:: SIN_COS9 semantic presentation begin theorem Th1: :: SIN_COS9:1 ].(- (PI / 2)),(PI / 2).[ c= dom tan proof ].(- (PI / 2)),(PI / 2).[ /\ (cos " {0}) = {} proof assume ].(- (PI / 2)),(PI / 2).[ /\ (cos " {0}) <> {} ; ::_thesis: contradiction then consider rr being set such that A1: rr in ].(- (PI / 2)),(PI / 2).[ /\ (cos " {0}) by XBOOLE_0:7; rr in cos " {0} by A1, XBOOLE_0:def_4; then A2: cos . rr in {0} by FUNCT_1:def_7; rr in ].(- (PI / 2)),(PI / 2).[ by A1, XBOOLE_0:def_4; then cos . rr <> 0 by COMPTRIG:11; hence contradiction by A2, TARSKI:def_1; ::_thesis: verum end; then A3: ].(- (PI / 2)),(PI / 2).[ misses cos " {0} by XBOOLE_0:def_7; ].(- (PI / 2)),(PI / 2).[ \ (cos " {0}) c= (dom cos) \ (cos " {0}) by SIN_COS:24, XBOOLE_1:33; then ].(- (PI / 2)),(PI / 2).[ c= (dom cos) \ (cos " {0}) by A3, XBOOLE_1:83; then ].(- (PI / 2)),(PI / 2).[ c= (dom sin) /\ ((dom cos) \ (cos " {0})) by SIN_COS:24, XBOOLE_1:19; hence ].(- (PI / 2)),(PI / 2).[ c= dom tan by RFUNCT_1:def_1; ::_thesis: verum end; theorem Th2: :: SIN_COS9:2 ].0,PI.[ c= dom cot proof ].0,PI.[ /\ (sin " {0}) = {} proof assume ].0,PI.[ /\ (sin " {0}) <> {} ; ::_thesis: contradiction then consider rr being set such that A1: rr in ].0,PI.[ /\ (sin " {0}) by XBOOLE_0:7; rr in sin " {0} by A1, XBOOLE_0:def_4; then A2: sin . rr in {0} by FUNCT_1:def_7; rr in ].0,PI.[ by A1, XBOOLE_0:def_4; then sin . rr <> 0 by COMPTRIG:7; hence contradiction by A2, TARSKI:def_1; ::_thesis: verum end; then A3: ].0,PI.[ misses sin " {0} by XBOOLE_0:def_7; ].0,PI.[ \ (sin " {0}) c= (dom sin) \ (sin " {0}) by SIN_COS:24, XBOOLE_1:33; then ].0,PI.[ c= (dom sin) \ (sin " {0}) by A3, XBOOLE_1:83; then ].0,PI.[ c= (dom cos) /\ ((dom sin) \ (sin " {0})) by SIN_COS:24, XBOOLE_1:19; hence ].0,PI.[ c= dom cot by RFUNCT_1:def_1; ::_thesis: verum end; Lm1: tan is_differentiable_on ].(- (PI / 2)),(PI / 2).[ proof for x being Real st x in ].(- (PI / 2)),(PI / 2).[ holds tan is_differentiable_in x proof let x be Real; ::_thesis: ( x in ].(- (PI / 2)),(PI / 2).[ implies tan is_differentiable_in x ) assume x in ].(- (PI / 2)),(PI / 2).[ ; ::_thesis: tan is_differentiable_in x then cos . x <> 0 by COMPTRIG:11; hence tan is_differentiable_in x by FDIFF_7:46; ::_thesis: verum end; hence tan is_differentiable_on ].(- (PI / 2)),(PI / 2).[ by Th1, FDIFF_1:9; ::_thesis: verum end; Lm2: cot is_differentiable_on ].0,PI.[ proof for x being Real st x in ].0,PI.[ holds cot is_differentiable_in x proof let x be Real; ::_thesis: ( x in ].0,PI.[ implies cot is_differentiable_in x ) assume x in ].0,PI.[ ; ::_thesis: cot is_differentiable_in x then sin . x <> 0 by COMPTRIG:7; hence cot is_differentiable_in x by FDIFF_7:47; ::_thesis: verum end; hence cot is_differentiable_on ].0,PI.[ by Th2, FDIFF_1:9; ::_thesis: verum end; Lm3: for x being Real st x in ].(- (PI / 2)),(PI / 2).[ holds diff (tan,x) = 1 / ((cos . x) ^2) proof let x be Real; ::_thesis: ( x in ].(- (PI / 2)),(PI / 2).[ implies diff (tan,x) = 1 / ((cos . x) ^2) ) assume x in ].(- (PI / 2)),(PI / 2).[ ; ::_thesis: diff (tan,x) = 1 / ((cos . x) ^2) then cos . x <> 0 by COMPTRIG:11; hence diff (tan,x) = 1 / ((cos . x) ^2) by FDIFF_7:46; ::_thesis: verum end; Lm4: for x being Real st x in ].0,PI.[ holds diff (cot,x) = - (1 / ((sin . x) ^2)) proof let x be Real; ::_thesis: ( x in ].0,PI.[ implies diff (cot,x) = - (1 / ((sin . x) ^2)) ) assume x in ].0,PI.[ ; ::_thesis: diff (cot,x) = - (1 / ((sin . x) ^2)) then sin . x <> 0 by COMPTRIG:7; hence diff (cot,x) = - (1 / ((sin . x) ^2)) by FDIFF_7:47; ::_thesis: verum end; theorem :: SIN_COS9:3 ( tan is_differentiable_on ].(- (PI / 2)),(PI / 2).[ & ( for x being Real st x in ].(- (PI / 2)),(PI / 2).[ holds diff (tan,x) = 1 / ((cos . x) ^2) ) ) by Lm1, Lm3; theorem :: SIN_COS9:4 ( cot is_differentiable_on ].0,PI.[ & ( for x being Real st x in ].0,PI.[ holds diff (cot,x) = - (1 / ((sin . x) ^2)) ) ) by Lm2, Lm4; theorem :: SIN_COS9:5 tan | ].(- (PI / 2)),(PI / 2).[ is continuous by Lm1, FDIFF_1:25; theorem :: SIN_COS9:6 cot | ].0,PI.[ is continuous by Lm2, FDIFF_1:25; theorem Th7: :: SIN_COS9:7 tan | ].(- (PI / 2)),(PI / 2).[ is increasing proof A1: for x being Real st x in ].(- (PI / 2)),(PI / 2).[ holds diff (tan,x) > 0 proof let x be Real; ::_thesis: ( x in ].(- (PI / 2)),(PI / 2).[ implies diff (tan,x) > 0 ) assume A2: x in ].(- (PI / 2)),(PI / 2).[ ; ::_thesis: diff (tan,x) > 0 then 0 < cos . x by COMPTRIG:11; then (cos . x) ^2 > 0 by SQUARE_1:12; then 1 / ((cos . x) ^2) > 0 / ((cos . x) ^2) by XREAL_1:74; hence diff (tan,x) > 0 by A2, Lm3; ::_thesis: verum end; ].(- (PI / 2)),(PI / 2).[ c= dom tan by Lm1, FDIFF_1:def_6; hence tan | ].(- (PI / 2)),(PI / 2).[ is increasing by A1, Lm1, ROLLE:9; ::_thesis: verum end; theorem Th8: :: SIN_COS9:8 cot | ].0,PI.[ is decreasing proof A1: for x being Real st x in ].0,PI.[ holds diff (cot,x) < 0 proof let x be Real; ::_thesis: ( x in ].0,PI.[ implies diff (cot,x) < 0 ) assume A2: x in ].0,PI.[ ; ::_thesis: diff (cot,x) < 0 then 0 < sin . x by COMPTRIG:7; then (sin . x) ^2 > 0 by SQUARE_1:12; then 1 / ((sin . x) ^2) > 0 / ((sin . x) ^2) by XREAL_1:74; then - (1 / ((sin . x) ^2)) < - 0 by XREAL_1:24; hence diff (cot,x) < 0 by A2, Lm4; ::_thesis: verum end; ].0,PI.[ c= dom cot by Lm2, FDIFF_1:def_6; hence cot | ].0,PI.[ is decreasing by A1, Lm2, ROLLE:10; ::_thesis: verum end; theorem :: SIN_COS9:9 tan | ].(- (PI / 2)),(PI / 2).[ is one-to-one by Th7, FCONT_3:8; theorem :: SIN_COS9:10 cot | ].0,PI.[ is one-to-one by Th8, FCONT_3:8; registration clusterK77(tan,].(- (PI / 2)),(PI / 2).[) -> one-to-one ; coherence tan | ].(- (PI / 2)),(PI / 2).[ is one-to-one by Th7, FCONT_3:8; clusterK77(cot,].0,PI.[) -> one-to-one ; coherence cot | ].0,PI.[ is one-to-one by Th8, FCONT_3:8; end; definition func arctan -> PartFunc of REAL,REAL equals :: SIN_COS9:def 1 (tan | ].(- (PI / 2)),(PI / 2).[) " ; coherence (tan | ].(- (PI / 2)),(PI / 2).[) " is PartFunc of REAL,REAL ; func arccot -> PartFunc of REAL,REAL equals :: SIN_COS9:def 2 (cot | ].0,PI.[) " ; coherence (cot | ].0,PI.[) " is PartFunc of REAL,REAL ; end; :: deftheorem defines arctan SIN_COS9:def_1_:_ arctan = (tan | ].(- (PI / 2)),(PI / 2).[) " ; :: deftheorem defines arccot SIN_COS9:def_2_:_ arccot = (cot | ].0,PI.[) " ; definition let r be Real; func arctan r -> set equals :: SIN_COS9:def 3 arctan . r; coherence arctan . r is set ; func arccot r -> set equals :: SIN_COS9:def 4 arccot . r; coherence arccot . r is set ; end; :: deftheorem defines arctan SIN_COS9:def_3_:_ for r being Real holds arctan r = arctan . r; :: deftheorem defines arccot SIN_COS9:def_4_:_ for r being Real holds arccot r = arccot . r; definition let r be Real; :: original: arctan redefine func arctan r -> Real; coherence arctan r is Real ; :: original: arccot redefine func arccot r -> Real; coherence arccot r is Real ; end; Lm5: arctan " = tan | ].(- (PI / 2)),(PI / 2).[ by FUNCT_1:43; Lm6: arccot " = cot | ].0,PI.[ by FUNCT_1:43; theorem Th11: :: SIN_COS9:11 rng arctan = ].(- (PI / 2)),(PI / 2).[ proof dom (tan | ].(- (PI / 2)),(PI / 2).[) = ].(- (PI / 2)),(PI / 2).[ by Th1, RELAT_1:62; hence rng arctan = ].(- (PI / 2)),(PI / 2).[ by FUNCT_1:33; ::_thesis: verum end; theorem Th12: :: SIN_COS9:12 rng arccot = ].0,PI.[ proof dom (cot | ].0,PI.[) = ].0,PI.[ by Th2, RELAT_1:62; hence rng arccot = ].0,PI.[ by FUNCT_1:33; ::_thesis: verum end; registration cluster arctan -> one-to-one ; coherence arctan is one-to-one ; cluster arccot -> one-to-one ; coherence arccot is one-to-one ; end; Lm7: - (PI / 4) in ].(- (PI / 2)),(PI / 2).[ proof PI in ].0,4.[ by SIN_COS:def_28; then PI > 0 by XXREAL_1:4; then PI / (- 4) > PI / (- 2) by XREAL_1:99; then - (PI / 4) in { s where s is Real : ( - (PI / 2) < s & s < PI / 2 ) } ; hence - (PI / 4) in ].(- (PI / 2)),(PI / 2).[ by RCOMP_1:def_2; ::_thesis: verum end; Lm8: PI / 4 in ].(- (PI / 2)),(PI / 2).[ proof PI in ].0,4.[ by SIN_COS:def_28; then PI > 0 by XXREAL_1:4; then PI / 4 < PI / 2 by XREAL_1:76; then PI / 4 in { s where s is Real : ( - (PI / 2) < s & s < PI / 2 ) } ; hence PI / 4 in ].(- (PI / 2)),(PI / 2).[ by RCOMP_1:def_2; ::_thesis: verum end; Lm9: PI / 4 in ].0,PI.[ proof PI in ].0,4.[ by SIN_COS:def_28; then A1: PI > 0 by XXREAL_1:4; then A2: PI / 4 < PI / 1 by XREAL_1:76; PI / 4 > 0 / 4 by A1, XREAL_1:74; hence PI / 4 in ].0,PI.[ by A2, XXREAL_1:4; ::_thesis: verum end; Lm10: (3 / 4) * PI in ].0,PI.[ proof PI in ].0,4.[ by SIN_COS:def_28; then A1: PI > 0 by XXREAL_1:4; then A2: (3 / 4) * PI < PI by XREAL_1:157; (3 / 4) * PI > (3 / 4) * 0 by A1, XREAL_1:68; hence (3 / 4) * PI in ].0,PI.[ by A2, XXREAL_1:4; ::_thesis: verum end; Lm11: dom (tan | [.(- (PI / 4)),(PI / 4).]) = [.(- (PI / 4)),(PI / 4).] proof [.(- (PI / 4)),(PI / 4).] c= ].(- (PI / 2)),(PI / 2).[ by Lm7, Lm8, XXREAL_2:def_12; hence dom (tan | [.(- (PI / 4)),(PI / 4).]) = [.(- (PI / 4)),(PI / 4).] by Th1, RELAT_1:62, XBOOLE_1:1; ::_thesis: verum end; Lm12: dom (cot | [.(PI / 4),((3 / 4) * PI).]) = [.(PI / 4),((3 / 4) * PI).] proof [.(PI / 4),((3 / 4) * PI).] c= ].0,PI.[ by Lm9, Lm10, XXREAL_2:def_12; hence dom (cot | [.(PI / 4),((3 / 4) * PI).]) = [.(PI / 4),((3 / 4) * PI).] by Th2, RELAT_1:62, XBOOLE_1:1; ::_thesis: verum end; theorem Th13: :: SIN_COS9:13 for x being real number st x in ].(- (PI / 2)),(PI / 2).[ holds tan . x = tan x proof let x be real number ; ::_thesis: ( x in ].(- (PI / 2)),(PI / 2).[ implies tan . x = tan x ) assume x in ].(- (PI / 2)),(PI / 2).[ ; ::_thesis: tan . x = tan x then tan . x = (sin x) / (cos x) by Th1, RFUNCT_1:def_1 .= tan x by SIN_COS4:def_1 ; hence tan . x = tan x ; ::_thesis: verum end; theorem Th14: :: SIN_COS9:14 for x being real number st x in ].0,PI.[ holds cot . x = cot x proof let x be real number ; ::_thesis: ( x in ].0,PI.[ implies cot . x = cot x ) assume x in ].0,PI.[ ; ::_thesis: cot . x = cot x then cot . x = (cos x) / (sin x) by Th2, RFUNCT_1:def_1 .= cot x by SIN_COS4:def_2 ; hence cot . x = cot x ; ::_thesis: verum end; theorem :: SIN_COS9:15 for x being Real st cos . x <> 0 holds tan . x = tan x proof let x be Real; ::_thesis: ( cos . x <> 0 implies tan . x = tan x ) assume A1: cos . x <> 0 ; ::_thesis: tan . x = tan x not x in cos " {0} proof assume x in cos " {0} ; ::_thesis: contradiction then cos . x in {0} by FUNCT_1:def_7; hence contradiction by A1, TARSKI:def_1; ::_thesis: verum end; then x in (dom cos) \ (cos " {0}) by SIN_COS:24, XBOOLE_0:def_5; then x in (dom sin) /\ ((dom cos) \ (cos " {0})) by SIN_COS:24, XBOOLE_0:def_4; then x in dom (sin / cos) by RFUNCT_1:def_1; then tan . x = (sin x) / (cos x) by RFUNCT_1:def_1 .= tan x by SIN_COS4:def_1 ; hence tan . x = tan x ; ::_thesis: verum end; theorem :: SIN_COS9:16 for x being Real st sin . x <> 0 holds cot . x = cot x proof let x be Real; ::_thesis: ( sin . x <> 0 implies cot . x = cot x ) assume A1: sin . x <> 0 ; ::_thesis: cot . x = cot x not x in sin " {0} proof assume x in sin " {0} ; ::_thesis: contradiction then sin . x in {0} by FUNCT_1:def_7; hence contradiction by A1, TARSKI:def_1; ::_thesis: verum end; then x in (dom sin) \ (sin " {0}) by SIN_COS:24, XBOOLE_0:def_5; then x in (dom cos) /\ ((dom sin) \ (sin " {0})) by SIN_COS:24, XBOOLE_0:def_4; then x in dom (cos / sin) by RFUNCT_1:def_1; then cot . x = (cos x) / (sin x) by RFUNCT_1:def_1 .= cot x by SIN_COS4:def_2 ; hence cot . x = cot x ; ::_thesis: verum end; theorem Th17: :: SIN_COS9:17 ( tan . (- (PI / 4)) = - 1 & tan (- (PI / 4)) = - 1 ) proof cos . (PI / 4) <> 0 by Lm8, COMPTRIG:11; then A1: (sin . (PI / 4)) / (cos . (PI / 4)) = 1 by SIN_COS:73, XCMPLX_1:60; tan . (- (PI / 4)) = (sin . (- (PI / 4))) / (cos . (- (PI / 4))) by Lm7, Th1, RFUNCT_1:def_1 .= (- (sin . (PI / 4))) / (cos . (- (PI / 4))) by SIN_COS:30 .= (- (sin . (PI / 4))) / (cos . (PI / 4)) by SIN_COS:30 .= - 1 by A1 ; hence ( tan . (- (PI / 4)) = - 1 & tan (- (PI / 4)) = - 1 ) by Lm7, Th13; ::_thesis: verum end; theorem Th18: :: SIN_COS9:18 ( cot . (PI / 4) = 1 & cot (PI / 4) = 1 & cot . ((3 / 4) * PI) = - 1 & cot ((3 / 4) * PI) = - 1 ) proof A1: sin . (PI / 4) <> 0 by Lm9, COMPTRIG:7; A2: cot . ((3 / 4) * PI) = (cos . ((3 / 4) * PI)) * ((sin . ((3 / 4) * PI)) ") by Lm10, Th2, RFUNCT_1:def_1 .= (- (sin . (PI / 4))) / (sin . ((PI / 2) + (PI / 4))) by SIN_COS:78 .= (- (sin . (PI / 4))) / (cos . (PI / 4)) by SIN_COS:78 .= - ((sin . (PI / 4)) / (cos . (PI / 4))) .= - 1 by A1, SIN_COS:73, XCMPLX_1:60 ; cot . (PI / 4) = (cos . (PI / 4)) / (sin . (PI / 4)) by Lm9, Th2, RFUNCT_1:def_1 .= 1 by A1, SIN_COS:73, XCMPLX_1:60 ; hence ( cot . (PI / 4) = 1 & cot (PI / 4) = 1 & cot . ((3 / 4) * PI) = - 1 & cot ((3 / 4) * PI) = - 1 ) by A2, Lm9, Lm10, Th14; ::_thesis: verum end; theorem Th19: :: SIN_COS9:19 for x being set st x in [.(- (PI / 4)),(PI / 4).] holds tan . x in [.(- 1),1.] proof let x be set ; ::_thesis: ( x in [.(- (PI / 4)),(PI / 4).] implies tan . x in [.(- 1),1.] ) assume x in [.(- (PI / 4)),(PI / 4).] ; ::_thesis: tan . x in [.(- 1),1.] then x in ].(- (PI / 4)),(PI / 4).[ \/ {(- (PI / 4)),(PI / 4)} by XXREAL_1:128; then A1: ( x in ].(- (PI / 4)),(PI / 4).[ or x in {(- (PI / 4)),(PI / 4)} ) by XBOOLE_0:def_3; percases ( x in ].(- (PI / 4)),(PI / 4).[ or x = - (PI / 4) or x = PI / 4 ) by A1, TARSKI:def_2; supposeA2: x in ].(- (PI / 4)),(PI / 4).[ ; ::_thesis: tan . x in [.(- 1),1.] then x in { s where s is Real : ( - (PI / 4) < s & s < PI / 4 ) } by RCOMP_1:def_2; then A3: ex s being Real st ( s = x & - (PI / 4) < s & s < PI / 4 ) ; A4: ].(- (PI / 4)),(PI / 4).[ c= [.(- (PI / 4)),(PI / 4).] by XXREAL_1:25; - (PI / 4) in { s where s is Real : ( - (PI / 4) <= s & s <= PI / 4 ) } ; then A5: - (PI / 4) in [.(- (PI / 4)),(PI / 4).] by RCOMP_1:def_1; A6: [.(- (PI / 4)),(PI / 4).] c= ].(- (PI / 2)),(PI / 2).[ by Lm7, Lm8, XXREAL_2:def_12; then A7: tan | [.(- (PI / 4)),(PI / 4).] is increasing by Th7, RFUNCT_2:28; A8: [.(- (PI / 4)),(PI / 4).] /\ (dom tan) = [.(- (PI / 4)),(PI / 4).] by A6, Th1, XBOOLE_1:1, XBOOLE_1:28; PI / 4 in { s where s is Real : ( - (PI / 4) <= s & s <= PI / 4 ) } ; then PI / 4 in [.(- (PI / 4)),(PI / 4).] /\ (dom tan) by A8, RCOMP_1:def_1; then tan . x < tan . (PI / 4) by A2, A7, A8, A4, A3, RFUNCT_2:20; then A9: tan . x < 1 by SIN_COS:def_28; x in { s where s is Real : ( - (PI / 4) < s & s < PI / 4 ) } by A2, RCOMP_1:def_2; then ex s being Real st ( s = x & - (PI / 4) < s & s < PI / 4 ) ; then - 1 < tan . x by A2, A7, A5, A8, A4, Th17, RFUNCT_2:20; then tan . x in { s where s is Real : ( - 1 < s & s < 1 ) } by A9; then A10: tan . x in ].(- 1),1.[ by RCOMP_1:def_2; ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25; hence tan . x in [.(- 1),1.] by A10; ::_thesis: verum end; suppose x = - (PI / 4) ; ::_thesis: tan . x in [.(- 1),1.] then tan . x in { s where s is Real : ( - 1 <= s & s <= 1 ) } by Th17; hence tan . x in [.(- 1),1.] by RCOMP_1:def_1; ::_thesis: verum end; suppose x = PI / 4 ; ::_thesis: tan . x in [.(- 1),1.] then tan . x = 1 by SIN_COS:def_28; then tan . x in { s where s is Real : ( - 1 <= s & s <= 1 ) } ; hence tan . x in [.(- 1),1.] by RCOMP_1:def_1; ::_thesis: verum end; end; end; theorem Th20: :: SIN_COS9:20 for x being set st x in [.(PI / 4),((3 / 4) * PI).] holds cot . x in [.(- 1),1.] proof let x be set ; ::_thesis: ( x in [.(PI / 4),((3 / 4) * PI).] implies cot . x in [.(- 1),1.] ) PI in ].0,4.[ by SIN_COS:def_28; then PI > 0 by XXREAL_1:4; then PI / 4 > 0 / 4 by XREAL_1:74; then A1: (PI / 4) * 3 > PI / 4 by XREAL_1:155; assume x in [.(PI / 4),((3 / 4) * PI).] ; ::_thesis: cot . x in [.(- 1),1.] then x in ].(PI / 4),((3 / 4) * PI).[ \/ {(PI / 4),((3 / 4) * PI)} by A1, XXREAL_1:128; then A2: ( x in ].(PI / 4),((3 / 4) * PI).[ or x in {(PI / 4),((3 / 4) * PI)} ) by XBOOLE_0:def_3; percases ( x in ].(PI / 4),((3 / 4) * PI).[ or x = PI / 4 or x = (3 / 4) * PI ) by A2, TARSKI:def_2; supposeA3: x in ].(PI / 4),((3 / 4) * PI).[ ; ::_thesis: cot . x in [.(- 1),1.] then x in { s where s is Real : ( PI / 4 < s & s < (3 / 4) * PI ) } by RCOMP_1:def_2; then A4: ex s being Real st ( s = x & PI / 4 < s & s < (3 / 4) * PI ) ; A5: [.(PI / 4),((3 / 4) * PI).] c= ].0,PI.[ by Lm9, Lm10, XXREAL_2:def_12; then A6: cot | [.(PI / 4),((3 / 4) * PI).] is decreasing by Th8, RFUNCT_2:29; x in { s where s is Real : ( PI / 4 < s & s < (3 / 4) * PI ) } by A3, RCOMP_1:def_2; then A7: ex s being Real st ( s = x & PI / 4 < s & s < (3 / 4) * PI ) ; A8: ].(PI / 4),((3 / 4) * PI).[ c= [.(PI / 4),((3 / 4) * PI).] by XXREAL_1:25; A9: [.(PI / 4),((3 / 4) * PI).] /\ (dom cot) = [.(PI / 4),((3 / 4) * PI).] by A5, Th2, XBOOLE_1:1, XBOOLE_1:28; (3 / 4) * PI in { s where s is Real : ( PI / 4 <= s & s <= (3 / 4) * PI ) } by A1; then (3 / 4) * PI in [.(PI / 4),((3 / 4) * PI).] /\ (dom cot) by A9, RCOMP_1:def_1; then A10: - 1 < cot . x by A3, A6, A9, A8, A7, Th18, RFUNCT_2:21; PI / 4 in { s where s is Real : ( PI / 4 <= s & s <= (3 / 4) * PI ) } by A1; then PI / 4 in [.(PI / 4),((3 / 4) * PI).] /\ (dom cot) by A9, RCOMP_1:def_1; then cot . x < 1 by A3, A6, A9, A8, A4, Th18, RFUNCT_2:21; then cot . x in { s where s is Real : ( - 1 < s & s < 1 ) } by A10; then A11: cot . x in ].(- 1),1.[ by RCOMP_1:def_2; ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25; hence cot . x in [.(- 1),1.] by A11; ::_thesis: verum end; suppose x = PI / 4 ; ::_thesis: cot . x in [.(- 1),1.] then cot . x in { s where s is Real : ( - 1 <= s & s <= 1 ) } by Th18; hence cot . x in [.(- 1),1.] by RCOMP_1:def_1; ::_thesis: verum end; suppose x = (3 / 4) * PI ; ::_thesis: cot . x in [.(- 1),1.] then cot . x in { s where s is Real : ( - 1 <= s & s <= 1 ) } by Th18; hence cot . x in [.(- 1),1.] by RCOMP_1:def_1; ::_thesis: verum end; end; end; theorem Th21: :: SIN_COS9:21 rng (tan | [.(- (PI / 4)),(PI / 4).]) = [.(- 1),1.] proof now__::_thesis:_for_y_being_set_holds_ (_(_y_in_[.(-_1),1.]_implies_ex_x_being_set_st_ (_x_in_dom_(tan_|_[.(-_(PI_/_4)),(PI_/_4).])_&_y_=_(tan_|_[.(-_(PI_/_4)),(PI_/_4).])_._x_)_)_&_(_ex_x_being_set_st_ (_x_in_dom_(tan_|_[.(-_(PI_/_4)),(PI_/_4).])_&_y_=_(tan_|_[.(-_(PI_/_4)),(PI_/_4).])_._x_)_implies_y_in_[.(-_1),1.]_)_) let y be set ; ::_thesis: ( ( y in [.(- 1),1.] implies ex x being set st ( x in dom (tan | [.(- (PI / 4)),(PI / 4).]) & y = (tan | [.(- (PI / 4)),(PI / 4).]) . x ) ) & ( ex x being set st ( x in dom (tan | [.(- (PI / 4)),(PI / 4).]) & y = (tan | [.(- (PI / 4)),(PI / 4).]) . x ) implies y in [.(- 1),1.] ) ) thus ( y in [.(- 1),1.] implies ex x being set st ( x in dom (tan | [.(- (PI / 4)),(PI / 4).]) & y = (tan | [.(- (PI / 4)),(PI / 4).]) . x ) ) ::_thesis: ( ex x being set st ( x in dom (tan | [.(- (PI / 4)),(PI / 4).]) & y = (tan | [.(- (PI / 4)),(PI / 4).]) . x ) implies y in [.(- 1),1.] ) proof assume A1: y in [.(- 1),1.] ; ::_thesis: ex x being set st ( x in dom (tan | [.(- (PI / 4)),(PI / 4).]) & y = (tan | [.(- (PI / 4)),(PI / 4).]) . x ) then reconsider y1 = y as Real ; y1 in [.(tan . (- (PI / 4))),(tan . (PI / 4)).] by A1, Th17, SIN_COS:def_28; then A2: y1 in [.(tan . (- (PI / 4))),(tan . (PI / 4)).] \/ [.(tan . (PI / 4)),(tan . (- (PI / 4))).] by XBOOLE_0:def_3; A3: [.(- (PI / 4)),(PI / 4).] c= ].(- (PI / 2)),(PI / 2).[ by Lm7, Lm8, XXREAL_2:def_12; tan | ].(- (PI / 2)),(PI / 2).[ is continuous by Lm1, FDIFF_1:25; then tan | [.(- (PI / 4)),(PI / 4).] is continuous by A3, FCONT_1:16; then consider x being Real such that A4: x in [.(- (PI / 4)),(PI / 4).] and A5: y1 = tan . x by A3, A2, Th1, FCONT_2:15, XBOOLE_1:1; take x ; ::_thesis: ( x in dom (tan | [.(- (PI / 4)),(PI / 4).]) & y = (tan | [.(- (PI / 4)),(PI / 4).]) . x ) thus ( x in dom (tan | [.(- (PI / 4)),(PI / 4).]) & y = (tan | [.(- (PI / 4)),(PI / 4).]) . x ) by A4, A5, Lm11, FUNCT_1:49; ::_thesis: verum end; thus ( ex x being set st ( x in dom (tan | [.(- (PI / 4)),(PI / 4).]) & y = (tan | [.(- (PI / 4)),(PI / 4).]) . x ) implies y in [.(- 1),1.] ) ::_thesis: verum proof given x being set such that A6: x in dom (tan | [.(- (PI / 4)),(PI / 4).]) and A7: y = (tan | [.(- (PI / 4)),(PI / 4).]) . x ; ::_thesis: y in [.(- 1),1.] reconsider x1 = x as Real by A6; y = tan . x1 by A6, A7, Lm11, FUNCT_1:49; hence y in [.(- 1),1.] by A6, Lm11, Th19; ::_thesis: verum end; end; hence rng (tan | [.(- (PI / 4)),(PI / 4).]) = [.(- 1),1.] by FUNCT_1:def_3; ::_thesis: verum end; theorem Th22: :: SIN_COS9:22 rng (cot | [.(PI / 4),((3 / 4) * PI).]) = [.(- 1),1.] proof now__::_thesis:_for_y_being_set_holds_ (_(_y_in_[.(-_1),1.]_implies_ex_x_being_set_st_ (_x_in_dom_(cot_|_[.(PI_/_4),((3_/_4)_*_PI).])_&_y_=_(cot_|_[.(PI_/_4),((3_/_4)_*_PI).])_._x_)_)_&_(_ex_x_being_set_st_ (_x_in_dom_(cot_|_[.(PI_/_4),((3_/_4)_*_PI).])_&_y_=_(cot_|_[.(PI_/_4),((3_/_4)_*_PI).])_._x_)_implies_y_in_[.(-_1),1.]_)_) let y be set ; ::_thesis: ( ( y in [.(- 1),1.] implies ex x being set st ( x in dom (cot | [.(PI / 4),((3 / 4) * PI).]) & y = (cot | [.(PI / 4),((3 / 4) * PI).]) . x ) ) & ( ex x being set st ( x in dom (cot | [.(PI / 4),((3 / 4) * PI).]) & y = (cot | [.(PI / 4),((3 / 4) * PI).]) . x ) implies y in [.(- 1),1.] ) ) thus ( y in [.(- 1),1.] implies ex x being set st ( x in dom (cot | [.(PI / 4),((3 / 4) * PI).]) & y = (cot | [.(PI / 4),((3 / 4) * PI).]) . x ) ) ::_thesis: ( ex x being set st ( x in dom (cot | [.(PI / 4),((3 / 4) * PI).]) & y = (cot | [.(PI / 4),((3 / 4) * PI).]) . x ) implies y in [.(- 1),1.] ) proof PI in ].0,4.[ by SIN_COS:def_28; then PI > 0 by XXREAL_1:4; then PI / 4 > 0 / 4 by XREAL_1:74; then A1: (PI / 4) * 3 > PI / 4 by XREAL_1:155; assume A2: y in [.(- 1),1.] ; ::_thesis: ex x being set st ( x in dom (cot | [.(PI / 4),((3 / 4) * PI).]) & y = (cot | [.(PI / 4),((3 / 4) * PI).]) . x ) then reconsider y1 = y as Real ; A3: y1 in [.(cot . ((3 / 4) * PI)),(cot . (PI / 4)).] \/ [.(cot . (PI / 4)),(cot . ((3 / 4) * PI)).] by A2, Th18, XBOOLE_0:def_3; A4: [.(PI / 4),((3 / 4) * PI).] c= ].0,PI.[ by Lm9, Lm10, XXREAL_2:def_12; cot | ].0,PI.[ is continuous by Lm2, FDIFF_1:25; then cot | [.(PI / 4),((3 / 4) * PI).] is continuous by A4, FCONT_1:16; then consider x being Real such that A5: x in [.(PI / 4),((3 / 4) * PI).] and A6: y1 = cot . x by A1, A4, A3, Th2, FCONT_2:15, XBOOLE_1:1; take x ; ::_thesis: ( x in dom (cot | [.(PI / 4),((3 / 4) * PI).]) & y = (cot | [.(PI / 4),((3 / 4) * PI).]) . x ) thus ( x in dom (cot | [.(PI / 4),((3 / 4) * PI).]) & y = (cot | [.(PI / 4),((3 / 4) * PI).]) . x ) by A5, A6, Lm12, FUNCT_1:49; ::_thesis: verum end; thus ( ex x being set st ( x in dom (cot | [.(PI / 4),((3 / 4) * PI).]) & y = (cot | [.(PI / 4),((3 / 4) * PI).]) . x ) implies y in [.(- 1),1.] ) ::_thesis: verum proof given x being set such that A7: x in dom (cot | [.(PI / 4),((3 / 4) * PI).]) and A8: y = (cot | [.(PI / 4),((3 / 4) * PI).]) . x ; ::_thesis: y in [.(- 1),1.] reconsider x1 = x as Real by A7; y = cot . x1 by A7, A8, Lm12, FUNCT_1:49; hence y in [.(- 1),1.] by A7, Lm12, Th20; ::_thesis: verum end; end; hence rng (cot | [.(PI / 4),((3 / 4) * PI).]) = [.(- 1),1.] by FUNCT_1:def_3; ::_thesis: verum end; theorem Th23: :: SIN_COS9:23 [.(- 1),1.] c= dom arctan proof A1: [.(- (PI / 4)),(PI / 4).] c= ].(- (PI / 2)),(PI / 2).[ by Lm7, Lm8, XXREAL_2:def_12; rng (tan | [.(- (PI / 4)),(PI / 4).]) c= rng (tan | ].(- (PI / 2)),(PI / 2).[) proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (tan | [.(- (PI / 4)),(PI / 4).]) or y in rng (tan | ].(- (PI / 2)),(PI / 2).[) ) assume y in rng (tan | [.(- (PI / 4)),(PI / 4).]) ; ::_thesis: y in rng (tan | ].(- (PI / 2)),(PI / 2).[) then y in tan .: [.(- (PI / 4)),(PI / 4).] by RELAT_1:115; then ex x being set st ( x in dom tan & x in [.(- (PI / 4)),(PI / 4).] & y = tan . x ) by FUNCT_1:def_6; then y in tan .: ].(- (PI / 2)),(PI / 2).[ by A1, FUNCT_1:def_6; hence y in rng (tan | ].(- (PI / 2)),(PI / 2).[) by RELAT_1:115; ::_thesis: verum end; hence [.(- 1),1.] c= dom arctan by Th21, FUNCT_1:33; ::_thesis: verum end; theorem Th24: :: SIN_COS9:24 [.(- 1),1.] c= dom arccot proof A1: [.(PI / 4),((3 / 4) * PI).] c= ].0,PI.[ by Lm9, Lm10, XXREAL_2:def_12; rng (cot | [.(PI / 4),((3 / 4) * PI).]) c= rng (cot | ].0,PI.[) proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (cot | [.(PI / 4),((3 / 4) * PI).]) or y in rng (cot | ].0,PI.[) ) assume y in rng (cot | [.(PI / 4),((3 / 4) * PI).]) ; ::_thesis: y in rng (cot | ].0,PI.[) then y in cot .: [.(PI / 4),((3 / 4) * PI).] by RELAT_1:115; then ex x being set st ( x in dom cot & x in [.(PI / 4),((3 / 4) * PI).] & y = cot . x ) by FUNCT_1:def_6; then y in cot .: ].0,PI.[ by A1, FUNCT_1:def_6; hence y in rng (cot | ].0,PI.[) by RELAT_1:115; ::_thesis: verum end; hence [.(- 1),1.] c= dom arccot by Th22, FUNCT_1:33; ::_thesis: verum end; Lm13: (tan | [.(- (PI / 4)),(PI / 4).]) | [.(- (PI / 4)),(PI / 4).] is increasing proof [.(- (PI / 4)),(PI / 4).] c= ].(- (PI / 2)),(PI / 2).[ by Lm7, Lm8, XXREAL_2:def_12; then tan | [.(- (PI / 4)),(PI / 4).] is increasing by Th7, RFUNCT_2:28; hence (tan | [.(- (PI / 4)),(PI / 4).]) | [.(- (PI / 4)),(PI / 4).] is increasing ; ::_thesis: verum end; Lm14: (cot | [.(PI / 4),((3 / 4) * PI).]) | [.(PI / 4),((3 / 4) * PI).] is decreasing proof [.(PI / 4),((3 / 4) * PI).] c= ].0,PI.[ by Lm9, Lm10, XXREAL_2:def_12; then cot | [.(PI / 4),((3 / 4) * PI).] is decreasing by Th8, RFUNCT_2:29; hence (cot | [.(PI / 4),((3 / 4) * PI).]) | [.(PI / 4),((3 / 4) * PI).] is decreasing ; ::_thesis: verum end; Lm15: tan | [.(- (PI / 4)),(PI / 4).] is one-to-one proof [.(- (PI / 4)),(PI / 4).] c= ].(- (PI / 2)),(PI / 2).[ by Lm7, Lm8, XXREAL_2:def_12; then (tan | ].(- (PI / 2)),(PI / 2).[) | [.(- (PI / 4)),(PI / 4).] = tan | [.(- (PI / 4)),(PI / 4).] by RELAT_1:74; hence tan | [.(- (PI / 4)),(PI / 4).] is one-to-one ; ::_thesis: verum end; Lm16: cot | [.(PI / 4),((3 / 4) * PI).] is one-to-one proof [.(PI / 4),((3 / 4) * PI).] c= ].0,PI.[ by Lm9, Lm10, XXREAL_2:def_12; then (cot | ].0,PI.[) | [.(PI / 4),((3 / 4) * PI).] = cot | [.(PI / 4),((3 / 4) * PI).] by RELAT_1:74; hence cot | [.(PI / 4),((3 / 4) * PI).] is one-to-one ; ::_thesis: verum end; registration clusterK77(tan,[.(- (PI / 4)),(PI / 4).]) -> one-to-one ; coherence tan | [.(- (PI / 4)),(PI / 4).] is one-to-one by Lm15; clusterK77(cot,[.(PI / 4),((3 / 4) * PI).]) -> one-to-one ; coherence cot | [.(PI / 4),((3 / 4) * PI).] is one-to-one by Lm16; end; theorem Th25: :: SIN_COS9:25 arctan | [.(- 1),1.] = (tan | [.(- (PI / 4)),(PI / 4).]) " proof set h = tan | ].(- (PI / 2)),(PI / 2).[; A1: [.(- (PI / 4)),(PI / 4).] c= ].(- (PI / 2)),(PI / 2).[ by Lm7, Lm8, XXREAL_2:def_12; then (tan | [.(- (PI / 4)),(PI / 4).]) " = ((tan | ].(- (PI / 2)),(PI / 2).[) | [.(- (PI / 4)),(PI / 4).]) " by RELAT_1:74 .= ((tan | ].(- (PI / 2)),(PI / 2).[) ") | ((tan | ].(- (PI / 2)),(PI / 2).[) .: [.(- (PI / 4)),(PI / 4).]) by RFUNCT_2:17 .= ((tan | ].(- (PI / 2)),(PI / 2).[) ") | (rng ((tan | ].(- (PI / 2)),(PI / 2).[) | [.(- (PI / 4)),(PI / 4).])) by RELAT_1:115 .= ((tan | ].(- (PI / 2)),(PI / 2).[) ") | [.(- 1),1.] by A1, Th21, RELAT_1:74 ; hence arctan | [.(- 1),1.] = (tan | [.(- (PI / 4)),(PI / 4).]) " ; ::_thesis: verum end; theorem Th26: :: SIN_COS9:26 arccot | [.(- 1),1.] = (cot | [.(PI / 4),((3 / 4) * PI).]) " proof set h = cot | ].0,PI.[; A1: [.(PI / 4),((3 / 4) * PI).] c= ].0,PI.[ by Lm9, Lm10, XXREAL_2:def_12; then (cot | [.(PI / 4),((3 / 4) * PI).]) " = ((cot | ].0,PI.[) | [.(PI / 4),((3 / 4) * PI).]) " by RELAT_1:74 .= ((cot | ].0,PI.[) ") | ((cot | ].0,PI.[) .: [.(PI / 4),((3 / 4) * PI).]) by RFUNCT_2:17 .= ((cot | ].0,PI.[) ") | (rng ((cot | ].0,PI.[) | [.(PI / 4),((3 / 4) * PI).])) by RELAT_1:115 .= ((cot | ].0,PI.[) ") | [.(- 1),1.] by A1, Th22, RELAT_1:74 ; hence arccot | [.(- 1),1.] = (cot | [.(PI / 4),((3 / 4) * PI).]) " ; ::_thesis: verum end; theorem :: SIN_COS9:27 (tan | [.(- (PI / 4)),(PI / 4).]) * (arctan | [.(- 1),1.]) = id [.(- 1),1.] by Th21, Th25, FUNCT_1:39; theorem :: SIN_COS9:28 (cot | [.(PI / 4),((3 / 4) * PI).]) * (arccot | [.(- 1),1.]) = id [.(- 1),1.] by Th22, Th26, FUNCT_1:39; theorem :: SIN_COS9:29 (tan | [.(- (PI / 4)),(PI / 4).]) * (arctan | [.(- 1),1.]) = id [.(- 1),1.] by Th21, Th25, FUNCT_1:39; theorem :: SIN_COS9:30 (cot | [.(PI / 4),((3 / 4) * PI).]) * (arccot | [.(- 1),1.]) = id [.(- 1),1.] by Th22, Th26, FUNCT_1:39; theorem Th31: :: SIN_COS9:31 arctan * (tan | ].(- (PI / 2)),(PI / 2).[) = id ].(- (PI / 2)),(PI / 2).[ by Lm5, Th11, FUNCT_1:39; theorem Th32: :: SIN_COS9:32 arccot * (cot | ].0,PI.[) = id ].0,PI.[ by Lm6, Th12, FUNCT_1:39; theorem :: SIN_COS9:33 arctan * (tan | ].(- (PI / 2)),(PI / 2).[) = id ].(- (PI / 2)),(PI / 2).[ by Lm5, Th11, FUNCT_1:39; theorem :: SIN_COS9:34 arccot * (cot | ].0,PI.[) = id ].0,PI.[ by Lm6, Th12, FUNCT_1:39; theorem Th35: :: SIN_COS9:35 for r being Real st - (PI / 2) < r & r < PI / 2 holds ( arctan (tan . r) = r & arctan (tan r) = r ) proof let r be Real; ::_thesis: ( - (PI / 2) < r & r < PI / 2 implies ( arctan (tan . r) = r & arctan (tan r) = r ) ) assume that A1: - (PI / 2) < r and A2: r < PI / 2 ; ::_thesis: ( arctan (tan . r) = r & arctan (tan r) = r ) A3: dom (tan | ].(- (PI / 2)),(PI / 2).[) = ].(- (PI / 2)),(PI / 2).[ by Th1, RELAT_1:62; A4: r in ].(- (PI / 2)),(PI / 2).[ by A1, A2, XXREAL_1:4; then arctan (tan . r) = arctan . ((tan | ].(- (PI / 2)),(PI / 2).[) . r) by FUNCT_1:49 .= (id ].(- (PI / 2)),(PI / 2).[) . r by A4, A3, Th31, FUNCT_1:13 .= r by A4, FUNCT_1:18 ; hence ( arctan (tan . r) = r & arctan (tan r) = r ) by A4, Th13; ::_thesis: verum end; theorem Th36: :: SIN_COS9:36 for r being Real st 0 < r & r < PI holds ( arccot (cot . r) = r & arccot (cot r) = r ) proof let r be Real; ::_thesis: ( 0 < r & r < PI implies ( arccot (cot . r) = r & arccot (cot r) = r ) ) assume that A1: 0 < r and A2: r < PI ; ::_thesis: ( arccot (cot . r) = r & arccot (cot r) = r ) A3: dom (cot | ].0,PI.[) = ].0,PI.[ by Th2, RELAT_1:62; A4: r in ].0,PI.[ by A1, A2, XXREAL_1:4; then arccot (cot . r) = arccot . ((cot | ].0,PI.[) . r) by FUNCT_1:49 .= (id ].0,PI.[) . r by A4, A3, Th32, FUNCT_1:13 .= r by A4, FUNCT_1:18 ; hence ( arccot (cot . r) = r & arccot (cot r) = r ) by A4, Th14; ::_thesis: verum end; theorem Th37: :: SIN_COS9:37 ( arctan (- 1) = - (PI / 4) & arctan . (- 1) = - (PI / 4) ) proof - (PI / 2) < - (PI / 4) by Lm7, XXREAL_1:4; then arctan (- 1) = - (PI / 4) by Th17, Th35; hence ( arctan (- 1) = - (PI / 4) & arctan . (- 1) = - (PI / 4) ) ; ::_thesis: verum end; theorem Th38: :: SIN_COS9:38 ( arccot (- 1) = (3 / 4) * PI & arccot . (- 1) = (3 / 4) * PI ) proof A1: (3 / 4) * PI < PI by Lm10, XXREAL_1:4; 0 < (3 / 4) * PI by Lm10, XXREAL_1:4; then arccot (- 1) = (3 / 4) * PI by A1, Th18, Th36; hence ( arccot (- 1) = (3 / 4) * PI & arccot . (- 1) = (3 / 4) * PI ) ; ::_thesis: verum end; theorem Th39: :: SIN_COS9:39 ( arctan 1 = PI / 4 & arctan . 1 = PI / 4 ) proof A1: arctan 1 = arctan (tan . (PI / 4)) by SIN_COS:def_28; PI / 4 < PI / 2 by Lm8, XXREAL_1:4; hence ( arctan 1 = PI / 4 & arctan . 1 = PI / 4 ) by A1, Th35; ::_thesis: verum end; theorem Th40: :: SIN_COS9:40 ( arccot 1 = PI / 4 & arccot . 1 = PI / 4 ) proof A1: PI / 4 < PI by Lm9, XXREAL_1:4; 0 < PI / 4 by Lm9, XXREAL_1:4; then arccot 1 = PI / 4 by A1, Th18, Th36; hence ( arccot 1 = PI / 4 & arccot . 1 = PI / 4 ) ; ::_thesis: verum end; theorem Th41: :: SIN_COS9:41 ( tan . 0 = 0 & tan 0 = 0 ) proof PI in ].0,4.[ by SIN_COS:def_28; then PI > 0 by XXREAL_1:4; then A1: PI / 2 > 0 / 2 by XREAL_1:74; 0 - (PI / 2) < 0 by XREAL_1:49; then A2: 0 in ].(- (PI / 2)),(PI / 2).[ by A1, XXREAL_1:4; then tan . 0 = 0 / (cos . 0) by Th1, RFUNCT_1:def_1, SIN_COS:30 .= 0 ; hence ( tan . 0 = 0 & tan 0 = 0 ) by A2, Th13; ::_thesis: verum end; theorem Th42: :: SIN_COS9:42 ( cot . (PI / 2) = 0 & cot (PI / 2) = 0 ) proof PI in ].0,4.[ by SIN_COS:def_28; then A1: PI > 0 by XXREAL_1:4; then A2: PI / 2 < PI / 1 by XREAL_1:76; PI / 2 > 0 / 2 by A1, XREAL_1:74; then A3: PI / 2 in ].0,PI.[ by A2, XXREAL_1:4; then cot . (PI / 2) = 0 / (sin . (PI / 2)) by Th2, RFUNCT_1:def_1, SIN_COS:76 .= 0 ; hence ( cot . (PI / 2) = 0 & cot (PI / 2) = 0 ) by A3, Th14; ::_thesis: verum end; theorem :: SIN_COS9:43 ( arctan 0 = 0 & arctan . 0 = 0 ) proof PI in ].0,4.[ by SIN_COS:def_28; then PI > 0 by XXREAL_1:4; then A1: PI / 2 > 0 / 2 by XREAL_1:74; 0 - (PI / 2) < 0 by XREAL_1:49; then arctan 0 = 0 by A1, Th35, Th41; hence ( arctan 0 = 0 & arctan . 0 = 0 ) ; ::_thesis: verum end; theorem :: SIN_COS9:44 ( arccot 0 = PI / 2 & arccot . 0 = PI / 2 ) proof PI in ].0,4.[ by SIN_COS:def_28; then A1: PI > 0 by XXREAL_1:4; then A2: PI / 2 < PI / 1 by XREAL_1:76; PI / 2 > 0 / 2 by A1, XREAL_1:74; then arccot 0 = PI / 2 by A2, Th36, Th42; hence ( arccot 0 = PI / 2 & arccot . 0 = PI / 2 ) ; ::_thesis: verum end; theorem Th45: :: SIN_COS9:45 arctan | (tan .: ].(- (PI / 2)),(PI / 2).[) is increasing proof set f = tan | ].(- (PI / 2)),(PI / 2).[; A1: (tan | ].(- (PI / 2)),(PI / 2).[) .: ].(- (PI / 2)),(PI / 2).[ = rng ((tan | ].(- (PI / 2)),(PI / 2).[) | ].(- (PI / 2)),(PI / 2).[) by RELAT_1:115 .= rng (tan | ].(- (PI / 2)),(PI / 2).[) by RELAT_1:73 .= tan .: ].(- (PI / 2)),(PI / 2).[ by RELAT_1:115 ; (tan | ].(- (PI / 2)),(PI / 2).[) | ].(- (PI / 2)),(PI / 2).[ = tan | ].(- (PI / 2)),(PI / 2).[ by RELAT_1:73; hence arctan | (tan .: ].(- (PI / 2)),(PI / 2).[) is increasing by A1, Th7, FCONT_3:9; ::_thesis: verum end; theorem Th46: :: SIN_COS9:46 arccot | (cot .: ].0,PI.[) is decreasing proof set f = cot | ].0,PI.[; A1: (cot | ].0,PI.[) .: ].0,PI.[ = rng ((cot | ].0,PI.[) | ].0,PI.[) by RELAT_1:115 .= rng (cot | ].0,PI.[) by RELAT_1:73 .= cot .: ].0,PI.[ by RELAT_1:115 ; (cot | ].0,PI.[) | ].0,PI.[ = cot | ].0,PI.[ by RELAT_1:73; hence arccot | (cot .: ].0,PI.[) is decreasing by A1, Th8, FCONT_3:10; ::_thesis: verum end; theorem Th47: :: SIN_COS9:47 arctan | [.(- 1),1.] is increasing proof A1: [.(- 1),1.] = tan .: [.(- (PI / 4)),(PI / 4).] by Th21, RELAT_1:115; [.(- (PI / 4)),(PI / 4).] c= ].(- (PI / 2)),(PI / 2).[ by Lm7, Lm8, XXREAL_2:def_12; hence arctan | [.(- 1),1.] is increasing by A1, Th45, RELAT_1:123, RFUNCT_2:28; ::_thesis: verum end; theorem Th48: :: SIN_COS9:48 arccot | [.(- 1),1.] is decreasing proof A1: [.(- 1),1.] = cot .: [.(PI / 4),((3 / 4) * PI).] by Th22, RELAT_1:115; [.(PI / 4),((3 / 4) * PI).] c= ].0,PI.[ by Lm9, Lm10, XXREAL_2:def_12; hence arccot | [.(- 1),1.] is decreasing by A1, Th46, RELAT_1:123, RFUNCT_2:29; ::_thesis: verum end; theorem Th49: :: SIN_COS9:49 for x being set st x in [.(- 1),1.] holds arctan . x in [.(- (PI / 4)),(PI / 4).] proof let x be set ; ::_thesis: ( x in [.(- 1),1.] implies arctan . x in [.(- (PI / 4)),(PI / 4).] ) assume x in [.(- 1),1.] ; ::_thesis: arctan . x in [.(- (PI / 4)),(PI / 4).] then x in ].(- 1),1.[ \/ {(- 1),1} by XXREAL_1:128; then A1: ( x in ].(- 1),1.[ or x in {(- 1),1} ) by XBOOLE_0:def_3; percases ( x in ].(- 1),1.[ or x = - 1 or x = 1 ) by A1, TARSKI:def_2; supposeA2: x in ].(- 1),1.[ ; ::_thesis: arctan . x in [.(- (PI / 4)),(PI / 4).] then x in { s where s is Real : ( - 1 < s & s < 1 ) } by RCOMP_1:def_2; then A3: ex s being Real st ( s = x & - 1 < s & s < 1 ) ; A4: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25; A5: [.(- 1),1.] /\ (dom arctan) = [.(- 1),1.] by Th23, XBOOLE_1:28; then 1 in [.(- 1),1.] /\ (dom arctan) by XXREAL_1:1; then A6: arctan . x < PI / 4 by A2, A5, A4, A3, Th39, Th47, RFUNCT_2:20; - 1 in [.(- 1),1.] by XXREAL_1:1; then - (PI / 4) < arctan . x by A2, A5, A4, A3, Th37, Th47, RFUNCT_2:20; hence arctan . x in [.(- (PI / 4)),(PI / 4).] by A6, XXREAL_1:1; ::_thesis: verum end; suppose x = - 1 ; ::_thesis: arctan . x in [.(- (PI / 4)),(PI / 4).] hence arctan . x in [.(- (PI / 4)),(PI / 4).] by Th37, XXREAL_1:1; ::_thesis: verum end; suppose x = 1 ; ::_thesis: arctan . x in [.(- (PI / 4)),(PI / 4).] hence arctan . x in [.(- (PI / 4)),(PI / 4).] by Th39, XXREAL_1:1; ::_thesis: verum end; end; end; theorem Th50: :: SIN_COS9:50 for x being set st x in [.(- 1),1.] holds arccot . x in [.(PI / 4),((3 / 4) * PI).] proof let x be set ; ::_thesis: ( x in [.(- 1),1.] implies arccot . x in [.(PI / 4),((3 / 4) * PI).] ) assume x in [.(- 1),1.] ; ::_thesis: arccot . x in [.(PI / 4),((3 / 4) * PI).] then x in ].(- 1),1.[ \/ {(- 1),1} by XXREAL_1:128; then A1: ( x in ].(- 1),1.[ or x in {(- 1),1} ) by XBOOLE_0:def_3; percases ( x in ].(- 1),1.[ or x = - 1 or x = 1 ) by A1, TARSKI:def_2; supposeA2: x in ].(- 1),1.[ ; ::_thesis: arccot . x in [.(PI / 4),((3 / 4) * PI).] then x in { s where s is Real : ( - 1 < s & s < 1 ) } by RCOMP_1:def_2; then A3: ex s being Real st ( s = x & - 1 < s & s < 1 ) ; A4: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25; A5: [.(- 1),1.] /\ (dom arccot) = [.(- 1),1.] by Th24, XBOOLE_1:28; then 1 in [.(- 1),1.] /\ (dom arccot) by XXREAL_1:1; then A6: arccot . x > PI / 4 by A2, A5, A4, A3, Th40, Th48, RFUNCT_2:21; - 1 in [.(- 1),1.] by XXREAL_1:1; then (3 / 4) * PI > arccot . x by A2, A5, A4, A3, Th38, Th48, RFUNCT_2:21; hence arccot . x in [.(PI / 4),((3 / 4) * PI).] by A6, XXREAL_1:1; ::_thesis: verum end; supposeA7: x = - 1 ; ::_thesis: arccot . x in [.(PI / 4),((3 / 4) * PI).] PI in ].0,4.[ by SIN_COS:def_28; then PI > 0 by XXREAL_1:4; then PI / 4 > 0 / 4 by XREAL_1:74; then PI / 4 < (PI / 4) * 3 by XREAL_1:155; hence arccot . x in [.(PI / 4),((3 / 4) * PI).] by A7, Th38, XXREAL_1:1; ::_thesis: verum end; supposeA8: x = 1 ; ::_thesis: arccot . x in [.(PI / 4),((3 / 4) * PI).] PI in ].0,4.[ by SIN_COS:def_28; then PI > 0 by XXREAL_1:4; then PI / 4 > 0 / 4 by XREAL_1:74; then PI / 4 < (PI / 4) * 3 by XREAL_1:155; hence arccot . x in [.(PI / 4),((3 / 4) * PI).] by A8, Th40, XXREAL_1:1; ::_thesis: verum end; end; end; theorem Th51: :: SIN_COS9:51 for r being Real st - 1 <= r & r <= 1 holds tan (arctan r) = r proof let r be Real; ::_thesis: ( - 1 <= r & r <= 1 implies tan (arctan r) = r ) A1: [.(- (PI / 4)),(PI / 4).] c= ].(- (PI / 2)),(PI / 2).[ by Lm7, Lm8, XXREAL_2:def_12; assume that A2: - 1 <= r and A3: r <= 1 ; ::_thesis: tan (arctan r) = r A4: r in [.(- 1),1.] by A2, A3, XXREAL_1:1; then A5: r in dom (arctan | [.(- 1),1.]) by Th23, RELAT_1:62; arctan . r in [.(- (PI / 4)),(PI / 4).] by A4, Th49; hence tan (arctan r) = tan . (arctan . r) by A1, Th13 .= (tan | [.(- (PI / 4)),(PI / 4).]) . (arctan . r) by A4, Th49, FUNCT_1:49 .= (tan | [.(- (PI / 4)),(PI / 4).]) . ((arctan | [.(- 1),1.]) . r) by A4, FUNCT_1:49 .= ((tan | [.(- (PI / 4)),(PI / 4).]) * (arctan | [.(- 1),1.])) . r by A5, FUNCT_1:13 .= (id [.(- 1),1.]) . r by Th21, Th25, FUNCT_1:39 .= r by A4, FUNCT_1:18 ; ::_thesis: verum end; theorem Th52: :: SIN_COS9:52 for r being Real st - 1 <= r & r <= 1 holds cot (arccot r) = r proof let r be Real; ::_thesis: ( - 1 <= r & r <= 1 implies cot (arccot r) = r ) A1: [.(PI / 4),((3 / 4) * PI).] c= ].0,PI.[ by Lm9, Lm10, XXREAL_2:def_12; assume that A2: - 1 <= r and A3: r <= 1 ; ::_thesis: cot (arccot r) = r A4: r in [.(- 1),1.] by A2, A3, XXREAL_1:1; then A5: r in dom (arccot | [.(- 1),1.]) by Th24, RELAT_1:62; arccot . r in [.(PI / 4),((3 / 4) * PI).] by A4, Th50; hence cot (arccot r) = cot . (arccot . r) by A1, Th14 .= (cot | [.(PI / 4),((3 / 4) * PI).]) . (arccot . r) by A4, Th50, FUNCT_1:49 .= (cot | [.(PI / 4),((3 / 4) * PI).]) . ((arccot | [.(- 1),1.]) . r) by A4, FUNCT_1:49 .= ((cot | [.(PI / 4),((3 / 4) * PI).]) * (arccot | [.(- 1),1.])) . r by A5, FUNCT_1:13 .= (id [.(- 1),1.]) . r by Th22, Th26, FUNCT_1:39 .= r by A4, FUNCT_1:18 ; ::_thesis: verum end; theorem Th53: :: SIN_COS9:53 arctan | [.(- 1),1.] is continuous proof set f = tan | [.(- (PI / 4)),(PI / 4).]; A1: (tan | [.(- (PI / 4)),(PI / 4).]) | [.(- (PI / 4)),(PI / 4).] = tan | [.(- (PI / 4)),(PI / 4).] by RELAT_1:72; (((tan | [.(- (PI / 4)),(PI / 4).]) | [.(- (PI / 4)),(PI / 4).]) ") | ((tan | [.(- (PI / 4)),(PI / 4).]) .: [.(- (PI / 4)),(PI / 4).]) is continuous by Lm11, Lm13, FCONT_1:47; then (arctan | [.(- 1),1.]) | [.(- 1),1.] is continuous by A1, Th21, Th25, RELAT_1:115; hence arctan | [.(- 1),1.] is continuous by FCONT_1:15; ::_thesis: verum end; theorem Th54: :: SIN_COS9:54 arccot | [.(- 1),1.] is continuous proof set f = cot | [.(PI / 4),((3 / 4) * PI).]; PI in ].0,4.[ by SIN_COS:def_28; then PI > 0 by XXREAL_1:4; then PI / 4 > 0 / 4 by XREAL_1:74; then PI / 4 < (PI / 4) * 3 by XREAL_1:155; then A1: (((cot | [.(PI / 4),((3 / 4) * PI).]) | [.(PI / 4),((3 / 4) * PI).]) ") | ((cot | [.(PI / 4),((3 / 4) * PI).]) .: [.(PI / 4),((3 / 4) * PI).]) is continuous by Lm12, Lm14, FCONT_1:47; (cot | [.(PI / 4),((3 / 4) * PI).]) | [.(PI / 4),((3 / 4) * PI).] = cot | [.(PI / 4),((3 / 4) * PI).] by RELAT_1:72; then (arccot | [.(- 1),1.]) | [.(- 1),1.] is continuous by A1, Th22, Th26, RELAT_1:115; hence arccot | [.(- 1),1.] is continuous by FCONT_1:15; ::_thesis: verum end; theorem Th55: :: SIN_COS9:55 rng (arctan | [.(- 1),1.]) = [.(- (PI / 4)),(PI / 4).] proof now__::_thesis:_for_y_being_set_holds_ (_(_y_in_[.(-_(PI_/_4)),(PI_/_4).]_implies_ex_x_being_set_st_ (_x_in_dom_(arctan_|_[.(-_1),1.])_&_y_=_(arctan_|_[.(-_1),1.])_._x_)_)_&_(_ex_x_being_set_st_ (_x_in_dom_(arctan_|_[.(-_1),1.])_&_y_=_(arctan_|_[.(-_1),1.])_._x_)_implies_y_in_[.(-_(PI_/_4)),(PI_/_4).]_)_) let y be set ; ::_thesis: ( ( y in [.(- (PI / 4)),(PI / 4).] implies ex x being set st ( x in dom (arctan | [.(- 1),1.]) & y = (arctan | [.(- 1),1.]) . x ) ) & ( ex x being set st ( x in dom (arctan | [.(- 1),1.]) & y = (arctan | [.(- 1),1.]) . x ) implies y in [.(- (PI / 4)),(PI / 4).] ) ) thus ( y in [.(- (PI / 4)),(PI / 4).] implies ex x being set st ( x in dom (arctan | [.(- 1),1.]) & y = (arctan | [.(- 1),1.]) . x ) ) ::_thesis: ( ex x being set st ( x in dom (arctan | [.(- 1),1.]) & y = (arctan | [.(- 1),1.]) . x ) implies y in [.(- (PI / 4)),(PI / 4).] ) proof assume A1: y in [.(- (PI / 4)),(PI / 4).] ; ::_thesis: ex x being set st ( x in dom (arctan | [.(- 1),1.]) & y = (arctan | [.(- 1),1.]) . x ) then reconsider y1 = y as Real ; y1 in [.(arctan . (- 1)),(arctan . 1).] \/ [.(arctan . 1),(arctan . (- 1)).] by A1, Th37, Th39, XBOOLE_0:def_3; then consider x being Real such that A2: x in [.(- 1),1.] and A3: y1 = arctan . x by Th23, Th53, FCONT_2:15; take x ; ::_thesis: ( x in dom (arctan | [.(- 1),1.]) & y = (arctan | [.(- 1),1.]) . x ) thus ( x in dom (arctan | [.(- 1),1.]) & y = (arctan | [.(- 1),1.]) . x ) by A2, A3, Th23, FUNCT_1:49, RELAT_1:62; ::_thesis: verum end; thus ( ex x being set st ( x in dom (arctan | [.(- 1),1.]) & y = (arctan | [.(- 1),1.]) . x ) implies y in [.(- (PI / 4)),(PI / 4).] ) ::_thesis: verum proof given x being set such that A4: x in dom (arctan | [.(- 1),1.]) and A5: y = (arctan | [.(- 1),1.]) . x ; ::_thesis: y in [.(- (PI / 4)),(PI / 4).] A6: dom (arctan | [.(- 1),1.]) = [.(- 1),1.] by Th23, RELAT_1:62; then y = arctan . x by A4, A5, FUNCT_1:49; hence y in [.(- (PI / 4)),(PI / 4).] by A4, A6, Th49; ::_thesis: verum end; end; hence rng (arctan | [.(- 1),1.]) = [.(- (PI / 4)),(PI / 4).] by FUNCT_1:def_3; ::_thesis: verum end; theorem Th56: :: SIN_COS9:56 rng (arccot | [.(- 1),1.]) = [.(PI / 4),((3 / 4) * PI).] proof now__::_thesis:_for_y_being_set_holds_ (_(_y_in_[.(PI_/_4),((3_/_4)_*_PI).]_implies_ex_x_being_set_st_ (_x_in_dom_(arccot_|_[.(-_1),1.])_&_y_=_(arccot_|_[.(-_1),1.])_._x_)_)_&_(_ex_x_being_set_st_ (_x_in_dom_(arccot_|_[.(-_1),1.])_&_y_=_(arccot_|_[.(-_1),1.])_._x_)_implies_y_in_[.(PI_/_4),((3_/_4)_*_PI).]_)_) let y be set ; ::_thesis: ( ( y in [.(PI / 4),((3 / 4) * PI).] implies ex x being set st ( x in dom (arccot | [.(- 1),1.]) & y = (arccot | [.(- 1),1.]) . x ) ) & ( ex x being set st ( x in dom (arccot | [.(- 1),1.]) & y = (arccot | [.(- 1),1.]) . x ) implies y in [.(PI / 4),((3 / 4) * PI).] ) ) thus ( y in [.(PI / 4),((3 / 4) * PI).] implies ex x being set st ( x in dom (arccot | [.(- 1),1.]) & y = (arccot | [.(- 1),1.]) . x ) ) ::_thesis: ( ex x being set st ( x in dom (arccot | [.(- 1),1.]) & y = (arccot | [.(- 1),1.]) . x ) implies y in [.(PI / 4),((3 / 4) * PI).] ) proof assume A1: y in [.(PI / 4),((3 / 4) * PI).] ; ::_thesis: ex x being set st ( x in dom (arccot | [.(- 1),1.]) & y = (arccot | [.(- 1),1.]) . x ) then reconsider y1 = y as Real ; y1 in [.(arccot . 1),(arccot . (- 1)).] \/ [.(arccot . (- 1)),(arccot . 1).] by A1, Th38, Th40, XBOOLE_0:def_3; then consider x being Real such that A2: x in [.(- 1),1.] and A3: y1 = arccot . x by Th24, Th54, FCONT_2:15; take x ; ::_thesis: ( x in dom (arccot | [.(- 1),1.]) & y = (arccot | [.(- 1),1.]) . x ) thus ( x in dom (arccot | [.(- 1),1.]) & y = (arccot | [.(- 1),1.]) . x ) by A2, A3, Th24, FUNCT_1:49, RELAT_1:62; ::_thesis: verum end; thus ( ex x being set st ( x in dom (arccot | [.(- 1),1.]) & y = (arccot | [.(- 1),1.]) . x ) implies y in [.(PI / 4),((3 / 4) * PI).] ) ::_thesis: verum proof given x being set such that A4: x in dom (arccot | [.(- 1),1.]) and A5: y = (arccot | [.(- 1),1.]) . x ; ::_thesis: y in [.(PI / 4),((3 / 4) * PI).] A6: dom (arccot | [.(- 1),1.]) = [.(- 1),1.] by Th24, RELAT_1:62; then y = arccot . x by A4, A5, FUNCT_1:49; hence y in [.(PI / 4),((3 / 4) * PI).] by A4, A6, Th50; ::_thesis: verum end; end; hence rng (arccot | [.(- 1),1.]) = [.(PI / 4),((3 / 4) * PI).] by FUNCT_1:def_3; ::_thesis: verum end; theorem :: SIN_COS9:57 for r being Real st - 1 <= r & r <= 1 & arctan r = - (PI / 4) holds r = - 1 by Th17, Th51; theorem :: SIN_COS9:58 for r being Real st - 1 <= r & r <= 1 & arccot r = (3 / 4) * PI holds r = - 1 by Th18, Th52; theorem :: SIN_COS9:59 for r being Real st - 1 <= r & r <= 1 & arctan r = 0 holds r = 0 by Th41, Th51; theorem :: SIN_COS9:60 for r being Real st - 1 <= r & r <= 1 & arccot r = PI / 2 holds r = 0 by Th42, Th52; theorem :: SIN_COS9:61 for r being Real st - 1 <= r & r <= 1 & arctan r = PI / 4 holds r = 1 proof let r be Real; ::_thesis: ( - 1 <= r & r <= 1 & arctan r = PI / 4 implies r = 1 ) assume that A1: - 1 <= r and A2: r <= 1 and A3: arctan r = PI / 4 ; ::_thesis: r = 1 thus r = tan (PI / 4) by A1, A2, A3, Th51 .= tan . (PI / 4) by Lm8, Th13 .= 1 by SIN_COS:def_28 ; ::_thesis: verum end; theorem :: SIN_COS9:62 for r being Real st - 1 <= r & r <= 1 & arccot r = PI / 4 holds r = 1 by Th18, Th52; theorem Th63: :: SIN_COS9:63 for r being Real st - 1 <= r & r <= 1 holds ( - (PI / 4) <= arctan r & arctan r <= PI / 4 ) proof let r be Real; ::_thesis: ( - 1 <= r & r <= 1 implies ( - (PI / 4) <= arctan r & arctan r <= PI / 4 ) ) assume that A1: - 1 <= r and A2: r <= 1 ; ::_thesis: ( - (PI / 4) <= arctan r & arctan r <= PI / 4 ) A3: r in [.(- 1),1.] by A1, A2, XXREAL_1:1; then r in dom (arctan | [.(- 1),1.]) by Th23, RELAT_1:62; then (arctan | [.(- 1),1.]) . r in rng (arctan | [.(- 1),1.]) by FUNCT_1:def_3; then arctan r in rng (arctan | [.(- 1),1.]) by A3, FUNCT_1:49; hence ( - (PI / 4) <= arctan r & arctan r <= PI / 4 ) by Th55, XXREAL_1:1; ::_thesis: verum end; theorem Th64: :: SIN_COS9:64 for r being Real st - 1 <= r & r <= 1 holds ( PI / 4 <= arccot r & arccot r <= (3 / 4) * PI ) proof let r be Real; ::_thesis: ( - 1 <= r & r <= 1 implies ( PI / 4 <= arccot r & arccot r <= (3 / 4) * PI ) ) assume that A1: - 1 <= r and A2: r <= 1 ; ::_thesis: ( PI / 4 <= arccot r & arccot r <= (3 / 4) * PI ) A3: r in [.(- 1),1.] by A1, A2, XXREAL_1:1; then r in dom (arccot | [.(- 1),1.]) by Th24, RELAT_1:62; then (arccot | [.(- 1),1.]) . r in rng (arccot | [.(- 1),1.]) by FUNCT_1:def_3; then arccot r in rng (arccot | [.(- 1),1.]) by A3, FUNCT_1:49; hence ( PI / 4 <= arccot r & arccot r <= (3 / 4) * PI ) by Th56, XXREAL_1:1; ::_thesis: verum end; theorem :: SIN_COS9:65 for r being Real st - 1 < r & r < 1 holds ( - (PI / 4) < arctan r & arctan r < PI / 4 ) proof let r be Real; ::_thesis: ( - 1 < r & r < 1 implies ( - (PI / 4) < arctan r & arctan r < PI / 4 ) ) A1: tan (PI / 4) = tan . (PI / 4) by Lm8, Th13 .= 1 by SIN_COS:def_28 ; assume that A2: - 1 < r and A3: r < 1 ; ::_thesis: ( - (PI / 4) < arctan r & arctan r < PI / 4 ) A4: arctan r <= PI / 4 by A2, A3, Th63; - (PI / 4) <= arctan r by A2, A3, Th63; then ( ( - (PI / 4) < arctan r & arctan r < PI / 4 ) or - (PI / 4) = arctan r or arctan r = PI / 4 ) by A4, XXREAL_0:1; hence ( - (PI / 4) < arctan r & arctan r < PI / 4 ) by A2, A3, A1, Th17, Th51; ::_thesis: verum end; theorem :: SIN_COS9:66 for r being Real st - 1 < r & r < 1 holds ( PI / 4 < arccot r & arccot r < (3 / 4) * PI ) proof let r be Real; ::_thesis: ( - 1 < r & r < 1 implies ( PI / 4 < arccot r & arccot r < (3 / 4) * PI ) ) assume that A1: - 1 < r and A2: r < 1 ; ::_thesis: ( PI / 4 < arccot r & arccot r < (3 / 4) * PI ) A3: arccot r <= (3 / 4) * PI by A1, A2, Th64; PI / 4 <= arccot r by A1, A2, Th64; then ( ( PI / 4 < arccot r & arccot r < (3 / 4) * PI ) or PI / 4 = arccot r or arccot r = (3 / 4) * PI ) by A3, XXREAL_0:1; hence ( PI / 4 < arccot r & arccot r < (3 / 4) * PI ) by A1, A2, Th18, Th52; ::_thesis: verum end; theorem :: SIN_COS9:67 for r being Real st - 1 <= r & r <= 1 holds arctan r = - (arctan (- r)) proof let r be Real; ::_thesis: ( - 1 <= r & r <= 1 implies arctan r = - (arctan (- r)) ) A1: [.(- (PI / 4)),(PI / 4).] c= ].(- (PI / 2)),(PI / 2).[ by Lm7, Lm8, XXREAL_2:def_12; assume that A2: - 1 <= r and A3: r <= 1 ; ::_thesis: arctan r = - (arctan (- r)) A4: - r >= - 1 by A3, XREAL_1:24; A5: - (- 1) >= - r by A2, XREAL_1:24; then A6: arctan (- r) <= PI / 4 by A4, Th63; - (PI / 4) <= arctan (- r) by A5, A4, Th63; then A7: - (arctan (- r)) <= - (- (PI / 4)) by XREAL_1:24; arctan (- r) <= PI / 4 by A5, A4, Th63; then - (PI / 4) <= - (arctan (- r)) by XREAL_1:24; then A8: - (arctan (- r)) in [.(- (PI / 4)),(PI / 4).] by A7, XXREAL_1:1; - (PI / 4) <= arctan (- r) by A5, A4, Th63; then arctan (- r) in [.(- (PI / 4)),(PI / 4).] by A6, XXREAL_1:1; then A9: cos (arctan (- r)) <> 0 by A1, COMPTRIG:11; tan (arctan (- r)) = - r by A5, A4, Th51; then A10: r = ((tan 0) - (tan (arctan (- r)))) / (1 + ((tan 0) * (tan (arctan (- r))))) by Th41 .= tan (0 - (arctan (- r))) by A9, SIN_COS:31, SIN_COS4:8 ; A11: [.(- (PI / 4)),(PI / 4).] c= ].(- (PI / 2)),(PI / 2).[ by Lm7, Lm8, XXREAL_2:def_12; then A12: - (arctan (- r)) < PI / 2 by A8, XXREAL_1:4; - (PI / 2) < - (arctan (- r)) by A8, A11, XXREAL_1:4; hence arctan r = - (arctan (- r)) by A10, A12, Th35; ::_thesis: verum end; theorem :: SIN_COS9:68 for r being Real st - 1 <= r & r <= 1 holds arccot r = PI - (arccot (- r)) proof let r be Real; ::_thesis: ( - 1 <= r & r <= 1 implies arccot r = PI - (arccot (- r)) ) set x = arccot (- r); assume that A1: - 1 <= r and A2: r <= 1 ; ::_thesis: arccot r = PI - (arccot (- r)) A3: - r >= - 1 by A2, XREAL_1:24; A4: - (- 1) >= - r by A1, XREAL_1:24; then - r = cot (arccot (- r)) by A3, Th52; then A5: r = - (cot (arccot (- r))) .= - ((cos (arccot (- r))) / (sin (arccot (- r)))) by SIN_COS4:def_2 .= (cos (arccot (- r))) / (- (sin (arccot (- r)))) by XCMPLX_1:188 .= (cos (arccot (- r))) / (sin (- (arccot (- r)))) by SIN_COS:31 .= (cos (- (arccot (- r)))) / (sin (- (arccot (- r)))) by SIN_COS:31 .= cot (- (arccot (- r))) by SIN_COS4:def_2 ; - r in [.(- 1),1.] by A4, A3, XXREAL_1:1; then A6: arccot (- r) in [.(PI / 4),((3 / 4) * PI).] by Th50; then arccot (- r) <= (3 / 4) * PI by XXREAL_1:1; then - (arccot (- r)) >= - ((3 / 4) * PI) by XREAL_1:24; then A7: PI + (- (arccot (- r))) >= PI + (- ((3 / 4) * PI)) by XREAL_1:6; PI / 4 <= arccot (- r) by A6, XXREAL_1:1; then - (PI / 4) >= - (arccot (- r)) by XREAL_1:24; then PI + (- (PI / 4)) >= PI + (- (arccot (- r))) by XREAL_1:6; then A8: PI + (- (arccot (- r))) in [.(PI / 4),((3 / 4) * PI).] by A7, XXREAL_1:1; A9: [.(PI / 4),((3 / 4) * PI).] c= ].0,PI.[ by Lm9, Lm10, XXREAL_2:def_12; then A10: PI + (- (arccot (- r))) < PI by A8, XXREAL_1:4; A11: cot (PI + (- (arccot (- r)))) = (cos (PI + (- (arccot (- r))))) / (sin (PI + (- (arccot (- r))))) by SIN_COS4:def_2 .= (- (cos (- (arccot (- r))))) / (sin (PI + (- (arccot (- r))))) by SIN_COS:79 .= (- (cos (- (arccot (- r))))) / (- (sin (- (arccot (- r))))) by SIN_COS:79 .= (cos (- (arccot (- r)))) / (sin (- (arccot (- r)))) by XCMPLX_1:191 .= cot (- (arccot (- r))) by SIN_COS4:def_2 ; 0 < PI + (- (arccot (- r))) by A8, A9, XXREAL_1:4; hence arccot r = PI - (arccot (- r)) by A5, A10, A11, Th36; ::_thesis: verum end; theorem :: SIN_COS9:69 for r being Real st - 1 <= r & r <= 1 holds cot (arctan r) = 1 / r proof let r be Real; ::_thesis: ( - 1 <= r & r <= 1 implies cot (arctan r) = 1 / r ) set x = arctan r; assume that A1: - 1 <= r and A2: r <= 1 ; ::_thesis: cot (arctan r) = 1 / r A3: (sin (arctan r)) / (cos (arctan r)) = tan (arctan r) by SIN_COS4:def_1 .= r by A1, A2, Th51 ; cot (arctan r) = (cos (arctan r)) / (sin (arctan r)) by SIN_COS4:def_2 .= 1 / r by A3, XCMPLX_1:57 ; hence cot (arctan r) = 1 / r ; ::_thesis: verum end; theorem :: SIN_COS9:70 for r being Real st - 1 <= r & r <= 1 holds tan (arccot r) = 1 / r proof let r be Real; ::_thesis: ( - 1 <= r & r <= 1 implies tan (arccot r) = 1 / r ) set x = arccot r; assume that A1: - 1 <= r and A2: r <= 1 ; ::_thesis: tan (arccot r) = 1 / r A3: (cos (arccot r)) / (sin (arccot r)) = cot (arccot r) by SIN_COS4:def_2 .= r by A1, A2, Th52 ; tan (arccot r) = (sin (arccot r)) / (cos (arccot r)) by SIN_COS4:def_1 .= 1 / r by A3, XCMPLX_1:57 ; hence tan (arccot r) = 1 / r ; ::_thesis: verum end; theorem Th71: :: SIN_COS9:71 arctan is_differentiable_on tan .: ].(- (PI / 2)),(PI / 2).[ proof set f = tan | ].(- (PI / 2)),(PI / 2).[; A1: dom ((tan | ].(- (PI / 2)),(PI / 2).[) ") = rng (tan | ].(- (PI / 2)),(PI / 2).[) by FUNCT_1:33 .= tan .: ].(- (PI / 2)),(PI / 2).[ by RELAT_1:115 ; dom (tan | ].(- (PI / 2)),(PI / 2).[) = (dom tan) /\ ].(- (PI / 2)),(PI / 2).[ by RELAT_1:61; then A2: ].(- (PI / 2)),(PI / 2).[ c= dom (tan | ].(- (PI / 2)),(PI / 2).[) by Th1, XBOOLE_1:19; A3: tan | ].(- (PI / 2)),(PI / 2).[ is_differentiable_on ].(- (PI / 2)),(PI / 2).[ by Lm1, FDIFF_2:16; A4: now__::_thesis:_for_x0_being_Real_st_x0_in_].(-_(PI_/_2)),(PI_/_2).[_holds_ 0_<_diff_((tan_|_].(-_(PI_/_2)),(PI_/_2).[),x0) A5: for x0 being Real st x0 in ].(- (PI / 2)),(PI / 2).[ holds 1 / ((cos . x0) ^2) > 0 proof let x0 be Real; ::_thesis: ( x0 in ].(- (PI / 2)),(PI / 2).[ implies 1 / ((cos . x0) ^2) > 0 ) assume x0 in ].(- (PI / 2)),(PI / 2).[ ; ::_thesis: 1 / ((cos . x0) ^2) > 0 then 0 < cos . x0 by COMPTRIG:11; then (cos . x0) ^2 > 0 by SQUARE_1:12; then 1 / ((cos . x0) ^2) > 0 / ((cos . x0) ^2) by XREAL_1:74; hence 1 / ((cos . x0) ^2) > 0 ; ::_thesis: verum end; let x0 be Real; ::_thesis: ( x0 in ].(- (PI / 2)),(PI / 2).[ implies 0 < diff ((tan | ].(- (PI / 2)),(PI / 2).[),x0) ) assume A6: x0 in ].(- (PI / 2)),(PI / 2).[ ; ::_thesis: 0 < diff ((tan | ].(- (PI / 2)),(PI / 2).[),x0) diff ((tan | ].(- (PI / 2)),(PI / 2).[),x0) = ((tan | ].(- (PI / 2)),(PI / 2).[) `| ].(- (PI / 2)),(PI / 2).[) . x0 by A3, A6, FDIFF_1:def_7 .= (tan `| ].(- (PI / 2)),(PI / 2).[) . x0 by Lm1, FDIFF_2:16 .= diff (tan,x0) by A6, Lm1, FDIFF_1:def_7 .= 1 / ((cos . x0) ^2) by A6, Lm3 ; hence 0 < diff ((tan | ].(- (PI / 2)),(PI / 2).[),x0) by A6, A5; ::_thesis: verum end; (tan | ].(- (PI / 2)),(PI / 2).[) | ].(- (PI / 2)),(PI / 2).[ = tan | ].(- (PI / 2)),(PI / 2).[ by RELAT_1:72; hence arctan is_differentiable_on tan .: ].(- (PI / 2)),(PI / 2).[ by A1, A3, A2, A4, FDIFF_2:48; ::_thesis: verum end; theorem Th72: :: SIN_COS9:72 arccot is_differentiable_on cot .: ].0,PI.[ proof set f = cot | ].0,PI.[; A1: dom ((cot | ].0,PI.[) ") = rng (cot | ].0,PI.[) by FUNCT_1:33 .= cot .: ].0,PI.[ by RELAT_1:115 ; dom (cot | ].0,PI.[) = (dom cot) /\ ].0,PI.[ by RELAT_1:61; then A2: ].0,PI.[ c= dom (cot | ].0,PI.[) by Th2, XBOOLE_1:19; A3: cot | ].0,PI.[ is_differentiable_on ].0,PI.[ by Lm2, FDIFF_2:16; A4: now__::_thesis:_for_x0_being_Real_st_x0_in_].0,PI.[_holds_ diff_((cot_|_].0,PI.[),x0)_<_0 A5: for x0 being Real st x0 in ].0,PI.[ holds - (1 / ((sin . x0) ^2)) < 0 proof let x0 be Real; ::_thesis: ( x0 in ].0,PI.[ implies - (1 / ((sin . x0) ^2)) < 0 ) assume x0 in ].0,PI.[ ; ::_thesis: - (1 / ((sin . x0) ^2)) < 0 then 0 < sin . x0 by COMPTRIG:7; then (sin . x0) ^2 > 0 by SQUARE_1:12; then 1 / ((sin . x0) ^2) > 0 / ((sin . x0) ^2) by XREAL_1:74; then - (1 / ((sin . x0) ^2)) < - 0 by XREAL_1:24; hence - (1 / ((sin . x0) ^2)) < 0 ; ::_thesis: verum end; let x0 be Real; ::_thesis: ( x0 in ].0,PI.[ implies diff ((cot | ].0,PI.[),x0) < 0 ) assume A6: x0 in ].0,PI.[ ; ::_thesis: diff ((cot | ].0,PI.[),x0) < 0 diff ((cot | ].0,PI.[),x0) = ((cot | ].0,PI.[) `| ].0,PI.[) . x0 by A3, A6, FDIFF_1:def_7 .= (cot `| ].0,PI.[) . x0 by Lm2, FDIFF_2:16 .= diff (cot,x0) by A6, Lm2, FDIFF_1:def_7 .= - (1 / ((sin . x0) ^2)) by A6, Lm4 ; hence diff ((cot | ].0,PI.[),x0) < 0 by A6, A5; ::_thesis: verum end; (cot | ].0,PI.[) | ].0,PI.[ = cot | ].0,PI.[ by RELAT_1:72; hence arccot is_differentiable_on cot .: ].0,PI.[ by A1, A2, A3, A4, FDIFF_2:48; ::_thesis: verum end; theorem Th73: :: SIN_COS9:73 arctan is_differentiable_on ].(- 1),1.[ proof ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25; then A1: ].(- 1),1.[ c= dom arctan by Th23, XBOOLE_1:1; for x being Real st x in ].(- 1),1.[ holds arctan is_differentiable_in x proof let x be Real; ::_thesis: ( x in ].(- 1),1.[ implies arctan is_differentiable_in x ) A2: dom arctan = rng (tan | ].(- (PI / 2)),(PI / 2).[) by FUNCT_1:33 .= tan .: ].(- (PI / 2)),(PI / 2).[ by RELAT_1:115 ; assume x in ].(- 1),1.[ ; ::_thesis: arctan is_differentiable_in x then arctan | (dom arctan) is_differentiable_in x by A1, A2, Th71, FDIFF_1:def_6; hence arctan is_differentiable_in x by RELAT_1:69; ::_thesis: verum end; hence arctan is_differentiable_on ].(- 1),1.[ by A1, FDIFF_1:9; ::_thesis: verum end; theorem Th74: :: SIN_COS9:74 arccot is_differentiable_on ].(- 1),1.[ proof ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25; then A1: ].(- 1),1.[ c= dom arccot by Th24, XBOOLE_1:1; for x being Real st x in ].(- 1),1.[ holds arccot is_differentiable_in x proof let x be Real; ::_thesis: ( x in ].(- 1),1.[ implies arccot is_differentiable_in x ) A2: dom arccot = rng (cot | ].0,PI.[) by FUNCT_1:33 .= cot .: ].0,PI.[ by RELAT_1:115 ; assume x in ].(- 1),1.[ ; ::_thesis: arccot is_differentiable_in x then arccot | (dom arccot) is_differentiable_in x by A1, A2, Th72, FDIFF_1:def_6; hence arccot is_differentiable_in x by RELAT_1:69; ::_thesis: verum end; hence arccot is_differentiable_on ].(- 1),1.[ by A1, FDIFF_1:9; ::_thesis: verum end; theorem Th75: :: SIN_COS9:75 for r being Real st - 1 <= r & r <= 1 holds diff (arctan,r) = 1 / (1 + (r ^2)) proof let r be Real; ::_thesis: ( - 1 <= r & r <= 1 implies diff (arctan,r) = 1 / (1 + (r ^2)) ) set g = arctan ; set f = tan | ].(- (PI / 2)),(PI / 2).[; set x = arctan . r; assume that A1: - 1 <= r and A2: r <= 1 ; ::_thesis: diff (arctan,r) = 1 / (1 + (r ^2)) A3: ((sin . (arctan . r)) ^2) + ((cos . (arctan . r)) ^2) = 1 by SIN_COS:28; A4: tan | ].(- (PI / 2)),(PI / 2).[ is_differentiable_on ].(- (PI / 2)),(PI / 2).[ by Lm1, FDIFF_2:16; A5: now__::_thesis:_for_x0_being_Real_st_x0_in_].(-_(PI_/_2)),(PI_/_2).[_holds_ 0_<_diff_((tan_|_].(-_(PI_/_2)),(PI_/_2).[),x0) A6: for x0 being Real st x0 in ].(- (PI / 2)),(PI / 2).[ holds 1 / ((cos . x0) ^2) > 0 proof let x0 be Real; ::_thesis: ( x0 in ].(- (PI / 2)),(PI / 2).[ implies 1 / ((cos . x0) ^2) > 0 ) assume x0 in ].(- (PI / 2)),(PI / 2).[ ; ::_thesis: 1 / ((cos . x0) ^2) > 0 then 0 < cos . x0 by COMPTRIG:11; then (cos . x0) ^2 > 0 by SQUARE_1:12; then 1 / ((cos . x0) ^2) > 0 / ((cos . x0) ^2) by XREAL_1:74; hence 1 / ((cos . x0) ^2) > 0 ; ::_thesis: verum end; let x0 be Real; ::_thesis: ( x0 in ].(- (PI / 2)),(PI / 2).[ implies 0 < diff ((tan | ].(- (PI / 2)),(PI / 2).[),x0) ) assume A7: x0 in ].(- (PI / 2)),(PI / 2).[ ; ::_thesis: 0 < diff ((tan | ].(- (PI / 2)),(PI / 2).[),x0) diff ((tan | ].(- (PI / 2)),(PI / 2).[),x0) = ((tan | ].(- (PI / 2)),(PI / 2).[) `| ].(- (PI / 2)),(PI / 2).[) . x0 by A4, A7, FDIFF_1:def_7 .= (tan `| ].(- (PI / 2)),(PI / 2).[) . x0 by Lm1, FDIFF_2:16 .= diff (tan,x0) by A7, Lm1, FDIFF_1:def_7 .= 1 / ((cos . x0) ^2) by A7, Lm3 ; hence 0 < diff ((tan | ].(- (PI / 2)),(PI / 2).[),x0) by A7, A6; ::_thesis: verum end; A8: r in [.(- 1),1.] by A1, A2, XXREAL_1:1; then A9: arctan . r in [.(- (PI / 4)),(PI / 4).] by Th49; arctan . r = arctan r ; then A10: r = tan (arctan . r) by A1, A2, Th51 .= (sin (arctan . r)) / (cos (arctan . r)) by SIN_COS4:def_1 ; dom (tan | ].(- (PI / 2)),(PI / 2).[) = (dom tan) /\ ].(- (PI / 2)),(PI / 2).[ by RELAT_1:61; then A11: ].(- (PI / 2)),(PI / 2).[ c= dom (tan | ].(- (PI / 2)),(PI / 2).[) by Th1, XBOOLE_1:19; A12: (tan | ].(- (PI / 2)),(PI / 2).[) | ].(- (PI / 2)),(PI / 2).[ = tan | ].(- (PI / 2)),(PI / 2).[ by RELAT_1:72; A13: [.(- (PI / 4)),(PI / 4).] c= ].(- (PI / 2)),(PI / 2).[ by Lm7, Lm8, XXREAL_2:def_12; then cos (arctan . r) <> 0 by A9, COMPTRIG:11; then r * (cos (arctan . r)) = sin (arctan . r) by A10, XCMPLX_1:87; then A14: 1 = ((cos (arctan . r)) ^2) * ((r ^2) + 1) by A3; tan | ].(- (PI / 2)),(PI / 2).[ is_differentiable_on ].(- (PI / 2)),(PI / 2).[ by Lm1, FDIFF_2:16; then diff ((tan | ].(- (PI / 2)),(PI / 2).[),(arctan . r)) = ((tan | ].(- (PI / 2)),(PI / 2).[) `| ].(- (PI / 2)),(PI / 2).[) . (arctan . r) by A9, A13, FDIFF_1:def_7 .= (tan `| ].(- (PI / 2)),(PI / 2).[) . (arctan . r) by Lm1, FDIFF_2:16 .= diff (tan,(arctan . r)) by A9, A13, Lm1, FDIFF_1:def_7 .= 1 / ((cos (arctan . r)) ^2) by A9, A13, Lm3 ; then diff (arctan,r) = 1 / (1 / ((cos (arctan . r)) ^2)) by A8, A4, A5, A12, A11, Th23, FDIFF_2:48 .= 1 / ((r ^2) + 1) by A14, XCMPLX_1:73 ; hence diff (arctan,r) = 1 / (1 + (r ^2)) ; ::_thesis: verum end; theorem Th76: :: SIN_COS9:76 for r being Real st - 1 <= r & r <= 1 holds diff (arccot,r) = - (1 / (1 + (r ^2))) proof let r be Real; ::_thesis: ( - 1 <= r & r <= 1 implies diff (arccot,r) = - (1 / (1 + (r ^2))) ) set g = arccot ; set f = cot | ].0,PI.[; set x = arccot . r; assume that A1: - 1 <= r and A2: r <= 1 ; ::_thesis: diff (arccot,r) = - (1 / (1 + (r ^2))) A3: ((sin . (arccot . r)) ^2) + ((cos . (arccot . r)) ^2) = 1 by SIN_COS:28; A4: cot | ].0,PI.[ is_differentiable_on ].0,PI.[ by Lm2, FDIFF_2:16; A5: now__::_thesis:_for_x0_being_Real_st_x0_in_].0,PI.[_holds_ diff_((cot_|_].0,PI.[),x0)_<_0 A6: for x0 being Real st x0 in ].0,PI.[ holds - (1 / ((sin . x0) ^2)) < 0 proof let x0 be Real; ::_thesis: ( x0 in ].0,PI.[ implies - (1 / ((sin . x0) ^2)) < 0 ) assume x0 in ].0,PI.[ ; ::_thesis: - (1 / ((sin . x0) ^2)) < 0 then 0 < sin . x0 by COMPTRIG:7; then (sin . x0) ^2 > 0 by SQUARE_1:12; then 1 / ((sin . x0) ^2) > 0 / ((sin . x0) ^2) by XREAL_1:74; then - (1 / ((sin . x0) ^2)) < - 0 by XREAL_1:24; hence - (1 / ((sin . x0) ^2)) < 0 ; ::_thesis: verum end; let x0 be Real; ::_thesis: ( x0 in ].0,PI.[ implies diff ((cot | ].0,PI.[),x0) < 0 ) assume A7: x0 in ].0,PI.[ ; ::_thesis: diff ((cot | ].0,PI.[),x0) < 0 diff ((cot | ].0,PI.[),x0) = ((cot | ].0,PI.[) `| ].0,PI.[) . x0 by A4, A7, FDIFF_1:def_7 .= (cot `| ].0,PI.[) . x0 by Lm2, FDIFF_2:16 .= diff (cot,x0) by A7, Lm2, FDIFF_1:def_7 .= - (1 / ((sin . x0) ^2)) by A7, Lm4 ; hence diff ((cot | ].0,PI.[),x0) < 0 by A7, A6; ::_thesis: verum end; A8: r in [.(- 1),1.] by A1, A2, XXREAL_1:1; then A9: arccot . r in [.(PI / 4),((3 / 4) * PI).] by Th50; arccot . r = arccot r ; then A10: r = cot (arccot . r) by A1, A2, Th52 .= (cos (arccot . r)) / (sin (arccot . r)) by SIN_COS4:def_2 ; dom (cot | ].0,PI.[) = (dom cot) /\ ].0,PI.[ by RELAT_1:61; then A11: ].0,PI.[ c= dom (cot | ].0,PI.[) by Th2, XBOOLE_1:19; A12: (cot | ].0,PI.[) | ].0,PI.[ = cot | ].0,PI.[ by RELAT_1:72; A13: [.(PI / 4),((3 / 4) * PI).] c= ].0,PI.[ by Lm9, Lm10, XXREAL_2:def_12; then sin (arccot . r) <> 0 by A9, COMPTRIG:7; then r * (sin (arccot . r)) = cos (arccot . r) by A10, XCMPLX_1:87; then A14: 1 = ((sin (arccot . r)) ^2) * ((r ^2) + 1) by A3; cot | ].0,PI.[ is_differentiable_on ].0,PI.[ by Lm2, FDIFF_2:16; then diff ((cot | ].0,PI.[),(arccot . r)) = ((cot | ].0,PI.[) `| ].0,PI.[) . (arccot . r) by A9, A13, FDIFF_1:def_7 .= (cot `| ].0,PI.[) . (arccot . r) by Lm2, FDIFF_2:16 .= diff (cot,(arccot . r)) by A9, A13, Lm2, FDIFF_1:def_7 .= - (1 / ((sin (arccot . r)) ^2)) by A9, A13, Lm4 ; then diff (arccot,r) = 1 / (- (1 / ((sin (arccot . r)) ^2))) by A8, A4, A5, A12, A11, Th24, FDIFF_2:48 .= - (1 / (1 / ((sin (arccot . r)) ^2))) by XCMPLX_1:188 .= - (1 / ((r ^2) + 1)) by A14, XCMPLX_1:73 ; hence diff (arccot,r) = - (1 / (1 + (r ^2))) ; ::_thesis: verum end; theorem :: SIN_COS9:77 arctan | (tan .: ].(- (PI / 2)),(PI / 2).[) is continuous by Th71, FDIFF_1:25; theorem :: SIN_COS9:78 arccot | (cot .: ].0,PI.[) is continuous by Th72, FDIFF_1:25; theorem :: SIN_COS9:79 dom arctan is open proof for x0 being Real st x0 in ].(- (PI / 2)),(PI / 2).[ holds 0 < diff (tan,x0) proof let x0 be Real; ::_thesis: ( x0 in ].(- (PI / 2)),(PI / 2).[ implies 0 < diff (tan,x0) ) assume A1: x0 in ].(- (PI / 2)),(PI / 2).[ ; ::_thesis: 0 < diff (tan,x0) then 0 < cos . x0 by COMPTRIG:11; then (cos . x0) ^2 > 0 by SQUARE_1:12; then 1 / ((cos . x0) ^2) > 0 / ((cos . x0) ^2) by XREAL_1:74; hence 0 < diff (tan,x0) by A1, Lm3; ::_thesis: verum end; then rng (tan | ].(- (PI / 2)),(PI / 2).[) is open by Lm1, Th1, FDIFF_2:41; hence dom arctan is open by FUNCT_1:33; ::_thesis: verum end; theorem :: SIN_COS9:80 dom arccot is open proof for x0 being Real st x0 in ].0,PI.[ holds diff (cot,x0) < 0 proof let x0 be Real; ::_thesis: ( x0 in ].0,PI.[ implies diff (cot,x0) < 0 ) assume A1: x0 in ].0,PI.[ ; ::_thesis: diff (cot,x0) < 0 then 0 < sin . x0 by COMPTRIG:7; then (sin . x0) ^2 > 0 by SQUARE_1:12; then 1 / ((sin . x0) ^2) > 0 / ((sin . x0) ^2) by XREAL_1:74; then - (1 / ((sin . x0) ^2)) < - 0 by XREAL_1:24; hence diff (cot,x0) < 0 by A1, Lm4; ::_thesis: verum end; then rng (cot | ].0,PI.[) is open by Lm2, Th2, FDIFF_2:41; hence dom arccot is open by FUNCT_1:33; ::_thesis: verum end; begin theorem Th81: :: SIN_COS9:81 for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds ( arctan is_differentiable_on Z & ( for x being Real st x in Z holds (arctan `| Z) . x = 1 / (1 + (x ^2)) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= ].(- 1),1.[ implies ( arctan is_differentiable_on Z & ( for x being Real st x in Z holds (arctan `| Z) . x = 1 / (1 + (x ^2)) ) ) ) assume A1: Z c= ].(- 1),1.[ ; ::_thesis: ( arctan is_differentiable_on Z & ( for x being Real st x in Z holds (arctan `| Z) . x = 1 / (1 + (x ^2)) ) ) then A2: arctan is_differentiable_on Z by Th73, FDIFF_1:26; for x being Real st x in Z holds (arctan `| Z) . x = 1 / (1 + (x ^2)) proof let x be Real; ::_thesis: ( x in Z implies (arctan `| Z) . x = 1 / (1 + (x ^2)) ) assume A3: x in Z ; ::_thesis: (arctan `| Z) . x = 1 / (1 + (x ^2)) then A4: - 1 <= x by A1, XXREAL_1:4; A5: x <= 1 by A1, A3, XXREAL_1:4; thus (arctan `| Z) . x = diff (arctan,x) by A2, A3, FDIFF_1:def_7 .= 1 / (1 + (x ^2)) by A4, A5, Th75 ; ::_thesis: verum end; hence ( arctan is_differentiable_on Z & ( for x being Real st x in Z holds (arctan `| Z) . x = 1 / (1 + (x ^2)) ) ) by A1, Th73, FDIFF_1:26; ::_thesis: verum end; theorem Th82: :: SIN_COS9:82 for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds ( arccot is_differentiable_on Z & ( for x being Real st x in Z holds (arccot `| Z) . x = - (1 / (1 + (x ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= ].(- 1),1.[ implies ( arccot is_differentiable_on Z & ( for x being Real st x in Z holds (arccot `| Z) . x = - (1 / (1 + (x ^2))) ) ) ) assume A1: Z c= ].(- 1),1.[ ; ::_thesis: ( arccot is_differentiable_on Z & ( for x being Real st x in Z holds (arccot `| Z) . x = - (1 / (1 + (x ^2))) ) ) then A2: arccot is_differentiable_on Z by Th74, FDIFF_1:26; for x being Real st x in Z holds (arccot `| Z) . x = - (1 / (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies (arccot `| Z) . x = - (1 / (1 + (x ^2))) ) assume A3: x in Z ; ::_thesis: (arccot `| Z) . x = - (1 / (1 + (x ^2))) then A4: - 1 <= x by A1, XXREAL_1:4; A5: x <= 1 by A1, A3, XXREAL_1:4; thus (arccot `| Z) . x = diff (arccot,x) by A2, A3, FDIFF_1:def_7 .= - (1 / (1 + (x ^2))) by A4, A5, Th76 ; ::_thesis: verum end; hence ( arccot is_differentiable_on Z & ( for x being Real st x in Z holds (arccot `| Z) . x = - (1 / (1 + (x ^2))) ) ) by A1, Th74, FDIFF_1:26; ::_thesis: verum end; theorem :: SIN_COS9:83 for r being Real for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds ( r (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((r (#) arctan) `| Z) . x = r / (1 + (x ^2)) ) ) proof let r be Real; ::_thesis: for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds ( r (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((r (#) arctan) `| Z) . x = r / (1 + (x ^2)) ) ) let Z be open Subset of REAL; ::_thesis: ( Z c= ].(- 1),1.[ implies ( r (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((r (#) arctan) `| Z) . x = r / (1 + (x ^2)) ) ) ) assume A1: Z c= ].(- 1),1.[ ; ::_thesis: ( r (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((r (#) arctan) `| Z) . x = r / (1 + (x ^2)) ) ) ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25; then ].(- 1),1.[ c= dom arctan by Th23, XBOOLE_1:1; then Z c= dom arctan by A1, XBOOLE_1:1; then A2: Z c= dom (r (#) arctan) by VALUED_1:def_5; A3: arctan is_differentiable_on Z by A1, Th81; for x being Real st x in Z holds ((r (#) arctan) `| Z) . x = r / (1 + (x ^2)) proof let x be Real; ::_thesis: ( x in Z implies ((r (#) arctan) `| Z) . x = r / (1 + (x ^2)) ) assume A4: x in Z ; ::_thesis: ((r (#) arctan) `| Z) . x = r / (1 + (x ^2)) then A5: - 1 < x by A1, XXREAL_1:4; A6: x < 1 by A1, A4, XXREAL_1:4; ((r (#) arctan) `| Z) . x = r * (diff (arctan,x)) by A2, A3, A4, FDIFF_1:20 .= r * (1 / (1 + (x ^2))) by A5, A6, Th75 .= r / (1 + (x ^2)) ; hence ((r (#) arctan) `| Z) . x = r / (1 + (x ^2)) ; ::_thesis: verum end; hence ( r (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((r (#) arctan) `| Z) . x = r / (1 + (x ^2)) ) ) by A2, A3, FDIFF_1:20; ::_thesis: verum end; theorem :: SIN_COS9:84 for r being Real for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds ( r (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((r (#) arccot) `| Z) . x = - (r / (1 + (x ^2))) ) ) proof let r be Real; ::_thesis: for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds ( r (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((r (#) arccot) `| Z) . x = - (r / (1 + (x ^2))) ) ) let Z be open Subset of REAL; ::_thesis: ( Z c= ].(- 1),1.[ implies ( r (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((r (#) arccot) `| Z) . x = - (r / (1 + (x ^2))) ) ) ) assume A1: Z c= ].(- 1),1.[ ; ::_thesis: ( r (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((r (#) arccot) `| Z) . x = - (r / (1 + (x ^2))) ) ) ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25; then ].(- 1),1.[ c= dom arccot by Th24, XBOOLE_1:1; then Z c= dom arccot by A1, XBOOLE_1:1; then A2: Z c= dom (r (#) arccot) by VALUED_1:def_5; A3: arccot is_differentiable_on Z by A1, Th82; for x being Real st x in Z holds ((r (#) arccot) `| Z) . x = - (r / (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((r (#) arccot) `| Z) . x = - (r / (1 + (x ^2))) ) assume A4: x in Z ; ::_thesis: ((r (#) arccot) `| Z) . x = - (r / (1 + (x ^2))) then A5: - 1 < x by A1, XXREAL_1:4; A6: x < 1 by A1, A4, XXREAL_1:4; ((r (#) arccot) `| Z) . x = r * (diff (arccot,x)) by A2, A3, A4, FDIFF_1:20 .= r * (- (1 / (1 + (x ^2)))) by A5, A6, Th76 .= - (r / (1 + (x ^2))) ; hence ((r (#) arccot) `| Z) . x = - (r / (1 + (x ^2))) ; ::_thesis: verum end; hence ( r (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((r (#) arccot) `| Z) . x = - (r / (1 + (x ^2))) ) ) by A2, A3, FDIFF_1:20; ::_thesis: verum end; theorem Th85: :: SIN_COS9:85 for x being Real for f being PartFunc of REAL,REAL st f is_differentiable_in x & f . x > - 1 & f . x < 1 holds ( arctan * f is_differentiable_in x & diff ((arctan * f),x) = (diff (f,x)) / (1 + ((f . x) ^2)) ) proof let x be Real; ::_thesis: for f being PartFunc of REAL,REAL st f is_differentiable_in x & f . x > - 1 & f . x < 1 holds ( arctan * f is_differentiable_in x & diff ((arctan * f),x) = (diff (f,x)) / (1 + ((f . x) ^2)) ) let f be PartFunc of REAL,REAL; ::_thesis: ( f is_differentiable_in x & f . x > - 1 & f . x < 1 implies ( arctan * f is_differentiable_in x & diff ((arctan * f),x) = (diff (f,x)) / (1 + ((f . x) ^2)) ) ) assume that A1: f is_differentiable_in x and A2: f . x > - 1 and A3: f . x < 1 ; ::_thesis: ( arctan * f is_differentiable_in x & diff ((arctan * f),x) = (diff (f,x)) / (1 + ((f . x) ^2)) ) f . x in ].(- 1),1.[ by A2, A3, XXREAL_1:4; then A4: arctan is_differentiable_in f . x by Th73, FDIFF_1:9; then diff ((arctan * f),x) = (diff (arctan,(f . x))) * (diff (f,x)) by A1, FDIFF_2:13 .= (diff (f,x)) * (1 / (1 + ((f . x) ^2))) by A2, A3, Th75 .= (diff (f,x)) / (1 + ((f . x) ^2)) ; hence ( arctan * f is_differentiable_in x & diff ((arctan * f),x) = (diff (f,x)) / (1 + ((f . x) ^2)) ) by A1, A4, FDIFF_2:13; ::_thesis: verum end; theorem Th86: :: SIN_COS9:86 for x being Real for f being PartFunc of REAL,REAL st f is_differentiable_in x & f . x > - 1 & f . x < 1 holds ( arccot * f is_differentiable_in x & diff ((arccot * f),x) = - ((diff (f,x)) / (1 + ((f . x) ^2))) ) proof let x be Real; ::_thesis: for f being PartFunc of REAL,REAL st f is_differentiable_in x & f . x > - 1 & f . x < 1 holds ( arccot * f is_differentiable_in x & diff ((arccot * f),x) = - ((diff (f,x)) / (1 + ((f . x) ^2))) ) let f be PartFunc of REAL,REAL; ::_thesis: ( f is_differentiable_in x & f . x > - 1 & f . x < 1 implies ( arccot * f is_differentiable_in x & diff ((arccot * f),x) = - ((diff (f,x)) / (1 + ((f . x) ^2))) ) ) assume that A1: f is_differentiable_in x and A2: f . x > - 1 and A3: f . x < 1 ; ::_thesis: ( arccot * f is_differentiable_in x & diff ((arccot * f),x) = - ((diff (f,x)) / (1 + ((f . x) ^2))) ) f . x in ].(- 1),1.[ by A2, A3, XXREAL_1:4; then A4: arccot is_differentiable_in f . x by Th74, FDIFF_1:9; then diff ((arccot * f),x) = (diff (arccot,(f . x))) * (diff (f,x)) by A1, FDIFF_2:13 .= (diff (f,x)) * (- (1 / (1 + ((f . x) ^2)))) by A2, A3, Th76 .= - ((diff (f,x)) / (1 + ((f . x) ^2))) ; hence ( arccot * f is_differentiable_in x & diff ((arccot * f),x) = - ((diff (f,x)) / (1 + ((f . x) ^2))) ) by A1, A4, FDIFF_2:13; ::_thesis: verum end; theorem Th87: :: SIN_COS9:87 for r, s being Real for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom (arctan * f) & ( for x being Real st x in Z holds ( f . x = (r * x) + s & f . x > - 1 & f . x < 1 ) ) holds ( arctan * f is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * f) `| Z) . x = r / (1 + (((r * x) + s) ^2)) ) ) proof let r, s be Real; ::_thesis: for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom (arctan * f) & ( for x being Real st x in Z holds ( f . x = (r * x) + s & f . x > - 1 & f . x < 1 ) ) holds ( arctan * f is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * f) `| Z) . x = r / (1 + (((r * x) + s) ^2)) ) ) let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st Z c= dom (arctan * f) & ( for x being Real st x in Z holds ( f . x = (r * x) + s & f . x > - 1 & f . x < 1 ) ) holds ( arctan * f is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * f) `| Z) . x = r / (1 + (((r * x) + s) ^2)) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (arctan * f) & ( for x being Real st x in Z holds ( f . x = (r * x) + s & f . x > - 1 & f . x < 1 ) ) implies ( arctan * f is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * f) `| Z) . x = r / (1 + (((r * x) + s) ^2)) ) ) ) assume that A1: Z c= dom (arctan * f) and A2: for x being Real st x in Z holds ( f . x = (r * x) + s & f . x > - 1 & f . x < 1 ) ; ::_thesis: ( arctan * f is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * f) `| Z) . x = r / (1 + (((r * x) + s) ^2)) ) ) for y being set st y in Z holds y in dom f by A1, FUNCT_1:11; then A3: Z c= dom f by TARSKI:def_3; A4: for x being Real st x in Z holds f . x = (r * x) + s by A2; then A5: f is_differentiable_on Z by A3, FDIFF_1:23; A6: for x being Real st x in Z holds arctan * f is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies arctan * f is_differentiable_in x ) assume A7: x in Z ; ::_thesis: arctan * f is_differentiable_in x then A8: f . x > - 1 by A2; A9: f . x < 1 by A2, A7; f is_differentiable_in x by A5, A7, FDIFF_1:9; hence arctan * f is_differentiable_in x by A8, A9, Th85; ::_thesis: verum end; then A10: arctan * f is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((arctan * f) `| Z) . x = r / (1 + (((r * x) + s) ^2)) proof let x be Real; ::_thesis: ( x in Z implies ((arctan * f) `| Z) . x = r / (1 + (((r * x) + s) ^2)) ) assume A11: x in Z ; ::_thesis: ((arctan * f) `| Z) . x = r / (1 + (((r * x) + s) ^2)) then A12: f . x > - 1 by A2; A13: f . x < 1 by A2, A11; f is_differentiable_in x by A5, A11, FDIFF_1:9; then diff ((arctan * f),x) = (diff (f,x)) / (1 + ((f . x) ^2)) by A12, A13, Th85 .= ((f `| Z) . x) / (1 + ((f . x) ^2)) by A5, A11, FDIFF_1:def_7 .= r / (1 + ((f . x) ^2)) by A4, A3, A11, FDIFF_1:23 .= r / (1 + (((r * x) + s) ^2)) by A2, A11 ; hence ((arctan * f) `| Z) . x = r / (1 + (((r * x) + s) ^2)) by A10, A11, FDIFF_1:def_7; ::_thesis: verum end; hence ( arctan * f is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * f) `| Z) . x = r / (1 + (((r * x) + s) ^2)) ) ) by A1, A6, FDIFF_1:9; ::_thesis: verum end; theorem Th88: :: SIN_COS9:88 for r, s being Real for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom (arccot * f) & ( for x being Real st x in Z holds ( f . x = (r * x) + s & f . x > - 1 & f . x < 1 ) ) holds ( arccot * f is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * f) `| Z) . x = - (r / (1 + (((r * x) + s) ^2))) ) ) proof let r, s be Real; ::_thesis: for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom (arccot * f) & ( for x being Real st x in Z holds ( f . x = (r * x) + s & f . x > - 1 & f . x < 1 ) ) holds ( arccot * f is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * f) `| Z) . x = - (r / (1 + (((r * x) + s) ^2))) ) ) let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st Z c= dom (arccot * f) & ( for x being Real st x in Z holds ( f . x = (r * x) + s & f . x > - 1 & f . x < 1 ) ) holds ( arccot * f is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * f) `| Z) . x = - (r / (1 + (((r * x) + s) ^2))) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (arccot * f) & ( for x being Real st x in Z holds ( f . x = (r * x) + s & f . x > - 1 & f . x < 1 ) ) implies ( arccot * f is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * f) `| Z) . x = - (r / (1 + (((r * x) + s) ^2))) ) ) ) assume that A1: Z c= dom (arccot * f) and A2: for x being Real st x in Z holds ( f . x = (r * x) + s & f . x > - 1 & f . x < 1 ) ; ::_thesis: ( arccot * f is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * f) `| Z) . x = - (r / (1 + (((r * x) + s) ^2))) ) ) for y being set st y in Z holds y in dom f by A1, FUNCT_1:11; then A3: Z c= dom f by TARSKI:def_3; A4: for x being Real st x in Z holds f . x = (r * x) + s by A2; then A5: f is_differentiable_on Z by A3, FDIFF_1:23; A6: for x being Real st x in Z holds arccot * f is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies arccot * f is_differentiable_in x ) assume A7: x in Z ; ::_thesis: arccot * f is_differentiable_in x then A8: f . x > - 1 by A2; A9: f . x < 1 by A2, A7; f is_differentiable_in x by A5, A7, FDIFF_1:9; hence arccot * f is_differentiable_in x by A8, A9, Th86; ::_thesis: verum end; then A10: arccot * f is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((arccot * f) `| Z) . x = - (r / (1 + (((r * x) + s) ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((arccot * f) `| Z) . x = - (r / (1 + (((r * x) + s) ^2))) ) assume A11: x in Z ; ::_thesis: ((arccot * f) `| Z) . x = - (r / (1 + (((r * x) + s) ^2))) then A12: f . x > - 1 by A2; A13: f . x < 1 by A2, A11; f is_differentiable_in x by A5, A11, FDIFF_1:9; then diff ((arccot * f),x) = - ((diff (f,x)) / (1 + ((f . x) ^2))) by A12, A13, Th86 .= - (((f `| Z) . x) / (1 + ((f . x) ^2))) by A5, A11, FDIFF_1:def_7 .= - (r / (1 + ((f . x) ^2))) by A4, A3, A11, FDIFF_1:23 .= - (r / (1 + (((r * x) + s) ^2))) by A2, A11 ; hence ((arccot * f) `| Z) . x = - (r / (1 + (((r * x) + s) ^2))) by A10, A11, FDIFF_1:def_7; ::_thesis: verum end; hence ( arccot * f is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * f) `| Z) . x = - (r / (1 + (((r * x) + s) ^2))) ) ) by A1, A6, FDIFF_1:9; ::_thesis: verum end; theorem :: SIN_COS9:89 for Z being open Subset of REAL st Z c= dom (ln * arctan) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds arctan . x > 0 ) holds ( ln * arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((ln * arctan) `| Z) . x = 1 / ((1 + (x ^2)) * (arctan . x)) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (ln * arctan) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds arctan . x > 0 ) implies ( ln * arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((ln * arctan) `| Z) . x = 1 / ((1 + (x ^2)) * (arctan . x)) ) ) ) assume that A1: Z c= dom (ln * arctan) and A2: Z c= ].(- 1),1.[ and A3: for x being Real st x in Z holds arctan . x > 0 ; ::_thesis: ( ln * arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((ln * arctan) `| Z) . x = 1 / ((1 + (x ^2)) * (arctan . x)) ) ) A4: for x being Real st x in Z holds ln * arctan is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies ln * arctan is_differentiable_in x ) assume A5: x in Z ; ::_thesis: ln * arctan is_differentiable_in x arctan is_differentiable_on Z by A2, Th81; then A6: arctan is_differentiable_in x by A5, FDIFF_1:9; arctan . x > 0 by A3, A5; hence ln * arctan is_differentiable_in x by A6, TAYLOR_1:20; ::_thesis: verum end; then A7: ln * arctan is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((ln * arctan) `| Z) . x = 1 / ((1 + (x ^2)) * (arctan . x)) proof let x be Real; ::_thesis: ( x in Z implies ((ln * arctan) `| Z) . x = 1 / ((1 + (x ^2)) * (arctan . x)) ) assume A8: x in Z ; ::_thesis: ((ln * arctan) `| Z) . x = 1 / ((1 + (x ^2)) * (arctan . x)) then A9: - 1 < x by A2, XXREAL_1:4; arctan is_differentiable_on Z by A2, Th81; then A10: arctan is_differentiable_in x by A8, FDIFF_1:9; A11: x < 1 by A2, A8, XXREAL_1:4; arctan . x > 0 by A3, A8; then diff ((ln * arctan),x) = (diff (arctan,x)) / (arctan . x) by A10, TAYLOR_1:20 .= (1 / (1 + (x ^2))) / (arctan . x) by A9, A11, Th75 .= 1 / ((1 + (x ^2)) * (arctan . x)) by XCMPLX_1:78 ; hence ((ln * arctan) `| Z) . x = 1 / ((1 + (x ^2)) * (arctan . x)) by A7, A8, FDIFF_1:def_7; ::_thesis: verum end; hence ( ln * arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((ln * arctan) `| Z) . x = 1 / ((1 + (x ^2)) * (arctan . x)) ) ) by A1, A4, FDIFF_1:9; ::_thesis: verum end; theorem :: SIN_COS9:90 for Z being open Subset of REAL st Z c= dom (ln * arccot) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds arccot . x > 0 ) holds ( ln * arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((ln * arccot) `| Z) . x = - (1 / ((1 + (x ^2)) * (arccot . x))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (ln * arccot) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds arccot . x > 0 ) implies ( ln * arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((ln * arccot) `| Z) . x = - (1 / ((1 + (x ^2)) * (arccot . x))) ) ) ) assume that A1: Z c= dom (ln * arccot) and A2: Z c= ].(- 1),1.[ and A3: for x being Real st x in Z holds arccot . x > 0 ; ::_thesis: ( ln * arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((ln * arccot) `| Z) . x = - (1 / ((1 + (x ^2)) * (arccot . x))) ) ) A4: for x being Real st x in Z holds ln * arccot is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies ln * arccot is_differentiable_in x ) assume A5: x in Z ; ::_thesis: ln * arccot is_differentiable_in x arccot is_differentiable_on Z by A2, Th82; then A6: arccot is_differentiable_in x by A5, FDIFF_1:9; arccot . x > 0 by A3, A5; hence ln * arccot is_differentiable_in x by A6, TAYLOR_1:20; ::_thesis: verum end; then A7: ln * arccot is_differentiable_on Z by A1, FDIFF_1:9; 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 A8: x in Z ; ::_thesis: ((ln * arccot) `| Z) . x = - (1 / ((1 + (x ^2)) * (arccot . x))) then A9: - 1 < x by A2, XXREAL_1:4; arccot is_differentiable_on Z by A2, Th82; then A10: arccot is_differentiable_in x by A8, FDIFF_1:9; A11: x < 1 by A2, A8, XXREAL_1:4; arccot . x > 0 by A3, A8; then diff ((ln * arccot),x) = (diff (arccot,x)) / (arccot . x) by A10, TAYLOR_1:20 .= (- (1 / (1 + (x ^2)))) / (arccot . x) by A9, A11, Th76 .= - ((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))) by A7, A8, FDIFF_1:def_7; ::_thesis: verum end; hence ( ln * arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((ln * arccot) `| Z) . x = - (1 / ((1 + (x ^2)) * (arccot . x))) ) ) by A1, A4, FDIFF_1:9; ::_thesis: verum end; theorem Th91: :: SIN_COS9:91 for n being Element of NAT for Z being open Subset of REAL st Z c= dom ((#Z n) * arctan) & Z c= ].(- 1),1.[ holds ( (#Z n) * arctan is_differentiable_on Z & ( for x being Real st x in Z holds (((#Z n) * arctan) `| Z) . x = (n * ((arctan . x) #Z (n - 1))) / (1 + (x ^2)) ) ) proof let n be Element of NAT ; ::_thesis: for Z being open Subset of REAL st Z c= dom ((#Z n) * arctan) & Z c= ].(- 1),1.[ holds ( (#Z n) * arctan is_differentiable_on Z & ( for x being Real st x in Z holds (((#Z n) * arctan) `| Z) . x = (n * ((arctan . x) #Z (n - 1))) / (1 + (x ^2)) ) ) let Z be open Subset of REAL; ::_thesis: ( Z c= dom ((#Z n) * arctan) & Z c= ].(- 1),1.[ implies ( (#Z n) * arctan is_differentiable_on Z & ( for x being Real st x in Z holds (((#Z n) * arctan) `| Z) . x = (n * ((arctan . x) #Z (n - 1))) / (1 + (x ^2)) ) ) ) assume that A1: Z c= dom ((#Z n) * arctan) and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( (#Z n) * arctan is_differentiable_on Z & ( for x being Real st x in Z holds (((#Z n) * arctan) `| Z) . x = (n * ((arctan . x) #Z (n - 1))) / (1 + (x ^2)) ) ) A3: for x being Real st x in Z holds (#Z n) * arctan is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies (#Z n) * arctan is_differentiable_in x ) assume A4: x in Z ; ::_thesis: (#Z n) * arctan is_differentiable_in x arctan is_differentiable_on Z by A2, Th81; then arctan is_differentiable_in x by A4, FDIFF_1:9; hence (#Z n) * arctan is_differentiable_in x by TAYLOR_1:3; ::_thesis: verum end; then A5: (#Z n) * arctan is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds (((#Z n) * arctan) `| Z) . x = (n * ((arctan . x) #Z (n - 1))) / (1 + (x ^2)) proof let x be Real; ::_thesis: ( x in Z implies (((#Z n) * arctan) `| Z) . x = (n * ((arctan . x) #Z (n - 1))) / (1 + (x ^2)) ) assume A6: x in Z ; ::_thesis: (((#Z n) * arctan) `| Z) . x = (n * ((arctan . x) #Z (n - 1))) / (1 + (x ^2)) then A7: - 1 < x by A2, XXREAL_1:4; arctan is_differentiable_on Z by A2, Th81; then A8: arctan is_differentiable_in x by A6, FDIFF_1:9; A9: x < 1 by A2, A6, XXREAL_1:4; (((#Z n) * arctan) `| Z) . x = diff (((#Z n) * arctan),x) by A5, A6, FDIFF_1:def_7 .= (n * ((arctan . x) #Z (n - 1))) * (diff (arctan,x)) by A8, TAYLOR_1:3 .= (n * ((arctan . x) #Z (n - 1))) * (1 / (1 + (x ^2))) by A7, A9, Th75 .= (n * ((arctan . x) #Z (n - 1))) / (1 + (x ^2)) ; hence (((#Z n) * arctan) `| Z) . x = (n * ((arctan . x) #Z (n - 1))) / (1 + (x ^2)) ; ::_thesis: verum end; hence ( (#Z n) * arctan is_differentiable_on Z & ( for x being Real st x in Z holds (((#Z n) * arctan) `| Z) . x = (n * ((arctan . x) #Z (n - 1))) / (1 + (x ^2)) ) ) by A1, A3, FDIFF_1:9; ::_thesis: verum end; theorem Th92: :: SIN_COS9:92 for n being Element of NAT for Z being open Subset of REAL st Z c= dom ((#Z n) * arccot) & Z c= ].(- 1),1.[ holds ( (#Z n) * arccot is_differentiable_on Z & ( for x being Real st x in Z holds (((#Z n) * arccot) `| Z) . x = - ((n * ((arccot . x) #Z (n - 1))) / (1 + (x ^2))) ) ) proof let n be Element of NAT ; ::_thesis: for Z being open Subset of REAL st Z c= dom ((#Z n) * arccot) & Z c= ].(- 1),1.[ holds ( (#Z n) * arccot is_differentiable_on Z & ( for x being Real st x in Z holds (((#Z n) * arccot) `| Z) . x = - ((n * ((arccot . x) #Z (n - 1))) / (1 + (x ^2))) ) ) let Z be open Subset of REAL; ::_thesis: ( Z c= dom ((#Z n) * arccot) & Z c= ].(- 1),1.[ implies ( (#Z n) * arccot is_differentiable_on Z & ( for x being Real st x in Z holds (((#Z n) * arccot) `| Z) . x = - ((n * ((arccot . x) #Z (n - 1))) / (1 + (x ^2))) ) ) ) assume that A1: Z c= dom ((#Z n) * arccot) and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( (#Z n) * arccot is_differentiable_on Z & ( for x being Real st x in Z holds (((#Z n) * arccot) `| Z) . x = - ((n * ((arccot . x) #Z (n - 1))) / (1 + (x ^2))) ) ) A3: for x being Real st x in Z holds (#Z n) * arccot is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies (#Z n) * arccot is_differentiable_in x ) assume A4: x in Z ; ::_thesis: (#Z n) * arccot is_differentiable_in x arccot is_differentiable_on Z by A2, Th82; then arccot is_differentiable_in x by A4, FDIFF_1:9; hence (#Z n) * arccot is_differentiable_in x by TAYLOR_1:3; ::_thesis: verum end; then A5: (#Z n) * arccot is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds (((#Z n) * arccot) `| Z) . x = - ((n * ((arccot . x) #Z (n - 1))) / (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies (((#Z n) * arccot) `| Z) . x = - ((n * ((arccot . x) #Z (n - 1))) / (1 + (x ^2))) ) assume A6: x in Z ; ::_thesis: (((#Z n) * arccot) `| Z) . x = - ((n * ((arccot . x) #Z (n - 1))) / (1 + (x ^2))) then A7: - 1 < x by A2, XXREAL_1:4; arccot is_differentiable_on Z by A2, Th82; then A8: arccot is_differentiable_in x by A6, FDIFF_1:9; A9: x < 1 by A2, A6, XXREAL_1:4; (((#Z n) * arccot) `| Z) . x = diff (((#Z n) * arccot),x) by A5, A6, FDIFF_1:def_7 .= (n * ((arccot . x) #Z (n - 1))) * (diff (arccot,x)) by A8, TAYLOR_1:3 .= (n * ((arccot . x) #Z (n - 1))) * (- (1 / (1 + (x ^2)))) by A7, A9, Th76 .= - ((n * ((arccot . x) #Z (n - 1))) / (1 + (x ^2))) ; hence (((#Z n) * arccot) `| Z) . x = - ((n * ((arccot . x) #Z (n - 1))) / (1 + (x ^2))) ; ::_thesis: verum end; hence ( (#Z n) * arccot is_differentiable_on Z & ( for x being Real st x in Z holds (((#Z n) * arccot) `| Z) . x = - ((n * ((arccot . x) #Z (n - 1))) / (1 + (x ^2))) ) ) by A1, A3, FDIFF_1:9; ::_thesis: verum end; theorem :: SIN_COS9:93 for Z being open Subset of REAL st Z c= dom ((1 / 2) (#) ((#Z 2) * arctan)) & Z c= ].(- 1),1.[ holds ( (1 / 2) (#) ((#Z 2) * arctan) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) ((#Z 2) * arctan)) `| Z) . x = (arctan . x) / (1 + (x ^2)) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom ((1 / 2) (#) ((#Z 2) * arctan)) & Z c= ].(- 1),1.[ implies ( (1 / 2) (#) ((#Z 2) * arctan) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) ((#Z 2) * arctan)) `| Z) . x = (arctan . x) / (1 + (x ^2)) ) ) ) assume that A1: Z c= dom ((1 / 2) (#) ((#Z 2) * arctan)) and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( (1 / 2) (#) ((#Z 2) * arctan) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) ((#Z 2) * arctan)) `| Z) . x = (arctan . x) / (1 + (x ^2)) ) ) A3: Z c= dom ((#Z 2) * arctan) by A1, VALUED_1:def_5; then A4: (#Z 2) * arctan is_differentiable_on Z by A2, Th91; for x being Real st x in Z holds (((1 / 2) (#) ((#Z 2) * arctan)) `| Z) . x = (arctan . x) / (1 + (x ^2)) proof let x be Real; ::_thesis: ( x in Z implies (((1 / 2) (#) ((#Z 2) * arctan)) `| Z) . x = (arctan . x) / (1 + (x ^2)) ) assume A5: x in Z ; ::_thesis: (((1 / 2) (#) ((#Z 2) * arctan)) `| Z) . x = (arctan . x) / (1 + (x ^2)) then (((1 / 2) (#) ((#Z 2) * arctan)) `| Z) . x = (1 / 2) * (diff (((#Z 2) * arctan),x)) by A1, A4, FDIFF_1:20 .= (1 / 2) * ((((#Z 2) * arctan) `| Z) . x) by A4, A5, FDIFF_1:def_7 .= (1 / 2) * ((2 * ((arctan . x) #Z (2 - 1))) / (1 + (x ^2))) by A2, A3, A5, Th91 .= (1 / 2) * ((2 * (arctan . x)) / (1 + (x ^2))) by PREPOWER:35 .= (arctan . x) / (1 + (x ^2)) ; hence (((1 / 2) (#) ((#Z 2) * arctan)) `| Z) . x = (arctan . x) / (1 + (x ^2)) ; ::_thesis: verum end; hence ( (1 / 2) (#) ((#Z 2) * arctan) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) ((#Z 2) * arctan)) `| Z) . x = (arctan . x) / (1 + (x ^2)) ) ) by A1, A4, FDIFF_1:20; ::_thesis: verum end; theorem :: SIN_COS9:94 for Z being open Subset of REAL st Z c= dom ((1 / 2) (#) ((#Z 2) * arccot)) & Z c= ].(- 1),1.[ holds ( (1 / 2) (#) ((#Z 2) * arccot) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) ((#Z 2) * arccot)) `| Z) . x = - ((arccot . x) / (1 + (x ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom ((1 / 2) (#) ((#Z 2) * arccot)) & Z c= ].(- 1),1.[ implies ( (1 / 2) (#) ((#Z 2) * arccot) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) ((#Z 2) * arccot)) `| Z) . x = - ((arccot . x) / (1 + (x ^2))) ) ) ) assume that A1: Z c= dom ((1 / 2) (#) ((#Z 2) * arccot)) and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( (1 / 2) (#) ((#Z 2) * arccot) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) ((#Z 2) * arccot)) `| Z) . x = - ((arccot . x) / (1 + (x ^2))) ) ) A3: Z c= dom ((#Z 2) * arccot) by A1, VALUED_1:def_5; then A4: (#Z 2) * arccot is_differentiable_on Z by A2, Th92; for x being Real st x in Z holds (((1 / 2) (#) ((#Z 2) * arccot)) `| Z) . x = - ((arccot . x) / (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies (((1 / 2) (#) ((#Z 2) * arccot)) `| Z) . x = - ((arccot . x) / (1 + (x ^2))) ) assume A5: x in Z ; ::_thesis: (((1 / 2) (#) ((#Z 2) * arccot)) `| Z) . x = - ((arccot . x) / (1 + (x ^2))) then (((1 / 2) (#) ((#Z 2) * arccot)) `| Z) . x = (1 / 2) * (diff (((#Z 2) * arccot),x)) by A1, A4, FDIFF_1:20 .= (1 / 2) * ((((#Z 2) * arccot) `| Z) . x) by A4, A5, FDIFF_1:def_7 .= (1 / 2) * (- ((2 * ((arccot . x) #Z (2 - 1))) / (1 + (x ^2)))) by A2, A3, A5, Th92 .= - ((1 / 2) * ((2 * ((arccot . x) #Z 1)) / (1 + (x ^2)))) .= - ((1 / 2) * ((2 * (arccot . x)) / (1 + (x ^2)))) by PREPOWER:35 .= - ((arccot . x) / (1 + (x ^2))) ; hence (((1 / 2) (#) ((#Z 2) * arccot)) `| Z) . x = - ((arccot . x) / (1 + (x ^2))) ; ::_thesis: verum end; hence ( (1 / 2) (#) ((#Z 2) * arccot) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) ((#Z 2) * arccot)) `| Z) . x = - ((arccot . x) / (1 + (x ^2))) ) ) by A1, A4, FDIFF_1:20; ::_thesis: verum end; theorem Th95: :: SIN_COS9:95 for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds ( (id Z) (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds (((id Z) (#) arctan) `| Z) . x = (arctan . x) + (x / (1 + (x ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= ].(- 1),1.[ implies ( (id Z) (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds (((id Z) (#) arctan) `| Z) . x = (arctan . x) + (x / (1 + (x ^2))) ) ) ) assume A1: Z c= ].(- 1),1.[ ; ::_thesis: ( (id Z) (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds (((id Z) (#) arctan) `| Z) . x = (arctan . x) + (x / (1 + (x ^2))) ) ) A2: Z c= dom (id Z) by FUNCT_1:17; ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25; then ].(- 1),1.[ c= dom arctan by Th23, XBOOLE_1:1; then Z c= dom arctan by A1, XBOOLE_1:1; then Z c= (dom (id Z)) /\ (dom arctan) by A2, XBOOLE_1:19; then A3: Z c= dom ((id Z) (#) arctan) by VALUED_1:def_4; A4: for x being Real st x in Z holds (id Z) . x = (1 * x) + 0 by FUNCT_1:18; then A5: id Z is_differentiable_on Z by A2, FDIFF_1:23; A6: arctan is_differentiable_on Z by A1, Th81; for x being Real st x in Z holds (((id Z) (#) arctan) `| Z) . x = (arctan . x) + (x / (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies (((id Z) (#) arctan) `| Z) . x = (arctan . x) + (x / (1 + (x ^2))) ) assume A7: x in Z ; ::_thesis: (((id Z) (#) arctan) `| Z) . x = (arctan . x) + (x / (1 + (x ^2))) then A8: - 1 < x by A1, XXREAL_1:4; A9: x < 1 by A1, A7, XXREAL_1:4; (((id Z) (#) arctan) `| Z) . x = ((arctan . x) * (diff ((id Z),x))) + (((id Z) . x) * (diff (arctan,x))) by A3, A5, A6, A7, FDIFF_1:21 .= ((arctan . x) * (((id Z) `| Z) . x)) + (((id Z) . x) * (diff (arctan,x))) by A5, A7, FDIFF_1:def_7 .= ((arctan . x) * 1) + (((id Z) . x) * (diff (arctan,x))) by A2, A4, A7, FDIFF_1:23 .= (arctan . x) + (x * (diff (arctan,x))) by A7, FUNCT_1:18 .= (arctan . x) + (x * (1 / (1 + (x ^2)))) by A8, A9, Th75 .= (arctan . x) + (x / (1 + (x ^2))) ; hence (((id Z) (#) arctan) `| Z) . x = (arctan . x) + (x / (1 + (x ^2))) ; ::_thesis: verum end; hence ( (id Z) (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds (((id Z) (#) arctan) `| Z) . x = (arctan . x) + (x / (1 + (x ^2))) ) ) by A3, A5, A6, FDIFF_1:21; ::_thesis: verum end; theorem Th96: :: SIN_COS9:96 for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds ( (id Z) (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds (((id Z) (#) arccot) `| Z) . x = (arccot . x) - (x / (1 + (x ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= ].(- 1),1.[ implies ( (id Z) (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds (((id Z) (#) arccot) `| Z) . x = (arccot . x) - (x / (1 + (x ^2))) ) ) ) assume A1: Z c= ].(- 1),1.[ ; ::_thesis: ( (id Z) (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds (((id Z) (#) arccot) `| Z) . x = (arccot . x) - (x / (1 + (x ^2))) ) ) A2: Z c= dom (id Z) by FUNCT_1:17; ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25; then ].(- 1),1.[ c= dom arccot by Th24, XBOOLE_1:1; then Z c= dom arccot by A1, XBOOLE_1:1; then Z c= (dom (id Z)) /\ (dom arccot) by A2, XBOOLE_1:19; then A3: Z c= dom ((id Z) (#) arccot) by VALUED_1:def_4; A4: for x being Real st x in Z holds (id Z) . x = (1 * x) + 0 by FUNCT_1:18; then A5: id Z is_differentiable_on Z by A2, FDIFF_1:23; A6: arccot is_differentiable_on Z by A1, Th82; for x being Real st x in Z holds (((id Z) (#) arccot) `| Z) . x = (arccot . x) - (x / (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies (((id Z) (#) arccot) `| Z) . x = (arccot . x) - (x / (1 + (x ^2))) ) assume A7: x in Z ; ::_thesis: (((id Z) (#) arccot) `| Z) . x = (arccot . x) - (x / (1 + (x ^2))) then A8: - 1 < x by A1, XXREAL_1:4; A9: x < 1 by A1, A7, XXREAL_1:4; (((id Z) (#) arccot) `| Z) . x = ((arccot . x) * (diff ((id Z),x))) + (((id Z) . x) * (diff (arccot,x))) by A3, A5, A6, A7, FDIFF_1:21 .= ((arccot . x) * (((id Z) `| Z) . x)) + (((id Z) . x) * (diff (arccot,x))) by A5, A7, FDIFF_1:def_7 .= ((arccot . x) * 1) + (((id Z) . x) * (diff (arccot,x))) by A2, A4, A7, FDIFF_1:23 .= (arccot . x) + (x * (diff (arccot,x))) by A7, FUNCT_1:18 .= (arccot . x) + (x * (- (1 / (1 + (x ^2))))) by A8, A9, Th76 .= (arccot . x) - (x / (1 + (x ^2))) ; hence (((id Z) (#) arccot) `| Z) . x = (arccot . x) - (x / (1 + (x ^2))) ; ::_thesis: verum end; hence ( (id Z) (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds (((id Z) (#) arccot) `| Z) . x = (arccot . x) - (x / (1 + (x ^2))) ) ) by A3, A5, A6, FDIFF_1:21; ::_thesis: verum end; theorem :: SIN_COS9:97 for r, s being Real for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom (f (#) arctan) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds f . x = (r * x) + s ) holds ( f (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((f (#) arctan) `| Z) . x = (r * (arctan . x)) + (((r * x) + s) / (1 + (x ^2))) ) ) proof let r, s be Real; ::_thesis: for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom (f (#) arctan) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds f . x = (r * x) + s ) holds ( f (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((f (#) arctan) `| Z) . x = (r * (arctan . x)) + (((r * x) + s) / (1 + (x ^2))) ) ) let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st Z c= dom (f (#) arctan) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds f . x = (r * x) + s ) holds ( f (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((f (#) arctan) `| Z) . x = (r * (arctan . x)) + (((r * x) + s) / (1 + (x ^2))) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (f (#) arctan) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds f . x = (r * x) + s ) implies ( f (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((f (#) arctan) `| Z) . x = (r * (arctan . x)) + (((r * x) + s) / (1 + (x ^2))) ) ) ) assume that A1: Z c= dom (f (#) arctan) and A2: Z c= ].(- 1),1.[ and A3: for x being Real st x in Z holds f . x = (r * x) + s ; ::_thesis: ( f (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((f (#) arctan) `| Z) . x = (r * (arctan . x)) + (((r * x) + s) / (1 + (x ^2))) ) ) Z c= (dom f) /\ (dom arctan) by A1, VALUED_1:def_4; then A4: Z c= dom f by XBOOLE_1:18; then A5: f is_differentiable_on Z by A3, FDIFF_1:23; A6: arctan is_differentiable_on Z by A2, Th81; for x being Real st x in Z holds ((f (#) arctan) `| Z) . x = (r * (arctan . x)) + (((r * x) + s) / (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((f (#) arctan) `| Z) . x = (r * (arctan . x)) + (((r * x) + s) / (1 + (x ^2))) ) assume A7: x in Z ; ::_thesis: ((f (#) arctan) `| Z) . x = (r * (arctan . x)) + (((r * x) + s) / (1 + (x ^2))) then A8: - 1 < x by A2, XXREAL_1:4; A9: x < 1 by A2, A7, XXREAL_1:4; ((f (#) arctan) `| Z) . x = ((arctan . x) * (diff (f,x))) + ((f . x) * (diff (arctan,x))) by A1, A5, A6, A7, FDIFF_1:21 .= ((arctan . x) * ((f `| Z) . x)) + ((f . x) * (diff (arctan,x))) by A5, A7, FDIFF_1:def_7 .= ((arctan . x) * r) + ((f . x) * (diff (arctan,x))) by A3, A4, A7, FDIFF_1:23 .= ((arctan . x) * r) + ((f . x) * (1 / (1 + (x ^2)))) by A8, A9, Th75 .= (r * (arctan . x)) + (((r * x) + s) / (1 + (x ^2))) by A3, A7 ; hence ((f (#) arctan) `| Z) . x = (r * (arctan . x)) + (((r * x) + s) / (1 + (x ^2))) ; ::_thesis: verum end; hence ( f (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((f (#) arctan) `| Z) . x = (r * (arctan . x)) + (((r * x) + s) / (1 + (x ^2))) ) ) by A1, A5, A6, FDIFF_1:21; ::_thesis: verum end; theorem :: SIN_COS9:98 for r, s being Real for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom (f (#) arccot) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds f . x = (r * x) + s ) holds ( f (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((f (#) arccot) `| Z) . x = (r * (arccot . x)) - (((r * x) + s) / (1 + (x ^2))) ) ) proof let r, s be Real; ::_thesis: for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom (f (#) arccot) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds f . x = (r * x) + s ) holds ( f (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((f (#) arccot) `| Z) . x = (r * (arccot . x)) - (((r * x) + s) / (1 + (x ^2))) ) ) let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st Z c= dom (f (#) arccot) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds f . x = (r * x) + s ) holds ( f (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((f (#) arccot) `| Z) . x = (r * (arccot . x)) - (((r * x) + s) / (1 + (x ^2))) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (f (#) arccot) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds f . x = (r * x) + s ) implies ( f (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((f (#) arccot) `| Z) . x = (r * (arccot . x)) - (((r * x) + s) / (1 + (x ^2))) ) ) ) assume that A1: Z c= dom (f (#) arccot) and A2: Z c= ].(- 1),1.[ and A3: for x being Real st x in Z holds f . x = (r * x) + s ; ::_thesis: ( f (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((f (#) arccot) `| Z) . x = (r * (arccot . x)) - (((r * x) + s) / (1 + (x ^2))) ) ) Z c= (dom f) /\ (dom arccot) by A1, VALUED_1:def_4; then A4: Z c= dom f by XBOOLE_1:18; then A5: f is_differentiable_on Z by A3, FDIFF_1:23; A6: arccot is_differentiable_on Z by A2, Th82; for x being Real st x in Z holds ((f (#) arccot) `| Z) . x = (r * (arccot . x)) - (((r * x) + s) / (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((f (#) arccot) `| Z) . x = (r * (arccot . x)) - (((r * x) + s) / (1 + (x ^2))) ) assume A7: x in Z ; ::_thesis: ((f (#) arccot) `| Z) . x = (r * (arccot . x)) - (((r * x) + s) / (1 + (x ^2))) then A8: - 1 < x by A2, XXREAL_1:4; A9: x < 1 by A2, A7, XXREAL_1:4; ((f (#) arccot) `| Z) . x = ((arccot . x) * (diff (f,x))) + ((f . x) * (diff (arccot,x))) by A1, A5, A6, A7, FDIFF_1:21 .= ((arccot . x) * ((f `| Z) . x)) + ((f . x) * (diff (arccot,x))) by A5, A7, FDIFF_1:def_7 .= ((arccot . x) * r) + ((f . x) * (diff (arccot,x))) by A3, A4, A7, FDIFF_1:23 .= ((arccot . x) * r) + ((f . x) * (- (1 / (1 + (x ^2))))) by A8, A9, Th76 .= ((arccot . x) * r) - ((f . x) * (1 / (1 + (x ^2)))) .= (r * (arccot . x)) - (((r * x) + s) / (1 + (x ^2))) by A3, A7 ; hence ((f (#) arccot) `| Z) . x = (r * (arccot . x)) - (((r * x) + s) / (1 + (x ^2))) ; ::_thesis: verum end; hence ( f (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((f (#) arccot) `| Z) . x = (r * (arccot . x)) - (((r * x) + s) / (1 + (x ^2))) ) ) by A1, A5, A6, FDIFF_1:21; ::_thesis: verum end; theorem :: SIN_COS9:99 for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom ((1 / 2) (#) (arctan * f)) & ( for x being Real st x in Z holds ( f . x = 2 * x & f . x > - 1 & f . x < 1 ) ) holds ( (1 / 2) (#) (arctan * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) (arctan * f)) `| Z) . x = 1 / (1 + ((2 * x) ^2)) ) ) proof let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st Z c= dom ((1 / 2) (#) (arctan * f)) & ( for x being Real st x in Z holds ( f . x = 2 * x & f . x > - 1 & f . x < 1 ) ) holds ( (1 / 2) (#) (arctan * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) (arctan * f)) `| Z) . x = 1 / (1 + ((2 * x) ^2)) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom ((1 / 2) (#) (arctan * f)) & ( for x being Real st x in Z holds ( f . x = 2 * x & f . x > - 1 & f . x < 1 ) ) implies ( (1 / 2) (#) (arctan * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) (arctan * f)) `| Z) . x = 1 / (1 + ((2 * x) ^2)) ) ) ) assume that A1: Z c= dom ((1 / 2) (#) (arctan * f)) and A2: for x being Real st x in Z holds ( f . x = 2 * x & f . x > - 1 & f . x < 1 ) ; ::_thesis: ( (1 / 2) (#) (arctan * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) (arctan * f)) `| Z) . x = 1 / (1 + ((2 * x) ^2)) ) ) A3: for x being Real st x in Z holds ( f . x = (2 * x) + 0 & f . x > - 1 & f . x < 1 ) by A2; A4: Z c= dom (arctan * f) by A1, VALUED_1:def_5; then A5: arctan * f is_differentiable_on Z by A3, Th87; for x being Real st x in Z holds (((1 / 2) (#) (arctan * f)) `| Z) . x = 1 / (1 + ((2 * x) ^2)) proof let x be Real; ::_thesis: ( x in Z implies (((1 / 2) (#) (arctan * f)) `| Z) . x = 1 / (1 + ((2 * x) ^2)) ) assume A6: x in Z ; ::_thesis: (((1 / 2) (#) (arctan * f)) `| Z) . x = 1 / (1 + ((2 * x) ^2)) then (((1 / 2) (#) (arctan * f)) `| Z) . x = (1 / 2) * (diff ((arctan * f),x)) by A1, A5, FDIFF_1:20 .= (1 / 2) * (((arctan * f) `| Z) . x) by A5, A6, FDIFF_1:def_7 .= (1 / 2) * (2 / (1 + (((2 * x) + 0) ^2))) by A4, A3, A6, Th87 .= 1 / (1 + ((2 * x) ^2)) ; hence (((1 / 2) (#) (arctan * f)) `| Z) . x = 1 / (1 + ((2 * x) ^2)) ; ::_thesis: verum end; hence ( (1 / 2) (#) (arctan * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) (arctan * f)) `| Z) . x = 1 / (1 + ((2 * x) ^2)) ) ) by A1, A5, FDIFF_1:20; ::_thesis: verum end; theorem :: SIN_COS9:100 for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom ((1 / 2) (#) (arccot * f)) & ( for x being Real st x in Z holds ( f . x = 2 * x & f . x > - 1 & f . x < 1 ) ) holds ( (1 / 2) (#) (arccot * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) (arccot * f)) `| Z) . x = - (1 / (1 + ((2 * x) ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st Z c= dom ((1 / 2) (#) (arccot * f)) & ( for x being Real st x in Z holds ( f . x = 2 * x & f . x > - 1 & f . x < 1 ) ) holds ( (1 / 2) (#) (arccot * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) (arccot * f)) `| Z) . x = - (1 / (1 + ((2 * x) ^2))) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom ((1 / 2) (#) (arccot * f)) & ( for x being Real st x in Z holds ( f . x = 2 * x & f . x > - 1 & f . x < 1 ) ) implies ( (1 / 2) (#) (arccot * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) (arccot * f)) `| Z) . x = - (1 / (1 + ((2 * x) ^2))) ) ) ) assume that A1: Z c= dom ((1 / 2) (#) (arccot * f)) and A2: for x being Real st x in Z holds ( f . x = 2 * x & f . x > - 1 & f . x < 1 ) ; ::_thesis: ( (1 / 2) (#) (arccot * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) (arccot * f)) `| Z) . x = - (1 / (1 + ((2 * x) ^2))) ) ) A3: for x being Real st x in Z holds ( f . x = (2 * x) + 0 & f . x > - 1 & f . x < 1 ) by A2; A4: Z c= dom (arccot * f) by A1, VALUED_1:def_5; then A5: arccot * f is_differentiable_on Z by A3, Th88; for x being Real st x in Z holds (((1 / 2) (#) (arccot * f)) `| Z) . x = - (1 / (1 + ((2 * x) ^2))) proof let x be Real; ::_thesis: ( x in Z implies (((1 / 2) (#) (arccot * f)) `| Z) . x = - (1 / (1 + ((2 * x) ^2))) ) assume A6: x in Z ; ::_thesis: (((1 / 2) (#) (arccot * f)) `| Z) . x = - (1 / (1 + ((2 * x) ^2))) then (((1 / 2) (#) (arccot * f)) `| Z) . x = (1 / 2) * (diff ((arccot * f),x)) by A1, A5, FDIFF_1:20 .= (1 / 2) * (((arccot * f) `| Z) . x) by A5, A6, FDIFF_1:def_7 .= (1 / 2) * (- (2 / (1 + (((2 * x) + 0) ^2)))) by A4, A3, A6, Th88 .= - (1 / (1 + ((2 * x) ^2))) ; hence (((1 / 2) (#) (arccot * f)) `| Z) . x = - (1 / (1 + ((2 * x) ^2))) ; ::_thesis: verum end; hence ( (1 / 2) (#) (arccot * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) (arccot * f)) `| Z) . x = - (1 / (1 + ((2 * x) ^2))) ) ) by A1, A5, FDIFF_1:20; ::_thesis: verum end; theorem Th101: :: SIN_COS9:101 for Z being open Subset of REAL for f1, f2 being PartFunc of REAL,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 ) ) proof let Z be open Subset of REAL; ::_thesis: for f1, f2 being PartFunc of REAL,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 ) ) let f1, f2 be PartFunc of REAL,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 ) ) ) assume that A1: Z c= dom (f1 + f2) and A2: for x being Real st x in Z holds f1 . x = 1 and A3: 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 ) ) A4: for x being Real st x in Z holds f2 is_differentiable_in x by A3, TAYLOR_1:2; A5: Z c= (dom f1) /\ (dom f2) by A1, VALUED_1:def_1; then A6: Z c= dom f1 by XBOOLE_1:18; A7: for x being Real st x in Z holds f1 . x = (0 * x) + 1 by A2; then A8: f1 is_differentiable_on Z by A6, FDIFF_1:23; Z c= dom f2 by A5, XBOOLE_1:18; then A9: f2 is_differentiable_on Z by A4, FDIFF_1:9; A10: for x being Real st x in Z holds (f2 `| Z) . x = 2 * x proof let x be Real; ::_thesis: ( x in Z implies (f2 `| Z) . x = 2 * x ) 2 * (x #Z (2 - 1)) = 2 * x by PREPOWER:35; then A11: diff (f2,x) = 2 * x by A3, TAYLOR_1:2; assume x in Z ; ::_thesis: (f2 `| Z) . x = 2 * x hence (f2 `| Z) . x = 2 * x by A9, A11, FDIFF_1:def_7; ::_thesis: verum end; for x being Real st x in Z holds ((f1 + f2) `| Z) . x = 2 * x proof let x be Real; ::_thesis: ( x in Z implies ((f1 + f2) `| Z) . x = 2 * x ) assume A12: x in Z ; ::_thesis: ((f1 + f2) `| Z) . x = 2 * x then ((f1 + f2) `| Z) . x = (diff (f1,x)) + (diff (f2,x)) by A1, A8, A9, FDIFF_1:18 .= ((f1 `| Z) . x) + (diff (f2,x)) by A8, A12, FDIFF_1:def_7 .= ((f1 `| Z) . x) + ((f2 `| Z) . x) by A9, A12, FDIFF_1:def_7 .= 0 + ((f2 `| Z) . x) by A6, A7, A12, FDIFF_1:23 .= 2 * x by A10, A12 ; hence ((f1 + f2) `| Z) . x = 2 * x ; ::_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 ) ) by A1, A8, A9, FDIFF_1:18; ::_thesis: verum end; theorem Th102: :: SIN_COS9:102 for Z being open Subset of REAL for f1, f2 being PartFunc of REAL,REAL st Z c= dom ((1 / 2) (#) (ln * (f1 + f2))) & f2 = #Z 2 & ( for x being Real st x in Z holds f1 . x = 1 ) holds ( (1 / 2) (#) (ln * (f1 + f2)) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (1 + (x ^2)) ) ) proof let Z be open Subset of REAL; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st Z c= dom ((1 / 2) (#) (ln * (f1 + f2))) & f2 = #Z 2 & ( for x being Real st x in Z holds f1 . x = 1 ) holds ( (1 / 2) (#) (ln * (f1 + f2)) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (1 + (x ^2)) ) ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom ((1 / 2) (#) (ln * (f1 + f2))) & f2 = #Z 2 & ( for x being Real st x in Z holds f1 . x = 1 ) implies ( (1 / 2) (#) (ln * (f1 + f2)) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (1 + (x ^2)) ) ) ) assume that A1: Z c= dom ((1 / 2) (#) (ln * (f1 + f2))) and A2: f2 = #Z 2 and A3: for x being Real st x in Z holds f1 . x = 1 ; ::_thesis: ( (1 / 2) (#) (ln * (f1 + f2)) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (1 + (x ^2)) ) ) A4: Z c= dom (ln * (f1 + f2)) by A1, VALUED_1:def_5; then for y being set st y in Z holds y in dom (f1 + f2) by FUNCT_1:11; then A5: Z c= dom (f1 + f2) by TARSKI:def_3; then A6: f1 + f2 is_differentiable_on Z by A2, A3, Th101; for x being Real st x in Z holds ln * (f1 + f2) is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies ln * (f1 + f2) is_differentiable_in x ) assume A7: x in Z ; ::_thesis: ln * (f1 + f2) is_differentiable_in x then (f1 + f2) . x = (f1 . x) + (f2 . x) by A5, VALUED_1:def_1 .= 1 + (f2 . x) by A3, A7 .= 1 + (x #Z (1 + 1)) by A2, TAYLOR_1:def_1 .= 1 + ((x #Z 1) * (x #Z 1)) by TAYLOR_1:1 .= 1 + (x * (x #Z 1)) by PREPOWER:35 .= 1 + (x * x) by PREPOWER:35 ; then A8: (f1 + f2) . x > 0 by XREAL_1:34, XREAL_1:63; f1 + f2 is_differentiable_in x by A6, A7, FDIFF_1:9; hence ln * (f1 + f2) is_differentiable_in x by A8, TAYLOR_1:20; ::_thesis: verum end; then A9: ln * (f1 + f2) is_differentiable_on Z by A4, FDIFF_1:9; for x being Real st x in Z holds (((1 / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (1 + (x ^2)) proof let x be Real; ::_thesis: ( x in Z implies (((1 / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (1 + (x ^2)) ) assume A10: x in Z ; ::_thesis: (((1 / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (1 + (x ^2)) then A11: f1 + f2 is_differentiable_in x by A6, FDIFF_1:9; A12: (f1 + f2) . x = (f1 . x) + (f2 . x) by A5, A10, VALUED_1:def_1 .= 1 + (f2 . x) by A3, A10 .= 1 + (x #Z (1 + 1)) by A2, TAYLOR_1:def_1 .= 1 + ((x #Z 1) * (x #Z 1)) by TAYLOR_1:1 .= 1 + (x * (x #Z 1)) by PREPOWER:35 .= 1 + (x * x) by PREPOWER:35 ; then (f1 + f2) . x > 0 by XREAL_1:34, XREAL_1:63; then diff ((ln * (f1 + f2)),x) = (diff ((f1 + f2),x)) / ((f1 + f2) . x) by A11, TAYLOR_1:20 .= (((f1 + f2) `| Z) . x) / ((f1 + f2) . x) by A6, A10, FDIFF_1:def_7 .= (2 * x) / (1 + (x ^2)) by A2, A3, A5, A10, A12, Th101 ; hence (((1 / 2) (#) (ln * (f1 + f2))) `| Z) . x = (1 / 2) * ((2 * x) / (1 + (x ^2))) by A1, A9, A10, FDIFF_1:20 .= x / (1 + (x ^2)) ; ::_thesis: verum end; hence ( (1 / 2) (#) (ln * (f1 + f2)) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (1 + (x ^2)) ) ) by A1, A9, FDIFF_1:20; ::_thesis: verum end; theorem :: SIN_COS9:103 for Z being open Subset of REAL for f1, f2 being PartFunc of REAL,REAL st Z c= dom (((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) & Z c= ].(- 1),1.[ & f2 = #Z 2 & ( for x being Real st x in Z holds f1 . x = 1 ) holds ( ((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . x ) ) proof let Z be open Subset of REAL; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st Z c= dom (((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) & Z c= ].(- 1),1.[ & f2 = #Z 2 & ( for x being Real st x in Z holds f1 . x = 1 ) holds ( ((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . x ) ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) & Z c= ].(- 1),1.[ & f2 = #Z 2 & ( for x being Real st x in Z holds f1 . x = 1 ) implies ( ((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . x ) ) ) assume that A1: Z c= dom (((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) and A2: Z c= ].(- 1),1.[ and A3: f2 = #Z 2 and A4: for x being Real st x in Z holds f1 . x = 1 ; ::_thesis: ( ((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . x ) ) Z c= (dom ((id Z) (#) arctan)) /\ (dom ((1 / 2) (#) (ln * (f1 + f2)))) by A1, VALUED_1:12; then A5: Z c= dom ((1 / 2) (#) (ln * (f1 + f2))) by XBOOLE_1:18; then A6: (1 / 2) (#) (ln * (f1 + f2)) is_differentiable_on Z by A3, A4, Th102; A7: (id Z) (#) arctan is_differentiable_on Z by A2, Th95; for x being Real st x in Z holds ((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . x proof let x be Real; ::_thesis: ( x in Z implies ((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . x ) assume A8: x in Z ; ::_thesis: ((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . x hence ((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = (diff (((id Z) (#) arctan),x)) - (diff (((1 / 2) (#) (ln * (f1 + f2))),x)) by A1, A7, A6, FDIFF_1:19 .= ((((id Z) (#) arctan) `| Z) . x) - (diff (((1 / 2) (#) (ln * (f1 + f2))),x)) by A7, A8, FDIFF_1:def_7 .= ((((id Z) (#) arctan) `| Z) . x) - ((((1 / 2) (#) (ln * (f1 + f2))) `| Z) . x) by A6, A8, FDIFF_1:def_7 .= ((arctan . x) + (x / (1 + (x ^2)))) - ((((1 / 2) (#) (ln * (f1 + f2))) `| Z) . x) by A2, A8, Th95 .= ((arctan . x) + (x / (1 + (x ^2)))) - (x / (1 + (x ^2))) by A3, A4, A5, A8, Th102 .= arctan . x ; ::_thesis: verum end; hence ( ((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . x ) ) by A1, A7, A6, FDIFF_1:19; ::_thesis: verum end; theorem :: SIN_COS9:104 for Z being open Subset of REAL for f1, f2 being PartFunc of REAL,REAL st Z c= dom (((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) & Z c= ].(- 1),1.[ & f2 = #Z 2 & ( for x being Real st x in Z holds f1 . x = 1 ) holds ( ((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . x ) ) proof let Z be open Subset of REAL; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st Z c= dom (((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) & Z c= ].(- 1),1.[ & f2 = #Z 2 & ( for x being Real st x in Z holds f1 . x = 1 ) holds ( ((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . x ) ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) & Z c= ].(- 1),1.[ & f2 = #Z 2 & ( for x being Real st x in Z holds f1 . x = 1 ) implies ( ((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . x ) ) ) assume that A1: Z c= dom (((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) and A2: Z c= ].(- 1),1.[ and A3: f2 = #Z 2 and A4: for x being Real st x in Z holds f1 . x = 1 ; ::_thesis: ( ((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . x ) ) Z c= (dom ((id Z) (#) arccot)) /\ (dom ((1 / 2) (#) (ln * (f1 + f2)))) by A1, VALUED_1:def_1; then A5: Z c= dom ((1 / 2) (#) (ln * (f1 + f2))) by XBOOLE_1:18; then A6: (1 / 2) (#) (ln * (f1 + f2)) is_differentiable_on Z by A3, A4, Th102; A7: (id Z) (#) arccot is_differentiable_on Z by A2, Th96; for x being Real st x in Z holds ((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . x proof let x be Real; ::_thesis: ( x in Z implies ((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . x ) assume A8: x in Z ; ::_thesis: ((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . x hence ((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = (diff (((id Z) (#) arccot),x)) + (diff (((1 / 2) (#) (ln * (f1 + f2))),x)) by A1, A7, A6, FDIFF_1:18 .= ((((id Z) (#) arccot) `| Z) . x) + (diff (((1 / 2) (#) (ln * (f1 + f2))),x)) by A7, A8, FDIFF_1:def_7 .= ((((id Z) (#) arccot) `| Z) . x) + ((((1 / 2) (#) (ln * (f1 + f2))) `| Z) . x) by A6, A8, FDIFF_1:def_7 .= ((arccot . x) - (x / (1 + (x ^2)))) + ((((1 / 2) (#) (ln * (f1 + f2))) `| Z) . x) by A2, A8, Th96 .= ((arccot . x) - (x / (1 + (x ^2)))) + (x / (1 + (x ^2))) by A3, A4, A5, A8, Th102 .= arccot . x ; ::_thesis: verum end; hence ( ((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . x ) ) by A1, A7, A6, FDIFF_1:18; ::_thesis: verum end; theorem Th105: :: SIN_COS9:105 for r being Real for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom ((id Z) (#) (arctan * f)) & ( for x being Real st x in Z holds ( f . x = x / r & f . x > - 1 & f . x < 1 ) ) holds ( (id Z) (#) (arctan * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((id Z) (#) (arctan * f)) `| Z) . x = (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2)))) ) ) proof let r be Real; ::_thesis: for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom ((id Z) (#) (arctan * f)) & ( for x being Real st x in Z holds ( f . x = x / r & f . x > - 1 & f . x < 1 ) ) holds ( (id Z) (#) (arctan * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((id Z) (#) (arctan * f)) `| Z) . x = (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2)))) ) ) let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st Z c= dom ((id Z) (#) (arctan * f)) & ( for x being Real st x in Z holds ( f . x = x / r & f . x > - 1 & f . x < 1 ) ) holds ( (id Z) (#) (arctan * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((id Z) (#) (arctan * f)) `| Z) . x = (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2)))) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom ((id Z) (#) (arctan * f)) & ( for x being Real st x in Z holds ( f . x = x / r & f . x > - 1 & f . x < 1 ) ) implies ( (id Z) (#) (arctan * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((id Z) (#) (arctan * f)) `| Z) . x = (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2)))) ) ) ) assume that A1: Z c= dom ((id Z) (#) (arctan * f)) and A2: for x being Real st x in Z holds ( f . x = x / r & f . x > - 1 & f . x < 1 ) ; ::_thesis: ( (id Z) (#) (arctan * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((id Z) (#) (arctan * f)) `| Z) . x = (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2)))) ) ) A3: Z c= (dom (id Z)) /\ (dom (arctan * f)) by A1, VALUED_1:def_4; then A4: Z c= dom (id Z) by XBOOLE_1:18; A5: Z c= dom (arctan * f) by A3, XBOOLE_1:18; for x being Real st x in Z holds f . x = ((1 / r) * x) + 0 proof let x be Real; ::_thesis: ( x in Z implies f . x = ((1 / r) * x) + 0 ) assume x in Z ; ::_thesis: f . x = ((1 / r) * x) + 0 then f . x = x / r by A2; hence f . x = ((1 / r) * x) + 0 ; ::_thesis: verum end; then A6: for x being Real st x in Z holds ( f . x = ((1 / r) * x) + 0 & f . x > - 1 & f . x < 1 ) by A2; then A7: arctan * f is_differentiable_on Z by A5, Th87; A8: for x being Real st x in Z holds (id Z) . x = (1 * x) + 0 by FUNCT_1:18; then A9: id Z is_differentiable_on Z by A4, FDIFF_1:23; A10: for x being Real st x in Z holds ((arctan * f) `| Z) . x = 1 / (r * (1 + ((x / r) ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((arctan * f) `| Z) . x = 1 / (r * (1 + ((x / r) ^2))) ) assume x in Z ; ::_thesis: ((arctan * f) `| Z) . x = 1 / (r * (1 + ((x / r) ^2))) then ((arctan * f) `| Z) . x = (1 / r) / (1 + ((((1 / r) * x) + 0) ^2)) by A6, A5, Th87 .= 1 / (r * (1 + ((x / r) ^2))) by XCMPLX_1:78 ; hence ((arctan * f) `| Z) . x = 1 / (r * (1 + ((x / r) ^2))) ; ::_thesis: verum end; for x being Real st x in Z holds (((id Z) (#) (arctan * f)) `| Z) . x = (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2)))) proof let x be Real; ::_thesis: ( x in Z implies (((id Z) (#) (arctan * f)) `| Z) . x = (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2)))) ) assume A11: x in Z ; ::_thesis: (((id Z) (#) (arctan * f)) `| Z) . x = (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2)))) then A12: (arctan * f) . x = arctan . (f . x) by A5, FUNCT_1:12 .= arctan . (x / r) by A2, A11 ; (((id Z) (#) (arctan * f)) `| Z) . x = (((arctan * f) . x) * (diff ((id Z),x))) + (((id Z) . x) * (diff ((arctan * f),x))) by A1, A9, A7, A11, FDIFF_1:21 .= (((arctan * f) . x) * (((id Z) `| Z) . x)) + (((id Z) . x) * (diff ((arctan * f),x))) by A9, A11, FDIFF_1:def_7 .= (((arctan * f) . x) * 1) + (((id Z) . x) * (diff ((arctan * f),x))) by A4, A8, A11, FDIFF_1:23 .= (((arctan * f) . x) * 1) + (((id Z) . x) * (((arctan * f) `| Z) . x)) by A7, A11, FDIFF_1:def_7 .= ((arctan * f) . x) + (x * (((arctan * f) `| Z) . x)) by A11, FUNCT_1:18 .= (arctan . (x / r)) + (x * (1 / (r * (1 + ((x / r) ^2))))) by A10, A11, A12 .= (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2)))) ; hence (((id Z) (#) (arctan * f)) `| Z) . x = (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2)))) ; ::_thesis: verum end; hence ( (id Z) (#) (arctan * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((id Z) (#) (arctan * f)) `| Z) . x = (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2)))) ) ) by A1, A9, A7, FDIFF_1:21; ::_thesis: verum end; theorem Th106: :: SIN_COS9:106 for r being Real for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom ((id Z) (#) (arccot * f)) & ( for x being Real st x in Z holds ( f . x = x / r & f . x > - 1 & f . x < 1 ) ) holds ( (id Z) (#) (arccot * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((id Z) (#) (arccot * f)) `| Z) . x = (arccot . (x / r)) - (x / (r * (1 + ((x / r) ^2)))) ) ) proof let r be Real; ::_thesis: for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom ((id Z) (#) (arccot * f)) & ( for x being Real st x in Z holds ( f . x = x / r & f . x > - 1 & f . x < 1 ) ) holds ( (id Z) (#) (arccot * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((id Z) (#) (arccot * f)) `| Z) . x = (arccot . (x / r)) - (x / (r * (1 + ((x / r) ^2)))) ) ) let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st Z c= dom ((id Z) (#) (arccot * f)) & ( for x being Real st x in Z holds ( f . x = x / r & f . x > - 1 & f . x < 1 ) ) holds ( (id Z) (#) (arccot * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((id Z) (#) (arccot * f)) `| Z) . x = (arccot . (x / r)) - (x / (r * (1 + ((x / r) ^2)))) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom ((id Z) (#) (arccot * f)) & ( for x being Real st x in Z holds ( f . x = x / r & f . x > - 1 & f . x < 1 ) ) implies ( (id Z) (#) (arccot * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((id Z) (#) (arccot * f)) `| Z) . x = (arccot . (x / r)) - (x / (r * (1 + ((x / r) ^2)))) ) ) ) assume that A1: Z c= dom ((id Z) (#) (arccot * f)) and A2: for x being Real st x in Z holds ( f . x = x / r & f . x > - 1 & f . x < 1 ) ; ::_thesis: ( (id Z) (#) (arccot * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((id Z) (#) (arccot * f)) `| Z) . x = (arccot . (x / r)) - (x / (r * (1 + ((x / r) ^2)))) ) ) A3: Z c= (dom (id Z)) /\ (dom (arccot * f)) by A1, VALUED_1:def_4; then A4: Z c= dom (id Z) by XBOOLE_1:18; A5: Z c= dom (arccot * f) by A3, XBOOLE_1:18; for x being Real st x in Z holds f . x = ((1 / r) * x) + 0 proof let x be Real; ::_thesis: ( x in Z implies f . x = ((1 / r) * x) + 0 ) assume x in Z ; ::_thesis: f . x = ((1 / r) * x) + 0 then f . x = x / r by A2; hence f . x = ((1 / r) * x) + 0 ; ::_thesis: verum end; then A6: for x being Real st x in Z holds ( f . x = ((1 / r) * x) + 0 & f . x > - 1 & f . x < 1 ) by A2; then A7: arccot * f is_differentiable_on Z by A5, Th88; A8: for x being Real st x in Z holds (id Z) . x = (1 * x) + 0 by FUNCT_1:18; then A9: id Z is_differentiable_on Z by A4, FDIFF_1:23; A10: for x being Real st x in Z holds ((arccot * f) `| Z) . x = - (1 / (r * (1 + ((x / r) ^2)))) proof let x be Real; ::_thesis: ( x in Z implies ((arccot * f) `| Z) . x = - (1 / (r * (1 + ((x / r) ^2)))) ) assume x in Z ; ::_thesis: ((arccot * f) `| Z) . x = - (1 / (r * (1 + ((x / r) ^2)))) then ((arccot * f) `| Z) . x = - ((1 / r) / (1 + ((((1 / r) * x) + 0) ^2))) by A6, A5, Th88 .= - (1 / (r * (1 + ((x / r) ^2)))) by XCMPLX_1:78 ; hence ((arccot * f) `| Z) . x = - (1 / (r * (1 + ((x / r) ^2)))) ; ::_thesis: verum end; for x being Real st x in Z holds (((id Z) (#) (arccot * f)) `| Z) . x = (arccot . (x / r)) - (x / (r * (1 + ((x / r) ^2)))) proof let x be Real; ::_thesis: ( x in Z implies (((id Z) (#) (arccot * f)) `| Z) . x = (arccot . (x / r)) - (x / (r * (1 + ((x / r) ^2)))) ) assume A11: x in Z ; ::_thesis: (((id Z) (#) (arccot * f)) `| Z) . x = (arccot . (x / r)) - (x / (r * (1 + ((x / r) ^2)))) then A12: (arccot * f) . x = arccot . (f . x) by A5, FUNCT_1:12 .= arccot . (x / r) by A2, A11 ; (((id Z) (#) (arccot * f)) `| Z) . x = (((arccot * f) . x) * (diff ((id Z),x))) + (((id Z) . x) * (diff ((arccot * f),x))) by A1, A9, A7, A11, FDIFF_1:21 .= (((arccot * f) . x) * (((id Z) `| Z) . x)) + (((id Z) . x) * (diff ((arccot * f),x))) by A9, A11, FDIFF_1:def_7 .= (((arccot * f) . x) * 1) + (((id Z) . x) * (diff ((arccot * f),x))) by A4, A8, A11, FDIFF_1:23 .= (((arccot * f) . x) * 1) + (((id Z) . x) * (((arccot * f) `| Z) . x)) by A7, A11, FDIFF_1:def_7 .= ((arccot * f) . x) + (x * (((arccot * f) `| Z) . x)) by A11, FUNCT_1:18 .= (arccot . (x / r)) + (x * (- (1 / (r * (1 + ((x / r) ^2)))))) by A10, A11, A12 .= (arccot . (x / r)) - (x / (r * (1 + ((x / r) ^2)))) ; hence (((id Z) (#) (arccot * f)) `| Z) . x = (arccot . (x / r)) - (x / (r * (1 + ((x / r) ^2)))) ; ::_thesis: verum end; hence ( (id Z) (#) (arccot * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((id Z) (#) (arccot * f)) `| Z) . x = (arccot . (x / r)) - (x / (r * (1 + ((x / r) ^2)))) ) ) by A1, A9, A7, FDIFF_1:21; ::_thesis: verum end; theorem Th107: :: SIN_COS9:107 for r being Real for Z being open Subset of REAL for f1, f2, f being PartFunc of REAL,REAL st Z c= dom (f1 + f2) & ( for x being Real st x in Z holds f1 . x = 1 ) & f2 = (#Z 2) * f & ( for x being Real st x in Z holds f . x = x / r ) holds ( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 + f2) `| Z) . x = (2 * x) / (r ^2) ) ) proof let r be Real; ::_thesis: for Z being open Subset of REAL for f1, f2, f being PartFunc of REAL,REAL st Z c= dom (f1 + f2) & ( for x being Real st x in Z holds f1 . x = 1 ) & f2 = (#Z 2) * f & ( for x being Real st x in Z holds f . x = x / r ) holds ( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 + f2) `| Z) . x = (2 * x) / (r ^2) ) ) let Z be open Subset of REAL; ::_thesis: for f1, f2, f being PartFunc of REAL,REAL st Z c= dom (f1 + f2) & ( for x being Real st x in Z holds f1 . x = 1 ) & f2 = (#Z 2) * f & ( for x being Real st x in Z holds f . x = x / r ) holds ( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 + f2) `| Z) . x = (2 * x) / (r ^2) ) ) let f1, f2, f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (f1 + f2) & ( for x being Real st x in Z holds f1 . x = 1 ) & f2 = (#Z 2) * f & ( for x being Real st x in Z holds f . x = x / r ) implies ( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 + f2) `| Z) . x = (2 * x) / (r ^2) ) ) ) assume that A1: Z c= dom (f1 + f2) and A2: for x being Real st x in Z holds f1 . x = 1 and A3: f2 = (#Z 2) * f and A4: for x being Real st x in Z holds f . x = x / r ; ::_thesis: ( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 + f2) `| Z) . x = (2 * x) / (r ^2) ) ) A5: for x being Real st x in Z holds f1 . x = (0 * x) + 1 by A2; A6: Z c= (dom f1) /\ (dom f2) by A1, VALUED_1:def_1; then A7: Z c= dom f1 by XBOOLE_1:18; then A8: f1 is_differentiable_on Z by A5, FDIFF_1:23; A9: for x being Real st x in Z holds f . x = ((1 / r) * x) + 0 proof let x be Real; ::_thesis: ( x in Z implies f . x = ((1 / r) * x) + 0 ) assume x in Z ; ::_thesis: f . x = ((1 / r) * x) + 0 hence f . x = x / r by A4 .= ((1 / r) * x) + 0 ; ::_thesis: verum end; A10: for x being Real st x in Z holds f2 is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies f2 is_differentiable_in x ) Z c= dom ((#Z 2) * f) by A3, A6, XBOOLE_1:18; then for y being set st y in Z holds y in dom f by FUNCT_1:11; then Z c= dom f by TARSKI:def_3; then A11: f is_differentiable_on Z by A9, FDIFF_1:23; assume x in Z ; ::_thesis: f2 is_differentiable_in x then f is_differentiable_in x by A11, FDIFF_1:9; hence f2 is_differentiable_in x by A3, TAYLOR_1:3; ::_thesis: verum end; Z c= dom f2 by A6, XBOOLE_1:18; then A12: f2 is_differentiable_on Z by A10, FDIFF_1:9; A13: for x being Real st x in Z holds (f2 `| Z) . x = (2 * x) / (r ^2) proof let x be Real; ::_thesis: ( x in Z implies (f2 `| Z) . x = (2 * x) / (r ^2) ) assume A14: x in Z ; ::_thesis: (f2 `| Z) . x = (2 * x) / (r ^2) Z c= dom ((#Z 2) * f) by A3, A6, XBOOLE_1:18; then for y being set st y in Z holds y in dom f by FUNCT_1:11; then A15: Z c= dom f by TARSKI:def_3; then A16: f is_differentiable_on Z by A9, FDIFF_1:23; then A17: f is_differentiable_in x by A14, FDIFF_1:9; (f2 `| Z) . x = diff (((#Z 2) * f),x) by A3, A12, A14, FDIFF_1:def_7 .= (2 * ((f . x) #Z (2 - 1))) * (diff (f,x)) by A17, TAYLOR_1:3 .= (2 * (f . x)) * (diff (f,x)) by PREPOWER:35 .= (2 * (x / r)) * (diff (f,x)) by A4, A14 .= (2 * (x / r)) * ((f `| Z) . x) by A14, A16, FDIFF_1:def_7 .= (2 * (x / r)) * (1 / r) by A9, A14, A15, FDIFF_1:23 .= 2 * ((x / r) * (1 / r)) .= 2 * ((x * 1) / (r * r)) by XCMPLX_1:76 .= (2 * x) / (r ^2) ; hence (f2 `| Z) . x = (2 * x) / (r ^2) ; ::_thesis: verum end; for x being Real st x in Z holds ((f1 + f2) `| Z) . x = (2 * x) / (r ^2) proof let x be Real; ::_thesis: ( x in Z implies ((f1 + f2) `| Z) . x = (2 * x) / (r ^2) ) assume A18: x in Z ; ::_thesis: ((f1 + f2) `| Z) . x = (2 * x) / (r ^2) then ((f1 + f2) `| Z) . x = (diff (f1,x)) + (diff (f2,x)) by A1, A8, A12, FDIFF_1:18 .= ((f1 `| Z) . x) + (diff (f2,x)) by A8, A18, FDIFF_1:def_7 .= ((f1 `| Z) . x) + ((f2 `| Z) . x) by A12, A18, FDIFF_1:def_7 .= 0 + ((f2 `| Z) . x) by A7, A5, A18, FDIFF_1:23 .= (2 * x) / (r ^2) by A13, A18 ; hence ((f1 + f2) `| Z) . x = (2 * x) / (r ^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) / (r ^2) ) ) by A1, A8, A12, FDIFF_1:18; ::_thesis: verum end; theorem Th108: :: SIN_COS9:108 for r being Real for Z being open Subset of REAL for f1, f2, f being PartFunc of REAL,REAL st Z c= dom ((r / 2) (#) (ln * (f1 + f2))) & ( for x being Real st x in Z holds f1 . x = 1 ) & r <> 0 & f2 = (#Z 2) * f & ( for x being Real st x in Z holds f . x = x / r ) holds ( (r / 2) (#) (ln * (f1 + f2)) is_differentiable_on Z & ( for x being Real st x in Z holds (((r / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (r * (1 + ((x / r) ^2))) ) ) proof let r be Real; ::_thesis: for Z being open Subset of REAL for f1, f2, f being PartFunc of REAL,REAL st Z c= dom ((r / 2) (#) (ln * (f1 + f2))) & ( for x being Real st x in Z holds f1 . x = 1 ) & r <> 0 & f2 = (#Z 2) * f & ( for x being Real st x in Z holds f . x = x / r ) holds ( (r / 2) (#) (ln * (f1 + f2)) is_differentiable_on Z & ( for x being Real st x in Z holds (((r / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (r * (1 + ((x / r) ^2))) ) ) let Z be open Subset of REAL; ::_thesis: for f1, f2, f being PartFunc of REAL,REAL st Z c= dom ((r / 2) (#) (ln * (f1 + f2))) & ( for x being Real st x in Z holds f1 . x = 1 ) & r <> 0 & f2 = (#Z 2) * f & ( for x being Real st x in Z holds f . x = x / r ) holds ( (r / 2) (#) (ln * (f1 + f2)) is_differentiable_on Z & ( for x being Real st x in Z holds (((r / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (r * (1 + ((x / r) ^2))) ) ) let f1, f2, f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom ((r / 2) (#) (ln * (f1 + f2))) & ( for x being Real st x in Z holds f1 . x = 1 ) & r <> 0 & f2 = (#Z 2) * f & ( for x being Real st x in Z holds f . x = x / r ) implies ( (r / 2) (#) (ln * (f1 + f2)) is_differentiable_on Z & ( for x being Real st x in Z holds (((r / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (r * (1 + ((x / r) ^2))) ) ) ) assume that A1: Z c= dom ((r / 2) (#) (ln * (f1 + f2))) and A2: for x being Real st x in Z holds f1 . x = 1 and A3: r <> 0 and A4: f2 = (#Z 2) * f and A5: for x being Real st x in Z holds f . x = x / r ; ::_thesis: ( (r / 2) (#) (ln * (f1 + f2)) is_differentiable_on Z & ( for x being Real st x in Z holds (((r / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (r * (1 + ((x / r) ^2))) ) ) A6: Z c= dom (ln * (f1 + f2)) by A1, VALUED_1:def_5; then for y being set st y in Z holds y in dom (f1 + f2) by FUNCT_1:11; then A7: Z c= dom (f1 + f2) by TARSKI:def_3; then A8: f1 + f2 is_differentiable_on Z by A2, A4, A5, Th107; dom (f1 + f2) = (dom f1) /\ (dom f2) by VALUED_1:def_1; then A9: Z c= dom f2 by A7, XBOOLE_1:18; for x being Real st x in Z holds ln * (f1 + f2) is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies ln * (f1 + f2) is_differentiable_in x ) set g = #Z 2; assume A10: x in Z ; ::_thesis: ln * (f1 + f2) is_differentiable_in x then (f1 + f2) . x = (f1 . x) + (f2 . x) by A7, VALUED_1:def_1 .= 1 + (((#Z 2) * f) . x) by A2, A4, A10 .= 1 + ((#Z 2) . (f . x)) by A4, A9, A10, FUNCT_1:12 .= 1 + ((#Z 2) . (x / r)) by A5, A10 .= 1 + ((x / r) #Z (1 + 1)) by TAYLOR_1:def_1 .= 1 + (((x / r) #Z 1) * ((x / r) #Z 1)) by TAYLOR_1:1 .= 1 + ((x / r) * ((x / r) #Z 1)) by PREPOWER:35 .= 1 + ((x / r) * (x / r)) by PREPOWER:35 ; then A11: (f1 + f2) . x > 0 by XREAL_1:34, XREAL_1:63; f1 + f2 is_differentiable_in x by A8, A10, FDIFF_1:9; hence ln * (f1 + f2) is_differentiable_in x by A11, TAYLOR_1:20; ::_thesis: verum end; then A12: ln * (f1 + f2) is_differentiable_on Z by A6, FDIFF_1:9; for x being Real st x in Z holds (((r / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (r * (1 + ((x / r) ^2))) proof let x be Real; ::_thesis: ( x in Z implies (((r / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (r * (1 + ((x / r) ^2))) ) set g = #Z 2; assume A13: x in Z ; ::_thesis: (((r / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (r * (1 + ((x / r) ^2))) then A14: f1 + f2 is_differentiable_in x by A8, FDIFF_1:9; A15: (f1 + f2) . x = (f1 . x) + (f2 . x) by A7, A13, VALUED_1:def_1 .= 1 + (((#Z 2) * f) . x) by A2, A4, A13 .= 1 + ((#Z 2) . (f . x)) by A4, A9, A13, FUNCT_1:12 .= 1 + ((#Z 2) . (x / r)) by A5, A13 .= 1 + ((x / r) #Z (1 + 1)) by TAYLOR_1:def_1 .= 1 + (((x / r) #Z 1) * ((x / r) #Z 1)) by TAYLOR_1:1 .= 1 + ((x / r) * ((x / r) #Z 1)) by PREPOWER:35 .= 1 + ((x / r) * (x / r)) by PREPOWER:35 ; then (f1 + f2) . x > 0 by XREAL_1:34, XREAL_1:63; then A16: diff ((ln * (f1 + f2)),x) = (diff ((f1 + f2),x)) / ((f1 + f2) . x) by A14, TAYLOR_1:20 .= (((f1 + f2) `| Z) . x) / ((f1 + f2) . x) by A8, A13, FDIFF_1:def_7 .= ((2 * x) / (r ^2)) / (1 + ((x / r) ^2)) by A2, A4, A5, A7, A13, A15, Th107 ; thus (((r / 2) (#) (ln * (f1 + f2))) `| Z) . x = (r / 2) * (diff ((ln * (f1 + f2)),x)) by A1, A12, A13, FDIFF_1:20 .= ((r * x) / (r ^2)) / (1 + ((x / r) ^2)) by A16 .= ((r / r) * (x / r)) / (1 + ((x / r) ^2)) by XCMPLX_1:76 .= (1 * (x / r)) / (1 + ((x / r) ^2)) by A3, XCMPLX_1:60 .= x / (r * (1 + ((x / r) ^2))) by XCMPLX_1:78 ; ::_thesis: verum end; hence ( (r / 2) (#) (ln * (f1 + f2)) is_differentiable_on Z & ( for x being Real st x in Z holds (((r / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (r * (1 + ((x / r) ^2))) ) ) by A1, A12, FDIFF_1:20; ::_thesis: verum end; theorem :: SIN_COS9:109 for r being Real for Z being open Subset of REAL for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom (((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) & r <> 0 & ( for x being Real st x in Z holds ( f . x = x / r & f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds f1 . x = 1 ) & f2 = (#Z 2) * f & ( for x being Real st x in Z holds f . x = x / r ) holds ( ((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . (x / r) ) ) proof let r be Real; ::_thesis: for Z being open Subset of REAL for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom (((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) & r <> 0 & ( for x being Real st x in Z holds ( f . x = x / r & f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds f1 . x = 1 ) & f2 = (#Z 2) * f & ( for x being Real st x in Z holds f . x = x / r ) holds ( ((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . (x / r) ) ) let Z be open Subset of REAL; ::_thesis: for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom (((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) & r <> 0 & ( for x being Real st x in Z holds ( f . x = x / r & f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds f1 . x = 1 ) & f2 = (#Z 2) * f & ( for x being Real st x in Z holds f . x = x / r ) holds ( ((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . (x / r) ) ) let f, f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) & r <> 0 & ( for x being Real st x in Z holds ( f . x = x / r & f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds f1 . x = 1 ) & f2 = (#Z 2) * f & ( for x being Real st x in Z holds f . x = x / r ) implies ( ((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . (x / r) ) ) ) assume that A1: Z c= dom (((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) and A2: r <> 0 and A3: for x being Real st x in Z holds ( f . x = x / r & f . x > - 1 & f . x < 1 ) and A4: for x being Real st x in Z holds f1 . x = 1 and A5: f2 = (#Z 2) * f and A6: for x being Real st x in Z holds f . x = x / r ; ::_thesis: ( ((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . (x / r) ) ) A7: Z c= (dom ((id Z) (#) (arctan * f))) /\ (dom ((r / 2) (#) (ln * (f1 + f2)))) by A1, VALUED_1:12; then A8: Z c= dom ((r / 2) (#) (ln * (f1 + f2))) by XBOOLE_1:18; then A9: (r / 2) (#) (ln * (f1 + f2)) is_differentiable_on Z by A2, A4, A5, A6, Th108; A10: Z c= dom ((id Z) (#) (arctan * f)) by A7, XBOOLE_1:18; then A11: (id Z) (#) (arctan * f) is_differentiable_on Z by A3, Th105; for x being Real st x in Z holds ((((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . (x / r) proof let x be Real; ::_thesis: ( x in Z implies ((((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . (x / r) ) assume A12: x in Z ; ::_thesis: ((((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . (x / r) hence ((((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = (diff (((id Z) (#) (arctan * f)),x)) - (diff (((r / 2) (#) (ln * (f1 + f2))),x)) by A1, A11, A9, FDIFF_1:19 .= ((((id Z) (#) (arctan * f)) `| Z) . x) - (diff (((r / 2) (#) (ln * (f1 + f2))),x)) by A11, A12, FDIFF_1:def_7 .= ((((id Z) (#) (arctan * f)) `| Z) . x) - ((((r / 2) (#) (ln * (f1 + f2))) `| Z) . x) by A9, A12, FDIFF_1:def_7 .= ((arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2))))) - ((((r / 2) (#) (ln * (f1 + f2))) `| Z) . x) by A3, A10, A12, Th105 .= ((arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2))))) - (x / (r * (1 + ((x / r) ^2)))) by A2, A4, A5, A6, A8, A12, Th108 .= arctan . (x / r) ; ::_thesis: verum end; hence ( ((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . (x / r) ) ) by A1, A11, A9, FDIFF_1:19; ::_thesis: verum end; theorem :: SIN_COS9:110 for r being Real for Z being open Subset of REAL for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom (((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) & r <> 0 & ( for x being Real st x in Z holds ( f . x = x / r & f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds f1 . x = 1 ) & f2 = (#Z 2) * f & ( for x being Real st x in Z holds f . x = x / r ) holds ( ((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . (x / r) ) ) proof let r be Real; ::_thesis: for Z being open Subset of REAL for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom (((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) & r <> 0 & ( for x being Real st x in Z holds ( f . x = x / r & f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds f1 . x = 1 ) & f2 = (#Z 2) * f & ( for x being Real st x in Z holds f . x = x / r ) holds ( ((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . (x / r) ) ) let Z be open Subset of REAL; ::_thesis: for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom (((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) & r <> 0 & ( for x being Real st x in Z holds ( f . x = x / r & f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds f1 . x = 1 ) & f2 = (#Z 2) * f & ( for x being Real st x in Z holds f . x = x / r ) holds ( ((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . (x / r) ) ) let f, f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) & r <> 0 & ( for x being Real st x in Z holds ( f . x = x / r & f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds f1 . x = 1 ) & f2 = (#Z 2) * f & ( for x being Real st x in Z holds f . x = x / r ) implies ( ((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . (x / r) ) ) ) assume that A1: Z c= dom (((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) and A2: r <> 0 and A3: for x being Real st x in Z holds ( f . x = x / r & f . x > - 1 & f . x < 1 ) and A4: for x being Real st x in Z holds f1 . x = 1 and A5: f2 = (#Z 2) * f and A6: for x being Real st x in Z holds f . x = x / r ; ::_thesis: ( ((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . (x / r) ) ) A7: Z c= (dom ((id Z) (#) (arccot * f))) /\ (dom ((r / 2) (#) (ln * (f1 + f2)))) by A1, VALUED_1:def_1; then A8: Z c= dom ((r / 2) (#) (ln * (f1 + f2))) by XBOOLE_1:18; then A9: (r / 2) (#) (ln * (f1 + f2)) is_differentiable_on Z by A2, A4, A5, A6, Th108; A10: Z c= dom ((id Z) (#) (arccot * f)) by A7, XBOOLE_1:18; then A11: (id Z) (#) (arccot * f) is_differentiable_on Z by A3, Th106; for x being Real st x in Z holds ((((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . (x / r) proof let x be Real; ::_thesis: ( x in Z implies ((((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . (x / r) ) assume A12: x in Z ; ::_thesis: ((((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . (x / r) hence ((((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = (diff (((id Z) (#) (arccot * f)),x)) + (diff (((r / 2) (#) (ln * (f1 + f2))),x)) by A1, A11, A9, FDIFF_1:18 .= ((((id Z) (#) (arccot * f)) `| Z) . x) + (diff (((r / 2) (#) (ln * (f1 + f2))),x)) by A11, A12, FDIFF_1:def_7 .= ((((id Z) (#) (arccot * f)) `| Z) . x) + ((((r / 2) (#) (ln * (f1 + f2))) `| Z) . x) by A9, A12, FDIFF_1:def_7 .= ((arccot . (x / r)) - (x / (r * (1 + ((x / r) ^2))))) + ((((r / 2) (#) (ln * (f1 + f2))) `| Z) . x) by A3, A10, A12, Th106 .= ((arccot . (x / r)) - (x / (r * (1 + ((x / r) ^2))))) + (x / (r * (1 + ((x / r) ^2)))) by A2, A4, A5, A6, A8, A12, Th108 .= arccot . (x / r) ; ::_thesis: verum end; hence ( ((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . (x / r) ) ) by A1, A11, A9, FDIFF_1:18; ::_thesis: verum end; theorem :: SIN_COS9:111 for Z being open Subset of REAL st not 0 in Z & Z c= dom (arctan * ((id Z) ^)) & ( for x being Real st x in Z holds ( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ) holds ( arctan * ((id Z) ^) is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * ((id Z) ^)) `| Z) . x = - (1 / (1 + (x ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( not 0 in Z & Z c= dom (arctan * ((id Z) ^)) & ( for x being Real st x in Z holds ( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ) implies ( arctan * ((id Z) ^) is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * ((id Z) ^)) `| Z) . x = - (1 / (1 + (x ^2))) ) ) ) set f = id Z; assume that A1: not 0 in Z and A2: Z c= dom (arctan * ((id Z) ^)) and A3: for x being Real st x in Z holds ( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ; ::_thesis: ( arctan * ((id Z) ^) is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * ((id Z) ^)) `| Z) . x = - (1 / (1 + (x ^2))) ) ) dom (arctan * ((id Z) ^)) c= dom ((id Z) ^) by RELAT_1:25; then A4: Z c= dom ((id Z) ^) by A2, XBOOLE_1:1; A5: (id Z) ^ is_differentiable_on Z by A1, FDIFF_5:4; A6: for x being Real st x in Z holds arctan * ((id Z) ^) is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies arctan * ((id Z) ^) is_differentiable_in x ) assume A7: x in Z ; ::_thesis: arctan * ((id Z) ^) is_differentiable_in x then A8: ((id Z) ^) . x > - 1 by A3; A9: ((id Z) ^) . x < 1 by A3, A7; (id Z) ^ is_differentiable_in x by A5, A7, FDIFF_1:9; hence arctan * ((id Z) ^) is_differentiable_in x by A8, A9, Th85; ::_thesis: verum end; then A10: arctan * ((id Z) ^) is_differentiable_on Z by A2, FDIFF_1:9; for x being Real st x in Z holds ((arctan * ((id Z) ^)) `| Z) . x = - (1 / (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((arctan * ((id Z) ^)) `| Z) . x = - (1 / (1 + (x ^2))) ) assume A11: x in Z ; ::_thesis: ((arctan * ((id Z) ^)) `| Z) . x = - (1 / (1 + (x ^2))) then A12: (id Z) ^ is_differentiable_in x by A5, FDIFF_1:9; A13: ((id Z) ^) . x < 1 by A3, A11; A14: ((id Z) ^) . x > - 1 by A3, A11; (id Z) . x = x by A11, FUNCT_1:18; then x <> 0 by A4, A11, RFUNCT_1:3; then A15: x ^2 <> 0 by SQUARE_1:12; ((arctan * ((id Z) ^)) `| Z) . x = diff ((arctan * ((id Z) ^)),x) by A10, A11, FDIFF_1:def_7 .= (diff (((id Z) ^),x)) / (1 + ((((id Z) ^) . x) ^2)) by A12, A14, A13, Th85 .= ((((id Z) ^) `| Z) . x) / (1 + ((((id Z) ^) . x) ^2)) by A5, A11, FDIFF_1:def_7 .= (- (1 / (x ^2))) / (1 + ((((id Z) ^) . x) ^2)) by A1, A11, FDIFF_5:4 .= (- (1 / (x ^2))) / (1 + ((((id Z) . x) ") ^2)) by A4, A11, RFUNCT_1:def_2 .= (- (1 / (x ^2))) / (1 + ((1 / x) ^2)) by A11, FUNCT_1:18 .= - ((1 / (x ^2)) / (1 + ((1 / x) ^2))) .= - (1 / ((x ^2) * (1 + ((1 / x) ^2)))) by XCMPLX_1:78 .= - (1 / (((x ^2) * 1) + ((x ^2) * ((1 / x) ^2)))) .= - (1 / ((x ^2) + ((x ^2) * (1 / (x * x))))) by XCMPLX_1:102 .= - (1 / ((x ^2) + (((x ^2) * 1) / (x ^2)))) .= - (1 / (1 + (x ^2))) by A15, XCMPLX_1:60 ; hence ((arctan * ((id Z) ^)) `| Z) . x = - (1 / (1 + (x ^2))) ; ::_thesis: verum end; hence ( arctan * ((id Z) ^) is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * ((id Z) ^)) `| Z) . x = - (1 / (1 + (x ^2))) ) ) by A2, A6, FDIFF_1:9; ::_thesis: verum end; theorem :: SIN_COS9:112 for Z being open Subset of REAL st not 0 in Z & Z c= dom (arccot * ((id Z) ^)) & ( for x being Real st x in Z holds ( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ) holds ( arccot * ((id Z) ^) is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * ((id Z) ^)) `| Z) . x = 1 / (1 + (x ^2)) ) ) proof let Z be open Subset of REAL; ::_thesis: ( not 0 in Z & Z c= dom (arccot * ((id Z) ^)) & ( for x being Real st x in Z holds ( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ) implies ( arccot * ((id Z) ^) is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * ((id Z) ^)) `| Z) . x = 1 / (1 + (x ^2)) ) ) ) set f = id Z; assume that A1: not 0 in Z and A2: Z c= dom (arccot * ((id Z) ^)) and A3: for x being Real st x in Z holds ( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ; ::_thesis: ( arccot * ((id Z) ^) is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * ((id Z) ^)) `| Z) . x = 1 / (1 + (x ^2)) ) ) dom (arccot * ((id Z) ^)) c= dom ((id Z) ^) by RELAT_1:25; then A4: Z c= dom ((id Z) ^) by A2, XBOOLE_1:1; A5: (id Z) ^ is_differentiable_on Z by A1, FDIFF_5:4; A6: for x being Real st x in Z holds arccot * ((id Z) ^) is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies arccot * ((id Z) ^) is_differentiable_in x ) assume A7: x in Z ; ::_thesis: arccot * ((id Z) ^) is_differentiable_in x then A8: ((id Z) ^) . x > - 1 by A3; A9: ((id Z) ^) . x < 1 by A3, A7; (id Z) ^ is_differentiable_in x by A5, A7, FDIFF_1:9; hence arccot * ((id Z) ^) is_differentiable_in x by A8, A9, Th86; ::_thesis: verum end; then A10: arccot * ((id Z) ^) is_differentiable_on Z by A2, FDIFF_1:9; for x being Real st x in Z holds ((arccot * ((id Z) ^)) `| Z) . x = 1 / (1 + (x ^2)) proof let x be Real; ::_thesis: ( x in Z implies ((arccot * ((id Z) ^)) `| Z) . x = 1 / (1 + (x ^2)) ) assume A11: x in Z ; ::_thesis: ((arccot * ((id Z) ^)) `| Z) . x = 1 / (1 + (x ^2)) then A12: (id Z) ^ is_differentiable_in x by A5, FDIFF_1:9; A13: ((id Z) ^) . x < 1 by A3, A11; A14: ((id Z) ^) . x > - 1 by A3, A11; (id Z) . x = x by A11, FUNCT_1:18; then x <> 0 by A4, A11, RFUNCT_1:3; then A15: x ^2 <> 0 by SQUARE_1:12; ((arccot * ((id Z) ^)) `| Z) . x = diff ((arccot * ((id Z) ^)),x) by A10, A11, FDIFF_1:def_7 .= - ((diff (((id Z) ^),x)) / (1 + ((((id Z) ^) . x) ^2))) by A12, A14, A13, Th86 .= - (((((id Z) ^) `| Z) . x) / (1 + ((((id Z) ^) . x) ^2))) by A5, A11, FDIFF_1:def_7 .= - ((- (1 / (x ^2))) / (1 + ((((id Z) ^) . x) ^2))) by A1, A11, FDIFF_5:4 .= - ((- (1 / (x ^2))) / (1 + ((((id Z) . x) ") ^2))) by A4, A11, RFUNCT_1:def_2 .= - ((- (1 / (x ^2))) / (1 + ((1 / x) ^2))) by A11, FUNCT_1:18 .= (1 / (x ^2)) / (1 + ((1 / x) ^2)) .= 1 / ((x ^2) * (1 + ((1 / x) ^2))) by XCMPLX_1:78 .= 1 / (((x ^2) * 1) + ((x ^2) * ((1 / x) ^2))) .= 1 / ((x ^2) + ((x ^2) * (1 / (x * x)))) by XCMPLX_1:102 .= 1 / ((x ^2) + (((x ^2) * 1) / (x ^2))) .= 1 / (1 + (x ^2)) by A15, XCMPLX_1:60 ; hence ((arccot * ((id Z) ^)) `| Z) . x = 1 / (1 + (x ^2)) ; ::_thesis: verum end; hence ( arccot * ((id Z) ^) is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * ((id Z) ^)) `| Z) . x = 1 / (1 + (x ^2)) ) ) by A2, A6, FDIFF_1:9; ::_thesis: verum end; theorem :: SIN_COS9:113 for h, r, s being Real for Z being open Subset of REAL for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom (arctan * f) & f = f1 + (h (#) f2) & ( for x being Real st x in Z holds ( f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds f1 . x = r + (s * x) ) & f2 = #Z 2 holds ( arctan * (f1 + (h (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * (f1 + (h (#) f2))) `| Z) . x = (s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2)) ) ) proof let h, r, s be Real; ::_thesis: for Z being open Subset of REAL for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom (arctan * f) & f = f1 + (h (#) f2) & ( for x being Real st x in Z holds ( f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds f1 . x = r + (s * x) ) & f2 = #Z 2 holds ( arctan * (f1 + (h (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * (f1 + (h (#) f2))) `| Z) . x = (s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2)) ) ) let Z be open Subset of REAL; ::_thesis: for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom (arctan * f) & f = f1 + (h (#) f2) & ( for x being Real st x in Z holds ( f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds f1 . x = r + (s * x) ) & f2 = #Z 2 holds ( arctan * (f1 + (h (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * (f1 + (h (#) f2))) `| Z) . x = (s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2)) ) ) let f, f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (arctan * f) & f = f1 + (h (#) f2) & ( for x being Real st x in Z holds ( f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds f1 . x = r + (s * x) ) & f2 = #Z 2 implies ( arctan * (f1 + (h (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * (f1 + (h (#) f2))) `| Z) . x = (s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2)) ) ) ) assume that A1: Z c= dom (arctan * f) and A2: f = f1 + (h (#) f2) and A3: for x being Real st x in Z holds ( f . x > - 1 & f . x < 1 ) and A4: for x being Real st x in Z holds f1 . x = r + (s * x) and A5: f2 = #Z 2 ; ::_thesis: ( arctan * (f1 + (h (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * (f1 + (h (#) f2))) `| Z) . x = (s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2)) ) ) dom (arctan * f) c= dom f by RELAT_1:25; then A6: Z c= dom (f1 + (h (#) f2)) by A1, A2, XBOOLE_1:1; then Z c= (dom f1) /\ (dom (h (#) f2)) by VALUED_1:def_1; then A7: Z c= dom (h (#) f2) by XBOOLE_1:18; A8: f1 + (h (#) f2) is_differentiable_on Z by A4, A5, A6, FDIFF_4:12; A9: for x being Real st x in Z holds arctan * (f1 + (h (#) f2)) is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies arctan * (f1 + (h (#) f2)) is_differentiable_in x ) assume A10: x in Z ; ::_thesis: arctan * (f1 + (h (#) f2)) is_differentiable_in x then A11: f . x > - 1 by A3; A12: f . x < 1 by A3, A10; f is_differentiable_in x by A2, A8, A10, FDIFF_1:9; hence arctan * (f1 + (h (#) f2)) is_differentiable_in x by A2, A11, A12, Th85; ::_thesis: verum end; then A13: arctan * (f1 + (h (#) f2)) is_differentiable_on Z by A1, A2, FDIFF_1:9; for x being Real st x in Z holds ((arctan * (f1 + (h (#) f2))) `| Z) . x = (s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2)) proof let x be Real; ::_thesis: ( x in Z implies ((arctan * (f1 + (h (#) f2))) `| Z) . x = (s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2)) ) assume A14: x in Z ; ::_thesis: ((arctan * (f1 + (h (#) f2))) `| Z) . x = (s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2)) then A15: (f1 + (h (#) f2)) . x = (f1 . x) + ((h (#) f2) . x) by A6, VALUED_1:def_1 .= (f1 . x) + (h * (f2 . x)) by A7, A14, VALUED_1:def_5 .= (r + (s * x)) + (h * (f2 . x)) by A4, A14 .= (r + (s * x)) + (h * (x #Z (1 + 1))) by A5, TAYLOR_1:def_1 .= (r + (s * x)) + (h * ((x #Z 1) * (x #Z 1))) by TAYLOR_1:1 .= (r + (s * x)) + (h * (x * (x #Z 1))) by PREPOWER:35 .= (r + (s * x)) + (h * (x ^2)) by PREPOWER:35 ; A16: f is_differentiable_in x by A2, A8, A14, FDIFF_1:9; A17: f . x > - 1 by A3, A14; A18: f . x < 1 by A3, A14; ((arctan * (f1 + (h (#) f2))) `| Z) . x = diff ((arctan * f),x) by A2, A13, A14, FDIFF_1:def_7 .= (diff (f,x)) / (1 + ((f . x) ^2)) by A16, A17, A18, Th85 .= (((f1 + (h (#) f2)) `| Z) . x) / (1 + ((f . x) ^2)) by A2, A8, A14, FDIFF_1:def_7 .= (s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2)) by A2, A4, A5, A6, A14, A15, FDIFF_4:12 ; hence ((arctan * (f1 + (h (#) f2))) `| Z) . x = (s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2)) ; ::_thesis: verum end; hence ( arctan * (f1 + (h (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * (f1 + (h (#) f2))) `| Z) . x = (s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2)) ) ) by A1, A2, A9, FDIFF_1:9; ::_thesis: verum end; theorem :: SIN_COS9:114 for h, r, s being Real for Z being open Subset of REAL for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom (arccot * f) & f = f1 + (h (#) f2) & ( for x being Real st x in Z holds ( f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds f1 . x = r + (s * x) ) & f2 = #Z 2 holds ( arccot * (f1 + (h (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * (f1 + (h (#) f2))) `| Z) . x = - ((s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2))) ) ) proof let h, r, s be Real; ::_thesis: for Z being open Subset of REAL for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom (arccot * f) & f = f1 + (h (#) f2) & ( for x being Real st x in Z holds ( f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds f1 . x = r + (s * x) ) & f2 = #Z 2 holds ( arccot * (f1 + (h (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * (f1 + (h (#) f2))) `| Z) . x = - ((s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2))) ) ) let Z be open Subset of REAL; ::_thesis: for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom (arccot * f) & f = f1 + (h (#) f2) & ( for x being Real st x in Z holds ( f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds f1 . x = r + (s * x) ) & f2 = #Z 2 holds ( arccot * (f1 + (h (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * (f1 + (h (#) f2))) `| Z) . x = - ((s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2))) ) ) let f, f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (arccot * f) & f = f1 + (h (#) f2) & ( for x being Real st x in Z holds ( f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds f1 . x = r + (s * x) ) & f2 = #Z 2 implies ( arccot * (f1 + (h (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * (f1 + (h (#) f2))) `| Z) . x = - ((s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2))) ) ) ) assume that A1: Z c= dom (arccot * f) and A2: f = f1 + (h (#) f2) and A3: for x being Real st x in Z holds ( f . x > - 1 & f . x < 1 ) and A4: for x being Real st x in Z holds f1 . x = r + (s * x) and A5: f2 = #Z 2 ; ::_thesis: ( arccot * (f1 + (h (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * (f1 + (h (#) f2))) `| Z) . x = - ((s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2))) ) ) dom (arccot * f) c= dom f by RELAT_1:25; then A6: Z c= dom (f1 + (h (#) f2)) by A1, A2, XBOOLE_1:1; then Z c= (dom f1) /\ (dom (h (#) f2)) by VALUED_1:def_1; then A7: Z c= dom (h (#) f2) by XBOOLE_1:18; A8: f1 + (h (#) f2) is_differentiable_on Z by A4, A5, A6, FDIFF_4:12; A9: for x being Real st x in Z holds arccot * (f1 + (h (#) f2)) is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies arccot * (f1 + (h (#) f2)) is_differentiable_in x ) assume A10: x in Z ; ::_thesis: arccot * (f1 + (h (#) f2)) is_differentiable_in x then A11: f . x > - 1 by A3; A12: f . x < 1 by A3, A10; f is_differentiable_in x by A2, A8, A10, FDIFF_1:9; hence arccot * (f1 + (h (#) f2)) is_differentiable_in x by A2, A11, A12, Th86; ::_thesis: verum end; then A13: arccot * (f1 + (h (#) f2)) is_differentiable_on Z by A1, A2, FDIFF_1:9; for x being Real st x in Z holds ((arccot * (f1 + (h (#) f2))) `| Z) . x = - ((s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((arccot * (f1 + (h (#) f2))) `| Z) . x = - ((s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2))) ) assume A14: x in Z ; ::_thesis: ((arccot * (f1 + (h (#) f2))) `| Z) . x = - ((s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2))) then A15: (f1 + (h (#) f2)) . x = (f1 . x) + ((h (#) f2) . x) by A6, VALUED_1:def_1 .= (f1 . x) + (h * (f2 . x)) by A7, A14, VALUED_1:def_5 .= (r + (s * x)) + (h * (f2 . x)) by A4, A14 .= (r + (s * x)) + (h * (x #Z (1 + 1))) by A5, TAYLOR_1:def_1 .= (r + (s * x)) + (h * ((x #Z 1) * (x #Z 1))) by TAYLOR_1:1 .= (r + (s * x)) + (h * (x * (x #Z 1))) by PREPOWER:35 .= (r + (s * x)) + (h * (x ^2)) by PREPOWER:35 ; A16: f is_differentiable_in x by A2, A8, A14, FDIFF_1:9; A17: f . x > - 1 by A3, A14; A18: f . x < 1 by A3, A14; ((arccot * (f1 + (h (#) f2))) `| Z) . x = diff ((arccot * f),x) by A2, A13, A14, FDIFF_1:def_7 .= - ((diff (f,x)) / (1 + ((f . x) ^2))) by A16, A17, A18, Th86 .= - ((((f1 + (h (#) f2)) `| Z) . x) / (1 + ((f . x) ^2))) by A2, A8, A14, FDIFF_1:def_7 .= - ((s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2))) by A2, A4, A5, A6, A14, A15, FDIFF_4:12 ; hence ((arccot * (f1 + (h (#) f2))) `| Z) . x = - ((s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2))) ; ::_thesis: verum end; hence ( arccot * (f1 + (h (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * (f1 + (h (#) f2))) `| Z) . x = - ((s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2))) ) ) by A1, A2, A9, FDIFF_1:9; ::_thesis: verum end; theorem :: SIN_COS9:115 for Z being open Subset of REAL st Z c= dom (arctan * exp_R) & ( for x being Real st x in Z holds exp_R . x < 1 ) holds ( arctan * exp_R is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * exp_R) `| Z) . x = (exp_R . x) / (1 + ((exp_R . x) ^2)) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (arctan * exp_R) & ( for x being Real st x in Z holds exp_R . x < 1 ) implies ( arctan * exp_R is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * exp_R) `| Z) . x = (exp_R . x) / (1 + ((exp_R . x) ^2)) ) ) ) assume that A1: Z c= dom (arctan * exp_R) and A2: for x being Real st x in Z holds exp_R . x < 1 ; ::_thesis: ( arctan * exp_R is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * exp_R) `| Z) . x = (exp_R . x) / (1 + ((exp_R . x) ^2)) ) ) A3: for x being Real st x in Z holds arctan * exp_R is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies arctan * exp_R is_differentiable_in x ) A4: exp_R is_differentiable_in x by SIN_COS:65; assume x in Z ; ::_thesis: arctan * exp_R is_differentiable_in x then A5: exp_R . x < 1 by A2; (exp_R . x) + 0 > 0 + (- 1) by SIN_COS:54, XREAL_1:8; hence arctan * exp_R is_differentiable_in x by A5, A4, Th85; ::_thesis: verum end; then A6: arctan * exp_R is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((arctan * exp_R) `| Z) . x = (exp_R . x) / (1 + ((exp_R . x) ^2)) proof let x be Real; ::_thesis: ( x in Z implies ((arctan * exp_R) `| Z) . x = (exp_R . x) / (1 + ((exp_R . x) ^2)) ) A7: (exp_R . x) + 0 > 0 + (- 1) by SIN_COS:54, XREAL_1:8; A8: exp_R is_differentiable_in x by SIN_COS:65; assume A9: x in Z ; ::_thesis: ((arctan * exp_R) `| Z) . x = (exp_R . x) / (1 + ((exp_R . x) ^2)) then A10: exp_R . x < 1 by A2; ((arctan * exp_R) `| Z) . x = diff ((arctan * exp_R),x) by A6, A9, FDIFF_1:def_7 .= (diff (exp_R,x)) / (1 + ((exp_R . x) ^2)) by A7, A10, A8, Th85 .= (exp_R . x) / (1 + ((exp_R . x) ^2)) by SIN_COS:65 ; hence ((arctan * exp_R) `| Z) . x = (exp_R . x) / (1 + ((exp_R . x) ^2)) ; ::_thesis: verum end; hence ( arctan * exp_R is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * exp_R) `| Z) . x = (exp_R . x) / (1 + ((exp_R . x) ^2)) ) ) by A1, A3, FDIFF_1:9; ::_thesis: verum end; theorem :: SIN_COS9:116 for Z being open Subset of REAL st Z c= dom (arccot * exp_R) & ( for x being Real st x in Z holds exp_R . x < 1 ) holds ( arccot * exp_R is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * exp_R) `| Z) . x = - ((exp_R . x) / (1 + ((exp_R . x) ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (arccot * exp_R) & ( for x being Real st x in Z holds exp_R . x < 1 ) implies ( arccot * exp_R is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * exp_R) `| Z) . x = - ((exp_R . x) / (1 + ((exp_R . x) ^2))) ) ) ) assume that A1: Z c= dom (arccot * exp_R) and A2: for x being Real st x in Z holds exp_R . x < 1 ; ::_thesis: ( arccot * exp_R is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * exp_R) `| Z) . x = - ((exp_R . x) / (1 + ((exp_R . x) ^2))) ) ) A3: for x being Real st x in Z holds arccot * exp_R is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies arccot * exp_R is_differentiable_in x ) A4: exp_R is_differentiable_in x by SIN_COS:65; assume x in Z ; ::_thesis: arccot * exp_R is_differentiable_in x then A5: exp_R . x < 1 by A2; (exp_R . x) + 0 > 0 + (- 1) by SIN_COS:54, XREAL_1:8; hence arccot * exp_R is_differentiable_in x by A5, A4, Th86; ::_thesis: verum end; then A6: arccot * exp_R is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((arccot * exp_R) `| Z) . x = - ((exp_R . x) / (1 + ((exp_R . x) ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((arccot * exp_R) `| Z) . x = - ((exp_R . x) / (1 + ((exp_R . x) ^2))) ) A7: (exp_R . x) + 0 > 0 + (- 1) by SIN_COS:54, XREAL_1:8; A8: exp_R is_differentiable_in x by SIN_COS:65; assume A9: x in Z ; ::_thesis: ((arccot * exp_R) `| Z) . x = - ((exp_R . x) / (1 + ((exp_R . x) ^2))) then A10: exp_R . x < 1 by A2; ((arccot * exp_R) `| Z) . x = diff ((arccot * exp_R),x) by A6, A9, FDIFF_1:def_7 .= - ((diff (exp_R,x)) / (1 + ((exp_R . x) ^2))) by A7, A10, A8, Th86 .= - ((exp_R . x) / (1 + ((exp_R . x) ^2))) by SIN_COS:65 ; hence ((arccot * exp_R) `| Z) . x = - ((exp_R . x) / (1 + ((exp_R . x) ^2))) ; ::_thesis: verum end; hence ( arccot * exp_R is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * exp_R) `| Z) . x = - ((exp_R . x) / (1 + ((exp_R . x) ^2))) ) ) by A1, A3, FDIFF_1:9; ::_thesis: verum end; theorem :: SIN_COS9:117 for Z being open Subset of REAL st Z c= dom (arctan * ln) & ( for x being Real st x in Z holds ( ln . x > - 1 & ln . x < 1 ) ) holds ( arctan * ln is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * ln) `| Z) . x = 1 / (x * (1 + ((ln . x) ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (arctan * ln) & ( for x being Real st x in Z holds ( ln . x > - 1 & ln . x < 1 ) ) implies ( arctan * ln is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * ln) `| Z) . x = 1 / (x * (1 + ((ln . x) ^2))) ) ) ) A1: right_open_halfline 0 = { g where g is Real : 0 < g } by XXREAL_1:230; assume that A2: Z c= dom (arctan * ln) and A3: for x being Real st x in Z holds ( ln . x > - 1 & ln . x < 1 ) ; ::_thesis: ( arctan * ln is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * ln) `| Z) . x = 1 / (x * (1 + ((ln . x) ^2))) ) ) dom (arctan * ln) c= dom ln by RELAT_1:25; then A4: Z c= dom ln by A2, XBOOLE_1:1; A5: for x being Real st x in Z holds x > 0 proof let x be Real; ::_thesis: ( x in Z implies x > 0 ) assume x in Z ; ::_thesis: x > 0 then x in right_open_halfline 0 by A4, TAYLOR_1:18; then ex g being Real st ( x = g & 0 < g ) by A1; hence x > 0 ; ::_thesis: verum end; A6: for x being Real st x in Z holds arctan * ln is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies arctan * ln is_differentiable_in x ) assume A7: x in Z ; ::_thesis: arctan * ln is_differentiable_in x then A8: ln . x > - 1 by A3; A9: ln . x < 1 by A3, A7; ln is_differentiable_in x by A5, A7, TAYLOR_1:18; hence arctan * ln is_differentiable_in x by A8, A9, Th85; ::_thesis: verum end; then A10: arctan * ln is_differentiable_on Z by A2, FDIFF_1:9; for x being Real st x in Z holds ((arctan * ln) `| Z) . x = 1 / (x * (1 + ((ln . x) ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((arctan * ln) `| Z) . x = 1 / (x * (1 + ((ln . x) ^2))) ) assume A11: x in Z ; ::_thesis: ((arctan * ln) `| Z) . x = 1 / (x * (1 + ((ln . x) ^2))) then A12: ln is_differentiable_in x by A5, TAYLOR_1:18; A13: ln . x < 1 by A3, A11; A14: ln . x > - 1 by A3, A11; x > 0 by A5, A11; then A15: x in right_open_halfline 0 by A1; ((arctan * ln) `| Z) . x = diff ((arctan * ln),x) by A10, A11, FDIFF_1:def_7 .= (diff (ln,x)) / (1 + ((ln . x) ^2)) by A12, A14, A13, Th85 .= (1 / x) / (1 + ((ln . x) ^2)) by A15, TAYLOR_1:18 .= 1 / (x * (1 + ((ln . x) ^2))) by XCMPLX_1:78 ; hence ((arctan * ln) `| Z) . x = 1 / (x * (1 + ((ln . x) ^2))) ; ::_thesis: verum end; hence ( arctan * ln is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * ln) `| Z) . x = 1 / (x * (1 + ((ln . x) ^2))) ) ) by A2, A6, FDIFF_1:9; ::_thesis: verum end; theorem :: SIN_COS9:118 for Z being open Subset of REAL st Z c= dom (arccot * ln) & ( for x being Real st x in Z holds ( ln . x > - 1 & ln . x < 1 ) ) holds ( arccot * ln is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * ln) `| Z) . x = - (1 / (x * (1 + ((ln . x) ^2)))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (arccot * ln) & ( for x being Real st x in Z holds ( ln . x > - 1 & ln . x < 1 ) ) implies ( arccot * ln is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * ln) `| Z) . x = - (1 / (x * (1 + ((ln . x) ^2)))) ) ) ) A1: right_open_halfline 0 = { g where g is Real : 0 < g } by XXREAL_1:230; assume that A2: Z c= dom (arccot * ln) and A3: for x being Real st x in Z holds ( ln . x > - 1 & ln . x < 1 ) ; ::_thesis: ( arccot * ln is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * ln) `| Z) . x = - (1 / (x * (1 + ((ln . x) ^2)))) ) ) dom (arccot * ln) c= dom ln by RELAT_1:25; then A4: Z c= dom ln by A2, XBOOLE_1:1; A5: for x being Real st x in Z holds x > 0 proof let x be Real; ::_thesis: ( x in Z implies x > 0 ) assume x in Z ; ::_thesis: x > 0 then x in right_open_halfline 0 by A4, TAYLOR_1:18; then ex g being Real st ( x = g & 0 < g ) by A1; hence x > 0 ; ::_thesis: verum end; A6: for x being Real st x in Z holds arccot * ln is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies arccot * ln is_differentiable_in x ) assume A7: x in Z ; ::_thesis: arccot * ln is_differentiable_in x then A8: ln . x > - 1 by A3; A9: ln . x < 1 by A3, A7; ln is_differentiable_in x by A5, A7, TAYLOR_1:18; hence arccot * ln is_differentiable_in x by A8, A9, Th86; ::_thesis: verum end; then A10: arccot * ln is_differentiable_on Z by A2, FDIFF_1:9; for x being Real st x in Z holds ((arccot * ln) `| Z) . x = - (1 / (x * (1 + ((ln . x) ^2)))) proof let x be Real; ::_thesis: ( x in Z implies ((arccot * ln) `| Z) . x = - (1 / (x * (1 + ((ln . x) ^2)))) ) assume A11: x in Z ; ::_thesis: ((arccot * ln) `| Z) . x = - (1 / (x * (1 + ((ln . x) ^2)))) then A12: ln is_differentiable_in x by A5, TAYLOR_1:18; A13: ln . x < 1 by A3, A11; A14: ln . x > - 1 by A3, A11; x > 0 by A5, A11; then A15: x in right_open_halfline 0 by A1; ((arccot * ln) `| Z) . x = diff ((arccot * ln),x) by A10, A11, FDIFF_1:def_7 .= - ((diff (ln,x)) / (1 + ((ln . x) ^2))) by A12, A14, A13, Th86 .= - ((1 / x) / (1 + ((ln . x) ^2))) by A15, TAYLOR_1:18 .= - (1 / (x * (1 + ((ln . x) ^2)))) by XCMPLX_1:78 ; hence ((arccot * ln) `| Z) . x = - (1 / (x * (1 + ((ln . x) ^2)))) ; ::_thesis: verum end; hence ( arccot * ln is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * ln) `| Z) . x = - (1 / (x * (1 + ((ln . x) ^2)))) ) ) by A2, A6, FDIFF_1:9; ::_thesis: verum end; theorem :: SIN_COS9:119 for Z being open Subset of REAL st Z c= dom (exp_R * arctan) & Z c= ].(- 1),1.[ holds ( exp_R * arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R * arctan) `| Z) . x = (exp_R . (arctan . x)) / (1 + (x ^2)) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (exp_R * arctan) & Z c= ].(- 1),1.[ implies ( exp_R * arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R * arctan) `| Z) . x = (exp_R . (arctan . x)) / (1 + (x ^2)) ) ) ) assume that A1: Z c= dom (exp_R * arctan) and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( exp_R * arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R * arctan) `| Z) . x = (exp_R . (arctan . x)) / (1 + (x ^2)) ) ) A3: for x being Real st x in Z holds exp_R * arctan is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies exp_R * arctan is_differentiable_in x ) assume A4: x in Z ; ::_thesis: exp_R * arctan is_differentiable_in x arctan is_differentiable_on Z by A2, Th81; then A5: arctan is_differentiable_in x by A4, FDIFF_1:9; exp_R is_differentiable_in arctan . x by SIN_COS:65; hence exp_R * arctan is_differentiable_in x by A5, FDIFF_2:13; ::_thesis: verum end; then A6: exp_R * arctan is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((exp_R * arctan) `| Z) . x = (exp_R . (arctan . x)) / (1 + (x ^2)) proof let x be Real; ::_thesis: ( x in Z implies ((exp_R * arctan) `| Z) . x = (exp_R . (arctan . x)) / (1 + (x ^2)) ) assume A7: x in Z ; ::_thesis: ((exp_R * arctan) `| Z) . x = (exp_R . (arctan . x)) / (1 + (x ^2)) A8: exp_R is_differentiable_in arctan . x by SIN_COS:65; A9: arctan is_differentiable_on Z by A2, Th81; then arctan is_differentiable_in x by A7, FDIFF_1:9; then diff ((exp_R * arctan),x) = (diff (exp_R,(arctan . x))) * (diff (arctan,x)) by A8, FDIFF_2:13 .= (diff (exp_R,(arctan . x))) * ((arctan `| Z) . x) by A7, A9, FDIFF_1:def_7 .= (diff (exp_R,(arctan . x))) * (1 / (1 + (x ^2))) by A2, A7, Th81 .= (exp_R . (arctan . x)) / (1 + (x ^2)) by SIN_COS:65 ; hence ((exp_R * arctan) `| Z) . x = (exp_R . (arctan . x)) / (1 + (x ^2)) by A6, A7, FDIFF_1:def_7; ::_thesis: verum end; hence ( exp_R * arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R * arctan) `| Z) . x = (exp_R . (arctan . x)) / (1 + (x ^2)) ) ) by A1, A3, FDIFF_1:9; ::_thesis: verum end; theorem :: SIN_COS9:120 for Z being open Subset of REAL st Z c= dom (exp_R * arccot) & Z c= ].(- 1),1.[ holds ( exp_R * arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R * arccot) `| Z) . x = - ((exp_R . (arccot . x)) / (1 + (x ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (exp_R * arccot) & Z c= ].(- 1),1.[ implies ( exp_R * arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R * arccot) `| Z) . x = - ((exp_R . (arccot . x)) / (1 + (x ^2))) ) ) ) assume that A1: Z c= dom (exp_R * arccot) and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( exp_R * arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R * arccot) `| Z) . x = - ((exp_R . (arccot . x)) / (1 + (x ^2))) ) ) A3: for x being Real st x in Z holds exp_R * arccot is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies exp_R * arccot is_differentiable_in x ) assume A4: x in Z ; ::_thesis: exp_R * arccot is_differentiable_in x arccot is_differentiable_on Z by A2, Th82; then A5: arccot is_differentiable_in x by A4, FDIFF_1:9; exp_R is_differentiable_in arccot . x by SIN_COS:65; hence exp_R * arccot is_differentiable_in x by A5, FDIFF_2:13; ::_thesis: verum end; then A6: exp_R * arccot is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((exp_R * arccot) `| Z) . x = - ((exp_R . (arccot . x)) / (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((exp_R * arccot) `| Z) . x = - ((exp_R . (arccot . x)) / (1 + (x ^2))) ) assume A7: x in Z ; ::_thesis: ((exp_R * arccot) `| Z) . x = - ((exp_R . (arccot . x)) / (1 + (x ^2))) A8: exp_R is_differentiable_in arccot . x by SIN_COS:65; A9: arccot is_differentiable_on Z by A2, Th82; then arccot is_differentiable_in x by A7, FDIFF_1:9; then diff ((exp_R * arccot),x) = (diff (exp_R,(arccot . x))) * (diff (arccot,x)) by A8, FDIFF_2:13 .= (diff (exp_R,(arccot . x))) * ((arccot `| Z) . x) by A7, A9, FDIFF_1:def_7 .= (diff (exp_R,(arccot . x))) * (- (1 / (1 + (x ^2)))) by A2, A7, Th82 .= - ((diff (exp_R,(arccot . x))) * (1 / (1 + (x ^2)))) .= - ((exp_R . (arccot . x)) / (1 + (x ^2))) by SIN_COS:65 ; hence ((exp_R * arccot) `| Z) . x = - ((exp_R . (arccot . x)) / (1 + (x ^2))) by A6, A7, FDIFF_1:def_7; ::_thesis: verum end; hence ( exp_R * arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R * arccot) `| Z) . x = - ((exp_R . (arccot . x)) / (1 + (x ^2))) ) ) by A1, A3, FDIFF_1:9; ::_thesis: verum end; theorem :: SIN_COS9:121 for Z being open Subset of REAL st Z c= dom (arctan - (id Z)) & Z c= ].(- 1),1.[ holds ( arctan - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (arctan - (id Z)) & Z c= ].(- 1),1.[ implies ( arctan - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2))) ) ) ) assume that A1: Z c= dom (arctan - (id Z)) and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( arctan - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2))) ) ) A3: for x being Real st x in Z holds (id Z) . x = (1 * x) + 0 by FUNCT_1:18; Z c= (dom arctan) /\ (dom (id Z)) by A1, VALUED_1:12; then A4: Z c= dom (id Z) by XBOOLE_1:18; then A5: id Z is_differentiable_on Z by A3, FDIFF_1:23; A6: arctan is_differentiable_on Z by A2, Th81; for x being Real st x in Z holds ((arctan - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((arctan - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2))) ) A7: 1 + (x ^2) > 0 by XREAL_1:34, XREAL_1:63; assume A8: x in Z ; ::_thesis: ((arctan - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2))) then ((arctan - (id Z)) `| Z) . x = (diff (arctan,x)) - (diff ((id Z),x)) by A1, A5, A6, FDIFF_1:19 .= ((arctan `| Z) . x) - (diff ((id Z),x)) by A6, A8, FDIFF_1:def_7 .= (1 / (1 + (x ^2))) - (diff ((id Z),x)) by A2, A8, Th81 .= (1 / (1 + (x ^2))) - (((id Z) `| Z) . x) by A5, A8, FDIFF_1:def_7 .= (1 / (1 + (x ^2))) - 1 by A4, A3, A8, FDIFF_1:23 .= (1 / (1 + (x ^2))) - ((1 + (x ^2)) / (1 + (x ^2))) by A7, XCMPLX_1:60 .= - ((x ^2) / (1 + (x ^2))) ; hence ((arctan - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2))) ; ::_thesis: verum end; hence ( arctan - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2))) ) ) by A1, A5, A6, FDIFF_1:19; ::_thesis: verum end; theorem :: SIN_COS9:122 for Z being open Subset of REAL st Z c= dom ((- arccot) - (id Z)) & Z c= ].(- 1),1.[ holds ( (- arccot) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds (((- arccot) - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom ((- arccot) - (id Z)) & Z c= ].(- 1),1.[ implies ( (- arccot) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds (((- arccot) - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2))) ) ) ) assume that A1: Z c= dom ((- arccot) - (id Z)) and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( (- arccot) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds (((- arccot) - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2))) ) ) A3: arccot is_differentiable_on Z by A2, Th82; A4: Z c= (dom (- arccot)) /\ (dom (id Z)) by A1, VALUED_1:12; then A5: Z c= dom (id Z) by XBOOLE_1:18; A6: for x being Real st x in Z holds (id Z) . x = (1 * x) + 0 by FUNCT_1:18; then A7: id Z is_differentiable_on Z by A5, FDIFF_1:23; A8: Z c= dom ((- 1) (#) arccot) by A4, XBOOLE_1:18; then A9: - arccot is_differentiable_on Z by A3, FDIFF_1:20; for x being Real st x in Z holds (((- arccot) - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies (((- arccot) - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2))) ) A10: 1 + (x ^2) > 0 by XREAL_1:34, XREAL_1:63; assume A11: x in Z ; ::_thesis: (((- arccot) - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2))) then (((- arccot) - (id Z)) `| Z) . x = (diff ((- arccot),x)) - (diff ((id Z),x)) by A1, A7, A9, FDIFF_1:19 .= (((- arccot) `| Z) . x) - (diff ((id Z),x)) by A9, A11, FDIFF_1:def_7 .= ((- 1) * (diff (arccot,x))) - (diff ((id Z),x)) by A8, A3, A11, FDIFF_1:20 .= ((- 1) * ((arccot `| Z) . x)) - (diff ((id Z),x)) by A3, A11, FDIFF_1:def_7 .= ((- 1) * (- (1 / (1 + (x ^2))))) - (diff ((id Z),x)) by A2, A11, Th82 .= (1 / (1 + (x ^2))) - (((id Z) `| Z) . x) by A7, A11, FDIFF_1:def_7 .= (1 / (1 + (x ^2))) - 1 by A5, A6, A11, FDIFF_1:23 .= (1 / (1 + (x ^2))) - ((1 + (x ^2)) / (1 + (x ^2))) by A10, XCMPLX_1:60 .= - ((x ^2) / (1 + (x ^2))) ; hence (((- arccot) - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2))) ; ::_thesis: verum end; hence ( (- arccot) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds (((- arccot) - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2))) ) ) by A1, A7, A9, FDIFF_1:19; ::_thesis: verum end; theorem :: SIN_COS9:123 for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds ( exp_R (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R (#) arctan) `| Z) . x = ((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= ].(- 1),1.[ implies ( exp_R (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R (#) arctan) `| Z) . x = ((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2))) ) ) ) assume A1: Z c= ].(- 1),1.[ ; ::_thesis: ( exp_R (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R (#) arctan) `| Z) . x = ((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2))) ) ) then A2: arctan is_differentiable_on Z by Th81; ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25; then ].(- 1),1.[ c= dom arctan by Th23, XBOOLE_1:1; then Z c= dom arctan by A1, XBOOLE_1:1; then Z c= (dom exp_R) /\ (dom arctan) by SIN_COS:47, XBOOLE_1:19; then A3: Z c= dom (exp_R (#) arctan) by VALUED_1:def_4; for x being Real st x in Z holds exp_R is_differentiable_in x by SIN_COS:65; then A4: exp_R is_differentiable_on Z by FDIFF_1:9, SIN_COS:47; for x being Real st x in Z holds ((exp_R (#) arctan) `| Z) . x = ((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((exp_R (#) arctan) `| Z) . x = ((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2))) ) assume A5: x in Z ; ::_thesis: ((exp_R (#) arctan) `| Z) . x = ((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2))) then ((exp_R (#) arctan) `| Z) . x = ((arctan . x) * (diff (exp_R,x))) + ((exp_R . x) * (diff (arctan,x))) by A3, A4, A2, FDIFF_1:21 .= ((arctan . x) * (exp_R . x)) + ((exp_R . x) * (diff (arctan,x))) by SIN_COS:65 .= ((exp_R . x) * (arctan . x)) + ((exp_R . x) * ((arctan `| Z) . x)) by A2, A5, FDIFF_1:def_7 .= ((exp_R . x) * (arctan . x)) + ((exp_R . x) * (1 / (1 + (x ^2)))) by A1, A5, Th81 .= ((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2))) ; hence ((exp_R (#) arctan) `| Z) . x = ((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2))) ; ::_thesis: verum end; hence ( exp_R (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R (#) arctan) `| Z) . x = ((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2))) ) ) by A3, A4, A2, FDIFF_1:21; ::_thesis: verum end; theorem :: SIN_COS9:124 for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds ( exp_R (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R (#) arccot) `| Z) . x = ((exp_R . x) * (arccot . x)) - ((exp_R . x) / (1 + (x ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= ].(- 1),1.[ implies ( exp_R (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R (#) arccot) `| Z) . x = ((exp_R . x) * (arccot . x)) - ((exp_R . x) / (1 + (x ^2))) ) ) ) assume A1: Z c= ].(- 1),1.[ ; ::_thesis: ( exp_R (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R (#) arccot) `| Z) . x = ((exp_R . x) * (arccot . x)) - ((exp_R . x) / (1 + (x ^2))) ) ) then A2: arccot is_differentiable_on Z by Th82; ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25; then ].(- 1),1.[ c= dom arccot by Th24, XBOOLE_1:1; then Z c= dom arccot by A1, XBOOLE_1:1; then Z c= (dom exp_R) /\ (dom arccot) by SIN_COS:47, XBOOLE_1:19; then A3: Z c= dom (exp_R (#) arccot) by VALUED_1:def_4; for x being Real st x in Z holds exp_R is_differentiable_in x by SIN_COS:65; then A4: exp_R is_differentiable_on Z by FDIFF_1:9, SIN_COS:47; for x being Real st x in Z holds ((exp_R (#) arccot) `| Z) . x = ((exp_R . x) * (arccot . x)) - ((exp_R . x) / (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((exp_R (#) arccot) `| Z) . x = ((exp_R . x) * (arccot . x)) - ((exp_R . x) / (1 + (x ^2))) ) assume A5: x in Z ; ::_thesis: ((exp_R (#) arccot) `| Z) . x = ((exp_R . x) * (arccot . x)) - ((exp_R . x) / (1 + (x ^2))) then ((exp_R (#) arccot) `| Z) . x = ((arccot . x) * (diff (exp_R,x))) + ((exp_R . x) * (diff (arccot,x))) by A3, A4, A2, FDIFF_1:21 .= ((arccot . x) * (exp_R . x)) + ((exp_R . x) * (diff (arccot,x))) by SIN_COS:65 .= ((exp_R . x) * (arccot . x)) + ((exp_R . x) * ((arccot `| Z) . x)) by A2, A5, FDIFF_1:def_7 .= ((exp_R . x) * (arccot . x)) + ((exp_R . x) * (- (1 / (1 + (x ^2))))) by A1, A5, Th82 .= ((exp_R . x) * (arccot . x)) - ((exp_R . x) / (1 + (x ^2))) ; hence ((exp_R (#) arccot) `| Z) . x = ((exp_R . x) * (arccot . x)) - ((exp_R . x) / (1 + (x ^2))) ; ::_thesis: verum end; hence ( exp_R (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R (#) arccot) `| Z) . x = ((exp_R . x) * (arccot . x)) - ((exp_R . x) / (1 + (x ^2))) ) ) by A3, A4, A2, FDIFF_1:21; ::_thesis: verum end; theorem :: SIN_COS9:125 for r being Real for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom (((1 / r) (#) (arctan * f)) - (id Z)) & ( for x being Real st x in Z holds ( f . x = r * x & r <> 0 & f . x > - 1 & f . x < 1 ) ) holds ( ((1 / r) (#) (arctan * f)) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds ((((1 / r) (#) (arctan * f)) - (id Z)) `| Z) . x = - (((r * x) ^2) / (1 + ((r * x) ^2))) ) ) proof let r be Real; ::_thesis: for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom (((1 / r) (#) (arctan * f)) - (id Z)) & ( for x being Real st x in Z holds ( f . x = r * x & r <> 0 & f . x > - 1 & f . x < 1 ) ) holds ( ((1 / r) (#) (arctan * f)) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds ((((1 / r) (#) (arctan * f)) - (id Z)) `| Z) . x = - (((r * x) ^2) / (1 + ((r * x) ^2))) ) ) let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st Z c= dom (((1 / r) (#) (arctan * f)) - (id Z)) & ( for x being Real st x in Z holds ( f . x = r * x & r <> 0 & f . x > - 1 & f . x < 1 ) ) holds ( ((1 / r) (#) (arctan * f)) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds ((((1 / r) (#) (arctan * f)) - (id Z)) `| Z) . x = - (((r * x) ^2) / (1 + ((r * x) ^2))) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (((1 / r) (#) (arctan * f)) - (id Z)) & ( for x being Real st x in Z holds ( f . x = r * x & r <> 0 & f . x > - 1 & f . x < 1 ) ) implies ( ((1 / r) (#) (arctan * f)) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds ((((1 / r) (#) (arctan * f)) - (id Z)) `| Z) . x = - (((r * x) ^2) / (1 + ((r * x) ^2))) ) ) ) assume that A1: Z c= dom (((1 / r) (#) (arctan * f)) - (id Z)) and A2: for x being Real st x in Z holds ( f . x = r * x & r <> 0 & f . x > - 1 & f . x < 1 ) ; ::_thesis: ( ((1 / r) (#) (arctan * f)) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds ((((1 / r) (#) (arctan * f)) - (id Z)) `| Z) . x = - (((r * x) ^2) / (1 + ((r * x) ^2))) ) ) A3: for x being Real st x in Z holds ( f . x = (r * x) + 0 & f . x > - 1 & f . x < 1 ) by A2; set g = (1 / r) (#) (arctan * f); A4: for x being Real st x in Z holds (id Z) . x = (1 * x) + 0 by FUNCT_1:18; A5: Z c= (dom ((1 / r) (#) (arctan * f))) /\ (dom (id Z)) by A1, VALUED_1:12; then A6: Z c= dom ((1 / r) (#) (arctan * f)) by XBOOLE_1:18; A7: Z c= dom (id Z) by A5, XBOOLE_1:18; then A8: id Z is_differentiable_on Z by A4, FDIFF_1:23; A9: Z c= dom (arctan * f) by A6, VALUED_1:def_5; then A10: arctan * f is_differentiable_on Z by A3, Th87; then A11: (1 / r) (#) (arctan * f) is_differentiable_on Z by A6, FDIFF_1:20; for x being Real st x in Z holds ((((1 / r) (#) (arctan * f)) - (id Z)) `| Z) . x = - (((r * x) ^2) / (1 + ((r * x) ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((((1 / r) (#) (arctan * f)) - (id Z)) `| Z) . x = - (((r * x) ^2) / (1 + ((r * x) ^2))) ) A12: 1 + ((r * x) ^2) > 0 by XREAL_1:34, XREAL_1:63; assume A13: x in Z ; ::_thesis: ((((1 / r) (#) (arctan * f)) - (id Z)) `| Z) . x = - (((r * x) ^2) / (1 + ((r * x) ^2))) then A14: r <> 0 by A2; ((((1 / r) (#) (arctan * f)) - (id Z)) `| Z) . x = (diff (((1 / r) (#) (arctan * f)),x)) - (diff ((id Z),x)) by A1, A11, A8, A13, FDIFF_1:19 .= ((((1 / r) (#) (arctan * f)) `| Z) . x) - (diff ((id Z),x)) by A11, A13, FDIFF_1:def_7 .= ((1 / r) * (diff ((arctan * f),x))) - (diff ((id Z),x)) by A6, A10, A13, FDIFF_1:20 .= ((1 / r) * (((arctan * f) `| Z) . x)) - (diff ((id Z),x)) by A10, A13, FDIFF_1:def_7 .= ((1 / r) * (((arctan * f) `| Z) . x)) - (((id Z) `| Z) . x) by A8, A13, FDIFF_1:def_7 .= ((1 / r) * (r / (1 + (((r * x) + 0) ^2)))) - (((id Z) `| Z) . x) by A3, A9, A13, Th87 .= ((1 / r) * (r / (1 + ((r * x) ^2)))) - 1 by A7, A4, A13, FDIFF_1:23 .= ((1 * r) / (r * (1 + ((r * x) ^2)))) - 1 by XCMPLX_1:76 .= (1 / (1 + ((r * x) ^2))) - 1 by A14, XCMPLX_1:91 .= (1 / (1 + ((r * x) ^2))) - ((1 + ((r * x) ^2)) / (1 + ((r * x) ^2))) by A12, XCMPLX_1:60 .= - (((r * x) ^2) / (1 + ((r * x) ^2))) ; hence ((((1 / r) (#) (arctan * f)) - (id Z)) `| Z) . x = - (((r * x) ^2) / (1 + ((r * x) ^2))) ; ::_thesis: verum end; hence ( ((1 / r) (#) (arctan * f)) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds ((((1 / r) (#) (arctan * f)) - (id Z)) `| Z) . x = - (((r * x) ^2) / (1 + ((r * x) ^2))) ) ) by A1, A11, A8, FDIFF_1:19; ::_thesis: verum end; theorem :: SIN_COS9:126 for r being Real for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom (((- (1 / r)) (#) (arccot * f)) - (id Z)) & ( for x being Real st x in Z holds ( f . x = r * x & r <> 0 & f . x > - 1 & f . x < 1 ) ) holds ( ((- (1 / r)) (#) (arccot * f)) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds ((((- (1 / r)) (#) (arccot * f)) - (id Z)) `| Z) . x = - (((r * x) ^2) / (1 + ((r * x) ^2))) ) ) proof let r be Real; ::_thesis: for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom (((- (1 / r)) (#) (arccot * f)) - (id Z)) & ( for x being Real st x in Z holds ( f . x = r * x & r <> 0 & f . x > - 1 & f . x < 1 ) ) holds ( ((- (1 / r)) (#) (arccot * f)) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds ((((- (1 / r)) (#) (arccot * f)) - (id Z)) `| Z) . x = - (((r * x) ^2) / (1 + ((r * x) ^2))) ) ) let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st Z c= dom (((- (1 / r)) (#) (arccot * f)) - (id Z)) & ( for x being Real st x in Z holds ( f . x = r * x & r <> 0 & f . x > - 1 & f . x < 1 ) ) holds ( ((- (1 / r)) (#) (arccot * f)) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds ((((- (1 / r)) (#) (arccot * f)) - (id Z)) `| Z) . x = - (((r * x) ^2) / (1 + ((r * x) ^2))) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (((- (1 / r)) (#) (arccot * f)) - (id Z)) & ( for x being Real st x in Z holds ( f . x = r * x & r <> 0 & f . x > - 1 & f . x < 1 ) ) implies ( ((- (1 / r)) (#) (arccot * f)) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds ((((- (1 / r)) (#) (arccot * f)) - (id Z)) `| Z) . x = - (((r * x) ^2) / (1 + ((r * x) ^2))) ) ) ) assume that A1: Z c= dom (((- (1 / r)) (#) (arccot * f)) - (id Z)) and A2: for x being Real st x in Z holds ( f . x = r * x & r <> 0 & f . x > - 1 & f . x < 1 ) ; ::_thesis: ( ((- (1 / r)) (#) (arccot * f)) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds ((((- (1 / r)) (#) (arccot * f)) - (id Z)) `| Z) . x = - (((r * x) ^2) / (1 + ((r * x) ^2))) ) ) A3: for x being Real st x in Z holds ( f . x = (r * x) + 0 & f . x > - 1 & f . x < 1 ) by A2; set g = (- (1 / r)) (#) (arccot * f); A4: for x being Real st x in Z holds (id Z) . x = (1 * x) + 0 by FUNCT_1:18; A5: Z c= (dom ((- (1 / r)) (#) (arccot * f))) /\ (dom (id Z)) by A1, VALUED_1:12; then A6: Z c= dom ((- (1 / r)) (#) (arccot * f)) by XBOOLE_1:18; A7: Z c= dom (id Z) by A5, XBOOLE_1:18; then A8: id Z is_differentiable_on Z by A4, FDIFF_1:23; A9: Z c= dom (arccot * f) by A6, VALUED_1:def_5; then A10: arccot * f is_differentiable_on Z by A3, Th88; then A11: (- (1 / r)) (#) (arccot * f) is_differentiable_on Z by A6, FDIFF_1:20; for x being Real st x in Z holds ((((- (1 / r)) (#) (arccot * f)) - (id Z)) `| Z) . x = - (((r * x) ^2) / (1 + ((r * x) ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((((- (1 / r)) (#) (arccot * f)) - (id Z)) `| Z) . x = - (((r * x) ^2) / (1 + ((r * x) ^2))) ) A12: 1 + ((r * x) ^2) > 0 by XREAL_1:34, XREAL_1:63; assume A13: x in Z ; ::_thesis: ((((- (1 / r)) (#) (arccot * f)) - (id Z)) `| Z) . x = - (((r * x) ^2) / (1 + ((r * x) ^2))) then A14: r <> 0 by A2; ((((- (1 / r)) (#) (arccot * f)) - (id Z)) `| Z) . x = (diff (((- (1 / r)) (#) (arccot * f)),x)) - (diff ((id Z),x)) by A1, A11, A8, A13, FDIFF_1:19 .= ((((- (1 / r)) (#) (arccot * f)) `| Z) . x) - (diff ((id Z),x)) by A11, A13, FDIFF_1:def_7 .= ((- (1 / r)) * (diff ((arccot * f),x))) - (diff ((id Z),x)) by A6, A10, A13, FDIFF_1:20 .= ((- (1 / r)) * (((arccot * f) `| Z) . x)) - (diff ((id Z),x)) by A10, A13, FDIFF_1:def_7 .= ((- (1 / r)) * (((arccot * f) `| Z) . x)) - (((id Z) `| Z) . x) by A8, A13, FDIFF_1:def_7 .= ((- (1 / r)) * (- (r / (1 + (((r * x) + 0) ^2))))) - (((id Z) `| Z) . x) by A3, A9, A13, Th88 .= (((- 1) / r) * ((- r) / (1 + ((r * x) ^2)))) - 1 by A7, A4, A13, FDIFF_1:23 .= (((- 1) * (- r)) / (r * (1 + ((r * x) ^2)))) - 1 by XCMPLX_1:76 .= ((1 * r) / (r * (1 + ((r * x) ^2)))) - 1 .= (1 / (1 + ((r * x) ^2))) - 1 by A14, XCMPLX_1:91 .= (1 / (1 + ((r * x) ^2))) - ((1 + ((r * x) ^2)) / (1 + ((r * x) ^2))) by A12, XCMPLX_1:60 .= - (((r * x) ^2) / (1 + ((r * x) ^2))) ; hence ((((- (1 / r)) (#) (arccot * f)) - (id Z)) `| Z) . x = - (((r * x) ^2) / (1 + ((r * x) ^2))) ; ::_thesis: verum end; hence ( ((- (1 / r)) (#) (arccot * f)) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds ((((- (1 / r)) (#) (arccot * f)) - (id Z)) `| Z) . x = - (((r * x) ^2) / (1 + ((r * x) ^2))) ) ) by A1, A11, A8, FDIFF_1:19; ::_thesis: verum end; theorem :: SIN_COS9:127 for Z being open Subset of REAL st Z c= dom (ln (#) arctan) & Z c= ].(- 1),1.[ holds ( ln (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((ln (#) arctan) `| Z) . x = ((arctan . x) / x) + ((ln . x) / (1 + (x ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (ln (#) arctan) & Z c= ].(- 1),1.[ implies ( ln (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((ln (#) arctan) `| Z) . x = ((arctan . x) / x) + ((ln . x) / (1 + (x ^2))) ) ) ) A1: right_open_halfline 0 = { g where g is Real : 0 < g } by XXREAL_1:230; assume that A2: Z c= dom (ln (#) arctan) and A3: Z c= ].(- 1),1.[ ; ::_thesis: ( ln (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((ln (#) arctan) `| Z) . x = ((arctan . x) / x) + ((ln . x) / (1 + (x ^2))) ) ) A4: arctan is_differentiable_on Z by A3, Th81; Z c= (dom ln) /\ (dom arctan) by A2, VALUED_1:def_4; then A5: Z c= dom ln by XBOOLE_1:18; A6: for x being Real st x in Z holds x > 0 proof let x be Real; ::_thesis: ( x in Z implies x > 0 ) assume x in Z ; ::_thesis: x > 0 then x in right_open_halfline 0 by A5, TAYLOR_1:18; then ex g being Real st ( x = g & 0 < g ) by A1; hence x > 0 ; ::_thesis: verum end; then for x being Real st x in Z holds ln is_differentiable_in x by TAYLOR_1:18; then A7: ln is_differentiable_on Z by A5, FDIFF_1:9; A8: for x being Real st x in Z holds diff (ln,x) = 1 / x proof let x be Real; ::_thesis: ( x in Z implies diff (ln,x) = 1 / x ) assume x in Z ; ::_thesis: diff (ln,x) = 1 / x then x > 0 by A6; then x in right_open_halfline 0 by A1; hence diff (ln,x) = 1 / x by TAYLOR_1:18; ::_thesis: verum end; for x being Real st x in Z holds ((ln (#) arctan) `| Z) . x = ((arctan . x) / x) + ((ln . x) / (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((ln (#) arctan) `| Z) . x = ((arctan . x) / x) + ((ln . x) / (1 + (x ^2))) ) assume A9: x in Z ; ::_thesis: ((ln (#) arctan) `| Z) . x = ((arctan . x) / x) + ((ln . x) / (1 + (x ^2))) then ((ln (#) arctan) `| Z) . x = ((arctan . x) * (diff (ln,x))) + ((ln . x) * (diff (arctan,x))) by A2, A7, A4, FDIFF_1:21 .= ((arctan . x) * (1 / x)) + ((ln . x) * (diff (arctan,x))) by A8, A9 .= ((arctan . x) * (1 / x)) + ((ln . x) * ((arctan `| Z) . x)) by A4, A9, FDIFF_1:def_7 .= (((arctan . x) * 1) / x) + ((ln . x) * (1 / (1 + (x ^2)))) by A3, A9, Th81 .= ((arctan . x) / x) + ((ln . x) / (1 + (x ^2))) ; hence ((ln (#) arctan) `| Z) . x = ((arctan . x) / x) + ((ln . x) / (1 + (x ^2))) ; ::_thesis: verum end; hence ( ln (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((ln (#) arctan) `| Z) . x = ((arctan . x) / x) + ((ln . x) / (1 + (x ^2))) ) ) by A2, A7, A4, FDIFF_1:21; ::_thesis: verum end; theorem :: SIN_COS9:128 for Z being open Subset of REAL st Z c= dom (ln (#) arccot) & Z c= ].(- 1),1.[ holds ( ln (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((ln (#) arccot) `| Z) . x = ((arccot . x) / x) - ((ln . x) / (1 + (x ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (ln (#) arccot) & Z c= ].(- 1),1.[ implies ( ln (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((ln (#) arccot) `| Z) . x = ((arccot . x) / x) - ((ln . x) / (1 + (x ^2))) ) ) ) A1: right_open_halfline 0 = { g where g is Real : 0 < g } by XXREAL_1:230; assume that A2: Z c= dom (ln (#) arccot) and A3: Z c= ].(- 1),1.[ ; ::_thesis: ( ln (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((ln (#) arccot) `| Z) . x = ((arccot . x) / x) - ((ln . x) / (1 + (x ^2))) ) ) A4: arccot is_differentiable_on Z by A3, Th82; Z c= (dom ln) /\ (dom arccot) by A2, VALUED_1:def_4; then A5: Z c= dom ln by XBOOLE_1:18; A6: for x being Real st x in Z holds x > 0 proof let x be Real; ::_thesis: ( x in Z implies x > 0 ) assume x in Z ; ::_thesis: x > 0 then x in right_open_halfline 0 by A5, TAYLOR_1:18; then ex g being Real st ( x = g & 0 < g ) by A1; hence x > 0 ; ::_thesis: verum end; then for x being Real st x in Z holds ln is_differentiable_in x by TAYLOR_1:18; then A7: ln is_differentiable_on Z by A5, FDIFF_1:9; A8: for x being Real st x in Z holds diff (ln,x) = 1 / x proof let x be Real; ::_thesis: ( x in Z implies diff (ln,x) = 1 / x ) assume x in Z ; ::_thesis: diff (ln,x) = 1 / x then x > 0 by A6; then x in right_open_halfline 0 by A1; hence diff (ln,x) = 1 / x by TAYLOR_1:18; ::_thesis: verum end; for x being Real st x in Z holds ((ln (#) arccot) `| Z) . x = ((arccot . x) / x) - ((ln . x) / (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((ln (#) arccot) `| Z) . x = ((arccot . x) / x) - ((ln . x) / (1 + (x ^2))) ) assume A9: x in Z ; ::_thesis: ((ln (#) arccot) `| Z) . x = ((arccot . x) / x) - ((ln . x) / (1 + (x ^2))) then ((ln (#) arccot) `| Z) . x = ((arccot . x) * (diff (ln,x))) + ((ln . x) * (diff (arccot,x))) by A2, A7, A4, FDIFF_1:21 .= ((arccot . x) * (1 / x)) + ((ln . x) * (diff (arccot,x))) by A8, A9 .= ((arccot . x) * (1 / x)) + ((ln . x) * ((arccot `| Z) . x)) by A4, A9, FDIFF_1:def_7 .= ((arccot . x) * (1 / x)) + ((ln . x) * (- (1 / (1 + (x ^2))))) by A3, A9, Th82 .= ((arccot . x) / x) - ((ln . x) / (1 + (x ^2))) ; hence ((ln (#) arccot) `| Z) . x = ((arccot . x) / x) - ((ln . x) / (1 + (x ^2))) ; ::_thesis: verum end; hence ( ln (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((ln (#) arccot) `| Z) . x = ((arccot . x) / x) - ((ln . x) / (1 + (x ^2))) ) ) by A2, A7, A4, FDIFF_1:21; ::_thesis: verum end; theorem :: SIN_COS9:129 for Z being open Subset of REAL st not 0 in Z & Z c= dom (((id Z) ^) (#) arctan) & Z c= ].(- 1),1.[ holds ( ((id Z) ^) (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) ^) (#) arctan) `| Z) . x = (- ((arctan . x) / (x ^2))) + (1 / (x * (1 + (x ^2)))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( not 0 in Z & Z c= dom (((id Z) ^) (#) arctan) & Z c= ].(- 1),1.[ implies ( ((id Z) ^) (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) ^) (#) arctan) `| Z) . x = (- ((arctan . x) / (x ^2))) + (1 / (x * (1 + (x ^2)))) ) ) ) set f = id Z; assume that A1: not 0 in Z and A2: Z c= dom (((id Z) ^) (#) arctan) and A3: Z c= ].(- 1),1.[ ; ::_thesis: ( ((id Z) ^) (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) ^) (#) arctan) `| Z) . x = (- ((arctan . x) / (x ^2))) + (1 / (x * (1 + (x ^2)))) ) ) A4: (id Z) ^ is_differentiable_on Z by A1, FDIFF_5:4; A5: arctan is_differentiable_on Z by A3, Th81; Z c= (dom ((id Z) ^)) /\ (dom arctan) by A2, VALUED_1:def_4; then A6: Z c= dom ((id Z) ^) by XBOOLE_1:18; for x being Real st x in Z holds ((((id Z) ^) (#) arctan) `| Z) . x = (- ((arctan . x) / (x ^2))) + (1 / (x * (1 + (x ^2)))) proof let x be Real; ::_thesis: ( x in Z implies ((((id Z) ^) (#) arctan) `| Z) . x = (- ((arctan . x) / (x ^2))) + (1 / (x * (1 + (x ^2)))) ) assume A7: x in Z ; ::_thesis: ((((id Z) ^) (#) arctan) `| Z) . x = (- ((arctan . x) / (x ^2))) + (1 / (x * (1 + (x ^2)))) then ((((id Z) ^) (#) arctan) `| Z) . x = ((arctan . x) * (diff (((id Z) ^),x))) + ((((id Z) ^) . x) * (diff (arctan,x))) by A2, A4, A5, FDIFF_1:21 .= ((arctan . x) * ((((id Z) ^) `| Z) . x)) + ((((id Z) ^) . x) * (diff (arctan,x))) by A4, A7, FDIFF_1:def_7 .= ((arctan . x) * (- (1 / (x ^2)))) + ((((id Z) ^) . x) * (diff (arctan,x))) by A1, A7, FDIFF_5:4 .= (- ((arctan . x) * (1 / (x ^2)))) + ((((id Z) ^) . x) * ((arctan `| Z) . x)) by A5, A7, FDIFF_1:def_7 .= (- (((arctan . x) * 1) / (x ^2))) + ((((id Z) ^) . x) * (1 / (1 + (x ^2)))) by A3, A7, Th81 .= (- ((arctan . x) / (x ^2))) + ((((id Z) . x) ") * (1 / (1 + (x ^2)))) by A6, A7, RFUNCT_1:def_2 .= (- ((arctan . x) / (x ^2))) + ((1 / x) * (1 / (1 + (x ^2)))) by A7, FUNCT_1:18 .= (- ((arctan . x) / (x ^2))) + ((1 * 1) / (x * (1 + (x ^2)))) by XCMPLX_1:76 .= (- ((arctan . x) / (x ^2))) + (1 / (x * (1 + (x ^2)))) ; hence ((((id Z) ^) (#) arctan) `| Z) . x = (- ((arctan . x) / (x ^2))) + (1 / (x * (1 + (x ^2)))) ; ::_thesis: verum end; hence ( ((id Z) ^) (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) ^) (#) arctan) `| Z) . x = (- ((arctan . x) / (x ^2))) + (1 / (x * (1 + (x ^2)))) ) ) by A2, A4, A5, FDIFF_1:21; ::_thesis: verum end; theorem :: SIN_COS9:130 for Z being open Subset of REAL st not 0 in Z & Z c= dom (((id Z) ^) (#) arccot) & Z c= ].(- 1),1.[ holds ( ((id Z) ^) (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) ^) (#) arccot) `| Z) . x = (- ((arccot . x) / (x ^2))) - (1 / (x * (1 + (x ^2)))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( not 0 in Z & Z c= dom (((id Z) ^) (#) arccot) & Z c= ].(- 1),1.[ implies ( ((id Z) ^) (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) ^) (#) arccot) `| Z) . x = (- ((arccot . x) / (x ^2))) - (1 / (x * (1 + (x ^2)))) ) ) ) set f = id Z; assume that A1: not 0 in Z and A2: Z c= dom (((id Z) ^) (#) arccot) and A3: Z c= ].(- 1),1.[ ; ::_thesis: ( ((id Z) ^) (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) ^) (#) arccot) `| Z) . x = (- ((arccot . x) / (x ^2))) - (1 / (x * (1 + (x ^2)))) ) ) A4: (id Z) ^ is_differentiable_on Z by A1, FDIFF_5:4; A5: arccot is_differentiable_on Z by A3, Th82; Z c= (dom ((id Z) ^)) /\ (dom arccot) by A2, VALUED_1:def_4; then A6: Z c= dom ((id Z) ^) by XBOOLE_1:18; for x being Real st x in Z holds ((((id Z) ^) (#) arccot) `| Z) . x = (- ((arccot . x) / (x ^2))) - (1 / (x * (1 + (x ^2)))) proof let x be Real; ::_thesis: ( x in Z implies ((((id Z) ^) (#) arccot) `| Z) . x = (- ((arccot . x) / (x ^2))) - (1 / (x * (1 + (x ^2)))) ) assume A7: x in Z ; ::_thesis: ((((id Z) ^) (#) arccot) `| Z) . x = (- ((arccot . x) / (x ^2))) - (1 / (x * (1 + (x ^2)))) then ((((id Z) ^) (#) arccot) `| Z) . x = ((arccot . x) * (diff (((id Z) ^),x))) + ((((id Z) ^) . x) * (diff (arccot,x))) by A2, A4, A5, FDIFF_1:21 .= ((arccot . x) * ((((id Z) ^) `| Z) . x)) + ((((id Z) ^) . x) * (diff (arccot,x))) by A4, A7, FDIFF_1:def_7 .= ((arccot . x) * (- (1 / (x ^2)))) + ((((id Z) ^) . x) * (diff (arccot,x))) by A1, A7, FDIFF_5:4 .= (- ((arccot . x) * (1 / (x ^2)))) + ((((id Z) ^) . x) * ((arccot `| Z) . x)) by A5, A7, FDIFF_1:def_7 .= (- ((arccot . x) * (1 / (x ^2)))) + ((((id Z) ^) . x) * (- (1 / (1 + (x ^2))))) by A3, A7, Th82 .= (- (((arccot . x) * 1) / (x ^2))) - ((((id Z) ^) . x) * (1 / (1 + (x ^2)))) .= (- ((arccot . x) / (x ^2))) - ((((id Z) . x) ") * (1 / (1 + (x ^2)))) by A6, A7, RFUNCT_1:def_2 .= (- ((arccot . x) / (x ^2))) - ((1 / x) * (1 / (1 + (x ^2)))) by A7, FUNCT_1:18 .= (- ((arccot . x) / (x ^2))) - ((1 * 1) / (x * (1 + (x ^2)))) by XCMPLX_1:76 .= (- ((arccot . x) / (x ^2))) - (1 / (x * (1 + (x ^2)))) ; hence ((((id Z) ^) (#) arccot) `| Z) . x = (- ((arccot . x) / (x ^2))) - (1 / (x * (1 + (x ^2)))) ; ::_thesis: verum end; hence ( ((id Z) ^) (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) ^) (#) arccot) `| Z) . x = (- ((arccot . x) / (x ^2))) - (1 / (x * (1 + (x ^2)))) ) ) by A2, A4, A5, FDIFF_1:21; ::_thesis: verum end;