:: 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;