum
end;
then A9: n (#) ((#Z (n - 1)) * cos) is_differentiable_on Z by A2, FDIFF_1:20;
A10: for x being Real holds (#Z (n + 1)) * sin is_differentiable_in x
proof
let x be Real; ::_thesis: (#Z (n + 1)) * sin is_differentiable_in x
 sin is_differentiable_in x by SIN_COS:64;
hence  (#Z (n + 1)) * sin is_differentiable_in x by TAYLOR_1:3; ::_thesis: verum
end;
 (#Z (n + 1)) * sin is_differentiable_on Z
proof
 for x being Real st x in Z holds
(#Z (n + 1)) * sin is_differentiable_in x by A10;
hence  (#Z (n + 1)) * sin is_differentiable_on Z by A4, FDIFF_1:9; ::_thesis: verum
end;
then  f | Z is continuous by A1, A5, A9, FDIFF_1:25, FDIFF_2:21;
then  f | A is continuous by A1, FCONT_1:16;
then A11: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A12: (#Z n) * cot is_differentiable_on Z by A1, FDIFF_8:21;
A13: dom ((#Z n) * cot) c= dom cot by RELAT_1:25;
A14: Z c= dom (- ((#Z n) * cot)) by A1, VALUED_1:8;
then A15: (- 1) (#) ((#Z n) * cot) is_differentiable_on Z by A12, Lm1, FDIFF_1:20;
A16: for x being Real st x in Z holds
sin . x <> 0
proof
let x be Real; ::_thesis: ( x in Z implies sin . x <> 0 )
assume  x in Z ; ::_thesis: sin . x <> 0
then  x in dom (cos / sin) by A1, FUNCT_1:11;
hence  sin . x <> 0 by FDIFF_8:2; ::_thesis: verum
end;
A17: for x being Real st x in Z holds
((- ((#Z n) * cot)) `| Z) . x = (n * ((cos . x) #Z (n - 1))) / ((sin . x) #Z (n + 1))
proof
let x be Real; ::_thesis: ( x in Z implies ((- ((#Z n) * cot)) `| Z) . x = (n * ((cos . x) #Z (n - 1))) / ((sin . x) #Z (n + 1)) )
assume A18: x in Z ; ::_thesis: ((- ((#Z n) * cot)) `| Z) . x = (n * ((cos . x) #Z (n - 1))) / ((sin . x) #Z (n + 1))
then A19: sin . x <> 0 by A16;
then A20: cot is_differentiable_in x by FDIFF_7:47;
consider m being Nat such that
A21: n = m + 1 by A1, NAT_1:6;
set m = n - 1;
A22: (#Z n) * cot is_differentiable_in x by A12, A18, FDIFF_1:9;
((- ((#Z n) * cot)) `| Z) . x = diff ((- ((#Z n) * cot)),x) by A15, A18, FDIFF_1:def_7
.= (- 1) * (diff (((#Z n) * cot),x)) by A22, Lm1, FDIFF_1:15
.= (- 1) * ((n * ((cot . x) #Z (n - 1))) * (diff (cot,x))) by A20, TAYLOR_1:3
.= (- 1) * ((n * ((cot . x) #Z (n - 1))) * (- (1 / ((sin . x) ^2)))) by A19, FDIFF_7:47
.= (- 1) * (- ((n * ((cot . x) #Z (n - 1))) / ((sin . x) ^2)))
.= (- 1) * (- ((n * (((cos . x) #Z (n - 1)) / ((sin . x) #Z (n - 1)))) / ((sin . x) ^2))) by A1, A13, A18, A21, FDIFF_8:3, XBOOLE_1:1
.= (- 1) * (- (((n * ((cos . x) #Z (n - 1))) / ((sin . x) #Z (n - 1))) / ((sin . x) ^2)))
.= (- 1) * (- ((n * ((cos . x) #Z (n - 1))) / (((sin . x) #Z (n - 1)) * ((sin . x) ^2)))) by XCMPLX_1:78
.= (- 1) * (- ((n * ((cos . x) #Z (n - 1))) / (((sin . x) #Z (n - 1)) * ((sin . x) #Z 2)))) by FDIFF_7:1
.= (- 1) * (- ((n * ((cos . x) #Z (n - 1))) / ((sin . x) #Z ((n - 1) + 2)))) by A16, A18, PREPOWER:44
.= (n * ((cos . x) #Z (n - 1))) / ((sin . x) #Z (n + 1)) ;
hence  ((- ((#Z n) * cot)) `| Z) . x = (n * ((cos . x) #Z (n - 1))) / ((sin . x) #Z (n + 1)) ; ::_thesis: verum
end;
A23: for x being Real st x in Z holds
f . x = (n * ((cos . x) #Z (n - 1))) / ((sin . x) #Z (n + 1))
proof
let x be Real; ::_thesis: ( x in Z implies f . x = (n * ((cos . x) #Z (n - 1))) / ((sin . x) #Z (n + 1)) )
assume A24: x in Z ; ::_thesis: f . x = (n * ((cos . x) #Z (n - 1))) / ((sin . x) #Z (n + 1))
then ((n (#) ((#Z (n - 1)) * cos)) / ((#Z (n + 1)) * sin)) . x = ((n (#) ((#Z (n - 1)) * cos)) . x) / (((#Z (n + 1)) * sin) . x) by A1, RFUNCT_1:def_1
.= (n * (((#Z (n - 1)) * cos) . x)) / (((#Z (n + 1)) * sin) . x) by VALUED_1:6
.= (n * ((#Z (n - 1)) . (cos . x))) / (((#Z (n + 1)) * sin) . x) by A6, A24, FUNCT_1:12
.= (n * ((cos . x) #Z (n - 1))) / (((#Z (n + 1)) * sin) . x) by TAYLOR_1:def_1
.= (n * ((cos . x) #Z (n - 1))) / ((#Z (n + 1)) . (sin . x)) by A4, A24, FUNCT_1:12
.= (n * ((cos . x) #Z (n - 1))) / ((sin . x) #Z (n + 1)) by TAYLOR_1:def_1 ;
hence  f . x = (n * ((cos . x) #Z (n - 1))) / ((sin . x) #Z (n + 1)) by A1; ::_thesis: verum
end;
A25: for x being Real st x in dom ((- ((#Z n) * cot)) `| Z) holds
((- ((#Z n) * cot)) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom ((- ((#Z n) * cot)) `| Z) implies ((- ((#Z n) * cot)) `| Z) . x = f . x )
assume  x in dom ((- ((#Z n) * cot)) `| Z) ; ::_thesis: ((- ((#Z n) * cot)) `| Z) . x = f . x
then A26: x in Z by A15, FDIFF_1:def_7;
then ((- ((#Z n) * cot)) `| Z) . x = (n * ((cos . x) #Z (n - 1))) / ((sin . x) #Z (n + 1)) by A17
.= f . x by A23, A26 ;
hence  ((- ((#Z n) * cot)) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom ((- ((#Z n) * cot)) `| Z) = dom f by A1, A15, FDIFF_1:def_7;
then  (- ((#Z n) * cot)) `| Z = f by A25, PARTFUN1:5;
hence  integral (f,A) = ((- ((#Z n) * cot)) . (upper_bound A)) - ((- ((#Z n) * cot)) . (lower_bound A)) by A1, A11, A12, A14, Lm1, FDIFF_1:20, INTEGRA5:13; ::_thesis: verum
end;

theorem :: INTEGR12:29
for a being Real
for A being non empty closed_interval Subset of REAL
for f1, f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z c= dom (tan * f1) & f = ((sin * f1) ^2) / ((cos * f1) ^2) & ( for x being Real st x in Z holds
( f1 . x = a * x & a <> 0 ) ) & Z = dom f holds
integral (f,A) = ((((1 / a) (#) (tan * f1)) - (id Z)) . (upper_bound A)) - ((((1 / a) (#) (tan * f1)) - (id Z)) . (lower_bound A))
proof
let a be Real; ::_thesis: for A being non empty closed_interval Subset of REAL
for f1, f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z c= dom (tan * f1) & f = ((sin * f1) ^2) / ((cos * f1) ^2) & ( for x being Real st x in Z holds
( f1 . x = a * x & a <> 0 ) ) & Z = dom f holds
integral (f,A) = ((((1 / a) (#) (tan * f1)) - (id Z)) . (upper_bound A)) - ((((1 / a) (#) (tan * f1)) - (id Z)) . (lower_bound A))
let A be non empty closed_interval Subset of REAL; ::_thesis: for f1, f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z c= dom (tan * f1) & f = ((sin * f1) ^2) / ((cos * f1) ^2) & ( for x being Real st x in Z holds
( f1 . x = a * x & a <> 0 ) ) & Z = dom f holds
integral (f,A) = ((((1 / a) (#) (tan * f1)) - (id Z)) . (upper_bound A)) - ((((1 / a) (#) (tan * f1)) - (id Z)) . (lower_bound A))
let f1, f be PartFunc of REAL,REAL; ::_thesis: for Z being open Subset of REAL st A c= Z & Z c= dom (tan * f1) & f = ((sin * f1) ^2) / ((cos * f1) ^2) & ( for x being Real st x in Z holds
( f1 . x = a * x & a <> 0 ) ) & Z = dom f holds
integral (f,A) = ((((1 / a) (#) (tan * f1)) - (id Z)) . (upper_bound A)) - ((((1 / a) (#) (tan * f1)) - (id Z)) . (lower_bound A))
let Z be open Subset of REAL; ::_thesis: ( A c= Z & Z c= dom (tan * f1) & f = ((sin * f1) ^2) / ((cos * f1) ^2) & ( for x being Real st x in Z holds
( f1 . x = a * x & a <> 0 ) ) & Z = dom f implies integral (f,A) = ((((1 / a) (#) (tan * f1)) - (id Z)) . (upper_bound A)) - ((((1 / a) (#) (tan * f1)) - (id Z)) . (lower_bound A)) )
assume A1: ( A c= Z & Z c= dom (tan * f1) & f = ((sin * f1) ^2) / ((cos * f1) ^2) & ( for x being Real st x in Z holds
( f1 . x = a * x & a <> 0 ) ) & Z = dom f ) ; ::_thesis: integral (f,A) = ((((1 / a) (#) (tan * f1)) - (id Z)) . (upper_bound A)) - ((((1 / a) (#) (tan * f1)) - (id Z)) . (lower_bound A))
then A2: Z c= dom ((1 / a) (#) (tan * f1)) by VALUED_1:def_5;
 Z c= (dom ((1 / a) (#) (tan * f1))) /\ (dom (id Z)) by A2, XBOOLE_1:19;
then A3: Z c= dom (((1 / a) (#) (tan * f1)) - (id Z)) by VALUED_1:12;
A4: for x being Real st x in Z holds
f1 . x = (a * x) + 0 by A1;
 Z = (dom ((sin * f1) ^2)) /\ ((dom ((cos * f1) ^2)) \ (((cos * f1) ^2) " {0})) by A1, RFUNCT_1:def_1;
then A5: ( Z c= dom ((sin * f1) ^2) & Z c= (dom ((cos * f1) ^2)) \ (((cos * f1) ^2) " {0}) ) by XBOOLE_1:18;
then A6: Z c= dom (sin * f1) by VALUED_1:11;
A7: Z c= dom (((cos * f1) ^2) ^) by A5, RFUNCT_1:def_2;
 dom (((cos * f1) ^2) ^) c= dom ((cos * f1) ^2) by RFUNCT_1:1;
then  Z c= dom ((cos * f1) ^2) by A7, XBOOLE_1:1;
then A8: Z c= dom (cos * f1) by VALUED_1:11;
A9: sin * f1 is_differentiable_on Z by A6, A4, FDIFF_4:37;
A10: cos * f1 is_differentiable_on Z by A4, A8, FDIFF_4:38;
A11: (sin * f1) ^2 is_differentiable_on Z by A9, FDIFF_2:20;
A12: (cos * f1) ^2 is_differentiable_on Z by A10, FDIFF_2:20;
 for x being Real st x in Z holds
((cos * f1) ^2) . x <> 0
proof
let x be Real; ::_thesis: ( x in Z implies ((cos * f1) ^2) . x <> 0 )
assume  x in Z ; ::_thesis: ((cos * f1) ^2) . x <> 0
then  x in (dom ((sin * f1) ^2)) /\ ((dom ((cos * f1) ^2)) \ (((cos * f1) ^2) " {0})) by A1, RFUNCT_1:def_1;
then  x in (dom ((cos * f1) ^2)) \ (((cos * f1) ^2) " {0}) by XBOOLE_0:def_4;
then  x in dom (((cos * f1) ^2) ^) by RFUNCT_1:def_2;
hence  ((cos * f1) ^2) . x <> 0 by RFUNCT_1:3; ::_thesis: verum
end;
then  f is_differentiable_on Z by A1, A11, A12, FDIFF_2:21;
then  f | Z is continuous by FDIFF_1:25;
then  f | A is continuous by A1, FCONT_1:16;
then A13: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A14: ((1 / a) (#) (tan * f1)) - (id Z) is_differentiable_on Z by A1, A3, FDIFF_8:26;
A15: for x being Real st x in Z holds
f . x = ((sin . (a * x)) ^2) / ((cos . (a * x)) ^2)
proof
let x be Real; ::_thesis: ( x in Z implies f . x = ((sin . (a * x)) ^2) / ((cos . (a * x)) ^2) )
assume A16: x in Z ; ::_thesis: f . x = ((sin . (a * x)) ^2) / ((cos . (a * x)) ^2)
then (((sin * f1) ^2) / ((cos * f1) ^2)) . x = (((sin * f1) ^2) . x) / (((cos * f1) ^2) . x) by A1, RFUNCT_1:def_1
.= (((sin * f1) . x) ^2) / (((cos * f1) ^2) . x) by VALUED_1:11
.= (((sin * f1) . x) ^2) / (((cos * f1) . x) ^2) by VALUED_1:11
.= ((sin . (f1 . x)) ^2) / (((cos * f1) . x) ^2) by A6, A16, FUNCT_1:12
.= ((sin . (f1 . x)) ^2) / ((cos . (f1 . x)) ^2) by A8, A16, FUNCT_1:12
.= ((sin . (a * x)) ^2) / ((cos . (f1 . x)) ^2) by A16, A1
.= ((sin . (a * x)) ^2) / ((cos . (a * x)) ^2) by A16, A1 ;
hence  f . x = ((sin . (a * x)) ^2) / ((cos . (a * x)) ^2) by A1; ::_thesis: verum
end;
A17: for x being Real st x in dom ((((1 / a) (#) (tan * f1)) - (id Z)) `| Z) holds
((((1 / a) (#) (tan * f1)) - (id Z)) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom ((((1 / a) (#) (tan * f1)) - (id Z)) `| Z) implies ((((1 / a) (#) (tan * f1)) - (id Z)) `| Z) . x = f . x )
assume  x in dom ((((1 / a) (#) (tan * f1)) - (id Z)) `| Z) ; ::_thesis: ((((1 / a) (#) (tan * f1)) - (id Z)) `| Z) . x = f . x
then A18: x in Z by A14, FDIFF_1:def_7;
then ((((1 / a) (#) (tan * f1)) - (id Z)) `| Z) . x = ((sin . (a * x)) ^2) / ((cos . (a * x)) ^2) by A1, A3, FDIFF_8:26
.= f . x by A15, A18 ;
hence  ((((1 / a) (#) (tan * f1)) - (id Z)) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom ((((1 / a) (#) (tan * f1)) - (id Z)) `| Z) = dom f by A1, A14, FDIFF_1:def_7;
then  (((1 / a) (#) (tan * f1)) - (id Z)) `| Z = f by A17, PARTFUN1:5;
hence  integral (f,A) = ((((1 / a) (#) (tan * f1)) - (id Z)) . (upper_bound A)) - ((((1 / a) (#) (tan * f1)) - (id Z)) . (lower_bound A)) by A1, A13, A14, INTEGRA5:13; ::_thesis: verum
end;

theorem :: INTEGR12:30
for a being Real
for A being non empty closed_interval Subset of REAL
for f1, f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z c= dom (cot * f1) & f = ((cos * f1) ^2) / ((sin * f1) ^2) & ( for x being Real st x in Z holds
( f1 . x = a * x & a <> 0 ) ) & Z = dom f holds
integral (f,A) = ((((- (1 / a)) (#) (cot * f1)) - (id Z)) . (upper_bound A)) - ((((- (1 / a)) (#) (cot * f1)) - (id Z)) . (lower_bound A))
proof
let a be Real; ::_thesis: for A being non empty closed_interval Subset of REAL
for f1, f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z c= dom (cot * f1) & f = ((cos * f1) ^2) / ((sin * f1) ^2) & ( for x being Real st x in Z holds
( f1 . x = a * x & a <> 0 ) ) & Z = dom f holds
integral (f,A) = ((((- (1 / a)) (#) (cot * f1)) - (id Z)) . (upper_bound A)) - ((((- (1 / a)) (#) (cot * f1)) - (id Z)) . (lower_bound A))
let A be non empty closed_interval Subset of REAL; ::_thesis: for f1, f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z c= dom (cot * f1) & f = ((cos * f1) ^2) / ((sin * f1) ^2) & ( for x being Real st x in Z holds
( f1 . x = a * x & a <> 0 ) ) & Z = dom f holds
integral (f,A) = ((((- (1 / a)) (#) (cot * f1)) - (id Z)) . (upper_bound A)) - ((((- (1 / a)) (#) (cot * f1)) - (id Z)) . (lower_bound A))
let f1, f be PartFunc of REAL,REAL; ::_thesis: for Z being open Subset of REAL st A c= Z & Z c= dom (cot * f1) & f = ((cos * f1) ^2) / ((sin * f1) ^2) & ( for x being Real st x in Z holds
( f1 . x = a * x & a <> 0 ) ) & Z = dom f holds
integral (f,A) = ((((- (1 / a)) (#) (cot * f1)) - (id Z)) . (upper_bound A)) - ((((- (1 / a)) (#) (cot * f1)) - (id Z)) . (lower_bound A))
let Z be open Subset of REAL; ::_thesis: ( A c= Z & Z c= dom (cot * f1) & f = ((cos * f1) ^2) / ((sin * f1) ^2) & ( for x being Real st x in Z holds
( f1 . x = a * x & a <> 0 ) ) & Z = dom f implies integral (f,A) = ((((- (1 / a)) (#) (cot * f1)) - (id Z)) . (upper_bound A)) - ((((- (1 / a)) (#) (cot * f1)) - (id Z)) . (lower_bound A)) )
assume A1: ( A c= Z & Z c= dom (cot * f1) & f = ((cos * f1) ^2) / ((sin * f1) ^2) & ( for x being Real st x in Z holds
( f1 . x = a * x & a <> 0 ) ) & Z = dom f ) ; ::_thesis: integral (f,A) = ((((- (1 / a)) (#) (cot * f1)) - (id Z)) . (upper_bound A)) - ((((- (1 / a)) (#) (cot * f1)) - (id Z)) . (lower_bound A))
then A2: Z c= dom ((- (1 / a)) (#) (cot * f1)) by VALUED_1:def_5;
 Z c= (dom ((- (1 / a)) (#) (cot * f1))) /\ (dom (id Z)) by A2, XBOOLE_1:19;
then A3: Z c= dom (((- (1 / a)) (#) (cot * f1)) - (id Z)) by VALUED_1:12;
A4: for x being Real st x in Z holds
f1 . x = (a * x) + 0 by A1;
 Z = (dom ((cos * f1) ^2)) /\ ((dom ((sin * f1) ^2)) \ (((sin * f1) ^2) " {0})) by A1, RFUNCT_1:def_1;
then A5: ( Z c= dom ((cos * f1) ^2) & Z c= (dom ((sin * f1) ^2)) \ (((sin * f1) ^2) " {0}) ) by XBOOLE_1:18;
then A6: Z c= dom (cos * f1) by VALUED_1:11;
A7: Z c= dom (((sin * f1) ^2) ^) by A5, RFUNCT_1:def_2;
 dom (((sin * f1) ^2) ^) c= dom ((sin * f1) ^2) by RFUNCT_1:1;
then  Z c= dom ((sin * f1) ^2) by A7, XBOOLE_1:1;
then A8: Z c= dom (sin * f1) by VALUED_1:11;
then A9: sin * f1 is_differentiable_on Z by A4, FDIFF_4:37;
A10: cos * f1 is_differentiable_on Z by A4, A6, FDIFF_4:38;
A11: (sin * f1) ^2 is_differentiable_on Z by A9, FDIFF_2:20;
A12: (cos * f1) ^2 is_differentiable_on Z by A10, FDIFF_2:20;
 for x being Real st x in Z holds
((sin * f1) ^2) . x <> 0
proof
let x be Real; ::_thesis: ( x in Z implies ((sin * f1) ^2) . x <> 0 )
assume  x in Z ; ::_thesis: ((sin * f1) ^2) . x <> 0
then  x in (dom ((cos * f1) ^2)) /\ ((dom ((sin * f1) ^2)) \ (((sin * f1) ^2) " {0})) by A1, RFUNCT_1:def_1;
then  x in (dom ((sin * f1) ^2)) \ (((sin * f1) ^2) " {0}) by XBOOLE_0:def_4;
then  x in dom (((sin * f1) ^2) ^) by RFUNCT_1:def_2;
hence  ((sin * f1) ^2) . x <> 0 by RFUNCT_1:3; ::_thesis: verum
end;
then  f is_differentiable_on Z by A1, A11, A12, FDIFF_2:21;
then  f | Z is continuous by FDIFF_1:25;
then  f | A is continuous by A1, FCONT_1:16;
then A13: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A14: ((- (1 / a)) (#) (cot * f1)) - (id Z) is_differentiable_on Z by A1, A3, FDIFF_8:27;
A15: for x being Real st x in Z holds
f . x = ((cos . (a * x)) ^2) / ((sin . (a * x)) ^2)
proof
let x be Real; ::_thesis: ( x in Z implies f . x = ((cos . (a * x)) ^2) / ((sin . (a * x)) ^2) )
assume A16: x in Z ; ::_thesis: f . x = ((cos . (a * x)) ^2) / ((sin . (a * x)) ^2)
then (((cos * f1) ^2) / ((sin * f1) ^2)) . x = (((cos * f1) ^2) . x) / (((sin * f1) ^2) . x) by A1, RFUNCT_1:def_1
.= (((cos * f1) . x) ^2) / (((sin * f1) ^2) . x) by VALUED_1:11
.= (((cos * f1) . x) ^2) / (((sin * f1) . x) ^2) by VALUED_1:11
.= ((cos . (f1 . x)) ^2) / (((sin * f1) . x) ^2) by A6, A16, FUNCT_1:12
.= ((cos . (f1 . x)) ^2) / ((sin . (f1 . x)) ^2) by A8, A16, FUNCT_1:12
.= ((cos . (a * x)) ^2) / ((sin . (f1 . x)) ^2) by A16, A1
.= ((cos . (a * x)) ^2) / ((sin . (a * x)) ^2) by A16, A1 ;
hence  f . x = ((cos . (a * x)) ^2) / ((sin . (a * x)) ^2) by A1; ::_thesis: verum
end;
A17: for x being Real st x in dom ((((- (1 / a)) (#) (cot * f1)) - (id Z)) `| Z) holds
((((- (1 / a)) (#) (cot * f1)) - (id Z)) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom ((((- (1 / a)) (#) (cot * f1)) - (id Z)) `| Z) implies ((((- (1 / a)) (#) (cot * f1)) - (id Z)) `| Z) . x = f . x )
assume  x in dom ((((- (1 / a)) (#) (cot * f1)) - (id Z)) `| Z) ; ::_thesis: ((((- (1 / a)) (#) (cot * f1)) - (id Z)) `| Z) . x = f . x
then A18: x in Z by A14, FDIFF_1:def_7;
then ((((- (1 / a)) (#) (cot * f1)) - (id Z)) `| Z) . x = ((cos . (a * x)) ^2) / ((sin . (a * x)) ^2) by A1, A3, FDIFF_8:27
.= f . x by A15, A18 ;
hence  ((((- (1 / a)) (#) (cot * f1)) - (id Z)) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom ((((- (1 / a)) (#) (cot * f1)) - (id Z)) `| Z) = dom f by A1, A14, FDIFF_1:def_7;
then  (((- (1 / a)) (#) (cot * f1)) - (id Z)) `| Z = f by A17, PARTFUN1:5;
hence  integral (f,A) = ((((- (1 / a)) (#) (cot * f1)) - (id Z)) . (upper_bound A)) - ((((- (1 / a)) (#) (cot * f1)) - (id Z)) . (lower_bound A)) by A1, A13, A14, INTEGRA5:13; ::_thesis: verum
end;

theorem :: INTEGR12:31
for a, b being Real
for A being non empty closed_interval Subset of REAL
for f1, f 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 = (a * x) + b ) & Z = dom f & f = (a (#) (sin / cos)) + (f1 / (cos ^2)) holds
integral (f,A) = ((f1 (#) tan) . (upper_bound A)) - ((f1 (#) tan) . (lower_bound A))
proof
let a, b be Real; ::_thesis: for A being non empty closed_interval Subset of REAL
for f1, f 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 = (a * x) + b ) & Z = dom f & f = (a (#) (sin / cos)) + (f1 / (cos ^2)) holds
integral (f,A) = ((f1 (#) tan) . (upper_bound A)) - ((f1 (#) tan) . (lower_bound A))
let A be non empty closed_interval Subset of REAL; ::_thesis: for f1, f 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 = (a * x) + b ) & Z = dom f & f = (a (#) (sin / cos)) + (f1 / (cos ^2)) holds
integral (f,A) = ((f1 (#) tan) . (upper_bound A)) - ((f1 (#) tan) . (lower_bound A))
let f1, f 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 = (a * x) + b ) & Z = dom f & f = (a (#) (sin / cos)) + (f1 / (cos ^2)) holds
integral (f,A) = ((f1 (#) tan) . (upper_bound A)) - ((f1 (#) tan) . (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 = (a * x) + b ) & Z = dom f & f = (a (#) (sin / cos)) + (f1 / (cos ^2)) implies integral (f,A) = ((f1 (#) tan) . (upper_bound A)) - ((f1 (#) tan) . (lower_bound A)) )
assume A1: ( A c= Z & ( for x being Real st x in Z holds
f1 . x = (a * x) + b ) & Z = dom f & f = (a (#) (sin / cos)) + (f1 / (cos ^2)) ) ; ::_thesis: integral (f,A) = ((f1 (#) tan) . (upper_bound A)) - ((f1 (#) tan) . (lower_bound A))
then A2: Z = (dom (a (#) (sin / cos))) /\ (dom (f1 / (cos ^2))) by VALUED_1:def_1;
then A3: Z c= dom (a (#) (sin / cos)) by XBOOLE_1:18;
then A4: Z c= dom (sin / cos) by VALUED_1:def_5;
A5: Z c= dom (f1 / (cos ^2)) by A2, XBOOLE_1:18;
 dom (f1 / (cos ^2)) = (dom f1) /\ ((dom (cos ^2)) \ ((cos ^2) " {0})) by RFUNCT_1:def_1;
then A6: Z c= dom f1 by A5, XBOOLE_1:18;
then  Z c= (dom f1) /\ (dom tan) by A4, XBOOLE_1:19;
then A7: Z c= dom (f1 (#) tan) by VALUED_1:def_4;
 for x being Real st x in Z holds
sin / cos is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies sin / cos is_differentiable_in x )
assume  x in Z ; ::_thesis: sin / cos is_differentiable_in x
then  cos . x <> 0 by A4, FDIFF_8:1;
hence  sin / cos is_differentiable_in x by FDIFF_7:46; ::_thesis: verum
end;
then  sin / cos is_differentiable_on Z by A4, FDIFF_1:9;
then A8: a (#) (sin / cos) is_differentiable_on Z by A3, FDIFF_1:20;
A9: f1 is_differentiable_on Z by A1, A6, FDIFF_1:23;
 cos is_differentiable_on Z by FDIFF_1:26, SIN_COS:67;
then A10: cos ^2 is_differentiable_on Z by FDIFF_2:20;
 for x being Real st x in Z holds
(cos ^2) . x <> 0
proof
let x be Real; ::_thesis: ( x in Z implies (cos ^2) . x <> 0 )
assume  x in Z ; ::_thesis: (cos ^2) . x <> 0
then  x in dom (f1 / (cos ^2)) by A5;
then  x in (dom f1) /\ ((dom (cos ^2)) \ ((cos ^2) " {0})) by RFUNCT_1:def_1;
then  x in (dom (cos ^2)) \ ((cos ^2) " {0}) by XBOOLE_0:def_4;
then  x in dom ((cos ^2) ^) by RFUNCT_1:def_2;
hence  (cos ^2) . x <> 0 by RFUNCT_1:3; ::_thesis: verum
end;
then  f1 / (cos ^2) is_differentiable_on Z by A9, A10, FDIFF_2:21;
then  f | Z is continuous by A1, A8, FDIFF_1:18, FDIFF_1:25;
then  f | A is continuous by A1, FCONT_1:16;
then A11: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A12: f1 (#) tan is_differentiable_on Z by A1, A7, FDIFF_8:28;
A13: for x being Real st x in Z holds
f . x = ((a * (sin . x)) / (cos . x)) + (((a * x) + b) / ((cos . x) ^2))
proof
let x be Real; ::_thesis: ( x in Z implies f . x = ((a * (sin . x)) / (cos . x)) + (((a * x) + b) / ((cos . x) ^2)) )
assume A14: x in Z ; ::_thesis: f . x = ((a * (sin . x)) / (cos . x)) + (((a * x) + b) / ((cos . x) ^2))
then ((a (#) (sin / cos)) + (f1 / (cos ^2))) . x = ((a (#) (sin / cos)) . x) + ((f1 / (cos ^2)) . x) by A1, VALUED_1:def_1
.= (a * ((sin / cos) . x)) + ((f1 / (cos ^2)) . x) by VALUED_1:6
.= (a * ((sin . x) / (cos . x))) + ((f1 / (cos ^2)) . x) by A14, A4, RFUNCT_1:def_1
.= ((a * (sin . x)) / (cos . x)) + ((f1 . x) / ((cos ^2) . x)) by A14, A5, RFUNCT_1:def_1
.= ((a * (sin . x)) / (cos . x)) + ((f1 . x) / ((cos . x) ^2)) by VALUED_1:11
.= ((a * (sin . x)) / (cos . x)) + (((a * x) + b) / ((cos . x) ^2)) by A1, A14 ;
hence  f . x = ((a * (sin . x)) / (cos . x)) + (((a * x) + b) / ((cos . x) ^2)) by A1; ::_thesis: verum
end;
A15: for x being Real st x in dom ((f1 (#) tan) `| Z) holds
((f1 (#) tan) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom ((f1 (#) tan) `| Z) implies ((f1 (#) tan) `| Z) . x = f . x )
assume  x in dom ((f1 (#) tan) `| Z) ; ::_thesis: ((f1 (#) tan) `| Z) . x = f . x
then A16: x in Z by A12, FDIFF_1:def_7;
then ((f1 (#) tan) `| Z) . x = ((a * (sin . x)) / (cos . x)) + (((a * x) + b) / ((cos . x) ^2)) by A1, A7, FDIFF_8:28
.= f . x by A13, A16 ;
hence  ((f1 (#) tan) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom ((f1 (#) tan) `| Z) = dom f by A1, A12, FDIFF_1:def_7;
then  (f1 (#) tan) `| Z = f by A15, PARTFUN1:5;
hence  integral (f,A) = ((f1 (#) tan) . (upper_bound A)) - ((f1 (#) tan) . (lower_bound A)) by A1, A11, A7, FDIFF_8:28, INTEGRA5:13; ::_thesis: verum
end;

theorem :: INTEGR12:32
for a, b being Real
for A being non empty closed_interval Subset of REAL
for f1, f 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 = (a * x) + b ) & Z = dom f & f = (a (#) (cos / sin)) - (f1 / (sin ^2)) holds
integral (f,A) = ((f1 (#) cot) . (upper_bound A)) - ((f1 (#) cot) . (lower_bound A))
proof
let a, b be Real; ::_thesis: for A being non empty closed_interval Subset of REAL
for f1, f 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 = (a * x) + b ) & Z = dom f & f = (a (#) (cos / sin)) - (f1 / (sin ^2)) holds
integral (f,A) = ((f1 (#) cot) . (upper_bound A)) - ((f1 (#) cot) . (lower_bound A))
let A be non empty closed_interval Subset of REAL; ::_thesis: for f1, f 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 = (a * x) + b ) & Z = dom f & f = (a (#) (cos / sin)) - (f1 / (sin ^2)) holds
integral (f,A) = ((f1 (#) cot) . (upper_bound A)) - ((f1 (#) cot) . (lower_bound A))
let f1, f 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 = (a * x) + b ) & Z = dom f & f = (a (#) (cos / sin)) - (f1 / (sin ^2)) holds
integral (f,A) = ((f1 (#) cot) . (upper_bound A)) - ((f1 (#) cot) . (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 = (a * x) + b ) & Z = dom f & f = (a (#) (cos / sin)) - (f1 / (sin ^2)) implies integral (f,A) = ((f1 (#) cot) . (upper_bound A)) - ((f1 (#) cot) . (lower_bound A)) )
assume A1: ( A c= Z & ( for x being Real st x in Z holds
f1 . x = (a * x) + b ) & Z = dom f & f = (a (#) (cos / sin)) - (f1 / (sin ^2)) ) ; ::_thesis: integral (f,A) = ((f1 (#) cot) . (upper_bound A)) - ((f1 (#) cot) . (lower_bound A))
then A2: Z = (dom (a (#) (cos / sin))) /\ (dom (- (f1 / (sin ^2)))) by VALUED_1:def_1;
then A3: Z c= dom (a (#) (cos / sin)) by XBOOLE_1:18;
then A4: Z c= dom (cos / sin) by VALUED_1:def_5;
 Z c= dom (- (f1 / (sin ^2))) by A2, XBOOLE_1:18;
then A5: Z c= dom (f1 / (sin ^2)) by VALUED_1:8;
 dom (f1 / (sin ^2)) = (dom f1) /\ ((dom (sin ^2)) \ ((sin ^2) " {0})) by RFUNCT_1:def_1;
then A6: Z c= dom f1 by A5, XBOOLE_1:18;
then  Z c= (dom f1) /\ (dom cot) by A4, XBOOLE_1:19;
then A7: Z c= dom (f1 (#) cot) by VALUED_1:def_4;
 for x being Real st x in Z holds
cos / sin is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies cos / sin is_differentiable_in x )
assume  x in Z ; ::_thesis: cos / sin is_differentiable_in x
then  sin . x <> 0 by A4, FDIFF_8:2;
hence  cos / sin is_differentiable_in x by FDIFF_7:47; ::_thesis: verum
end;
then  cos / sin is_differentiable_on Z by A4, FDIFF_1:9;
then A8: a (#) (cos / sin) is_differentiable_on Z by A3, FDIFF_1:20;
A9: f1 is_differentiable_on Z by A1, A6, FDIFF_1:23;
 sin is_differentiable_on Z by FDIFF_1:26, SIN_COS:68;
then A10: sin ^2 is_differentiable_on Z by FDIFF_2:20;
 for x being Real st x in Z holds
(sin ^2) . x <> 0
proof
let x be Real; ::_thesis: ( x in Z implies (sin ^2) . x <> 0 )
assume  x in Z ; ::_thesis: (sin ^2) . x <> 0
then  x in dom (f1 / (sin ^2)) by A5;
then  x in (dom f1) /\ ((dom (sin ^2)) \ ((sin ^2) " {0})) by RFUNCT_1:def_1;
then  x in (dom (sin ^2)) \ ((sin ^2) " {0}) by XBOOLE_0:def_4;
then  x in dom ((sin ^2) ^) by RFUNCT_1:def_2;
hence  (sin ^2) . x <> 0 by RFUNCT_1:3; ::_thesis: verum
end;
then  f1 / (sin ^2) is_differentiable_on Z by A9, A10, FDIFF_2:21;
then  f | Z is continuous by A1, A8, FDIFF_1:19, FDIFF_1:25;
then  f | A is continuous by A1, FCONT_1:16;
then A11: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A12: f1 (#) cot is_differentiable_on Z by A1, A7, FDIFF_8:29;
A13: for x being Real st x in Z holds
f . x = ((a * (cos . x)) / (sin . x)) - (((a * x) + b) / ((sin . x) ^2))
proof
let x be Real; ::_thesis: ( x in Z implies f . x = ((a * (cos . x)) / (sin . x)) - (((a * x) + b) / ((sin . x) ^2)) )
assume A14: x in Z ; ::_thesis: f . x = ((a * (cos . x)) / (sin . x)) - (((a * x) + b) / ((sin . x) ^2))
then ((a (#) (cos / sin)) - (f1 / (sin ^2))) . x = ((a (#) (cos / sin)) . x) - ((f1 / (sin ^2)) . x) by A1, VALUED_1:13
.= (a * ((cos / sin) . x)) - ((f1 / (sin ^2)) . x) by VALUED_1:6
.= (a * ((cos . x) / (sin . x))) - ((f1 / (sin ^2)) . x) by A14, A4, RFUNCT_1:def_1
.= ((a * (cos . x)) / (sin . x)) - ((f1 . x) / ((sin ^2) . x)) by A14, A5, RFUNCT_1:def_1
.= ((a * (cos . x)) / (sin . x)) - ((f1 . x) / ((sin . x) ^2)) by VALUED_1:11
.= ((a * (cos . x)) / (sin . x)) - (((a * x) + b) / ((sin . x) ^2)) by A1, A14 ;
hence  f . x = ((a * (cos . x)) / (sin . x)) - (((a * x) + b) / ((sin . x) ^2)) by A1; ::_thesis: verum
end;
A15: for x being Real st x in dom ((f1 (#) cot) `| Z) holds
((f1 (#) cot) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom ((f1 (#) cot) `| Z) implies ((f1 (#) cot) `| Z) . x = f . x )
assume  x in dom ((f1 (#) cot) `| Z) ; ::_thesis: ((f1 (#) cot) `| Z) . x = f . x
then A16: x in Z by A12, FDIFF_1:def_7;
then ((f1 (#) cot) `| Z) . x = ((a * (cos . x)) / (sin . x)) - (((a * x) + b) / ((sin . x) ^2)) by A1, A7, FDIFF_8:29
.= f . x by A13, A16 ;
hence  ((f1 (#) cot) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom ((f1 (#) cot) `| Z) = dom f by A1, A12, FDIFF_1:def_7;
then  (f1 (#) cot) `| Z = f by A15, PARTFUN1:5;
hence  integral (f,A) = ((f1 (#) cot) . (upper_bound A)) - ((f1 (#) cot) . (lower_bound A)) by A1, A11, A7, FDIFF_8:29, INTEGRA5:13; ::_thesis: verum
end;

theorem :: INTEGR12:33
for A being non empty closed_interval Subset of REAL
for f 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
f . x = ((sin . x) ^2) / ((cos . x) ^2) ) & Z c= dom (tan - (id Z)) & Z = dom f & f | A is continuous holds
integral (f,A) = ((tan - (id Z)) . (upper_bound A)) - ((tan - (id Z)) . (lower_bound A))
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for f 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
f . x = ((sin . x) ^2) / ((cos . x) ^2) ) & Z c= dom (tan - (id Z)) & Z = dom f & f | A is continuous holds
integral (f,A) = ((tan - (id Z)) . (upper_bound A)) - ((tan - (id Z)) . (lower_bound A))
let f 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
f . x = ((sin . x) ^2) / ((cos . x) ^2) ) & Z c= dom (tan - (id Z)) & Z = dom f & f | A is continuous holds
integral (f,A) = ((tan - (id Z)) . (upper_bound A)) - ((tan - (id Z)) . (lower_bound A))
let Z be open Subset of REAL; ::_thesis: ( A c= Z & ( for x being Real st x in Z holds
f . x = ((sin . x) ^2) / ((cos . x) ^2) ) & Z c= dom (tan - (id Z)) & Z = dom f & f | A is continuous implies integral (f,A) = ((tan - (id Z)) . (upper_bound A)) - ((tan - (id Z)) . (lower_bound A)) )
assume A1: ( A c= Z & ( for x being Real st x in Z holds
f . x = ((sin . x) ^2) / ((cos . x) ^2) ) & Z c= dom (tan - (id Z)) & Z = dom f & f | A is continuous ) ; ::_thesis: integral (f,A) = ((tan - (id Z)) . (upper_bound A)) - ((tan - (id Z)) . (lower_bound A))
then A2: ( f is_integrable_on A & f | A is bounded ) by INTEGRA5:10, INTEGRA5:11;
A3: tan - (id Z) is_differentiable_on Z by A1, FDIFF_8:24;
A4: for x being Real st x in dom ((tan - (id Z)) `| Z) holds
((tan - (id Z)) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom ((tan - (id Z)) `| Z) implies ((tan - (id Z)) `| Z) . x = f . x )
assume  x in dom ((tan - (id Z)) `| Z) ; ::_thesis: ((tan - (id Z)) `| Z) . x = f . x
then A5: x in Z by A3, FDIFF_1:def_7;
then ((tan - (id Z)) `| Z) . x = ((sin . x) ^2) / ((cos . x) ^2) by A1, FDIFF_8:24
.= f . x by A1, A5 ;
hence  ((tan - (id Z)) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom ((tan - (id Z)) `| Z) = dom f by A1, A3, FDIFF_1:def_7;
then  (tan - (id Z)) `| Z = f by A4, PARTFUN1:5;
hence  integral (f,A) = ((tan - (id Z)) . (upper_bound A)) - ((tan - (id Z)) . (lower_bound A)) by A1, A2, FDIFF_8:24, INTEGRA5:13; ::_thesis: verum
end;

theorem :: INTEGR12:34
for A being non empty closed_interval Subset of REAL
for f 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
f . x = ((cos . x) ^2) / ((sin . x) ^2) ) & Z c= dom ((- cot) - (id Z)) & Z = dom f & f | A is continuous holds
integral (f,A) = (((- cot) - (id Z)) . (upper_bound A)) - (((- cot) - (id Z)) . (lower_bound A))
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for f 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
f . x = ((cos . x) ^2) / ((sin . x) ^2) ) & Z c= dom ((- cot) - (id Z)) & Z = dom f & f | A is continuous holds
integral (f,A) = (((- cot) - (id Z)) . (upper_bound A)) - (((- cot) - (id Z)) . (lower_bound A))
let f 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
f . x = ((cos . x) ^2) / ((sin . x) ^2) ) & Z c= dom ((- cot) - (id Z)) & Z = dom f & f | A is continuous holds
integral (f,A) = (((- cot) - (id Z)) . (upper_bound A)) - (((- cot) - (id Z)) . (lower_bound A))
let Z be open Subset of REAL; ::_thesis: ( A c= Z & ( for x being Real st x in Z holds
f . x = ((cos . x) ^2) / ((sin . x) ^2) ) & Z c= dom ((- cot) - (id Z)) & Z = dom f & f | A is continuous implies integral (f,A) = (((- cot) - (id Z)) . (upper_bound A)) - (((- cot) - (id Z)) . (lower_bound A)) )
assume A1: ( A c= Z & ( for x being Real st x in Z holds
f . x = ((cos . x) ^2) / ((sin . x) ^2) ) & Z c= dom ((- cot) - (id Z)) & Z = dom f & f | A is continuous ) ; ::_thesis: integral (f,A) = (((- cot) - (id Z)) . (upper_bound A)) - (((- cot) - (id Z)) . (lower_bound A))
then A2: ( f is_integrable_on A & f | A is bounded ) by INTEGRA5:10, INTEGRA5:11;
A3: (- cot) - (id Z) is_differentiable_on Z by A1, FDIFF_8:25;
A4: for x being Real st x in dom (((- cot) - (id Z)) `| Z) holds
(((- cot) - (id Z)) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom (((- cot) - (id Z)) `| Z) implies (((- cot) - (id Z)) `| Z) . x = f . x )
assume  x in dom (((- cot) - (id Z)) `| Z) ; ::_thesis: (((- cot) - (id Z)) `| Z) . x = f . x
then A5: x in Z by A3, FDIFF_1:def_7;
then (((- cot) - (id Z)) `| Z) . x = ((cos . x) ^2) / ((sin . x) ^2) by A1, FDIFF_8:25
.= f . x by A1, A5 ;
hence  (((- cot) - (id Z)) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom (((- cot) - (id Z)) `| Z) = dom f by A1, A3, FDIFF_1:def_7;
then  ((- cot) - (id Z)) `| Z = f by A4, PARTFUN1:5;
hence  integral (f,A) = (((- cot) - (id Z)) . (upper_bound A)) - (((- cot) - (id Z)) . (lower_bound A)) by A1, A2, FDIFF_8:25, INTEGRA5:13; ::_thesis: verum
end;

theorem :: INTEGR12:35
for A being non empty closed_interval Subset of REAL
for f 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
( f . x = 1 / (x * (1 + ((ln . x) ^2))) & ln . x > - 1 & ln . x < 1 ) ) & Z c= dom (arctan * ln) & Z = dom f & f | A is continuous holds
integral (f,A) = ((arctan * ln) . (upper_bound A)) - ((arctan * ln) . (lower_bound A))
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for f 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
( f . x = 1 / (x * (1 + ((ln . x) ^2))) & ln . x > - 1 & ln . x < 1 ) ) & Z c= dom (arctan * ln) & Z = dom f & f | A is continuous holds
integral (f,A) = ((arctan * ln) . (upper_bound A)) - ((arctan * ln) . (lower_bound A))
let f 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
( f . x = 1 / (x * (1 + ((ln . x) ^2))) & ln . x > - 1 & ln . x < 1 ) ) & Z c= dom (arctan * ln) & Z = dom f & f | A is continuous holds
integral (f,A) = ((arctan * ln) . (upper_bound A)) - ((arctan * ln) . (lower_bound A))
let Z be open Subset of REAL; ::_thesis: ( A c= Z & ( for x being Real st x in Z holds
( f . x = 1 / (x * (1 + ((ln . x) ^2))) & ln . x > - 1 & ln . x < 1 ) ) & Z c= dom (arctan * ln) & Z = dom f & f | A is continuous implies integral (f,A) = ((arctan * ln) . (upper_bound A)) - ((arctan * ln) . (lower_bound A)) )
assume A1: ( A c= Z & ( for x being Real st x in Z holds
( f . x = 1 / (x * (1 + ((ln . x) ^2))) & ln . x > - 1 & ln . x < 1 ) ) & Z c= dom (arctan * ln) & Z = dom f & f | A is continuous ) ; ::_thesis: integral (f,A) = ((arctan * ln) . (upper_bound A)) - ((arctan * ln) . (lower_bound A))
then A2: ( f is_integrable_on A & f | A is bounded ) by INTEGRA5:10, INTEGRA5:11;
A3: for x being Real st x in Z holds
( ln . x > - 1 & ln . x < 1 ) by A1;
then A4: arctan * ln is_differentiable_on Z by A1, SIN_COS9:117;
A5: for x being Real st x in dom ((arctan * ln) `| Z) holds
((arctan * ln) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom ((arctan * ln) `| Z) implies ((arctan * ln) `| Z) . x = f . x )
assume  x in dom ((arctan * ln) `| Z) ; ::_thesis: ((arctan * ln) `| Z) . x = f . x
then A6: x in Z by A4, FDIFF_1:def_7;
then ((arctan * ln) `| Z) . x = 1 / (x * (1 + ((ln . x) ^2))) by A1, A3, SIN_COS9:117
.= f . x by A1, A6 ;
hence  ((arctan * ln) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom ((arctan * ln) `| Z) = dom f by A1, A4, FDIFF_1:def_7;
then  (arctan * ln) `| Z = f by A5, PARTFUN1:5;
hence  integral (f,A) = ((arctan * ln) . (upper_bound A)) - ((arctan * ln) . (lower_bound A)) by A1, A2, A4, INTEGRA5:13; ::_thesis: verum
end;

theorem :: INTEGR12:36
for A being non empty closed_interval Subset of REAL
for f 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
( f . x = - (1 / (x * (1 + ((ln . x) ^2)))) & ln . x > - 1 & ln . x < 1 ) ) & Z c= dom (arccot * ln) & Z = dom f & f | A is continuous holds
integral (f,A) = ((arccot * ln) . (upper_bound A)) - ((arccot * ln) . (lower_bound A))
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for f 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
( f . x = - (1 / (x * (1 + ((ln . x) ^2)))) & ln . x > - 1 & ln . x < 1 ) ) & Z c= dom (arccot * ln) & Z = dom f & f | A is continuous holds
integral (f,A) = ((arccot * ln) . (upper_bound A)) - ((arccot * ln) . (lower_bound A))
let f 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
( f . x = - (1 / (x * (1 + ((ln . x) ^2)))) & ln . x > - 1 & ln . x < 1 ) ) & Z c= dom (arccot * ln) & Z = dom f & f | A is continuous holds
integral (f,A) = ((arccot * ln) . (upper_bound A)) - ((arccot * ln) . (lower_bound A))
let Z be open Subset of REAL; ::_thesis: ( A c= Z & ( for x being Real st x in Z holds
( f . x = - (1 / (x * (1 + ((ln . x) ^2)))) & ln . x > - 1 & ln . x < 1 ) ) & Z c= dom (arccot * ln) & Z = dom f & f | A is continuous implies integral (f,A) = ((arccot * ln) . (upper_bound A)) - ((arccot * ln) . (lower_bound A)) )
assume A1: ( A c= Z & ( for x being Real st x in Z holds
( f . x = - (1 / (x * (1 + ((ln . x) ^2)))) & ln . x > - 1 & ln . x < 1 ) ) & Z c= dom (arccot * ln) & Z = dom f & f | A is continuous ) ; ::_thesis: integral (f,A) = ((arccot * ln) . (upper_bound A)) - ((arccot * ln) . (lower_bound A))
then A2: ( f is_integrable_on A & f | A is bounded ) by INTEGRA5:10, INTEGRA5:11;
A3: for x being Real st x in Z holds
( ln . x > - 1 & ln . x < 1 ) by A1;
then A4: arccot * ln is_differentiable_on Z by A1, SIN_COS9:118;
A5: for x being Real st x in dom ((arccot * ln) `| Z) holds
((arccot * ln) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom ((arccot * ln) `| Z) implies ((arccot * ln) `| Z) . x = f . x )
assume  x in dom ((arccot * ln) `| Z) ; ::_thesis: ((arccot * ln) `| Z) . x = f . x
then A6: x in Z by A4, FDIFF_1:def_7;
then ((arccot * ln) `| Z) . x = - (1 / (x * (1 + ((ln . x) ^2)))) by A1, A3, SIN_COS9:118
.= f . x by A1, A6 ;
hence  ((arccot * ln) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom ((arccot * ln) `| Z) = dom f by A1, A4, FDIFF_1:def_7;
then  (arccot * ln) `| Z = f by A5, PARTFUN1:5;
hence  integral (f,A) = ((arccot * ln) . (upper_bound A)) - ((arccot * ln) . (lower_bound A)) by A1, A2, A4, INTEGRA5:13; ::_thesis: verum
end;

theorem :: INTEGR12:37
for a, b being Real
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
( f . x = a / (sqrt (1 - (((a * x) + b) ^2))) & f1 . x = (a * x) + b & f1 . x > - 1 & f1 . x < 1 ) ) & Z c= dom (arcsin * f1) & Z = dom f & f | A is continuous holds
integral (f,A) = ((arcsin * f1) . (upper_bound A)) - ((arcsin * f1) . (lower_bound A))
proof
let a, b be Real; ::_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
( f . x = a / (sqrt (1 - (((a * x) + b) ^2))) & f1 . x = (a * x) + b & f1 . x > - 1 & f1 . x < 1 ) ) & Z c= dom (arcsin * f1) & Z = dom f & f | A is continuous holds
integral (f,A) = ((arcsin * f1) . (upper_bound A)) - ((arcsin * f1) . (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
( f . x = a / (sqrt (1 - (((a * x) + b) ^2))) & f1 . x = (a * x) + b & f1 . x > - 1 & f1 . x < 1 ) ) & Z c= dom (arcsin * f1) & Z = dom f & f | A is continuous holds
integral (f,A) = ((arcsin * f1) . (upper_bound A)) - ((arcsin * f1) . (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
( f . x = a / (sqrt (1 - (((a * x) + b) ^2))) & f1 . x = (a * x) + b & f1 . x > - 1 & f1 . x < 1 ) ) & Z c= dom (arcsin * f1) & Z = dom f & f | A is continuous holds
integral (f,A) = ((arcsin * f1) . (upper_bound A)) - ((arcsin * f1) . (lower_bound A))
let Z be open Subset of REAL; ::_thesis: ( A c= Z & ( for x being Real st x in Z holds
( f . x = a / (sqrt (1 - (((a * x) + b) ^2))) & f1 . x = (a * x) + b & f1 . x > - 1 & f1 . x < 1 ) ) & Z c= dom (arcsin * f1) & Z = dom f & f | A is continuous implies integral (f,A) = ((arcsin * f1) . (upper_bound A)) - ((arcsin * f1) . (lower_bound A)) )
assume A1: ( A c= Z & ( for x being Real st x in Z holds
( f . x = a / (sqrt (1 - (((a * x) + b) ^2))) & f1 . x = (a * x) + b & f1 . x > - 1 & f1 . x < 1 ) ) & Z c= dom (arcsin * f1) & Z = dom f & f | A is continuous ) ; ::_thesis: integral (f,A) = ((arcsin * f1) . (upper_bound A)) - ((arcsin * f1) . (lower_bound A))
then A2: ( f is_integrable_on A & f | A is bounded ) by INTEGRA5:10, INTEGRA5:11;
A3: for x being Real st x in Z holds
( f1 . x = (a * x) + b & f1 . x > - 1 & f1 . x < 1 ) by A1;
then A4: arcsin * f1 is_differentiable_on Z by A1, FDIFF_7:14;
A5: for x being Real st x in dom ((arcsin * f1) `| Z) holds
((arcsin * f1) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom ((arcsin * f1) `| Z) implies ((arcsin * f1) `| Z) . x = f . x )
assume  x in dom ((arcsin * f1) `| Z) ; ::_thesis: ((arcsin * f1) `| Z) . x = f . x
then A6: x in Z by A4, FDIFF_1:def_7;
then ((arcsin * f1) `| Z) . x = a / (sqrt (1 - (((a * x) + b) ^2))) by A1, A3, FDIFF_7:14
.= f . x by A1, A6 ;
hence  ((arcsin * f1) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom ((arcsin * f1) `| Z) = dom f by A1, A4, FDIFF_1:def_7;
then  (arcsin * f1) `| Z = f by A5, PARTFUN1:5;
hence  integral (f,A) = ((arcsin * f1) . (upper_bound A)) - ((arcsin * f1) . (lower_bound A)) by A1, A2, A4, INTEGRA5:13; ::_thesis: verum
end;

theorem :: INTEGR12:38
for a, b being Real
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
( f . x = a / (sqrt (1 - (((a * x) + b) ^2))) & f1 . x = (a * x) + b & f1 . x > - 1 & f1 . x < 1 ) ) & Z c= dom (arccos * f1) & Z = dom f & f | A is continuous holds
integral (f,A) = ((- (arccos * f1)) . (upper_bound A)) - ((- (arccos * f1)) . (lower_bound A))
proof
let a, b be Real; ::_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
( f . x = a / (sqrt (1 - (((a * x) + b) ^2))) & f1 . x = (a * x) + b & f1 . x > - 1 & f1 . x < 1 ) ) & Z c= dom (arccos * f1) & Z = dom f & f | A is continuous holds
integral (f,A) = ((- (arccos * f1)) . (upper_bound A)) - ((- (arccos * f1)) . (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
( f . x = a / (sqrt (1 - (((a * x) + b) ^2))) & f1 . x = (a * x) + b & f1 . x > - 1 & f1 . x < 1 ) ) & Z c= dom (arccos * f1) & Z = dom f & f | A is continuous holds
integral (f,A) = ((- (arccos * f1)) . (upper_bound A)) - ((- (arccos * f1)) . (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
( f . x = a / (sqrt (1 - (((a * x) + b) ^2))) & f1 . x = (a * x) + b & f1 . x > - 1 & f1 . x < 1 ) ) & Z c= dom (arccos * f1) & Z = dom f & f | A is continuous holds
integral (f,A) = ((- (arccos * f1)) . (upper_bound A)) - ((- (arccos * f1)) . (lower_bound A))
let Z be open Subset of REAL; ::_thesis: ( A c= Z & ( for x being Real st x in Z holds
( f . x = a / (sqrt (1 - (((a * x) + b) ^2))) & f1 . x = (a * x) + b & f1 . x > - 1 & f1 . x < 1 ) ) & Z c= dom (arccos * f1) & Z = dom f & f | A is continuous implies integral (f,A) = ((- (arccos * f1)) . (upper_bound A)) - ((- (arccos * f1)) . (lower_bound A)) )
assume A1: ( A c= Z & ( for x being Real st x in Z holds
( f . x = a / (sqrt (1 - (((a * x) + b) ^2))) & f1 . x = (a * x) + b & f1 . x > - 1 & f1 . x < 1 ) ) & Z c= dom (arccos * f1) & Z = dom f & f | A is continuous ) ; ::_thesis: integral (f,A) = ((- (arccos * f1)) . (upper_bound A)) - ((- (arccos * f1)) . (lower_bound A))
then A2: ( f is_integrable_on A & f | A is bounded ) by INTEGRA5:10, INTEGRA5:11;
A3: for x being Real st x in Z holds
f1 . x = (a * x) + b by A1;
A4: for x being Real st x in Z holds
( f1 . x = (a * x) + b & f1 . x > - 1 & f1 . x < 1 ) by A1;
A5: Z c= dom (- (arccos * f1)) by A1, VALUED_1:8;
A6: arccos * f1 is_differentiable_on Z by A1, A4, FDIFF_7:15;
then A7: (- 1) (#) (arccos * f1) is_differentiable_on Z by A5, Lm1, FDIFF_1:20;
 for y being set st y in Z holds
y in dom f1 by A1, FUNCT_1:11;
then A8: Z c= dom f1 by TARSKI:def_3;
then A9: ( f1 is_differentiable_on Z & ( for x being Real st x in Z holds
(f1 `| Z) . x = a ) ) by A3, FDIFF_1:23;
A10: for x being Real st x in Z holds
((- (arccos * f1)) `| Z) . x = a / (sqrt (1 - (((a * x) + b) ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((- (arccos * f1)) `| Z) . x = a / (sqrt (1 - (((a * x) + b) ^2))) )
assume A11: x in Z ; ::_thesis: ((- (arccos * f1)) `| Z) . x = a / (sqrt (1 - (((a * x) + b) ^2)))
then A12: f1 is_differentiable_in x by A9, FDIFF_1:9;
A13: ( f1 . x > - 1 & f1 . x < 1 ) by A1, A11;
A14: arccos * f1 is_differentiable_in x by A6, A11, FDIFF_1:9;
((- (arccos * f1)) `| Z) . x = diff ((- (arccos * f1)),x) by A7, A11, FDIFF_1:def_7
.= (- 1) * (diff ((arccos * f1),x)) by A14, Lm1, FDIFF_1:15
.= (- 1) * (- ((diff (f1,x)) / (sqrt (1 - ((f1 . x) ^2))))) by A12, A13, FDIFF_7:7
.= (- 1) * (- (((f1 `| Z) . x) / (sqrt (1 - ((f1 . x) ^2))))) by A9, A11, FDIFF_1:def_7
.= (- 1) * (- (a / (sqrt (1 - ((f1 . x) ^2))))) by A3, A8, A11, FDIFF_1:23
.= a / (sqrt (1 - (((a * x) + b) ^2))) by A1, A11 ;
hence  ((- (arccos * f1)) `| Z) . x = a / (sqrt (1 - (((a * x) + b) ^2))) ; ::_thesis: verum
end;
A15: for x being Real st x in dom ((- (arccos * f1)) `| Z) holds
((- (arccos * f1)) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom ((- (arccos * f1)) `| Z) implies ((- (arccos * f1)) `| Z) . x = f . x )
assume  x in dom ((- (arccos * f1)) `| Z) ; ::_thesis: ((- (arccos * f1)) `| Z) . x = f . x
then A16: x in Z by A7, FDIFF_1:def_7;
then ((- (arccos * f1)) `| Z) . x = a / (sqrt (1 - (((a * x) + b) ^2))) by A10
.= f . x by A1, A16 ;
hence  ((- (arccos * f1)) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom ((- (arccos * f1)) `| Z) = dom f by A1, A7, FDIFF_1:def_7;
then  (- (arccos * f1)) `| Z = f by A15, PARTFUN1:5;
hence  integral (f,A) = ((- (arccos * f1)) . (upper_bound A)) - ((- (arccos * f1)) . (lower_bound A)) by A1, A2, A7, INTEGRA5:13; ::_thesis: verum
end;

theorem :: INTEGR12:39
for A being non empty closed_interval Subset of REAL
for f1, g, f2, f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & f1 = g - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds
( f . x = x * ((1 - (x #Z 2)) #R (- (1 / 2))) & g . x = 1 & f1 . x > 0 ) ) & Z c= dom ((#R (1 / 2)) * f1) & Z = dom f & f | A is continuous holds
integral (f,A) = ((- ((#R (1 / 2)) * f1)) . (upper_bound A)) - ((- ((#R (1 / 2)) * f1)) . (lower_bound A))
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for f1, g, f2, f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & f1 = g - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds
( f . x = x * ((1 - (x #Z 2)) #R (- (1 / 2))) & g . x = 1 & f1 . x > 0 ) ) & Z c= dom ((#R (1 / 2)) * f1) & Z = dom f & f | A is continuous holds
integral (f,A) = ((- ((#R (1 / 2)) * f1)) . (upper_bound A)) - ((- ((#R (1 / 2)) * f1)) . (lower_bound A))
let f1, g, f2, f be PartFunc of REAL,REAL; ::_thesis: for Z being open Subset of REAL st A c= Z & f1 = g - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds
( f . x = x * ((1 - (x #Z 2)) #R (- (1 / 2))) & g . x = 1 & f1 . x > 0 ) ) & Z c= dom ((#R (1 / 2)) * f1) & Z = dom f & f | A is continuous holds
integral (f,A) = ((- ((#R (1 / 2)) * f1)) . (upper_bound A)) - ((- ((#R (1 / 2)) * f1)) . (lower_bound A))
let Z be open Subset of REAL; ::_thesis: ( A c= Z & f1 = g - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds
( f . x = x * ((1 - (x #Z 2)) #R (- (1 / 2))) & g . x = 1 & f1 . x > 0 ) ) & Z c= dom ((#R (1 / 2)) * f1) & Z = dom f & f | A is continuous implies integral (f,A) = ((- ((#R (1 / 2)) * f1)) . (upper_bound A)) - ((- ((#R (1 / 2)) * f1)) . (lower_bound A)) )
assume A1: ( A c= Z & f1 = g - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds
( f . x = x * ((1 - (x #Z 2)) #R (- (1 / 2))) & g . x = 1 & f1 . x > 0 ) ) & Z c= dom ((#R (1 / 2)) * f1) & Z = dom f & f | A is continuous ) ; ::_thesis: integral (f,A) = ((- ((#R (1 / 2)) * f1)) . (upper_bound A)) - ((- ((#R (1 / 2)) * f1)) . (lower_bound A))
then A2: ( f is_integrable_on A & f | A is bounded ) by INTEGRA5:10, INTEGRA5:11;
A3: for x being Real st x in Z holds
( g . x = 1 & f1 . x > 0 ) by A1;
A4: Z c= dom (- ((#R (1 / 2)) * f1)) by A1, VALUED_1:8;
 for y being set st y in Z holds
y in dom f1 by A1, FUNCT_1:11;
then A5: Z c= dom (g + ((- 1) (#) f2)) by A1, TARSKI:def_3;
A6: (#R (1 / 2)) * f1 is_differentiable_on Z by A1, A3, FDIFF_7:22;
then A7: (- 1) (#) ((#R (1 / 2)) * f1) is_differentiable_on Z by A4, Lm1, FDIFF_1:20;
A8: ( f2 = #Z 2 & ( for x being Real st x in Z holds
g . x = 1 + (0 * x) ) ) by A1;
then A9: ( f1 is_differentiable_on Z & ( for x being Real st x in Z holds
(f1 `| Z) . x = 0 + ((2 * (- 1)) * x) ) ) by A1, A5, Lm1, FDIFF_4:12;
A10: for x being Real st x in Z holds
((- ((#R (1 / 2)) * f1)) `| Z) . x = x * ((1 - (x #Z 2)) #R (- (1 / 2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((- ((#R (1 / 2)) * f1)) `| Z) . x = x * ((1 - (x #Z 2)) #R (- (1 / 2))) )
assume A11: x in Z ; ::_thesis: ((- ((#R (1 / 2)) * f1)) `| Z) . x = x * ((1 - (x #Z 2)) #R (- (1 / 2)))
then A12: x in dom (g - f2) by A1, FUNCT_1:11;
A13: f1 is_differentiable_in x by A9, A11, FDIFF_1:9;
A14: (g - f2) . x = (g . x) - (f2 . x) by A12, VALUED_1:13
.= 1 - (f2 . x) by A1, A11
.= 1 - (x #Z 2) by A1, TAYLOR_1:def_1 ;
then A15: ( f1 . x = 1 - (x #Z 2) & f1 . x > 0 ) by A1, A11;
A16: (#R (1 / 2)) * f1 is_differentiable_in x by A6, A11, FDIFF_1:9;
((- ((#R (1 / 2)) * f1)) `| Z) . x = diff ((- ((#R (1 / 2)) * f1)),x) by A7, A11, FDIFF_1:def_7
.= (- 1) * (diff (((#R (1 / 2)) * f1),x)) by A16, Lm1, FDIFF_1:15
.= (- 1) * (((1 / 2) * ((f1 . x) #R ((1 / 2) - 1))) * (diff (f1,x))) by A13, A15, TAYLOR_1:22
.= (- 1) * (((1 / 2) * ((f1 . x) #R ((1 / 2) - 1))) * ((f1 `| Z) . x)) by A9, A11, FDIFF_1:def_7
.= (- 1) * (((1 / 2) * ((f1 . x) #R ((1 / 2) - 1))) * (0 + ((2 * (- 1)) * x))) by A1, A5, A8, Lm1, A11, FDIFF_4:12
.= x * ((1 - (x #Z 2)) #R (- (1 / 2))) by A1, A14 ;
hence  ((- ((#R (1 / 2)) * f1)) `| Z) . x = x * ((1 - (x #Z 2)) #R (- (1 / 2))) ; ::_thesis: verum
end;
A17: for x being Real st x in dom ((- ((#R (1 / 2)) * f1)) `| Z) holds
((- ((#R (1 / 2)) * f1)) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom ((- ((#R (1 / 2)) * f1)) `| Z) implies ((- ((#R (1 / 2)) * f1)) `| Z) . x = f . x )
assume  x in dom ((- ((#R (1 / 2)) * f1)) `| Z) ; ::_thesis: ((- ((#R (1 / 2)) * f1)) `| Z) . x = f . x
then A18: x in Z by A7, FDIFF_1:def_7;
then ((- ((#R (1 / 2)) * f1)) `| Z) . x = x * ((1 - (x #Z 2)) #R (- (1 / 2))) by A10
.= f . x by A1, A18 ;
hence  ((- ((#R (1 / 2)) * f1)) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom ((- ((#R (1 / 2)) * f1)) `| Z) = dom f by A1, A7, FDIFF_1:def_7;
then  (- ((#R (1 / 2)) * f1)) `| Z = f by A17, PARTFUN1:5;
hence  integral (f,A) = ((- ((#R (1 / 2)) * f1)) . (upper_bound A)) - ((- ((#R (1 / 2)) * f1)) . (lower_bound A)) by A1, A2, A7, INTEGRA5:13; ::_thesis: verum
end;

theorem :: INTEGR12:40
for a being Real
for A being non empty closed_interval Subset of REAL
for g, f1, f2, f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & g = f1 - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds
( f . x = x * (((a ^2) - (x #Z 2)) #R (- (1 / 2))) & f1 . x = a ^2 & g . x > 0 ) ) & Z c= dom ((#R (1 / 2)) * g) & Z = dom f & f | A is continuous holds
integral (f,A) = ((- ((#R (1 / 2)) * g)) . (upper_bound A)) - ((- ((#R (1 / 2)) * g)) . (lower_bound A))
proof
let a be Real; ::_thesis: for A being non empty closed_interval Subset of REAL
for g, f1, f2, f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & g = f1 - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds
( f . x = x * (((a ^2) - (x #Z 2)) #R (- (1 / 2))) & f1 . x = a ^2 & g . x > 0 ) ) & Z c= dom ((#R (1 / 2)) * g) & Z = dom f & f | A is continuous holds
integral (f,A) = ((- ((#R (1 / 2)) * g)) . (upper_bound A)) - ((- ((#R (1 / 2)) * g)) . (lower_bound A))
let A be non empty closed_interval Subset of REAL; ::_thesis: for g, f1, f2, f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & g = f1 - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds
( f . x = x * (((a ^2) - (x #Z 2)) #R (- (1 / 2))) & f1 . x = a ^2 & g . x > 0 ) ) & Z c= dom ((#R (1 / 2)) * g) & Z = dom f & f | A is continuous holds
integral (f,A) = ((- ((#R (1 / 2)) * g)) . (upper_bound A)) - ((- ((#R (1 / 2)) * g)) . (lower_bound A))
let g, f1, f2, f be PartFunc of REAL,REAL; ::_thesis: for Z being open Subset of REAL st A c= Z & g = f1 - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds
( f . x = x * (((a ^2) - (x #Z 2)) #R (- (1 / 2))) & f1 . x = a ^2 & g . x > 0 ) ) & Z c= dom ((#R (1 / 2)) * g) & Z = dom f & f | A is continuous holds
integral (f,A) = ((- ((#R (1 / 2)) * g)) . (upper_bound A)) - ((- ((#R (1 / 2)) * g)) . (lower_bound A))
let Z be open Subset of REAL; ::_thesis: ( A c= Z & g = f1 - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds
( f . x = x * (((a ^2) - (x #Z 2)) #R (- (1 / 2))) & f1 . x = a ^2 & g . x > 0 ) ) & Z c= dom ((#R (1 / 2)) * g) & Z = dom f & f | A is continuous implies integral (f,A) = ((- ((#R (1 / 2)) * g)) . (upper_bound A)) - ((- ((#R (1 / 2)) * g)) . (lower_bound A)) )
assume A1: ( A c= Z & g = f1 - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds
( f . x = x * (((a ^2) - (x #Z 2)) #R (- (1 / 2))) & f1 . x = a ^2 & g . x > 0 ) ) & Z c= dom ((#R (1 / 2)) * g) & Z = dom f & f | A is continuous ) ; ::_thesis: integral (f,A) = ((- ((#R (1 / 2)) * g)) . (upper_bound A)) - ((- ((#R (1 / 2)) * g)) . (lower_bound A))
then A2: ( f is_integrable_on A & f | A is bounded ) by INTEGRA5:10, INTEGRA5:11;
A3: for x being Real st x in Z holds
( f1 . x = a ^2 & g . x > 0 ) by A1;
A4: Z c= dom (- ((#R (1 / 2)) * g)) by A1, VALUED_1:8;
 for y being set st y in Z holds
y in dom g by A1, FUNCT_1:11;
then A5: Z c= dom (f1 + ((- 1) (#) f2)) by A1, TARSKI:def_3;
A6: (#R (1 / 2)) * g is_differentiable_on Z by A1, A3, FDIFF_7:27;
then A7: (- 1) (#) ((#R (1 / 2)) * g) is_differentiable_on Z by A4, Lm1, FDIFF_1:20;
A8: ( f2 = #Z 2 & ( for x being Real st x in Z holds
f1 . x = (a ^2) + (0 * x) ) ) by A1;
then A9: ( g is_differentiable_on Z & ( for x being Real st x in Z holds
(g `| Z) . x = 0 + ((2 * (- 1)) * x) ) ) by A1, A5, Lm1, FDIFF_4:12;
A10: for x being Real st x in Z holds
((- ((#R (1 / 2)) * g)) `| Z) . x = x * (((a ^2) - (x #Z 2)) #R (- (1 / 2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((- ((#R (1 / 2)) * g)) `| Z) . x = x * (((a ^2) - (x #Z 2)) #R (- (1 / 2))) )
assume A11: x in Z ; ::_thesis: ((- ((#R (1 / 2)) * g)) `| Z) . x = x * (((a ^2) - (x #Z 2)) #R (- (1 / 2)))
then A12: x in dom (f1 - f2) by A1, FUNCT_1:11;
A13: g is_differentiable_in x by A9, A11, FDIFF_1:9;
A14: (f1 - f2) . x = (f1 . x) - (f2 . x) by A12, VALUED_1:13
.= (a ^2) - (f2 . x) by A1, A11
.= (a ^2) - (x #Z 2) by A1, TAYLOR_1:def_1 ;
then A15: ( g . x = (a ^2) - (x #Z 2) & g . x > 0 ) by A1, A11;
A16: (#R (1 / 2)) * g is_differentiable_in x by A6, A11, FDIFF_1:9;
((- ((#R (1 / 2)) * g)) `| Z) . x = diff ((- ((#R (1 / 2)) * g)),x) by A7, A11, FDIFF_1:def_7
.= (- 1) * (diff (((#R (1 / 2)) * g),x)) by A16, Lm1, FDIFF_1:15
.= (- 1) * (((1 / 2) * ((g . x) #R ((1 / 2) - 1))) * (diff (g,x))) by A13, A15, TAYLOR_1:22
.= (- 1) * (((1 / 2) * ((g . x) #R ((1 / 2) - 1))) * ((g `| Z) . x)) by A9, A11, FDIFF_1:def_7
.= (- 1) * (((1 / 2) * ((g . x) #R ((1 / 2) - 1))) * (0 + ((2 * (- 1)) * x))) by A1, A5, A8, Lm1, A11, FDIFF_4:12
.= x * (((a ^2) - (x #Z 2)) #R (- (1 / 2))) by A1, A14 ;
hence  ((- ((#R (1 / 2)) * g)) `| Z) . x = x * (((a ^2) - (x #Z 2)) #R (- (1 / 2))) ; ::_thesis: verum
end;
A17: for x being Real st x in dom ((- ((#R (1 / 2)) * g)) `| Z) holds
((- ((#R (1 / 2)) * g)) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom ((- ((#R (1 / 2)) * g)) `| Z) implies ((- ((#R (1 / 2)) * g)) `| Z) . x = f . x )
assume  x in dom ((- ((#R (1 / 2)) * g)) `| Z) ; ::_thesis: ((- ((#R (1 / 2)) * g)) `| Z) . x = f . x
then A18: x in Z by A7, FDIFF_1:def_7;
then ((- ((#R (1 / 2)) * g)) `| Z) . x = x * (((a ^2) - (x #Z 2)) #R (- (1 / 2))) by A10
.= f . x by A1, A18 ;
hence  ((- ((#R (1 / 2)) * g)) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom ((- ((#R (1 / 2)) * g)) `| Z) = dom f by A1, A7, FDIFF_1:def_7;
then  (- ((#R (1 / 2)) * g)) `| Z = f by A17, PARTFUN1:5;
hence  integral (f,A) = ((- ((#R (1 / 2)) * g)) . (upper_bound A)) - ((- ((#R (1 / 2)) * g)) . (lower_bound A)) by A1, A2, A7, INTEGRA5:13; ::_thesis: verum
end;

theorem :: INTEGR12:41
for n being Element of NAT
for A being non empty closed_interval Subset of REAL
for f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & n > 0 & ( for x being Real st x in Z holds
( f . x = (cos . x) / ((sin . x) #Z (n + 1)) & sin . x <> 0 ) ) & Z c= dom ((#Z n) * (sin ^)) & Z = dom f & f | A is continuous holds
integral (f,A) = (((- (1 / n)) (#) ((#Z n) * (sin ^))) . (upper_bound A)) - (((- (1 / n)) (#) ((#Z n) * (sin ^))) . (lower_bound A))
proof
let n be Element of NAT ; ::_thesis: for A being non empty closed_interval Subset of REAL
for f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & n > 0 & ( for x being Real st x in Z holds
( f . x = (cos . x) / ((sin . x) #Z (n + 1)) & sin . x <> 0 ) ) & Z c= dom ((#Z n) * (sin ^)) & Z = dom f & f | A is continuous holds
integral (f,A) = (((- (1 / n)) (#) ((#Z n) * (sin ^))) . (upper_bound A)) - (((- (1 / n)) (#) ((#Z n) * (sin ^))) . (lower_bound A))
let A be non empty closed_interval Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & n > 0 & ( for x being Real st x in Z holds
( f . x = (cos . x) / ((sin . x) #Z (n + 1)) & sin . x <> 0 ) ) & Z c= dom ((#Z n) * (sin ^)) & Z = dom f & f | A is continuous holds
integral (f,A) = (((- (1 / n)) (#) ((#Z n) * (sin ^))) . (upper_bound A)) - (((- (1 / n)) (#) ((#Z n) * (sin ^))) . (lower_bound A))
let f be PartFunc of REAL,REAL; ::_thesis: for Z being open Subset of REAL st A c= Z & n > 0 & ( for x being Real st x in Z holds
( f . x = (cos . x) / ((sin . x) #Z (n + 1)) & sin . x <> 0 ) ) & Z c= dom ((#Z n) * (sin ^)) & Z = dom f & f | A is continuous holds
integral (f,A) = (((- (1 / n)) (#) ((#Z n) * (sin ^))) . (upper_bound A)) - (((- (1 / n)) (#) ((#Z n) * (sin ^))) . (lower_bound A))
let Z be open Subset of REAL; ::_thesis: ( A c= Z & n > 0 & ( for x being Real st x in Z holds
( f . x = (cos . x) / ((sin . x) #Z (n + 1)) & sin . x <> 0 ) ) & Z c= dom ((#Z n) * (sin ^)) & Z = dom f & f | A is continuous implies integral (f,A) = (((- (1 / n)) (#) ((#Z n) * (sin ^))) . (upper_bound A)) - (((- (1 / n)) (#) ((#Z n) * (sin ^))) . (lower_bound A)) )
assume A1: ( A c= Z & n > 0 & ( for x being Real st x in Z holds
( f . x = (cos . x) / ((sin . x) #Z (n + 1)) & sin . x <> 0 ) ) & Z c= dom ((#Z n) * (sin ^)) & Z = dom f & f | A is continuous ) ; ::_thesis: integral (f,A) = (((- (1 / n)) (#) ((#Z n) * (sin ^))) . (upper_bound A)) - (((- (1 / n)) (#) ((#Z n) * (sin ^))) . (lower_bound A))
then A2: Z c= dom ((- (1 / n)) (#) ((#Z n) * (sin ^))) by VALUED_1:def_5;
A3: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A4: for x being Real st x in Z holds
sin . x <> 0 by A1;
then A5: (- (1 / n)) (#) ((#Z n) * (sin ^)) is_differentiable_on Z by A1, A2, FDIFF_7:30;
A6: for x being Real st x in dom (((- (1 / n)) (#) ((#Z n) * (sin ^))) `| Z) holds
(((- (1 / n)) (#) ((#Z n) * (sin ^))) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom (((- (1 / n)) (#) ((#Z n) * (sin ^))) `| Z) implies (((- (1 / n)) (#) ((#Z n) * (sin ^))) `| Z) . x = f . x )
assume  x in dom (((- (1 / n)) (#) ((#Z n) * (sin ^))) `| Z) ; ::_thesis: (((- (1 / n)) (#) ((#Z n) * (sin ^))) `| Z) . x = f . x
then A7: x in Z by A5, FDIFF_1:def_7;
then (((- (1 / n)) (#) ((#Z n) * (sin ^))) `| Z) . x = (cos . x) / ((sin . x) #Z (n + 1)) by A1, A2, A4, FDIFF_7:30
.= f . x by A1, A7 ;
hence  (((- (1 / n)) (#) ((#Z n) * (sin ^))) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom (((- (1 / n)) (#) ((#Z n) * (sin ^))) `| Z) = dom f by A1, A5, FDIFF_1:def_7;
then  ((- (1 / n)) (#) ((#Z n) * (sin ^))) `| Z = f by A6, PARTFUN1:5;
hence  integral (f,A) = (((- (1 / n)) (#) ((#Z n) * (sin ^))) . (upper_bound A)) - (((- (1 / n)) (#) ((#Z n) * (sin ^))) . (lower_bound A)) by A1, A3, A5, INTEGRA5:13; ::_thesis: verum
end;

theorem :: INTEGR12:42
for n being Element of NAT
for A being non empty closed_interval Subset of REAL
for f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & n > 0 & ( for x being Real st x in Z holds
( f . x = (sin . x) / ((cos . x) #Z (n + 1)) & cos . x <> 0 ) ) & Z c= dom ((#Z n) * (cos ^)) & Z = dom f & f | A is continuous holds
integral (f,A) = (((1 / n) (#) ((#Z n) * (cos ^))) . (upper_bound A)) - (((1 / n) (#) ((#Z n) * (cos ^))) . (lower_bound A))
proof
let n be Element of NAT ; ::_thesis: for A being non empty closed_interval Subset of REAL
for f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & n > 0 & ( for x being Real st x in Z holds
( f . x = (sin . x) / ((cos . x) #Z (n + 1)) & cos . x <> 0 ) ) & Z c= dom ((#Z n) * (cos ^)) & Z = dom f & f | A is continuous holds
integral (f,A) = (((1 / n) (#) ((#Z n) * (cos ^))) . (upper_bound A)) - (((1 / n) (#) ((#Z n) * (cos ^))) . (lower_bound A))
let A be non empty closed_interval Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & n > 0 & ( for x being Real st x in Z holds
( f . x = (sin . x) / ((cos . x) #Z (n + 1)) & cos . x <> 0 ) ) & Z c= dom ((#Z n) * (cos ^)) & Z = dom f & f | A is continuous holds
integral (f,A) = (((1 / n) (#) ((#Z n) * (cos ^))) . (upper_bound A)) - (((1 / n) (#) ((#Z n) * (cos ^))) . (lower_bound A))
let f be PartFunc of REAL,REAL; ::_thesis: for Z being open Subset of REAL st A c= Z & n > 0 & ( for x being Real st x in Z holds
( f . x = (sin . x) / ((cos . x) #Z (n + 1)) & cos . x <> 0 ) ) & Z c= dom ((#Z n) * (cos ^)) & Z = dom f & f | A is continuous holds
integral (f,A) = (((1 / n) (#) ((#Z n) * (cos ^))) . (upper_bound A)) - (((1 / n) (#) ((#Z n) * (cos ^))) . (lower_bound A))
let Z be open Subset of REAL; ::_thesis: ( A c= Z & n > 0 & ( for x being Real st x in Z holds
( f . x = (sin . x) / ((cos . x) #Z (n + 1)) & cos . x <> 0 ) ) & Z c= dom ((#Z n) * (cos ^)) & Z = dom f & f | A is continuous implies integral (f,A) = (((1 / n) (#) ((#Z n) * (cos ^))) . (upper_bound A)) - (((1 / n) (#) ((#Z n) * (cos ^))) . (lower_bound A)) )
assume A1: ( A c= Z & n > 0 & ( for x being Real st x in Z holds
( f . x = (sin . x) / ((cos . x) #Z (n + 1)) & cos . x <> 0 ) ) & Z c= dom ((#Z n) * (cos ^)) & Z = dom f & f | A is continuous ) ; ::_thesis: integral (f,A) = (((1 / n) (#) ((#Z n) * (cos ^))) . (upper_bound A)) - (((1 / n) (#) ((#Z n) * (cos ^))) . (lower_bound A))
then A2: Z c= dom ((1 / n) (#) ((#Z n) * (cos ^))) by VALUED_1:def_5;
A3: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A4: for x being Real st x in Z holds
cos . x <> 0 by A1;
then A5: (1 / n) (#) ((#Z n) * (cos ^)) is_differentiable_on Z by A1, A2, FDIFF_7:31;
A6: for x being Real st x in dom (((1 / n) (#) ((#Z n) * (cos ^))) `| Z) holds
(((1 / n) (#) ((#Z n) * (cos ^))) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom (((1 / n) (#) ((#Z n) * (cos ^))) `| Z) implies (((1 / n) (#) ((#Z n) * (cos ^))) `| Z) . x = f . x )
assume  x in dom (((1 / n) (#) ((#Z n) * (cos ^))) `| Z) ; ::_thesis: (((1 / n) (#) ((#Z n) * (cos ^))) `| Z) . x = f . x
then A7: x in Z by A5, FDIFF_1:def_7;
then (((1 / n) (#) ((#Z n) * (cos ^))) `| Z) . x = (sin . x) / ((cos . x) #Z (n + 1)) by A1, A2, A4, FDIFF_7:31
.= f . x by A1, A7 ;
hence  (((1 / n) (#) ((#Z n) * (cos ^))) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom (((1 / n) (#) ((#Z n) * (cos ^))) `| Z) = dom f by A1, A5, FDIFF_1:def_7;
then  ((1 / n) (#) ((#Z n) * (cos ^))) `| Z = f by A6, PARTFUN1:5;
hence  integral (f,A) = (((1 / n) (#) ((#Z n) * (cos ^))) . (upper_bound A)) - (((1 / n) (#) ((#Z n) * (cos ^))) . (lower_bound A)) by A1, A3, A5, INTEGRA5:13; ::_thesis: verum
end;

theorem :: INTEGR12:43
for A being non empty closed_interval Subset of REAL
for f, g1, g2, f2 being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & f = ((g1 + g2) ^) / f2 & f2 = arccot & Z c= ].(- 1),1.[ & g2 = #Z 2 & ( for x being Real st x in Z holds
( f . x = 1 / ((1 + (x ^2)) * (arccot . x)) & g1 . x = 1 & f2 . x > 0 ) ) & Z = dom f holds
integral (f,A) = ((- (ln * arccot)) . (upper_bound A)) - ((- (ln * arccot)) . (lower_bound A))
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for f, g1, g2, f2 being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & f = ((g1 + g2) ^) / f2 & f2 = arccot & Z c= ].(- 1),1.[ & g2 = #Z 2 & ( for x being Real st x in Z holds
( f . x = 1 / ((1 + (x ^2)) * (arccot . x)) & g1 . x = 1 & f2 . x > 0 ) ) & Z = dom f holds
integral (f,A) = ((- (ln * arccot)) . (upper_bound A)) - ((- (ln * arccot)) . (lower_bound A))
let f, g1, g2, f2 be PartFunc of REAL,REAL; ::_thesis: for Z being open Subset of REAL st A c= Z & f = ((g1 + g2) ^) / f2 & f2 = arccot & Z c= ].(- 1),1.[ & g2 = #Z 2 & ( for x being Real st x in Z holds
( f . x = 1 / ((1 + (x ^2)) * (arccot . x)) & g1 . x = 1 & f2 . x > 0 ) ) & Z = dom f holds
integral (f,A) = ((- (ln * arccot)) . (upper_bound A)) - ((- (ln * arccot)) . (lower_bound A))
let Z be open Subset of REAL; ::_thesis: ( A c= Z & f = ((g1 + g2) ^) / f2 & f2 = arccot & Z c= ].(- 1),1.[ & g2 = #Z 2 & ( for x being Real st x in Z holds
( f . x = 1 / ((1 + (x ^2)) * (arccot . x)) & g1 . x = 1 & f2 . x > 0 ) ) & Z = dom f implies integral (f,A) = ((- (ln * arccot)) . (upper_bound A)) - ((- (ln * arccot)) . (lower_bound A)) )
assume A1: ( A c= Z & f = ((g1 + g2) ^) / f2 & f2 = arccot & Z c= ].(- 1),1.[ & g2 = #Z 2 & ( for x being Real st x in Z holds
( f . x = 1 / ((1 + (x ^2)) * (arccot . x)) & g1 . x = 1 & f2 . x > 0 ) ) & Z = dom f ) ; ::_thesis: integral (f,A) = ((- (ln * arccot)) . (upper_bound A)) - ((- (ln * arccot)) . (lower_bound A))
then  Z = (dom ((g1 + g2) ^)) /\ ((dom f2) \ (f2 " {0})) by RFUNCT_1:def_1;
then A2: ( Z c= dom ((g1 + g2) ^) & Z c= (dom f2) \ (f2 " {0}) ) by XBOOLE_1:18;
 for x being Real st x in Z holds
g1 . x = 1 by A1;
then A3: (g1 + g2) ^ is_differentiable_on Z by A1, A2, Th1;
A4: f2 is_differentiable_on Z by A1, SIN_COS9:82;
 for x being Real st x in Z holds
f2 . x <> 0 by A1;
then  f is_differentiable_on Z by A1, A3, A4, FDIFF_2:21;
then  f | Z is continuous by FDIFF_1:25;
then A5: f | A is continuous by A1, FCONT_1:16;
A6: Z c= dom (f2 ^) by A2, RFUNCT_1:def_2;
 dom (f2 ^) c= dom f2 by RFUNCT_1:1;
then A7: Z c= dom f2 by A6, XBOOLE_1:1;
A8: for x being Real st x in Z holds
f2 . x > 0 by A1;
 rng (f2 | Z) c= right_open_halfline 0
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (f2 | Z) or x in right_open_halfline 0 )
assume  x in rng (f2 | Z) ; ::_thesis: x in right_open_halfline 0
then consider y being set  such that
A9: ( y in dom (f2 | Z) & x = (f2 | Z) . y ) by FUNCT_1:def_3;
 y in Z by A9;
then  f2 . y > 0 by A1;
then  (f2 | Z) . y > 0 by A9, FUNCT_1:47;
hence  x in right_open_halfline 0 by A9, XXREAL_1:235; ::_thesis: verum
end;
then  f2 .: Z c= dom ln by RELAT_1:115, TAYLOR_1:18;
then A10: Z c= dom (ln * arccot) by A1, A7, FUNCT_1:101;
A11: ( f is_integrable_on A & f | A is bounded ) by A1, A5, INTEGRA5:10, INTEGRA5:11;
A12: ln * arccot is_differentiable_on Z by A1, A10, A8, SIN_COS9:90;
 Z c= dom (- (ln * arccot)) by A10, VALUED_1:8;
then A13: - (ln * arccot) is_differentiable_on Z by A12, Lm1, FDIFF_1:20;
A14: for x being Real st x in Z holds
((- (ln * arccot)) `| Z) . x = 1 / ((1 + (x ^2)) * (arccot . x))
proof
let x be Real; ::_thesis: ( x in Z implies ((- (ln * arccot)) `| Z) . x = 1 / ((1 + (x ^2)) * (arccot . x)) )
assume A15: x in Z ; ::_thesis: ((- (ln * arccot)) `| Z) . x = 1 / ((1 + (x ^2)) * (arccot . x))
then A16: ( - 1 < x & x < 1 ) by A1, XXREAL_1:4;
 arccot is_differentiable_on Z by A1, SIN_COS9:82;
then A17: arccot is_differentiable_in x by A15, FDIFF_1:9;
A18: arccot . x > 0 by A1, A15;
A19: ln * arccot is_differentiable_in x by A12, A15, FDIFF_1:9;
((- (ln * arccot)) `| Z) . x = diff ((- (ln * arccot)),x) by A13, A15, FDIFF_1:def_7
.= (- 1) * (diff ((ln * arccot),x)) by A19, Lm1, FDIFF_1:15
.= (- 1) * ((diff (arccot,x)) / (arccot . x)) by A17, A18, TAYLOR_1:20
.= (- 1) * ((- (1 / (1 + (x ^2)))) / (arccot . x)) by A16, SIN_COS9:76
.= (1 / (1 + (x ^2))) / (arccot . x)
.= 1 / ((1 + (x ^2)) * (arccot . x)) by XCMPLX_1:78 ;
hence  ((- (ln * arccot)) `| Z) . x = 1 / ((1 + (x ^2)) * (arccot . x)) ; ::_thesis: verum
end;
A20: for x being Real st x in dom ((- (ln * arccot)) `| Z) holds
((- (ln * arccot)) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom ((- (ln * arccot)) `| Z) implies ((- (ln * arccot)) `| Z) . x = f . x )
assume  x in dom ((- (ln * arccot)) `| Z) ; ::_thesis: ((- (ln * arccot)) `| Z) . x = f . x
then A21: x in Z by A13, FDIFF_1:def_7;
then ((- (ln * arccot)) `| Z) . x = 1 / ((1 + (x ^2)) * (arccot . x)) by A14
.= f . x by A1, A21 ;
hence  ((- (ln * arccot)) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom ((- (ln * arccot)) `| Z) = dom f by A1, A13, FDIFF_1:def_7;
then  (- (ln * arccot)) `| Z = f by A20, PARTFUN1:5;
hence  integral (f,A) = ((- (ln * arccot)) . (upper_bound A)) - ((- (ln * arccot)) . (lower_bound A)) by A1, A11, A13, INTEGRA5:13; ::_thesis: verum
end;

theorem :: INTEGR12:44
for A being non empty closed_interval Subset of REAL
for f1, f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
( arcsin . x > 0 & f1 . x = 1 ) ) & Z c= dom (ln * arcsin) & Z = dom f & f = (((#R (1 / 2)) * (f1 - (#Z 2))) (#) arcsin) ^ holds
integral (f,A) = ((ln * arcsin) . (upper_bound A)) - ((ln * arcsin) . (lower_bound A))
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for f1, f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
( arcsin . x > 0 & f1 . x = 1 ) ) & Z c= dom (ln * arcsin) & Z = dom f & f = (((#R (1 / 2)) * (f1 - (#Z 2))) (#) arcsin) ^ holds
integral (f,A) = ((ln * arcsin) . (upper_bound A)) - ((ln * arcsin) . (lower_bound A))
let f1, f be PartFunc of REAL,REAL; ::_thesis: for Z being open Subset of REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
( arcsin . x > 0 & f1 . x = 1 ) ) & Z c= dom (ln * arcsin) & Z = dom f & f = (((#R (1 / 2)) * (f1 - (#Z 2))) (#) arcsin) ^ holds
integral (f,A) = ((ln * arcsin) . (upper_bound A)) - ((ln * arcsin) . (lower_bound A))
let Z be open Subset of REAL; ::_thesis: ( A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
( arcsin . x > 0 & f1 . x = 1 ) ) & Z c= dom (ln * arcsin) & Z = dom f & f = (((#R (1 / 2)) * (f1 - (#Z 2))) (#) arcsin) ^ implies integral (f,A) = ((ln * arcsin) . (upper_bound A)) - ((ln * arcsin) . (lower_bound A)) )
assume A1: ( A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
( arcsin . x > 0 & f1 . x = 1 ) ) & Z c= dom (ln * arcsin) & Z = dom f & f = (((#R (1 / 2)) * (f1 - (#Z 2))) (#) arcsin) ^ ) ; ::_thesis: integral (f,A) = ((ln * arcsin) . (upper_bound A)) - ((ln * arcsin) . (lower_bound A))
set g = ((#R (1 / 2)) * (f1 - (#Z 2))) (#) arcsin;
A2: Z c= dom (((#R (1 / 2)) * (f1 - (#Z 2))) (#) arcsin) by A1, RFUNCT_1:1;
 dom (((#R (1 / 2)) * (f1 - (#Z 2))) (#) arcsin) = (dom ((#R (1 / 2)) * (f1 - (#Z 2)))) /\ (dom arcsin) by VALUED_1:def_4;
then A3: ( Z c= dom ((#R (1 / 2)) * (f1 - (#Z 2))) & Z c= dom arcsin ) by A2, XBOOLE_1:18;
A4: arcsin is_differentiable_on Z by A1, FDIFF_1:26, SIN_COS6:83;
set f2 = #Z 2;
 for x being Real st x in Z holds
(f1 - (#Z 2)) . x > 0
proof
let x be Real; ::_thesis: ( x in Z implies (f1 - (#Z 2)) . x > 0 )
assume A5: x in Z ; ::_thesis: (f1 - (#Z 2)) . x > 0
then  ( - 1 < x & x < 1 ) by A1, XXREAL_1:4;
then  ( 0 < 1 + x & 0 < 1 - x ) by XREAL_1:50, XREAL_1:148;
then A6: 0 < (1 + x) * (1 - x) by XREAL_1:129;
 for x being Real st x in Z holds
x in dom (f1 - (#Z 2)) by A3, FUNCT_1:11;
then (f1 - (#Z 2)) . x = (f1 . x) - ((#Z 2) . x) by A5, VALUED_1:13
.= (f1 . x) - (x #Z (1 + 1)) by TAYLOR_1:def_1
.= (f1 . x) - ((x #Z 1) * (x #Z 1)) by TAYLOR_1:1
.= (f1 . x) - (x * (x #Z 1)) by PREPOWER:35
.= (f1 . x) - (x * x) by PREPOWER:35
.= 1 - (x * x) by A1, A5 ;
hence  (f1 - (#Z 2)) . x > 0 by A6; ::_thesis: verum
end;
then  for x being Real st x in Z holds
( f1 . x = 1 & (f1 - (#Z 2)) . x > 0 ) by A1;
then  (#R (1 / 2)) * (f1 - (#Z 2)) is_differentiable_on Z by A3, FDIFF_7:22;
then A7: ((#R (1 / 2)) * (f1 - (#Z 2))) (#) arcsin is_differentiable_on Z by A2, A4, FDIFF_1:21;
 for x being Real st x in Z holds
(((#R (1 / 2)) * (f1 - (#Z 2))) (#) arcsin) . x <> 0 by A1, RFUNCT_1:3;
then  f is_differentiable_on Z by A1, A7, FDIFF_2:22;
then  f | Z is continuous by FDIFF_1:25;
then  f | A is continuous by A1, FCONT_1:16;
then A8: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A9: for x being Real st x in Z holds
arcsin . x > 0 by A1;
then A10: ln * arcsin is_differentiable_on Z by A1, FDIFF_7:8;
A11: for x being Real st x in Z holds
f . x = 1 / ((sqrt (1 - (x ^2))) * (arcsin . x))
proof
let x be Real; ::_thesis: ( x in Z implies f . x = 1 / ((sqrt (1 - (x ^2))) * (arcsin . x)) )
assume A12: x in Z ; ::_thesis: f . x = 1 / ((sqrt (1 - (x ^2))) * (arcsin . x))
then A13: ( x in dom (f1 - (#Z 2)) & (f1 - (#Z 2)) . x in dom (#R (1 / 2)) ) by A3, FUNCT_1:11;
then A14: (f1 - (#Z 2)) . x in right_open_halfline 0 by TAYLOR_1:def_4;
 ( - 1 < x & x < 1 ) by A1, A12, XXREAL_1:4;
then  ( 0 < 1 + x & 0 < 1 - x ) by XREAL_1:50, XREAL_1:148;
then A15: 0 < (1 + x) * (1 - x) by XREAL_1:129;
((((#R (1 / 2)) * (f1 - (#Z 2))) (#) arcsin) ^) . x = 1 / ((((#R (1 / 2)) * (f1 - (#Z 2))) (#) arcsin) . x) by A1, A12, RFUNCT_1:def_2
.= 1 / ((((#R (1 / 2)) * (f1 - (#Z 2))) . x) * (arcsin . x)) by VALUED_1:5
.= 1 / (((#R (1 / 2)) . ((f1 - (#Z 2)) . x)) * (arcsin . x)) by A3, A12, FUNCT_1:12
.= 1 / ((((f1 - (#Z 2)) . x) #R (1 / 2)) * (arcsin . x)) by A14, TAYLOR_1:def_4
.= 1 / ((((f1 . x) - ((#Z 2) . x)) #R (1 / 2)) * (arcsin . x)) by A13, VALUED_1:13
.= 1 / ((((f1 . x) - (x #Z 2)) #R (1 / 2)) * (arcsin . x)) by TAYLOR_1:def_1
.= 1 / ((((f1 . x) - (x ^2)) #R (1 / 2)) * (arcsin . x)) by FDIFF_7:1
.= 1 / (((1 - (x ^2)) #R (1 / 2)) * (arcsin . x)) by A1, A12
.= 1 / ((sqrt (1 - (x ^2))) * (arcsin . x)) by A15, FDIFF_7:2 ;
hence  f . x = 1 / ((sqrt (1 - (x ^2))) * (arcsin . x)) by A1; ::_thesis: verum
end;
A16: for x being Real st x in dom ((ln * arcsin) `| Z) holds
((ln * arcsin) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom ((ln * arcsin) `| Z) implies ((ln * arcsin) `| Z) . x = f . x )
assume  x in dom ((ln * arcsin) `| Z) ; ::_thesis: ((ln * arcsin) `| Z) . x = f . x
then A17: x in Z by A10, FDIFF_1:def_7;
then ((ln * arcsin) `| Z) . x = 1 / ((sqrt (1 - (x ^2))) * (arcsin . x)) by A1, A9, FDIFF_7:8
.= f . x by A11, A17 ;
hence  ((ln * arcsin) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom ((ln * arcsin) `| Z) = dom f by A1, A10, FDIFF_1:def_7;
then  (ln * arcsin) `| Z = f by A16, PARTFUN1:5;
hence  integral (f,A) = ((ln * arcsin) . (upper_bound A)) - ((ln * arcsin) . (lower_bound A)) by A1, A8, A10, INTEGRA5:13; ::_thesis: verum
end;

theorem :: INTEGR12:45
for A being non empty closed_interval Subset of REAL
for f1, f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
( f1 . x = 1 & arccos . x > 0 ) ) & Z c= dom (ln * arccos) & Z = dom f & f = (((#R (1 / 2)) * (f1 - (#Z 2))) (#) arccos) ^ holds
integral (f,A) = ((- (ln * arccos)) . (upper_bound A)) - ((- (ln * arccos)) . (lower_bound A))
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for f1, f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
( f1 . x = 1 & arccos . x > 0 ) ) & Z c= dom (ln * arccos) & Z = dom f & f = (((#R (1 / 2)) * (f1 - (#Z 2))) (#) arccos) ^ holds
integral (f,A) = ((- (ln * arccos)) . (upper_bound A)) - ((- (ln * arccos)) . (lower_bound A))
let f1, f be PartFunc of REAL,REAL; ::_thesis: for Z being open Subset of REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
( f1 . x = 1 & arccos . x > 0 ) ) & Z c= dom (ln * arccos) & Z = dom f & f = (((#R (1 / 2)) * (f1 - (#Z 2))) (#) arccos) ^ holds
integral (f,A) = ((- (ln * arccos)) . (upper_bound A)) - ((- (ln * arccos)) . (lower_bound A))
let Z be open Subset of REAL; ::_thesis: ( A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
( f1 . x = 1 & arccos . x > 0 ) ) & Z c= dom (ln * arccos) & Z = dom f & f = (((#R (1 / 2)) * (f1 - (#Z 2))) (#) arccos) ^ implies integral (f,A) = ((- (ln * arccos)) . (upper_bound A)) - ((- (ln * arccos)) . (lower_bound A)) )
assume A1: ( A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
( f1 . x = 1 & arccos . x > 0 ) ) & Z c= dom (ln * arccos) & Z = dom f & f = (((#R (1 / 2)) * (f1 - (#Z 2))) (#) arccos) ^ ) ; ::_thesis: integral (f,A) = ((- (ln * arccos)) . (upper_bound A)) - ((- (ln * arccos)) . (lower_bound A))
set g = ((#R (1 / 2)) * (f1 - (#Z 2))) (#) arccos;
A2: Z c= dom (((#R (1 / 2)) * (f1 - (#Z 2))) (#) arccos) by A1, RFUNCT_1:1;
 dom (((#R (1 / 2)) * (f1 - (#Z 2))) (#) arccos) = (dom ((#R (1 / 2)) * (f1 - (#Z 2)))) /\ (dom arccos) by VALUED_1:def_4;
then A3: ( Z c= dom ((#R (1 / 2)) * (f1 - (#Z 2))) & Z c= dom arccos ) by A2, XBOOLE_1:18;
A4: arccos is_differentiable_on Z by A1, FDIFF_1:26, SIN_COS6:106;
set f2 = #Z 2;
 for x being Real st x in Z holds
(f1 - (#Z 2)) . x > 0
proof
let x be Real; ::_thesis: ( x in Z implies (f1 - (#Z 2)) . x > 0 )
assume A5: x in Z ; ::_thesis: (f1 - (#Z 2)) . x > 0
then  ( - 1 < x & x < 1 ) by A1, XXREAL_1:4;
then  ( 0 < 1 + x & 0 < 1 - x ) by XREAL_1:50, XREAL_1:148;
then A6: 0 < (1 + x) * (1 - x) by XREAL_1:129;
 for x being Real st x in Z holds
x in dom (f1 - (#Z 2)) by A3, FUNCT_1:11;
then (f1 - (#Z 2)) . x = (f1 . x) - ((#Z 2) . x) by A5, VALUED_1:13
.= (f1 . x) - (x #Z (1 + 1)) by TAYLOR_1:def_1
.= (f1 . x) - ((x #Z 1) * (x #Z 1)) by TAYLOR_1:1
.= (f1 . x) - (x * (x #Z 1)) by PREPOWER:35
.= (f1 . x) - (x * x) by PREPOWER:35
.= 1 - (x * x) by A1, A5 ;
hence  (f1 - (#Z 2)) . x > 0 by A6; ::_thesis: verum
end;
then  for x being Real st x in Z holds
( f1 . x = 1 & (f1 - (#Z 2)) . x > 0 ) by A1;
then  (#R (1 / 2)) * (f1 - (#Z 2)) is_differentiable_on Z by A3, FDIFF_7:22;
then A7: ((#R (1 / 2)) * (f1 - (#Z 2))) (#) arccos is_differentiable_on Z by A2, A4, FDIFF_1:21;
 for x being Real st x in Z holds
(((#R (1 / 2)) * (f1 - (#Z 2))) (#) arccos) . x <> 0 by A1, RFUNCT_1:3;
then  f is_differentiable_on Z by A1, A7, FDIFF_2:22;
then  f | Z is continuous by FDIFF_1:25;
then  f | A is continuous by A1, FCONT_1:16;
then A8: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
 for x being Real st x in Z holds
arccos . x > 0 by A1;
then A9: ln * arccos is_differentiable_on Z by A1, FDIFF_7:9;
 Z c= dom (- (ln * arccos)) by A1, VALUED_1:8;
then A10: (- 1) (#) (ln * arccos) is_differentiable_on Z by A9, Lm1, FDIFF_1:20;
A11: for x being Real st x in Z holds
((- (ln * arccos)) `| Z) . x = 1 / ((sqrt (1 - (x ^2))) * (arccos . x))
proof
let x be Real; ::_thesis: ( x in Z implies ((- (ln * arccos)) `| Z) . x = 1 / ((sqrt (1 - (x ^2))) * (arccos . x)) )
assume A12: x in Z ; ::_thesis: ((- (ln * arccos)) `| Z) . x = 1 / ((sqrt (1 - (x ^2))) * (arccos . x))
then A13: ( - 1 < x & x < 1 ) by A1, XXREAL_1:4;
A14: arccos is_differentiable_in x by A1, A12, FDIFF_1:9, SIN_COS6:106;
A15: arccos . x > 0 by A1, A12;
A16: ln * arccos is_differentiable_in x by A9, A12, FDIFF_1:9;
((- (ln * arccos)) `| Z) . x = diff ((- (ln * arccos)),x) by A10, A12, FDIFF_1:def_7
.= (- 1) * (diff ((ln * arccos),x)) by A16, Lm1, FDIFF_1:15
.= (- 1) * ((diff (arccos,x)) / (arccos . x)) by A14, A15, TAYLOR_1:20
.= (- 1) * ((- (1 / (sqrt (1 - (x ^2))))) / (arccos . x)) by A13, SIN_COS6:106
.= (- 1) * (- ((1 / (sqrt (1 - (x ^2)))) / (arccos . x)))
.= 1 / ((sqrt (1 - (x ^2))) * (arccos . x)) by XCMPLX_1:78 ;
hence  ((- (ln * arccos)) `| Z) . x = 1 / ((sqrt (1 - (x ^2))) * (arccos . x)) ; ::_thesis: verum
end;
A17: for x being Real st x in Z holds
f . x = 1 / ((sqrt (1 - (x ^2))) * (arccos . x))
proof
let x be Real; ::_thesis: ( x in Z implies f . x = 1 / ((sqrt (1 - (x ^2))) * (arccos . x)) )
assume A18: x in Z ; ::_thesis: f . x = 1 / ((sqrt (1 - (x ^2))) * (arccos . x))
then A19: ( x in dom (f1 - (#Z 2)) & (f1 - (#Z 2)) . x in dom (#R (1 / 2)) ) by A3, FUNCT_1:11;
then A20: (f1 - (#Z 2)) . x in right_open_halfline 0 by TAYLOR_1:def_4;
 ( - 1 < x & x < 1 ) by A1, A18, XXREAL_1:4;
then  ( 0 < 1 + x & 0 < 1 - x ) by XREAL_1:50, XREAL_1:148;
then A21: 0 < (1 + x) * (1 - x) by XREAL_1:129;
((((#R (1 / 2)) * (f1 - (#Z 2))) (#) arccos) ^) . x = 1 / ((((#R (1 / 2)) * (f1 - (#Z 2))) (#) arccos) . x) by A1, A18, RFUNCT_1:def_2
.= 1 / ((((#R (1 / 2)) * (f1 - (#Z 2))) . x) * (arccos . x)) by VALUED_1:5
.= 1 / (((#R (1 / 2)) . ((f1 - (#Z 2)) . x)) * (arccos . x)) by A3, A18, FUNCT_1:12
.= 1 / ((((f1 - (#Z 2)) . x) #R (1 / 2)) * (arccos . x)) by A20, TAYLOR_1:def_4
.= 1 / ((((f1 . x) - ((#Z 2) . x)) #R (1 / 2)) * (arccos . x)) by A19, VALUED_1:13
.= 1 / ((((f1 . x) - (x #Z 2)) #R (1 / 2)) * (arccos . x)) by TAYLOR_1:def_1
.= 1 / ((((f1 . x) - (x ^2)) #R (1 / 2)) * (arccos . x)) by FDIFF_7:1
.= 1 / (((1 - (x ^2)) #R (1 / 2)) * (arccos . x)) by A1, A18
.= 1 / ((sqrt (1 - (x ^2))) * (arccos . x)) by A21, FDIFF_7:2 ;
hence  f . x = 1 / ((sqrt (1 - (x ^2))) * (arccos . x)) by A1; ::_thesis: verum
end;
A22: for x being Real st x in dom ((- (ln * arccos)) `| Z) holds
((- (ln * arccos)) `| Z) . x = f . x
proof
let x be Real; ::_thesis: ( x in dom ((- (ln * arccos)) `| Z) implies ((- (ln * arccos)) `| Z) . x = f . x )
assume  x in dom ((- (ln * arccos)) `| Z) ; ::_thesis: ((- (ln * arccos)) `| Z) . x = f . x
then A23: x in Z by A10, FDIFF_1:def_7;
then ((- (ln * arccos)) `| Z) . x = 1 / ((sqrt (1 - (x ^2))) * (arccos . x)) by A11
.= f . x by A17, A23 ;
hence  ((- (ln * arccos)) `| Z) . x = f . x ; ::_thesis: verum
end;
 dom ((- (ln * arccos)) `| Z) = dom f by A1, A10, FDIFF_1:def_7;
then  (- (ln * arccos)) `| Z = f by A22, PARTFUN1:5;
hence  integral (f,A) = ((- (ln * arccos)) . (upper_bound A)) - ((- (ln * arccos)) . (lower_bound A)) by A1, A8, A10, INTEGRA5:13; ::_thesis: verum
end;