:: FDIFF_11 semantic presentation begin theorem :: FDIFF_11:1 for Z being open Subset of REAL st Z c= dom (arctan * sin) & ( for x being Real st x in Z holds ( sin . x > - 1 & sin . x < 1 ) ) holds ( arctan * sin is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * sin) `| Z) . x = (cos . x) / (1 + ((sin . x) ^2)) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (arctan * sin) & ( for x being Real st x in Z holds ( sin . x > - 1 & sin . x < 1 ) ) implies ( arctan * sin is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * sin) `| Z) . x = (cos . x) / (1 + ((sin . x) ^2)) ) ) ) assume that A1: Z c= dom (arctan * sin) and A2: for x being Real st x in Z holds ( sin . x > - 1 & sin . x < 1 ) ; ::_thesis: ( arctan * sin is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * sin) `| Z) . x = (cos . x) / (1 + ((sin . x) ^2)) ) ) A3: for x being Real st x in Z holds arctan * sin is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies arctan * sin is_differentiable_in x ) assume x in Z ; ::_thesis: arctan * sin is_differentiable_in x then A4: ( sin . x > - 1 & sin . x < 1 ) by A2; sin is_differentiable_in x by SIN_COS:64; hence arctan * sin is_differentiable_in x by A4, SIN_COS9:85; ::_thesis: verum end; then A5: arctan * sin is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((arctan * sin) `| Z) . x = (cos . x) / (1 + ((sin . x) ^2)) proof let x be Real; ::_thesis: ( x in Z implies ((arctan * sin) `| Z) . x = (cos . x) / (1 + ((sin . x) ^2)) ) A6: sin is_differentiable_in x by SIN_COS:64; assume A7: x in Z ; ::_thesis: ((arctan * sin) `| Z) . x = (cos . x) / (1 + ((sin . x) ^2)) then A8: ( sin . x > - 1 & sin . x < 1 ) by A2; ((arctan * sin) `| Z) . x = diff ((arctan * sin),x) by A5, A7, FDIFF_1:def_7 .= (diff (sin,x)) / (1 + ((sin . x) ^2)) by A6, A8, SIN_COS9:85 .= (cos . x) / (1 + ((sin . x) ^2)) by SIN_COS:64 ; hence ((arctan * sin) `| Z) . x = (cos . x) / (1 + ((sin . x) ^2)) ; ::_thesis: verum end; hence ( arctan * sin is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * sin) `| Z) . x = (cos . x) / (1 + ((sin . x) ^2)) ) ) by A1, A3, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_11:2 for Z being open Subset of REAL st Z c= dom (arccot * sin) & ( for x being Real st x in Z holds ( sin . x > - 1 & sin . x < 1 ) ) holds ( arccot * sin is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * sin) `| Z) . x = - ((cos . x) / (1 + ((sin . x) ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (arccot * sin) & ( for x being Real st x in Z holds ( sin . x > - 1 & sin . x < 1 ) ) implies ( arccot * sin is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * sin) `| Z) . x = - ((cos . x) / (1 + ((sin . x) ^2))) ) ) ) assume that A1: Z c= dom (arccot * sin) and A2: for x being Real st x in Z holds ( sin . x > - 1 & sin . x < 1 ) ; ::_thesis: ( arccot * sin is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * sin) `| Z) . x = - ((cos . x) / (1 + ((sin . x) ^2))) ) ) A3: for x being Real st x in Z holds arccot * sin is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies arccot * sin is_differentiable_in x ) assume x in Z ; ::_thesis: arccot * sin is_differentiable_in x then A4: ( sin . x > - 1 & sin . x < 1 ) by A2; sin is_differentiable_in x by SIN_COS:64; hence arccot * sin is_differentiable_in x by A4, SIN_COS9:86; ::_thesis: verum end; then A5: arccot * sin is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((arccot * sin) `| Z) . x = - ((cos . x) / (1 + ((sin . x) ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((arccot * sin) `| Z) . x = - ((cos . x) / (1 + ((sin . x) ^2))) ) A6: sin is_differentiable_in x by SIN_COS:64; assume A7: x in Z ; ::_thesis: ((arccot * sin) `| Z) . x = - ((cos . x) / (1 + ((sin . x) ^2))) then A8: ( sin . x > - 1 & sin . x < 1 ) by A2; ((arccot * sin) `| Z) . x = diff ((arccot * sin),x) by A5, A7, FDIFF_1:def_7 .= - ((diff (sin,x)) / (1 + ((sin . x) ^2))) by A6, A8, SIN_COS9:86 .= - ((cos . x) / (1 + ((sin . x) ^2))) by SIN_COS:64 ; hence ((arccot * sin) `| Z) . x = - ((cos . x) / (1 + ((sin . x) ^2))) ; ::_thesis: verum end; hence ( arccot * sin is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * sin) `| Z) . x = - ((cos . x) / (1 + ((sin . x) ^2))) ) ) by A1, A3, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_11:3 for Z being open Subset of REAL st Z c= dom (arctan * cos) & ( for x being Real st x in Z holds ( cos . x > - 1 & cos . x < 1 ) ) holds ( arctan * cos is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * cos) `| Z) . x = - ((sin . x) / (1 + ((cos . x) ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (arctan * cos) & ( for x being Real st x in Z holds ( cos . x > - 1 & cos . x < 1 ) ) implies ( arctan * cos is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * cos) `| Z) . x = - ((sin . x) / (1 + ((cos . x) ^2))) ) ) ) assume that A1: Z c= dom (arctan * cos) and A2: for x being Real st x in Z holds ( cos . x > - 1 & cos . x < 1 ) ; ::_thesis: ( arctan * cos is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * cos) `| Z) . x = - ((sin . x) / (1 + ((cos . x) ^2))) ) ) A3: for x being Real st x in Z holds arctan * cos is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies arctan * cos is_differentiable_in x ) assume x in Z ; ::_thesis: arctan * cos is_differentiable_in x then A4: ( cos . x > - 1 & cos . x < 1 ) by A2; cos is_differentiable_in x by SIN_COS:63; hence arctan * cos is_differentiable_in x by A4, SIN_COS9:85; ::_thesis: verum end; then A5: arctan * cos is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((arctan * cos) `| Z) . x = - ((sin . x) / (1 + ((cos . x) ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((arctan * cos) `| Z) . x = - ((sin . x) / (1 + ((cos . x) ^2))) ) A6: cos is_differentiable_in x by SIN_COS:63; assume A7: x in Z ; ::_thesis: ((arctan * cos) `| Z) . x = - ((sin . x) / (1 + ((cos . x) ^2))) then A8: ( cos . x > - 1 & cos . x < 1 ) by A2; ((arctan * cos) `| Z) . x = diff ((arctan * cos),x) by A5, A7, FDIFF_1:def_7 .= (diff (cos,x)) / (1 + ((cos . x) ^2)) by A6, A8, SIN_COS9:85 .= (- (sin . x)) / (1 + ((cos . x) ^2)) by SIN_COS:63 .= - ((sin . x) / (1 + ((cos . x) ^2))) ; hence ((arctan * cos) `| Z) . x = - ((sin . x) / (1 + ((cos . x) ^2))) ; ::_thesis: verum end; hence ( arctan * cos is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * cos) `| Z) . x = - ((sin . x) / (1 + ((cos . x) ^2))) ) ) by A1, A3, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_11:4 for Z being open Subset of REAL st Z c= dom (arccot * cos) & ( for x being Real st x in Z holds ( cos . x > - 1 & cos . x < 1 ) ) holds ( arccot * cos is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * cos) `| Z) . x = (sin . x) / (1 + ((cos . x) ^2)) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (arccot * cos) & ( for x being Real st x in Z holds ( cos . x > - 1 & cos . x < 1 ) ) implies ( arccot * cos is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * cos) `| Z) . x = (sin . x) / (1 + ((cos . x) ^2)) ) ) ) assume that A1: Z c= dom (arccot * cos) and A2: for x being Real st x in Z holds ( cos . x > - 1 & cos . x < 1 ) ; ::_thesis: ( arccot * cos is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * cos) `| Z) . x = (sin . x) / (1 + ((cos . x) ^2)) ) ) A3: for x being Real st x in Z holds arccot * cos is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies arccot * cos is_differentiable_in x ) assume x in Z ; ::_thesis: arccot * cos is_differentiable_in x then A4: ( cos . x > - 1 & cos . x < 1 ) by A2; cos is_differentiable_in x by SIN_COS:63; hence arccot * cos is_differentiable_in x by A4, SIN_COS9:86; ::_thesis: verum end; then A5: arccot * cos is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((arccot * cos) `| Z) . x = (sin . x) / (1 + ((cos . x) ^2)) proof let x be Real; ::_thesis: ( x in Z implies ((arccot * cos) `| Z) . x = (sin . x) / (1 + ((cos . x) ^2)) ) A6: cos is_differentiable_in x by SIN_COS:63; assume A7: x in Z ; ::_thesis: ((arccot * cos) `| Z) . x = (sin . x) / (1 + ((cos . x) ^2)) then A8: ( cos . x > - 1 & cos . x < 1 ) by A2; ((arccot * cos) `| Z) . x = diff ((arccot * cos),x) by A5, A7, FDIFF_1:def_7 .= - ((diff (cos,x)) / (1 + ((cos . x) ^2))) by A6, A8, SIN_COS9:86 .= - ((- (sin . x)) / (1 + ((cos . x) ^2))) by SIN_COS:63 .= (sin . x) / (1 + ((cos . x) ^2)) ; hence ((arccot * cos) `| Z) . x = (sin . x) / (1 + ((cos . x) ^2)) ; ::_thesis: verum end; hence ( arccot * cos is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * cos) `| Z) . x = (sin . x) / (1 + ((cos . x) ^2)) ) ) by A1, A3, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_11:5 for Z being open Subset of REAL st Z c= dom (arctan * tan) & ( for x being Real st x in Z holds ( tan . x > - 1 & tan . x < 1 ) ) holds ( arctan * tan is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * tan) `| Z) . x = 1 ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (arctan * tan) & ( for x being Real st x in Z holds ( tan . x > - 1 & tan . x < 1 ) ) implies ( arctan * tan is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * tan) `| Z) . x = 1 ) ) ) assume that A1: Z c= dom (arctan * tan) and A2: for x being Real st x in Z holds ( tan . x > - 1 & tan . x < 1 ) ; ::_thesis: ( arctan * tan is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * tan) `| Z) . x = 1 ) ) dom (arctan * tan) c= dom tan by RELAT_1:25; then A3: Z c= dom tan by A1, XBOOLE_1:1; A4: for x being Real st x in Z holds arctan * tan is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies arctan * tan is_differentiable_in x ) assume A5: x in Z ; ::_thesis: arctan * tan is_differentiable_in x then cos . x <> 0 by A3, FDIFF_8:1; then A6: tan is_differentiable_in x by FDIFF_7:46; ( tan . x > - 1 & tan . x < 1 ) by A2, A5; hence arctan * tan is_differentiable_in x by A6, SIN_COS9:85; ::_thesis: verum end; then A7: arctan * tan is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((arctan * tan) `| Z) . x = 1 proof let x be Real; ::_thesis: ( x in Z implies ((arctan * tan) `| Z) . x = 1 ) assume A8: x in Z ; ::_thesis: ((arctan * tan) `| Z) . x = 1 then A9: ( tan . x > - 1 & tan . x < 1 ) by A2; A10: tan . x = (sin . x) / (cos . x) by A3, A8, RFUNCT_1:def_1; A11: cos . x <> 0 by A3, A8, FDIFF_8:1; then A12: tan is_differentiable_in x by FDIFF_7:46; A13: (cos . x) ^2 <> 0 by A11, SQUARE_1:12; ((arctan * tan) `| Z) . x = diff ((arctan * tan),x) by A7, A8, FDIFF_1:def_7 .= (diff (tan,x)) / (1 + ((tan . x) ^2)) by A12, A9, SIN_COS9:85 .= (1 / ((cos . x) ^2)) / (1 + ((tan . x) ^2)) by A11, FDIFF_7:46 .= 1 / (((cos . x) ^2) * (1 + (((sin . x) / (cos . x)) * ((sin . x) / (cos . x))))) by A10, XCMPLX_1:78 .= 1 / (((cos . x) ^2) * (1 + (((sin . x) ^2) / ((cos . x) ^2)))) by XCMPLX_1:76 .= 1 / (((cos . x) ^2) + ((((cos . x) ^2) * ((sin . x) ^2)) / ((cos . x) ^2))) .= 1 / (((cos . x) ^2) + ((sin . x) ^2)) by A13, XCMPLX_1:89 .= 1 / 1 by SIN_COS:28 .= 1 ; hence ((arctan * tan) `| Z) . x = 1 ; ::_thesis: verum end; hence ( arctan * tan is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * tan) `| Z) . x = 1 ) ) by A1, A4, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_11:6 for Z being open Subset of REAL st Z c= dom (arccot * tan) & ( for x being Real st x in Z holds ( tan . x > - 1 & tan . x < 1 ) ) holds ( arccot * tan is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * tan) `| Z) . x = - 1 ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (arccot * tan) & ( for x being Real st x in Z holds ( tan . x > - 1 & tan . x < 1 ) ) implies ( arccot * tan is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * tan) `| Z) . x = - 1 ) ) ) assume that A1: Z c= dom (arccot * tan) and A2: for x being Real st x in Z holds ( tan . x > - 1 & tan . x < 1 ) ; ::_thesis: ( arccot * tan is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * tan) `| Z) . x = - 1 ) ) dom (arccot * tan) c= dom tan by RELAT_1:25; then A3: Z c= dom tan by A1, XBOOLE_1:1; A4: for x being Real st x in Z holds arccot * tan is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies arccot * tan is_differentiable_in x ) assume A5: x in Z ; ::_thesis: arccot * tan is_differentiable_in x then cos . x <> 0 by A3, FDIFF_8:1; then A6: tan is_differentiable_in x by FDIFF_7:46; ( tan . x > - 1 & tan . x < 1 ) by A2, A5; hence arccot * tan is_differentiable_in x by A6, SIN_COS9:86; ::_thesis: verum end; then A7: arccot * tan is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((arccot * tan) `| Z) . x = - 1 proof let x be Real; ::_thesis: ( x in Z implies ((arccot * tan) `| Z) . x = - 1 ) assume A8: x in Z ; ::_thesis: ((arccot * tan) `| Z) . x = - 1 then A9: ( tan . x > - 1 & tan . x < 1 ) by A2; A10: tan . x = (sin . x) / (cos . x) by A3, A8, RFUNCT_1:def_1; A11: cos . x <> 0 by A3, A8, FDIFF_8:1; then A12: tan is_differentiable_in x by FDIFF_7:46; A13: (cos . x) ^2 <> 0 by A11, SQUARE_1:12; ((arccot * tan) `| Z) . x = diff ((arccot * tan),x) by A7, A8, FDIFF_1:def_7 .= - ((diff (tan,x)) / (1 + ((tan . x) ^2))) by A12, A9, SIN_COS9:86 .= - ((1 / ((cos . x) ^2)) / (1 + ((tan . x) ^2))) by A11, FDIFF_7:46 .= - (1 / (((cos . x) ^2) * (1 + (((sin . x) / (cos . x)) * ((sin . x) / (cos . x)))))) by A10, XCMPLX_1:78 .= - (1 / (((cos . x) ^2) * (1 + (((sin . x) ^2) / ((cos . x) ^2))))) by XCMPLX_1:76 .= - (1 / (((cos . x) ^2) + ((((cos . x) ^2) * ((sin . x) ^2)) / ((cos . x) ^2)))) .= - (1 / (((cos . x) ^2) + ((sin . x) ^2))) by A13, XCMPLX_1:89 .= - (1 / 1) by SIN_COS:28 .= - 1 ; hence ((arccot * tan) `| Z) . x = - 1 ; ::_thesis: verum end; hence ( arccot * tan is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * tan) `| Z) . x = - 1 ) ) by A1, A4, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_11:7 for Z being open Subset of REAL st Z c= dom (arctan * cot) & ( for x being Real st x in Z holds ( cot . x > - 1 & cot . x < 1 ) ) holds ( arctan * cot is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * cot) `| Z) . x = - 1 ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (arctan * cot) & ( for x being Real st x in Z holds ( cot . x > - 1 & cot . x < 1 ) ) implies ( arctan * cot is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * cot) `| Z) . x = - 1 ) ) ) assume that A1: Z c= dom (arctan * cot) and A2: for x being Real st x in Z holds ( cot . x > - 1 & cot . x < 1 ) ; ::_thesis: ( arctan * cot is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * cot) `| Z) . x = - 1 ) ) dom (arctan * cot) c= dom cot by RELAT_1:25; then A3: Z c= dom cot by A1, XBOOLE_1:1; A4: for x being Real st x in Z holds arctan * cot is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies arctan * cot is_differentiable_in x ) assume A5: x in Z ; ::_thesis: arctan * cot is_differentiable_in x then sin . x <> 0 by A3, FDIFF_8:2; then A6: cot is_differentiable_in x by FDIFF_7:47; ( cot . x > - 1 & cot . x < 1 ) by A2, A5; hence arctan * cot is_differentiable_in x by A6, SIN_COS9:85; ::_thesis: verum end; then A7: arctan * cot is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((arctan * cot) `| Z) . x = - 1 proof let x be Real; ::_thesis: ( x in Z implies ((arctan * cot) `| Z) . x = - 1 ) assume A8: x in Z ; ::_thesis: ((arctan * cot) `| Z) . x = - 1 then A9: ( cot . x > - 1 & cot . x < 1 ) by A2; A10: cot . x = (cos . x) / (sin . x) by A3, A8, RFUNCT_1:def_1; A11: sin . x <> 0 by A3, A8, FDIFF_8:2; then A12: cot is_differentiable_in x by FDIFF_7:47; A13: (sin . x) ^2 <> 0 by A11, SQUARE_1:12; ((arctan * cot) `| Z) . x = diff ((arctan * cot),x) by A7, A8, FDIFF_1:def_7 .= (diff (cot,x)) / (1 + ((cot . x) ^2)) by A12, A9, SIN_COS9:85 .= (- (1 / ((sin . x) ^2))) / (1 + ((cot . x) ^2)) by A11, FDIFF_7:47 .= - ((1 / ((sin . x) ^2)) / (1 + ((cot . x) ^2))) .= - (1 / (((sin . x) ^2) * (1 + (((cos . x) / (sin . x)) * ((cos . x) / (sin . x)))))) by A10, XCMPLX_1:78 .= - (1 / (((sin . x) ^2) * (1 + (((cos . x) ^2) / ((sin . x) ^2))))) by XCMPLX_1:76 .= - (1 / (((sin . x) ^2) + ((((sin . x) ^2) * ((cos . x) ^2)) / ((sin . x) ^2)))) .= - (1 / (((sin . x) ^2) + ((cos . x) ^2))) by A13, XCMPLX_1:89 .= - (1 / 1) by SIN_COS:28 .= - 1 ; hence ((arctan * cot) `| Z) . x = - 1 ; ::_thesis: verum end; hence ( arctan * cot is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * cot) `| Z) . x = - 1 ) ) by A1, A4, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_11:8 for Z being open Subset of REAL st Z c= dom (arccot * cot) & ( for x being Real st x in Z holds ( cot . x > - 1 & cot . x < 1 ) ) holds ( arccot * cot is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * cot) `| Z) . x = 1 ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (arccot * cot) & ( for x being Real st x in Z holds ( cot . x > - 1 & cot . x < 1 ) ) implies ( arccot * cot is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * cot) `| Z) . x = 1 ) ) ) assume that A1: Z c= dom (arccot * cot) and A2: for x being Real st x in Z holds ( cot . x > - 1 & cot . x < 1 ) ; ::_thesis: ( arccot * cot is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * cot) `| Z) . x = 1 ) ) dom (arccot * cot) c= dom cot by RELAT_1:25; then A3: Z c= dom cot by A1, XBOOLE_1:1; A4: for x being Real st x in Z holds arccot * cot is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies arccot * cot is_differentiable_in x ) assume A5: x in Z ; ::_thesis: arccot * cot is_differentiable_in x then sin . x <> 0 by A3, FDIFF_8:2; then A6: cot is_differentiable_in x by FDIFF_7:47; ( cot . x > - 1 & cot . x < 1 ) by A2, A5; hence arccot * cot is_differentiable_in x by A6, SIN_COS9:86; ::_thesis: verum end; then A7: arccot * cot is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((arccot * cot) `| Z) . x = 1 proof let x be Real; ::_thesis: ( x in Z implies ((arccot * cot) `| Z) . x = 1 ) assume A8: x in Z ; ::_thesis: ((arccot * cot) `| Z) . x = 1 then A9: ( cot . x > - 1 & cot . x < 1 ) by A2; A10: cot . x = (cos . x) / (sin . x) by A3, A8, RFUNCT_1:def_1; A11: sin . x <> 0 by A3, A8, FDIFF_8:2; then A12: cot is_differentiable_in x by FDIFF_7:47; A13: (sin . x) ^2 <> 0 by A11, SQUARE_1:12; ((arccot * cot) `| Z) . x = diff ((arccot * cot),x) by A7, A8, FDIFF_1:def_7 .= - ((diff (cot,x)) / (1 + ((cot . x) ^2))) by A12, A9, SIN_COS9:86 .= - ((- (1 / ((sin . x) ^2))) / (1 + ((cot . x) ^2))) by A11, FDIFF_7:47 .= (1 / ((sin . x) ^2)) / (1 + ((cot . x) ^2)) .= 1 / (((sin . x) ^2) * (1 + (((cos . x) / (sin . x)) * ((cos . x) / (sin . x))))) by A10, XCMPLX_1:78 .= 1 / (((sin . x) ^2) * (1 + (((cos . x) ^2) / ((sin . x) ^2)))) by XCMPLX_1:76 .= 1 / (((sin . x) ^2) + ((((sin . x) ^2) * ((cos . x) ^2)) / ((sin . x) ^2))) .= 1 / (((sin . x) ^2) + ((cos . x) ^2)) by A13, XCMPLX_1:89 .= 1 / 1 by SIN_COS:28 .= 1 ; hence ((arccot * cot) `| Z) . x = 1 ; ::_thesis: verum end; hence ( arccot * cot is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * cot) `| Z) . x = 1 ) ) by A1, A4, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_11:9 for Z being open Subset of REAL st Z c= dom (arctan * arctan) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds ( arctan . x > - 1 & arctan . x < 1 ) ) holds ( arctan * arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * arctan) `| Z) . x = 1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (arctan * arctan) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds ( arctan . x > - 1 & arctan . x < 1 ) ) implies ( arctan * arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * arctan) `| Z) . x = 1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2))) ) ) ) assume that A1: Z c= dom (arctan * arctan) and A2: Z c= ].(- 1),1.[ and A3: for x being Real st x in Z holds ( arctan . x > - 1 & arctan . x < 1 ) ; ::_thesis: ( arctan * arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * arctan) `| Z) . x = 1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2))) ) ) A4: for x being Real st x in Z holds arctan * arctan is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies arctan * arctan is_differentiable_in x ) assume A5: x in Z ; ::_thesis: arctan * arctan is_differentiable_in x then A6: ( arctan . x > - 1 & arctan . x < 1 ) by A3; arctan is_differentiable_on Z by A2, SIN_COS9:81; then arctan is_differentiable_in x by A5, FDIFF_1:9; hence arctan * arctan is_differentiable_in x by A6, SIN_COS9:85; ::_thesis: verum end; then A7: arctan * arctan is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((arctan * arctan) `| Z) . x = 1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((arctan * arctan) `| Z) . x = 1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2))) ) assume A8: x in Z ; ::_thesis: ((arctan * arctan) `| Z) . x = 1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2))) then A9: ( arctan . x > - 1 & arctan . x < 1 ) by A3; A10: arctan is_differentiable_on Z by A2, SIN_COS9:81; then A11: arctan is_differentiable_in x by A8, FDIFF_1:9; ((arctan * arctan) `| Z) . x = diff ((arctan * arctan),x) by A7, A8, FDIFF_1:def_7 .= (diff (arctan,x)) / (1 + ((arctan . x) ^2)) by A11, A9, SIN_COS9:85 .= ((arctan `| Z) . x) / (1 + ((arctan . x) ^2)) by A8, A10, FDIFF_1:def_7 .= (1 / (1 + (x ^2))) / (1 + ((arctan . x) ^2)) by A2, A8, SIN_COS9:81 .= 1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2))) by XCMPLX_1:78 ; hence ((arctan * arctan) `| Z) . x = 1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2))) ; ::_thesis: verum end; hence ( arctan * arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * arctan) `| Z) . x = 1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2))) ) ) by A1, A4, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_11:10 for Z being open Subset of REAL st Z c= dom (arccot * arctan) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds ( arctan . x > - 1 & arctan . x < 1 ) ) holds ( arccot * arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * arctan) `| Z) . x = - (1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2)))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (arccot * arctan) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds ( arctan . x > - 1 & arctan . x < 1 ) ) implies ( arccot * arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * arctan) `| Z) . x = - (1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2)))) ) ) ) assume that A1: Z c= dom (arccot * arctan) and A2: Z c= ].(- 1),1.[ and A3: for x being Real st x in Z holds ( arctan . x > - 1 & arctan . x < 1 ) ; ::_thesis: ( arccot * arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * arctan) `| Z) . x = - (1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2)))) ) ) A4: for x being Real st x in Z holds arccot * arctan is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies arccot * arctan is_differentiable_in x ) assume A5: x in Z ; ::_thesis: arccot * arctan is_differentiable_in x then A6: ( arctan . x > - 1 & arctan . x < 1 ) by A3; arctan is_differentiable_on Z by A2, SIN_COS9:81; then arctan is_differentiable_in x by A5, FDIFF_1:9; hence arccot * arctan is_differentiable_in x by A6, SIN_COS9:86; ::_thesis: verum end; then A7: arccot * arctan is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((arccot * arctan) `| Z) . x = - (1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2)))) proof let x be Real; ::_thesis: ( x in Z implies ((arccot * arctan) `| Z) . x = - (1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2)))) ) assume A8: x in Z ; ::_thesis: ((arccot * arctan) `| Z) . x = - (1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2)))) then A9: ( arctan . x > - 1 & arctan . x < 1 ) by A3; A10: arctan is_differentiable_on Z by A2, SIN_COS9:81; then A11: arctan is_differentiable_in x by A8, FDIFF_1:9; ((arccot * arctan) `| Z) . x = diff ((arccot * arctan),x) by A7, A8, FDIFF_1:def_7 .= - ((diff (arctan,x)) / (1 + ((arctan . x) ^2))) by A11, A9, SIN_COS9:86 .= - (((arctan `| Z) . x) / (1 + ((arctan . x) ^2))) by A8, A10, FDIFF_1:def_7 .= - ((1 / (1 + (x ^2))) / (1 + ((arctan . x) ^2))) by A2, A8, SIN_COS9:81 .= - (1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2)))) by XCMPLX_1:78 ; hence ((arccot * arctan) `| Z) . x = - (1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2)))) ; ::_thesis: verum end; hence ( arccot * arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * arctan) `| Z) . x = - (1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2)))) ) ) by A1, A4, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_11:11 for Z being open Subset of REAL st Z c= dom (arctan * arccot) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds ( arccot . x > - 1 & arccot . x < 1 ) ) holds ( arctan * arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * arccot) `| Z) . x = - (1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2)))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (arctan * arccot) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds ( arccot . x > - 1 & arccot . x < 1 ) ) implies ( arctan * arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * arccot) `| Z) . x = - (1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2)))) ) ) ) assume that A1: Z c= dom (arctan * arccot) and A2: Z c= ].(- 1),1.[ and A3: for x being Real st x in Z holds ( arccot . x > - 1 & arccot . x < 1 ) ; ::_thesis: ( arctan * arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * arccot) `| Z) . x = - (1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2)))) ) ) A4: for x being Real st x in Z holds arctan * arccot is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies arctan * arccot is_differentiable_in x ) assume A5: x in Z ; ::_thesis: arctan * arccot is_differentiable_in x then A6: ( arccot . x > - 1 & arccot . x < 1 ) by A3; arccot is_differentiable_on Z by A2, SIN_COS9:82; then arccot is_differentiable_in x by A5, FDIFF_1:9; hence arctan * arccot is_differentiable_in x by A6, SIN_COS9:85; ::_thesis: verum end; then A7: arctan * arccot is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((arctan * arccot) `| Z) . x = - (1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2)))) proof let x be Real; ::_thesis: ( x in Z implies ((arctan * arccot) `| Z) . x = - (1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2)))) ) assume A8: x in Z ; ::_thesis: ((arctan * arccot) `| Z) . x = - (1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2)))) then A9: ( arccot . x > - 1 & arccot . x < 1 ) by A3; A10: arccot is_differentiable_on Z by A2, SIN_COS9:82; then A11: arccot is_differentiable_in x by A8, FDIFF_1:9; ((arctan * arccot) `| Z) . x = diff ((arctan * arccot),x) by A7, A8, FDIFF_1:def_7 .= (diff (arccot,x)) / (1 + ((arccot . x) ^2)) by A11, A9, SIN_COS9:85 .= ((arccot `| Z) . x) / (1 + ((arccot . x) ^2)) by A8, A10, FDIFF_1:def_7 .= (- (1 / (1 + (x ^2)))) / (1 + ((arccot . x) ^2)) by A2, A8, SIN_COS9:82 .= - ((1 / (1 + (x ^2))) / (1 + ((arccot . x) ^2))) .= - (1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2)))) by XCMPLX_1:78 ; hence ((arctan * arccot) `| Z) . x = - (1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2)))) ; ::_thesis: verum end; hence ( arctan * arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan * arccot) `| Z) . x = - (1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2)))) ) ) by A1, A4, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_11:12 for Z being open Subset of REAL st Z c= dom (arccot * arccot) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds ( arccot . x > - 1 & arccot . x < 1 ) ) holds ( arccot * arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * arccot) `| Z) . x = 1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (arccot * arccot) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds ( arccot . x > - 1 & arccot . x < 1 ) ) implies ( arccot * arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * arccot) `| Z) . x = 1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2))) ) ) ) assume that A1: Z c= dom (arccot * arccot) and A2: Z c= ].(- 1),1.[ and A3: for x being Real st x in Z holds ( arccot . x > - 1 & arccot . x < 1 ) ; ::_thesis: ( arccot * arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * arccot) `| Z) . x = 1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2))) ) ) A4: for x being Real st x in Z holds arccot * arccot is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies arccot * arccot is_differentiable_in x ) assume A5: x in Z ; ::_thesis: arccot * arccot is_differentiable_in x then A6: ( arccot . x > - 1 & arccot . x < 1 ) by A3; arccot is_differentiable_on Z by A2, SIN_COS9:82; then arccot is_differentiable_in x by A5, FDIFF_1:9; hence arccot * arccot is_differentiable_in x by A6, SIN_COS9:86; ::_thesis: verum end; then A7: arccot * arccot is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((arccot * arccot) `| Z) . x = 1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((arccot * arccot) `| Z) . x = 1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2))) ) assume A8: x in Z ; ::_thesis: ((arccot * arccot) `| Z) . x = 1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2))) then A9: ( arccot . x > - 1 & arccot . x < 1 ) by A3; A10: arccot is_differentiable_on Z by A2, SIN_COS9:82; then A11: arccot is_differentiable_in x by A8, FDIFF_1:9; ((arccot * arccot) `| Z) . x = diff ((arccot * arccot),x) by A7, A8, FDIFF_1:def_7 .= - ((diff (arccot,x)) / (1 + ((arccot . x) ^2))) by A11, A9, SIN_COS9:86 .= - (((arccot `| Z) . x) / (1 + ((arccot . x) ^2))) by A8, A10, FDIFF_1:def_7 .= - ((- (1 / (1 + (x ^2)))) / (1 + ((arccot . x) ^2))) by A2, A8, SIN_COS9:82 .= (1 / (1 + (x ^2))) / (1 + ((arccot . x) ^2)) .= 1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2))) by XCMPLX_1:78 ; hence ((arccot * arccot) `| Z) . x = 1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2))) ; ::_thesis: verum end; hence ( arccot * arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot * arccot) `| Z) . x = 1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2))) ) ) by A1, A4, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_11:13 for Z being open Subset of REAL st Z c= dom (sin * arctan) & Z c= ].(- 1),1.[ holds ( sin * arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((sin * arctan) `| Z) . x = (cos . (arctan . x)) / (1 + (x ^2)) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (sin * arctan) & Z c= ].(- 1),1.[ implies ( sin * arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((sin * arctan) `| Z) . x = (cos . (arctan . x)) / (1 + (x ^2)) ) ) ) assume that A1: Z c= dom (sin * arctan) and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( sin * arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((sin * arctan) `| Z) . x = (cos . (arctan . x)) / (1 + (x ^2)) ) ) A3: for x being Real st x in Z holds sin * arctan is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies sin * arctan is_differentiable_in x ) assume A4: x in Z ; ::_thesis: sin * arctan is_differentiable_in x A5: sin is_differentiable_in arctan . x by SIN_COS:64; arctan is_differentiable_on Z by A2, SIN_COS9:81; then arctan is_differentiable_in x by A4, FDIFF_1:9; hence sin * arctan is_differentiable_in x by A5, FDIFF_2:13; ::_thesis: verum end; then A6: sin * arctan is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((sin * arctan) `| Z) . x = (cos . (arctan . x)) / (1 + (x ^2)) proof let x be Real; ::_thesis: ( x in Z implies ((sin * arctan) `| Z) . x = (cos . (arctan . x)) / (1 + (x ^2)) ) assume A7: x in Z ; ::_thesis: ((sin * arctan) `| Z) . x = (cos . (arctan . x)) / (1 + (x ^2)) A8: arctan is_differentiable_on Z by A2, SIN_COS9:81; then A9: arctan is_differentiable_in x by A7, FDIFF_1:9; A10: sin is_differentiable_in arctan . x by SIN_COS:64; ((sin * arctan) `| Z) . x = diff ((sin * arctan),x) by A6, A7, FDIFF_1:def_7 .= (diff (sin,(arctan . x))) * (diff (arctan,x)) by A9, A10, FDIFF_2:13 .= (cos . (arctan . x)) * (diff (arctan,x)) by SIN_COS:64 .= (cos . (arctan . x)) * ((arctan `| Z) . x) by A7, A8, FDIFF_1:def_7 .= (cos . (arctan . x)) * (1 / (1 + (x ^2))) by A2, A7, SIN_COS9:81 .= (cos . (arctan . x)) / (1 + (x ^2)) ; hence ((sin * arctan) `| Z) . x = (cos . (arctan . x)) / (1 + (x ^2)) ; ::_thesis: verum end; hence ( sin * arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((sin * arctan) `| Z) . x = (cos . (arctan . x)) / (1 + (x ^2)) ) ) by A1, A3, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_11:14 for Z being open Subset of REAL st Z c= dom (sin * arccot) & Z c= ].(- 1),1.[ holds ( sin * arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((sin * arccot) `| Z) . x = - ((cos . (arccot . x)) / (1 + (x ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (sin * arccot) & Z c= ].(- 1),1.[ implies ( sin * arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((sin * arccot) `| Z) . x = - ((cos . (arccot . x)) / (1 + (x ^2))) ) ) ) assume that A1: Z c= dom (sin * arccot) and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( sin * arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((sin * arccot) `| Z) . x = - ((cos . (arccot . x)) / (1 + (x ^2))) ) ) A3: for x being Real st x in Z holds sin * arccot is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies sin * arccot is_differentiable_in x ) assume A4: x in Z ; ::_thesis: sin * arccot is_differentiable_in x A5: sin is_differentiable_in arccot . x by SIN_COS:64; arccot is_differentiable_on Z by A2, SIN_COS9:82; then arccot is_differentiable_in x by A4, FDIFF_1:9; hence sin * arccot is_differentiable_in x by A5, FDIFF_2:13; ::_thesis: verum end; then A6: sin * arccot is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((sin * arccot) `| Z) . x = - ((cos . (arccot . x)) / (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((sin * arccot) `| Z) . x = - ((cos . (arccot . x)) / (1 + (x ^2))) ) assume A7: x in Z ; ::_thesis: ((sin * arccot) `| Z) . x = - ((cos . (arccot . x)) / (1 + (x ^2))) A8: arccot is_differentiable_on Z by A2, SIN_COS9:82; then A9: arccot is_differentiable_in x by A7, FDIFF_1:9; A10: sin is_differentiable_in arccot . x by SIN_COS:64; ((sin * arccot) `| Z) . x = diff ((sin * arccot),x) by A6, A7, FDIFF_1:def_7 .= (diff (sin,(arccot . x))) * (diff (arccot,x)) by A9, A10, FDIFF_2:13 .= (cos . (arccot . x)) * (diff (arccot,x)) by SIN_COS:64 .= (cos . (arccot . x)) * ((arccot `| Z) . x) by A7, A8, FDIFF_1:def_7 .= (cos . (arccot . x)) * (- (1 / (1 + (x ^2)))) by A2, A7, SIN_COS9:82 .= - ((cos . (arccot . x)) / (1 + (x ^2))) ; hence ((sin * arccot) `| Z) . x = - ((cos . (arccot . x)) / (1 + (x ^2))) ; ::_thesis: verum end; hence ( sin * arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((sin * arccot) `| Z) . x = - ((cos . (arccot . x)) / (1 + (x ^2))) ) ) by A1, A3, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_11:15 for Z being open Subset of REAL st Z c= dom (cos * arctan) & Z c= ].(- 1),1.[ holds ( cos * arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((cos * arctan) `| Z) . x = - ((sin . (arctan . x)) / (1 + (x ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (cos * arctan) & Z c= ].(- 1),1.[ implies ( cos * arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((cos * arctan) `| Z) . x = - ((sin . (arctan . x)) / (1 + (x ^2))) ) ) ) assume that A1: Z c= dom (cos * arctan) and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( cos * arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((cos * arctan) `| Z) . x = - ((sin . (arctan . x)) / (1 + (x ^2))) ) ) A3: for x being Real st x in Z holds cos * arctan is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies cos * arctan is_differentiable_in x ) assume A4: x in Z ; ::_thesis: cos * arctan is_differentiable_in x A5: cos is_differentiable_in arctan . x by SIN_COS:63; arctan is_differentiable_on Z by A2, SIN_COS9:81; then arctan is_differentiable_in x by A4, FDIFF_1:9; hence cos * arctan is_differentiable_in x by A5, FDIFF_2:13; ::_thesis: verum end; then A6: cos * arctan is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((cos * arctan) `| Z) . x = - ((sin . (arctan . x)) / (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((cos * arctan) `| Z) . x = - ((sin . (arctan . x)) / (1 + (x ^2))) ) assume A7: x in Z ; ::_thesis: ((cos * arctan) `| Z) . x = - ((sin . (arctan . x)) / (1 + (x ^2))) A8: arctan is_differentiable_on Z by A2, SIN_COS9:81; then A9: arctan is_differentiable_in x by A7, FDIFF_1:9; A10: cos is_differentiable_in arctan . x by SIN_COS:63; ((cos * arctan) `| Z) . x = diff ((cos * arctan),x) by A6, A7, FDIFF_1:def_7 .= (diff (cos,(arctan . x))) * (diff (arctan,x)) by A9, A10, FDIFF_2:13 .= (- (sin . (arctan . x))) * (diff (arctan,x)) by SIN_COS:63 .= - ((sin . (arctan . x)) * (diff (arctan,x))) .= - ((sin . (arctan . x)) * ((arctan `| Z) . x)) by A7, A8, FDIFF_1:def_7 .= - ((sin . (arctan . x)) * (1 / (1 + (x ^2)))) by A2, A7, SIN_COS9:81 .= - ((sin . (arctan . x)) / (1 + (x ^2))) ; hence ((cos * arctan) `| Z) . x = - ((sin . (arctan . x)) / (1 + (x ^2))) ; ::_thesis: verum end; hence ( cos * arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((cos * arctan) `| Z) . x = - ((sin . (arctan . x)) / (1 + (x ^2))) ) ) by A1, A3, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_11:16 for Z being open Subset of REAL st Z c= dom (cos * arccot) & Z c= ].(- 1),1.[ holds ( cos * arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((cos * arccot) `| Z) . x = (sin . (arccot . x)) / (1 + (x ^2)) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (cos * arccot) & Z c= ].(- 1),1.[ implies ( cos * arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((cos * arccot) `| Z) . x = (sin . (arccot . x)) / (1 + (x ^2)) ) ) ) assume that A1: Z c= dom (cos * arccot) and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( cos * arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((cos * arccot) `| Z) . x = (sin . (arccot . x)) / (1 + (x ^2)) ) ) A3: for x being Real st x in Z holds cos * arccot is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies cos * arccot is_differentiable_in x ) assume A4: x in Z ; ::_thesis: cos * arccot is_differentiable_in x A5: cos is_differentiable_in arccot . x by SIN_COS:63; arccot is_differentiable_on Z by A2, SIN_COS9:82; then arccot is_differentiable_in x by A4, FDIFF_1:9; hence cos * arccot is_differentiable_in x by A5, FDIFF_2:13; ::_thesis: verum end; then A6: cos * arccot is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((cos * arccot) `| Z) . x = (sin . (arccot . x)) / (1 + (x ^2)) proof let x be Real; ::_thesis: ( x in Z implies ((cos * arccot) `| Z) . x = (sin . (arccot . x)) / (1 + (x ^2)) ) assume A7: x in Z ; ::_thesis: ((cos * arccot) `| Z) . x = (sin . (arccot . x)) / (1 + (x ^2)) A8: arccot is_differentiable_on Z by A2, SIN_COS9:82; then A9: arccot is_differentiable_in x by A7, FDIFF_1:9; A10: cos is_differentiable_in arccot . x by SIN_COS:63; ((cos * arccot) `| Z) . x = diff ((cos * arccot),x) by A6, A7, FDIFF_1:def_7 .= (diff (cos,(arccot . x))) * (diff (arccot,x)) by A9, A10, FDIFF_2:13 .= (- (sin . (arccot . x))) * (diff (arccot,x)) by SIN_COS:63 .= - ((sin . (arccot . x)) * (diff (arccot,x))) .= - ((sin . (arccot . x)) * ((arccot `| Z) . x)) by A7, A8, FDIFF_1:def_7 .= - ((sin . (arccot . x)) * (- (1 / (1 + (x ^2))))) by A2, A7, SIN_COS9:82 .= (sin . (arccot . x)) / (1 + (x ^2)) ; hence ((cos * arccot) `| Z) . x = (sin . (arccot . x)) / (1 + (x ^2)) ; ::_thesis: verum end; hence ( cos * arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((cos * arccot) `| Z) . x = (sin . (arccot . x)) / (1 + (x ^2)) ) ) by A1, A3, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_11:17 for Z being open Subset of REAL st Z c= dom (tan * arctan) & Z c= ].(- 1),1.[ holds ( tan * arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((tan * arctan) `| Z) . x = 1 / (((cos . (arctan . x)) ^2) * (1 + (x ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (tan * arctan) & Z c= ].(- 1),1.[ implies ( tan * arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((tan * arctan) `| Z) . x = 1 / (((cos . (arctan . x)) ^2) * (1 + (x ^2))) ) ) ) assume that A1: Z c= dom (tan * arctan) and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( tan * arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((tan * arctan) `| Z) . x = 1 / (((cos . (arctan . x)) ^2) * (1 + (x ^2))) ) ) A3: for x being Real st x in Z holds tan * arctan is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies tan * arctan is_differentiable_in x ) assume A4: x in Z ; ::_thesis: tan * arctan is_differentiable_in x then arctan . x in dom tan by A1, FUNCT_1:11; then cos . (arctan . x) <> 0 by FDIFF_8:1; then A5: tan is_differentiable_in arctan . x by FDIFF_7:46; arctan is_differentiable_on Z by A2, SIN_COS9:81; then arctan is_differentiable_in x by A4, FDIFF_1:9; hence tan * arctan is_differentiable_in x by A5, FDIFF_2:13; ::_thesis: verum end; then A6: tan * arctan is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((tan * arctan) `| Z) . x = 1 / (((cos . (arctan . x)) ^2) * (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((tan * arctan) `| Z) . x = 1 / (((cos . (arctan . x)) ^2) * (1 + (x ^2))) ) assume A7: x in Z ; ::_thesis: ((tan * arctan) `| Z) . x = 1 / (((cos . (arctan . x)) ^2) * (1 + (x ^2))) then arctan . x in dom tan by A1, FUNCT_1:11; then A8: cos . (arctan . x) <> 0 by FDIFF_8:1; then A9: tan is_differentiable_in arctan . x by FDIFF_7:46; A10: arctan is_differentiable_on Z by A2, SIN_COS9:81; then A11: arctan is_differentiable_in x by A7, FDIFF_1:9; ((tan * arctan) `| Z) . x = diff ((tan * arctan),x) by A6, A7, FDIFF_1:def_7 .= (diff (tan,(arctan . x))) * (diff (arctan,x)) by A11, A9, FDIFF_2:13 .= (1 / ((cos . (arctan . x)) ^2)) * (diff (arctan,x)) by A8, FDIFF_7:46 .= (1 / ((cos . (arctan . x)) ^2)) * ((arctan `| Z) . x) by A7, A10, FDIFF_1:def_7 .= (1 / ((cos . (arctan . x)) ^2)) * (1 / (1 + (x ^2))) by A2, A7, SIN_COS9:81 .= 1 / (((cos . (arctan . x)) ^2) * (1 + (x ^2))) by XCMPLX_1:102 ; hence ((tan * arctan) `| Z) . x = 1 / (((cos . (arctan . x)) ^2) * (1 + (x ^2))) ; ::_thesis: verum end; hence ( tan * arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((tan * arctan) `| Z) . x = 1 / (((cos . (arctan . x)) ^2) * (1 + (x ^2))) ) ) by A1, A3, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_11:18 for Z being open Subset of REAL st Z c= dom (tan * arccot) & Z c= ].(- 1),1.[ holds ( tan * arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((tan * arccot) `| Z) . x = - (1 / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (tan * arccot) & Z c= ].(- 1),1.[ implies ( tan * arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((tan * arccot) `| Z) . x = - (1 / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) ) ) ) assume that A1: Z c= dom (tan * arccot) and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( tan * arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((tan * arccot) `| Z) . x = - (1 / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) ) ) A3: for x being Real st x in Z holds tan * arccot is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies tan * arccot is_differentiable_in x ) assume A4: x in Z ; ::_thesis: tan * arccot is_differentiable_in x then arccot . x in dom tan by A1, FUNCT_1:11; then cos . (arccot . x) <> 0 by FDIFF_8:1; then A5: tan is_differentiable_in arccot . x by FDIFF_7:46; arccot is_differentiable_on Z by A2, SIN_COS9:82; then arccot is_differentiable_in x by A4, FDIFF_1:9; hence tan * arccot is_differentiable_in x by A5, FDIFF_2:13; ::_thesis: verum end; then A6: tan * arccot is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((tan * arccot) `| Z) . x = - (1 / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) proof let x be Real; ::_thesis: ( x in Z implies ((tan * arccot) `| Z) . x = - (1 / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) ) assume A7: x in Z ; ::_thesis: ((tan * arccot) `| Z) . x = - (1 / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) then arccot . x in dom tan by A1, FUNCT_1:11; then A8: cos . (arccot . x) <> 0 by FDIFF_8:1; then A9: tan is_differentiable_in arccot . x by FDIFF_7:46; A10: arccot is_differentiable_on Z by A2, SIN_COS9:82; then A11: arccot is_differentiable_in x by A7, FDIFF_1:9; ((tan * arccot) `| Z) . x = diff ((tan * arccot),x) by A6, A7, FDIFF_1:def_7 .= (diff (tan,(arccot . x))) * (diff (arccot,x)) by A11, A9, FDIFF_2:13 .= (1 / ((cos . (arccot . x)) ^2)) * (diff (arccot,x)) by A8, FDIFF_7:46 .= (1 / ((cos . (arccot . x)) ^2)) * ((arccot `| Z) . x) by A7, A10, FDIFF_1:def_7 .= (1 / ((cos . (arccot . x)) ^2)) * (- (1 / (1 + (x ^2)))) by A2, A7, SIN_COS9:82 .= - ((1 / ((cos . (arccot . x)) ^2)) * (1 / (1 + (x ^2)))) .= - (1 / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) by XCMPLX_1:102 ; hence ((tan * arccot) `| Z) . x = - (1 / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) ; ::_thesis: verum end; hence ( tan * arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((tan * arccot) `| Z) . x = - (1 / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) ) ) by A1, A3, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_11:19 for Z being open Subset of REAL st Z c= dom (cot * arctan) & Z c= ].(- 1),1.[ holds ( cot * arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((cot * arctan) `| Z) . x = - (1 / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (cot * arctan) & Z c= ].(- 1),1.[ implies ( cot * arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((cot * arctan) `| Z) . x = - (1 / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) ) ) ) assume that A1: Z c= dom (cot * arctan) and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( cot * arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((cot * arctan) `| Z) . x = - (1 / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) ) ) A3: for x being Real st x in Z holds cot * arctan is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies cot * arctan is_differentiable_in x ) assume A4: x in Z ; ::_thesis: cot * arctan is_differentiable_in x then arctan . x in dom cot by A1, FUNCT_1:11; then sin . (arctan . x) <> 0 by FDIFF_8:2; then A5: cot is_differentiable_in arctan . x by FDIFF_7:47; arctan is_differentiable_on Z by A2, SIN_COS9:81; then arctan is_differentiable_in x by A4, FDIFF_1:9; hence cot * arctan is_differentiable_in x by A5, FDIFF_2:13; ::_thesis: verum end; then A6: cot * arctan is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((cot * arctan) `| Z) . x = - (1 / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) proof let x be Real; ::_thesis: ( x in Z implies ((cot * arctan) `| Z) . x = - (1 / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) ) assume A7: x in Z ; ::_thesis: ((cot * arctan) `| Z) . x = - (1 / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) then arctan . x in dom cot by A1, FUNCT_1:11; then A8: sin . (arctan . x) <> 0 by FDIFF_8:2; then A9: cot is_differentiable_in arctan . x by FDIFF_7:47; A10: arctan is_differentiable_on Z by A2, SIN_COS9:81; then A11: arctan is_differentiable_in x by A7, FDIFF_1:9; ((cot * arctan) `| Z) . x = diff ((cot * arctan),x) by A6, A7, FDIFF_1:def_7 .= (diff (cot,(arctan . x))) * (diff (arctan,x)) by A11, A9, FDIFF_2:13 .= (- (1 / ((sin . (arctan . x)) ^2))) * (diff (arctan,x)) by A8, FDIFF_7:47 .= - ((1 / ((sin . (arctan . x)) ^2)) * (diff (arctan,x))) .= - ((1 / ((sin . (arctan . x)) ^2)) * ((arctan `| Z) . x)) by A7, A10, FDIFF_1:def_7 .= - ((1 / ((sin . (arctan . x)) ^2)) * (1 / (1 + (x ^2)))) by A2, A7, SIN_COS9:81 .= - (1 / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) by XCMPLX_1:102 ; hence ((cot * arctan) `| Z) . x = - (1 / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) ; ::_thesis: verum end; hence ( cot * arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((cot * arctan) `| Z) . x = - (1 / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) ) ) by A1, A3, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_11:20 for Z being open Subset of REAL st Z c= dom (cot * arccot) & Z c= ].(- 1),1.[ holds ( cot * arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((cot * arccot) `| Z) . x = 1 / (((sin . (arccot . x)) ^2) * (1 + (x ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (cot * arccot) & Z c= ].(- 1),1.[ implies ( cot * arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((cot * arccot) `| Z) . x = 1 / (((sin . (arccot . x)) ^2) * (1 + (x ^2))) ) ) ) assume that A1: Z c= dom (cot * arccot) and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( cot * arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((cot * arccot) `| Z) . x = 1 / (((sin . (arccot . x)) ^2) * (1 + (x ^2))) ) ) A3: for x being Real st x in Z holds cot * arccot is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies cot * arccot is_differentiable_in x ) assume A4: x in Z ; ::_thesis: cot * arccot is_differentiable_in x then arccot . x in dom cot by A1, FUNCT_1:11; then sin . (arccot . x) <> 0 by FDIFF_8:2; then A5: cot is_differentiable_in arccot . x by FDIFF_7:47; arccot is_differentiable_on Z by A2, SIN_COS9:82; then arccot is_differentiable_in x by A4, FDIFF_1:9; hence cot * arccot is_differentiable_in x by A5, FDIFF_2:13; ::_thesis: verum end; then A6: cot * arccot is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((cot * arccot) `| Z) . x = 1 / (((sin . (arccot . x)) ^2) * (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((cot * arccot) `| Z) . x = 1 / (((sin . (arccot . x)) ^2) * (1 + (x ^2))) ) assume A7: x in Z ; ::_thesis: ((cot * arccot) `| Z) . x = 1 / (((sin . (arccot . x)) ^2) * (1 + (x ^2))) then arccot . x in dom cot by A1, FUNCT_1:11; then A8: sin . (arccot . x) <> 0 by FDIFF_8:2; then A9: cot is_differentiable_in arccot . x by FDIFF_7:47; A10: arccot is_differentiable_on Z by A2, SIN_COS9:82; then A11: arccot is_differentiable_in x by A7, FDIFF_1:9; ((cot * arccot) `| Z) . x = diff ((cot * arccot),x) by A6, A7, FDIFF_1:def_7 .= (diff (cot,(arccot . x))) * (diff (arccot,x)) by A11, A9, FDIFF_2:13 .= (- (1 / ((sin . (arccot . x)) ^2))) * (diff (arccot,x)) by A8, FDIFF_7:47 .= - ((1 / ((sin . (arccot . x)) ^2)) * (diff (arccot,x))) .= - ((1 / ((sin . (arccot . x)) ^2)) * ((arccot `| Z) . x)) by A7, A10, FDIFF_1:def_7 .= - ((1 / ((sin . (arccot . x)) ^2)) * (- (1 / (1 + (x ^2))))) by A2, A7, SIN_COS9:82 .= (1 / ((sin . (arccot . x)) ^2)) * (1 / (1 + (x ^2))) .= 1 / (((sin . (arccot . x)) ^2) * (1 + (x ^2))) by XCMPLX_1:102 ; hence ((cot * arccot) `| Z) . x = 1 / (((sin . (arccot . x)) ^2) * (1 + (x ^2))) ; ::_thesis: verum end; hence ( cot * arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((cot * arccot) `| Z) . x = 1 / (((sin . (arccot . x)) ^2) * (1 + (x ^2))) ) ) by A1, A3, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_11:21 for Z being open Subset of REAL st Z c= dom (sec * arctan) & Z c= ].(- 1),1.[ holds ( sec * arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((sec * arctan) `| Z) . x = (sin . (arctan . x)) / (((cos . (arctan . x)) ^2) * (1 + (x ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (sec * arctan) & Z c= ].(- 1),1.[ implies ( sec * arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((sec * arctan) `| Z) . x = (sin . (arctan . x)) / (((cos . (arctan . x)) ^2) * (1 + (x ^2))) ) ) ) assume that A1: Z c= dom (sec * arctan) and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( sec * arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((sec * arctan) `| Z) . x = (sin . (arctan . x)) / (((cos . (arctan . x)) ^2) * (1 + (x ^2))) ) ) A3: for x being Real st x in Z holds cos . (arctan . x) <> 0 proof let x be Real; ::_thesis: ( x in Z implies cos . (arctan . x) <> 0 ) assume x in Z ; ::_thesis: cos . (arctan . x) <> 0 then arctan . x in dom sec by A1, FUNCT_1:11; hence cos . (arctan . x) <> 0 by RFUNCT_1:3; ::_thesis: verum end; A4: for x being Real st x in Z holds sec * arctan is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies sec * arctan is_differentiable_in x ) assume A5: x in Z ; ::_thesis: sec * arctan is_differentiable_in x then cos . (arctan . x) <> 0 by A3; then A6: sec is_differentiable_in arctan . x by FDIFF_9:1; arctan is_differentiable_on Z by A2, SIN_COS9:81; then arctan is_differentiable_in x by A5, FDIFF_1:9; hence sec * arctan is_differentiable_in x by A6, FDIFF_2:13; ::_thesis: verum end; then A7: sec * arctan is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((sec * arctan) `| Z) . x = (sin . (arctan . x)) / (((cos . (arctan . x)) ^2) * (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((sec * arctan) `| Z) . x = (sin . (arctan . x)) / (((cos . (arctan . x)) ^2) * (1 + (x ^2))) ) assume A8: x in Z ; ::_thesis: ((sec * arctan) `| Z) . x = (sin . (arctan . x)) / (((cos . (arctan . x)) ^2) * (1 + (x ^2))) then A9: cos . (arctan . x) <> 0 by A3; cos . (arctan . x) <> 0 by A3, A8; then A10: sec is_differentiable_in arctan . x by FDIFF_9:1; A11: arctan is_differentiable_on Z by A2, SIN_COS9:81; then A12: arctan is_differentiable_in x by A8, FDIFF_1:9; ((sec * arctan) `| Z) . x = diff ((sec * arctan),x) by A7, A8, FDIFF_1:def_7 .= (diff (sec,(arctan . x))) * (diff (arctan,x)) by A12, A10, FDIFF_2:13 .= ((sin . (arctan . x)) / ((cos . (arctan . x)) ^2)) * (diff (arctan,x)) by A9, FDIFF_9:1 .= ((sin . (arctan . x)) / ((cos . (arctan . x)) ^2)) * ((arctan `| Z) . x) by A8, A11, FDIFF_1:def_7 .= ((sin . (arctan . x)) / ((cos . (arctan . x)) ^2)) * (1 / (1 + (x ^2))) by A2, A8, SIN_COS9:81 .= ((sin . (arctan . x)) * 1) / (((cos . (arctan . x)) ^2) * (1 + (x ^2))) by XCMPLX_1:76 .= (sin . (arctan . x)) / (((cos . (arctan . x)) ^2) * (1 + (x ^2))) ; hence ((sec * arctan) `| Z) . x = (sin . (arctan . x)) / (((cos . (arctan . x)) ^2) * (1 + (x ^2))) ; ::_thesis: verum end; hence ( sec * arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((sec * arctan) `| Z) . x = (sin . (arctan . x)) / (((cos . (arctan . x)) ^2) * (1 + (x ^2))) ) ) by A1, A4, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_11:22 for Z being open Subset of REAL st Z c= dom (sec * arccot) & Z c= ].(- 1),1.[ holds ( sec * arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((sec * arccot) `| Z) . x = - ((sin . (arccot . x)) / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (sec * arccot) & Z c= ].(- 1),1.[ implies ( sec * arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((sec * arccot) `| Z) . x = - ((sin . (arccot . x)) / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) ) ) ) assume that A1: Z c= dom (sec * arccot) and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( sec * arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((sec * arccot) `| Z) . x = - ((sin . (arccot . x)) / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) ) ) A3: for x being Real st x in Z holds cos . (arccot . x) <> 0 proof let x be Real; ::_thesis: ( x in Z implies cos . (arccot . x) <> 0 ) assume x in Z ; ::_thesis: cos . (arccot . x) <> 0 then arccot . x in dom sec by A1, FUNCT_1:11; hence cos . (arccot . x) <> 0 by RFUNCT_1:3; ::_thesis: verum end; A4: for x being Real st x in Z holds sec * arccot is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies sec * arccot is_differentiable_in x ) assume A5: x in Z ; ::_thesis: sec * arccot is_differentiable_in x then cos . (arccot . x) <> 0 by A3; then A6: sec is_differentiable_in arccot . x by FDIFF_9:1; arccot is_differentiable_on Z by A2, SIN_COS9:82; then arccot is_differentiable_in x by A5, FDIFF_1:9; hence sec * arccot is_differentiable_in x by A6, FDIFF_2:13; ::_thesis: verum end; then A7: sec * arccot is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((sec * arccot) `| Z) . x = - ((sin . (arccot . x)) / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) proof let x be Real; ::_thesis: ( x in Z implies ((sec * arccot) `| Z) . x = - ((sin . (arccot . x)) / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) ) assume A8: x in Z ; ::_thesis: ((sec * arccot) `| Z) . x = - ((sin . (arccot . x)) / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) then A9: cos . (arccot . x) <> 0 by A3; cos . (arccot . x) <> 0 by A3, A8; then A10: sec is_differentiable_in arccot . x by FDIFF_9:1; A11: arccot is_differentiable_on Z by A2, SIN_COS9:82; then A12: arccot is_differentiable_in x by A8, FDIFF_1:9; ((sec * arccot) `| Z) . x = diff ((sec * arccot),x) by A7, A8, FDIFF_1:def_7 .= (diff (sec,(arccot . x))) * (diff (arccot,x)) by A12, A10, FDIFF_2:13 .= ((sin . (arccot . x)) / ((cos . (arccot . x)) ^2)) * (diff (arccot,x)) by A9, FDIFF_9:1 .= ((sin . (arccot . x)) / ((cos . (arccot . x)) ^2)) * ((arccot `| Z) . x) by A8, A11, FDIFF_1:def_7 .= ((sin . (arccot . x)) / ((cos . (arccot . x)) ^2)) * (- (1 / (1 + (x ^2)))) by A2, A8, SIN_COS9:82 .= - (((sin . (arccot . x)) / ((cos . (arccot . x)) ^2)) * (1 / (1 + (x ^2)))) .= - (((sin . (arccot . x)) * 1) / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) by XCMPLX_1:76 .= - ((sin . (arccot . x)) / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) ; hence ((sec * arccot) `| Z) . x = - ((sin . (arccot . x)) / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) ; ::_thesis: verum end; hence ( sec * arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((sec * arccot) `| Z) . x = - ((sin . (arccot . x)) / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) ) ) by A1, A4, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_11:23 for Z being open Subset of REAL st Z c= dom (cosec * arctan) & Z c= ].(- 1),1.[ holds ( cosec * arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((cosec * arctan) `| Z) . x = - ((cos . (arctan . x)) / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (cosec * arctan) & Z c= ].(- 1),1.[ implies ( cosec * arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((cosec * arctan) `| Z) . x = - ((cos . (arctan . x)) / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) ) ) ) assume that A1: Z c= dom (cosec * arctan) and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( cosec * arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((cosec * arctan) `| Z) . x = - ((cos . (arctan . x)) / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) ) ) A3: for x being Real st x in Z holds sin . (arctan . x) <> 0 proof let x be Real; ::_thesis: ( x in Z implies sin . (arctan . x) <> 0 ) assume x in Z ; ::_thesis: sin . (arctan . x) <> 0 then arctan . x in dom cosec by A1, FUNCT_1:11; hence sin . (arctan . x) <> 0 by RFUNCT_1:3; ::_thesis: verum end; A4: for x being Real st x in Z holds cosec * arctan is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies cosec * arctan is_differentiable_in x ) assume A5: x in Z ; ::_thesis: cosec * arctan is_differentiable_in x then sin . (arctan . x) <> 0 by A3; then A6: cosec is_differentiable_in arctan . x by FDIFF_9:2; arctan is_differentiable_on Z by A2, SIN_COS9:81; then arctan is_differentiable_in x by A5, FDIFF_1:9; hence cosec * arctan is_differentiable_in x by A6, FDIFF_2:13; ::_thesis: verum end; then A7: cosec * arctan is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((cosec * arctan) `| Z) . x = - ((cos . (arctan . x)) / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) proof let x be Real; ::_thesis: ( x in Z implies ((cosec * arctan) `| Z) . x = - ((cos . (arctan . x)) / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) ) assume A8: x in Z ; ::_thesis: ((cosec * arctan) `| Z) . x = - ((cos . (arctan . x)) / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) then A9: sin . (arctan . x) <> 0 by A3; sin . (arctan . x) <> 0 by A3, A8; then A10: cosec is_differentiable_in arctan . x by FDIFF_9:2; A11: arctan is_differentiable_on Z by A2, SIN_COS9:81; then A12: arctan is_differentiable_in x by A8, FDIFF_1:9; ((cosec * arctan) `| Z) . x = diff ((cosec * arctan),x) by A7, A8, FDIFF_1:def_7 .= (diff (cosec,(arctan . x))) * (diff (arctan,x)) by A12, A10, FDIFF_2:13 .= (- ((cos . (arctan . x)) / ((sin . (arctan . x)) ^2))) * (diff (arctan,x)) by A9, FDIFF_9:2 .= - (((cos . (arctan . x)) / ((sin . (arctan . x)) ^2)) * (diff (arctan,x))) .= - (((cos . (arctan . x)) / ((sin . (arctan . x)) ^2)) * ((arctan `| Z) . x)) by A8, A11, FDIFF_1:def_7 .= - (((cos . (arctan . x)) / ((sin . (arctan . x)) ^2)) * (1 / (1 + (x ^2)))) by A2, A8, SIN_COS9:81 .= - (((cos . (arctan . x)) * 1) / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) by XCMPLX_1:76 .= - ((cos . (arctan . x)) / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) ; hence ((cosec * arctan) `| Z) . x = - ((cos . (arctan . x)) / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) ; ::_thesis: verum end; hence ( cosec * arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((cosec * arctan) `| Z) . x = - ((cos . (arctan . x)) / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) ) ) by A1, A4, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_11:24 for Z being open Subset of REAL st Z c= dom (cosec * arccot) & Z c= ].(- 1),1.[ holds ( cosec * arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((cosec * arccot) `| Z) . x = (cos . (arccot . x)) / (((sin . (arccot . x)) ^2) * (1 + (x ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (cosec * arccot) & Z c= ].(- 1),1.[ implies ( cosec * arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((cosec * arccot) `| Z) . x = (cos . (arccot . x)) / (((sin . (arccot . x)) ^2) * (1 + (x ^2))) ) ) ) assume that A1: Z c= dom (cosec * arccot) and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( cosec * arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((cosec * arccot) `| Z) . x = (cos . (arccot . x)) / (((sin . (arccot . x)) ^2) * (1 + (x ^2))) ) ) A3: for x being Real st x in Z holds sin . (arccot . x) <> 0 proof let x be Real; ::_thesis: ( x in Z implies sin . (arccot . x) <> 0 ) assume x in Z ; ::_thesis: sin . (arccot . x) <> 0 then arccot . x in dom cosec by A1, FUNCT_1:11; hence sin . (arccot . x) <> 0 by RFUNCT_1:3; ::_thesis: verum end; A4: for x being Real st x in Z holds cosec * arccot is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies cosec * arccot is_differentiable_in x ) assume A5: x in Z ; ::_thesis: cosec * arccot is_differentiable_in x then sin . (arccot . x) <> 0 by A3; then A6: cosec is_differentiable_in arccot . x by FDIFF_9:2; arccot is_differentiable_on Z by A2, SIN_COS9:82; then arccot is_differentiable_in x by A5, FDIFF_1:9; hence cosec * arccot is_differentiable_in x by A6, FDIFF_2:13; ::_thesis: verum end; then A7: cosec * arccot is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((cosec * arccot) `| Z) . x = (cos . (arccot . x)) / (((sin . (arccot . x)) ^2) * (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((cosec * arccot) `| Z) . x = (cos . (arccot . x)) / (((sin . (arccot . x)) ^2) * (1 + (x ^2))) ) assume A8: x in Z ; ::_thesis: ((cosec * arccot) `| Z) . x = (cos . (arccot . x)) / (((sin . (arccot . x)) ^2) * (1 + (x ^2))) then A9: sin . (arccot . x) <> 0 by A3; sin . (arccot . x) <> 0 by A3, A8; then A10: cosec is_differentiable_in arccot . x by FDIFF_9:2; A11: arccot is_differentiable_on Z by A2, SIN_COS9:82; then A12: arccot is_differentiable_in x by A8, FDIFF_1:9; ((cosec * arccot) `| Z) . x = diff ((cosec * arccot),x) by A7, A8, FDIFF_1:def_7 .= (diff (cosec,(arccot . x))) * (diff (arccot,x)) by A12, A10, FDIFF_2:13 .= (- ((cos . (arccot . x)) / ((sin . (arccot . x)) ^2))) * (diff (arccot,x)) by A9, FDIFF_9:2 .= - (((cos . (arccot . x)) / ((sin . (arccot . x)) ^2)) * (diff (arccot,x))) .= - (((cos . (arccot . x)) / ((sin . (arccot . x)) ^2)) * ((arccot `| Z) . x)) by A8, A11, FDIFF_1:def_7 .= - (((cos . (arccot . x)) / ((sin . (arccot . x)) ^2)) * (- (1 / (1 + (x ^2))))) by A2, A8, SIN_COS9:82 .= ((cos . (arccot . x)) / ((sin . (arccot . x)) ^2)) * (1 / (1 + (x ^2))) .= ((cos . (arccot . x)) * 1) / (((sin . (arccot . x)) ^2) * (1 + (x ^2))) by XCMPLX_1:76 .= (cos . (arccot . x)) / (((sin . (arccot . x)) ^2) * (1 + (x ^2))) ; hence ((cosec * arccot) `| Z) . x = (cos . (arccot . x)) / (((sin . (arccot . x)) ^2) * (1 + (x ^2))) ; ::_thesis: verum end; hence ( cosec * arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((cosec * arccot) `| Z) . x = (cos . (arccot . x)) / (((sin . (arccot . x)) ^2) * (1 + (x ^2))) ) ) by A1, A4, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_11:25 for Z being open Subset of REAL st Z c= dom (sin (#) arctan) & Z c= ].(- 1),1.[ holds ( sin (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((sin (#) arctan) `| Z) . x = ((cos . x) * (arctan . x)) + ((sin . x) / (1 + (x ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (sin (#) arctan) & Z c= ].(- 1),1.[ implies ( sin (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((sin (#) arctan) `| Z) . x = ((cos . x) * (arctan . x)) + ((sin . x) / (1 + (x ^2))) ) ) ) assume that A1: Z c= dom (sin (#) arctan) and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( sin (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((sin (#) arctan) `| Z) . x = ((cos . x) * (arctan . x)) + ((sin . x) / (1 + (x ^2))) ) ) A3: arctan is_differentiable_on Z by A2, SIN_COS9:81; A4: for x being Real st x in Z holds sin is_differentiable_in x by SIN_COS:64; Z c= (dom sin) /\ (dom arctan) by A1, VALUED_1:def_4; then Z c= dom sin by XBOOLE_1:18; then A5: sin is_differentiable_on Z by A4, FDIFF_1:9; for x being Real st x in Z holds ((sin (#) arctan) `| Z) . x = ((cos . x) * (arctan . x)) + ((sin . x) / (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((sin (#) arctan) `| Z) . x = ((cos . x) * (arctan . x)) + ((sin . x) / (1 + (x ^2))) ) assume A6: x in Z ; ::_thesis: ((sin (#) arctan) `| Z) . x = ((cos . x) * (arctan . x)) + ((sin . x) / (1 + (x ^2))) then ((sin (#) arctan) `| Z) . x = ((arctan . x) * (diff (sin,x))) + ((sin . x) * (diff (arctan,x))) by A1, A5, A3, FDIFF_1:21 .= ((arctan . x) * (cos . x)) + ((sin . x) * (diff (arctan,x))) by SIN_COS:64 .= ((cos . x) * (arctan . x)) + ((sin . x) * ((arctan `| Z) . x)) by A3, A6, FDIFF_1:def_7 .= ((cos . x) * (arctan . x)) + ((sin . x) * (1 / (1 + (x ^2)))) by A2, A6, SIN_COS9:81 .= ((cos . x) * (arctan . x)) + ((sin . x) / (1 + (x ^2))) ; hence ((sin (#) arctan) `| Z) . x = ((cos . x) * (arctan . x)) + ((sin . x) / (1 + (x ^2))) ; ::_thesis: verum end; hence ( sin (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((sin (#) arctan) `| Z) . x = ((cos . x) * (arctan . x)) + ((sin . x) / (1 + (x ^2))) ) ) by A1, A5, A3, FDIFF_1:21; ::_thesis: verum end; theorem :: FDIFF_11:26 for Z being open Subset of REAL st Z c= dom (sin (#) arccot) & Z c= ].(- 1),1.[ holds ( sin (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((sin (#) arccot) `| Z) . x = ((cos . x) * (arccot . x)) - ((sin . x) / (1 + (x ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (sin (#) arccot) & Z c= ].(- 1),1.[ implies ( sin (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((sin (#) arccot) `| Z) . x = ((cos . x) * (arccot . x)) - ((sin . x) / (1 + (x ^2))) ) ) ) assume that A1: Z c= dom (sin (#) arccot) and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( sin (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((sin (#) arccot) `| Z) . x = ((cos . x) * (arccot . x)) - ((sin . x) / (1 + (x ^2))) ) ) A3: arccot is_differentiable_on Z by A2, SIN_COS9:82; A4: for x being Real st x in Z holds sin is_differentiable_in x by SIN_COS:64; Z c= (dom sin) /\ (dom arccot) by A1, VALUED_1:def_4; then Z c= dom sin by XBOOLE_1:18; then A5: sin is_differentiable_on Z by A4, FDIFF_1:9; for x being Real st x in Z holds ((sin (#) arccot) `| Z) . x = ((cos . x) * (arccot . x)) - ((sin . x) / (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((sin (#) arccot) `| Z) . x = ((cos . x) * (arccot . x)) - ((sin . x) / (1 + (x ^2))) ) assume A6: x in Z ; ::_thesis: ((sin (#) arccot) `| Z) . x = ((cos . x) * (arccot . x)) - ((sin . x) / (1 + (x ^2))) then ((sin (#) arccot) `| Z) . x = ((arccot . x) * (diff (sin,x))) + ((sin . x) * (diff (arccot,x))) by A1, A5, A3, FDIFF_1:21 .= ((arccot . x) * (cos . x)) + ((sin . x) * (diff (arccot,x))) by SIN_COS:64 .= ((cos . x) * (arccot . x)) + ((sin . x) * ((arccot `| Z) . x)) by A3, A6, FDIFF_1:def_7 .= ((cos . x) * (arccot . x)) + ((sin . x) * (- (1 / (1 + (x ^2))))) by A2, A6, SIN_COS9:82 .= ((cos . x) * (arccot . x)) - ((sin . x) / (1 + (x ^2))) ; hence ((sin (#) arccot) `| Z) . x = ((cos . x) * (arccot . x)) - ((sin . x) / (1 + (x ^2))) ; ::_thesis: verum end; hence ( sin (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((sin (#) arccot) `| Z) . x = ((cos . x) * (arccot . x)) - ((sin . x) / (1 + (x ^2))) ) ) by A1, A5, A3, FDIFF_1:21; ::_thesis: verum end; theorem :: FDIFF_11:27 for Z being open Subset of REAL st Z c= dom (cos (#) arctan) & Z c= ].(- 1),1.[ holds ( cos (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((cos (#) arctan) `| Z) . x = (- ((sin . x) * (arctan . x))) + ((cos . x) / (1 + (x ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (cos (#) arctan) & Z c= ].(- 1),1.[ implies ( cos (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((cos (#) arctan) `| Z) . x = (- ((sin . x) * (arctan . x))) + ((cos . x) / (1 + (x ^2))) ) ) ) assume that A1: Z c= dom (cos (#) arctan) and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( cos (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((cos (#) arctan) `| Z) . x = (- ((sin . x) * (arctan . x))) + ((cos . x) / (1 + (x ^2))) ) ) A3: arctan is_differentiable_on Z by A2, SIN_COS9:81; A4: for x being Real st x in Z holds cos is_differentiable_in x by SIN_COS:63; Z c= (dom cos) /\ (dom arctan) by A1, VALUED_1:def_4; then Z c= dom cos by XBOOLE_1:18; then A5: cos is_differentiable_on Z by A4, FDIFF_1:9; for x being Real st x in Z holds ((cos (#) arctan) `| Z) . x = (- ((sin . x) * (arctan . x))) + ((cos . x) / (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((cos (#) arctan) `| Z) . x = (- ((sin . x) * (arctan . x))) + ((cos . x) / (1 + (x ^2))) ) assume A6: x in Z ; ::_thesis: ((cos (#) arctan) `| Z) . x = (- ((sin . x) * (arctan . x))) + ((cos . x) / (1 + (x ^2))) then ((cos (#) arctan) `| Z) . x = ((arctan . x) * (diff (cos,x))) + ((cos . x) * (diff (arctan,x))) by A1, A5, A3, FDIFF_1:21 .= ((arctan . x) * (- (sin . x))) + ((cos . x) * (diff (arctan,x))) by SIN_COS:63 .= (- ((sin . x) * (arctan . x))) + ((cos . x) * ((arctan `| Z) . x)) by A3, A6, FDIFF_1:def_7 .= (- ((sin . x) * (arctan . x))) + ((cos . x) * (1 / (1 + (x ^2)))) by A2, A6, SIN_COS9:81 .= (- ((sin . x) * (arctan . x))) + ((cos . x) / (1 + (x ^2))) ; hence ((cos (#) arctan) `| Z) . x = (- ((sin . x) * (arctan . x))) + ((cos . x) / (1 + (x ^2))) ; ::_thesis: verum end; hence ( cos (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((cos (#) arctan) `| Z) . x = (- ((sin . x) * (arctan . x))) + ((cos . x) / (1 + (x ^2))) ) ) by A1, A5, A3, FDIFF_1:21; ::_thesis: verum end; theorem :: FDIFF_11:28 for Z being open Subset of REAL st Z c= dom (cos (#) arccot) & Z c= ].(- 1),1.[ holds ( cos (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((cos (#) arccot) `| Z) . x = (- ((sin . x) * (arccot . x))) - ((cos . x) / (1 + (x ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (cos (#) arccot) & Z c= ].(- 1),1.[ implies ( cos (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((cos (#) arccot) `| Z) . x = (- ((sin . x) * (arccot . x))) - ((cos . x) / (1 + (x ^2))) ) ) ) assume that A1: Z c= dom (cos (#) arccot) and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( cos (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((cos (#) arccot) `| Z) . x = (- ((sin . x) * (arccot . x))) - ((cos . x) / (1 + (x ^2))) ) ) A3: arccot is_differentiable_on Z by A2, SIN_COS9:82; A4: for x being Real st x in Z holds cos is_differentiable_in x by SIN_COS:63; Z c= (dom cos) /\ (dom arccot) by A1, VALUED_1:def_4; then Z c= dom cos by XBOOLE_1:18; then A5: cos is_differentiable_on Z by A4, FDIFF_1:9; for x being Real st x in Z holds ((cos (#) arccot) `| Z) . x = (- ((sin . x) * (arccot . x))) - ((cos . x) / (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((cos (#) arccot) `| Z) . x = (- ((sin . x) * (arccot . x))) - ((cos . x) / (1 + (x ^2))) ) assume A6: x in Z ; ::_thesis: ((cos (#) arccot) `| Z) . x = (- ((sin . x) * (arccot . x))) - ((cos . x) / (1 + (x ^2))) then ((cos (#) arccot) `| Z) . x = ((arccot . x) * (diff (cos,x))) + ((cos . x) * (diff (arccot,x))) by A1, A5, A3, FDIFF_1:21 .= ((arccot . x) * (- (sin . x))) + ((cos . x) * (diff (arccot,x))) by SIN_COS:63 .= (- ((sin . x) * (arccot . x))) + ((cos . x) * ((arccot `| Z) . x)) by A3, A6, FDIFF_1:def_7 .= (- ((sin . x) * (arccot . x))) + ((cos . x) * (- (1 / (1 + (x ^2))))) by A2, A6, SIN_COS9:82 .= (- ((sin . x) * (arccot . x))) - ((cos . x) / (1 + (x ^2))) ; hence ((cos (#) arccot) `| Z) . x = (- ((sin . x) * (arccot . x))) - ((cos . x) / (1 + (x ^2))) ; ::_thesis: verum end; hence ( cos (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((cos (#) arccot) `| Z) . x = (- ((sin . x) * (arccot . x))) - ((cos . x) / (1 + (x ^2))) ) ) by A1, A5, A3, FDIFF_1:21; ::_thesis: verum end; theorem :: FDIFF_11:29 for Z being open Subset of REAL st Z c= dom (tan (#) arctan) & Z c= ].(- 1),1.[ holds ( tan (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((tan (#) arctan) `| Z) . x = ((arctan . x) / ((cos . x) ^2)) + ((tan . x) / (1 + (x ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (tan (#) arctan) & Z c= ].(- 1),1.[ implies ( tan (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((tan (#) arctan) `| Z) . x = ((arctan . x) / ((cos . x) ^2)) + ((tan . x) / (1 + (x ^2))) ) ) ) assume that A1: Z c= dom (tan (#) arctan) and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( tan (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((tan (#) arctan) `| Z) . x = ((arctan . x) / ((cos . x) ^2)) + ((tan . x) / (1 + (x ^2))) ) ) A3: arctan is_differentiable_on Z by A2, SIN_COS9:81; Z c= (dom tan) /\ (dom arctan) by A1, VALUED_1:def_4; then A4: Z c= dom tan by XBOOLE_1:18; for x being Real st x in Z holds tan is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies tan is_differentiable_in x ) assume x in Z ; ::_thesis: tan is_differentiable_in x then cos . x <> 0 by A4, FDIFF_8:1; hence tan is_differentiable_in x by FDIFF_7:46; ::_thesis: verum end; then A5: tan is_differentiable_on Z by A4, FDIFF_1:9; for x being Real st x in Z holds ((tan (#) arctan) `| Z) . x = ((arctan . x) / ((cos . x) ^2)) + ((tan . x) / (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((tan (#) arctan) `| Z) . x = ((arctan . x) / ((cos . x) ^2)) + ((tan . x) / (1 + (x ^2))) ) assume A6: x in Z ; ::_thesis: ((tan (#) arctan) `| Z) . x = ((arctan . x) / ((cos . x) ^2)) + ((tan . x) / (1 + (x ^2))) then A7: cos . x <> 0 by A4, FDIFF_8:1; ((tan (#) arctan) `| Z) . x = ((arctan . x) * (diff (tan,x))) + ((tan . x) * (diff (arctan,x))) by A1, A5, A3, A6, FDIFF_1:21 .= ((arctan . x) * (1 / ((cos . x) ^2))) + ((tan . x) * (diff (arctan,x))) by A7, FDIFF_7:46 .= ((arctan . x) / ((cos . x) ^2)) + ((tan . x) * ((arctan `| Z) . x)) by A3, A6, FDIFF_1:def_7 .= ((arctan . x) / ((cos . x) ^2)) + ((tan . x) * (1 / (1 + (x ^2)))) by A2, A6, SIN_COS9:81 .= ((arctan . x) / ((cos . x) ^2)) + ((tan . x) / (1 + (x ^2))) ; hence ((tan (#) arctan) `| Z) . x = ((arctan . x) / ((cos . x) ^2)) + ((tan . x) / (1 + (x ^2))) ; ::_thesis: verum end; hence ( tan (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((tan (#) arctan) `| Z) . x = ((arctan . x) / ((cos . x) ^2)) + ((tan . x) / (1 + (x ^2))) ) ) by A1, A5, A3, FDIFF_1:21; ::_thesis: verum end; theorem :: FDIFF_11:30 for Z being open Subset of REAL st Z c= dom (tan (#) arccot) & Z c= ].(- 1),1.[ holds ( tan (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((tan (#) arccot) `| Z) . x = ((arccot . x) / ((cos . x) ^2)) - ((tan . x) / (1 + (x ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (tan (#) arccot) & Z c= ].(- 1),1.[ implies ( tan (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((tan (#) arccot) `| Z) . x = ((arccot . x) / ((cos . x) ^2)) - ((tan . x) / (1 + (x ^2))) ) ) ) assume that A1: Z c= dom (tan (#) arccot) and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( tan (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((tan (#) arccot) `| Z) . x = ((arccot . x) / ((cos . x) ^2)) - ((tan . x) / (1 + (x ^2))) ) ) A3: arccot is_differentiable_on Z by A2, SIN_COS9:82; Z c= (dom tan) /\ (dom arccot) by A1, VALUED_1:def_4; then A4: Z c= dom tan by XBOOLE_1:18; for x being Real st x in Z holds tan is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies tan is_differentiable_in x ) assume x in Z ; ::_thesis: tan is_differentiable_in x then cos . x <> 0 by A4, FDIFF_8:1; hence tan is_differentiable_in x by FDIFF_7:46; ::_thesis: verum end; then A5: tan is_differentiable_on Z by A4, FDIFF_1:9; for x being Real st x in Z holds ((tan (#) arccot) `| Z) . x = ((arccot . x) / ((cos . x) ^2)) - ((tan . x) / (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((tan (#) arccot) `| Z) . x = ((arccot . x) / ((cos . x) ^2)) - ((tan . x) / (1 + (x ^2))) ) assume A6: x in Z ; ::_thesis: ((tan (#) arccot) `| Z) . x = ((arccot . x) / ((cos . x) ^2)) - ((tan . x) / (1 + (x ^2))) then A7: cos . x <> 0 by A4, FDIFF_8:1; ((tan (#) arccot) `| Z) . x = ((arccot . x) * (diff (tan,x))) + ((tan . x) * (diff (arccot,x))) by A1, A5, A3, A6, FDIFF_1:21 .= ((arccot . x) * (1 / ((cos . x) ^2))) + ((tan . x) * (diff (arccot,x))) by A7, FDIFF_7:46 .= ((arccot . x) / ((cos . x) ^2)) + ((tan . x) * ((arccot `| Z) . x)) by A3, A6, FDIFF_1:def_7 .= ((arccot . x) / ((cos . x) ^2)) + ((tan . x) * (- (1 / (1 + (x ^2))))) by A2, A6, SIN_COS9:82 .= ((arccot . x) / ((cos . x) ^2)) - ((tan . x) / (1 + (x ^2))) ; hence ((tan (#) arccot) `| Z) . x = ((arccot . x) / ((cos . x) ^2)) - ((tan . x) / (1 + (x ^2))) ; ::_thesis: verum end; hence ( tan (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((tan (#) arccot) `| Z) . x = ((arccot . x) / ((cos . x) ^2)) - ((tan . x) / (1 + (x ^2))) ) ) by A1, A5, A3, FDIFF_1:21; ::_thesis: verum end; theorem :: FDIFF_11:31 for Z being open Subset of REAL st Z c= dom (cot (#) arctan) & Z c= ].(- 1),1.[ holds ( cot (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((cot (#) arctan) `| Z) . x = (- ((arctan . x) / ((sin . x) ^2))) + ((cot . x) / (1 + (x ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (cot (#) arctan) & Z c= ].(- 1),1.[ implies ( cot (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((cot (#) arctan) `| Z) . x = (- ((arctan . x) / ((sin . x) ^2))) + ((cot . x) / (1 + (x ^2))) ) ) ) assume that A1: Z c= dom (cot (#) arctan) and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( cot (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((cot (#) arctan) `| Z) . x = (- ((arctan . x) / ((sin . x) ^2))) + ((cot . x) / (1 + (x ^2))) ) ) A3: arctan is_differentiable_on Z by A2, SIN_COS9:81; Z c= (dom cot) /\ (dom arctan) by A1, VALUED_1:def_4; then A4: Z c= dom cot by XBOOLE_1:18; for x being Real st x in Z holds cot is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies cot is_differentiable_in x ) assume x in Z ; ::_thesis: cot is_differentiable_in x then sin . x <> 0 by A4, FDIFF_8:2; hence cot is_differentiable_in x by FDIFF_7:47; ::_thesis: verum end; then A5: cot is_differentiable_on Z by A4, FDIFF_1:9; for x being Real st x in Z holds ((cot (#) arctan) `| Z) . x = (- ((arctan . x) / ((sin . x) ^2))) + ((cot . x) / (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((cot (#) arctan) `| Z) . x = (- ((arctan . x) / ((sin . x) ^2))) + ((cot . x) / (1 + (x ^2))) ) assume A6: x in Z ; ::_thesis: ((cot (#) arctan) `| Z) . x = (- ((arctan . x) / ((sin . x) ^2))) + ((cot . x) / (1 + (x ^2))) then A7: sin . x <> 0 by A4, FDIFF_8:2; ((cot (#) arctan) `| Z) . x = ((arctan . x) * (diff (cot,x))) + ((cot . x) * (diff (arctan,x))) by A1, A5, A3, A6, FDIFF_1:21 .= ((arctan . x) * (- (1 / ((sin . x) ^2)))) + ((cot . x) * (diff (arctan,x))) by A7, FDIFF_7:47 .= (- ((arctan . x) / ((sin . x) ^2))) + ((cot . x) * ((arctan `| Z) . x)) by A3, A6, FDIFF_1:def_7 .= (- ((arctan . x) / ((sin . x) ^2))) + ((cot . x) * (1 / (1 + (x ^2)))) by A2, A6, SIN_COS9:81 .= (- ((arctan . x) / ((sin . x) ^2))) + ((cot . x) / (1 + (x ^2))) ; hence ((cot (#) arctan) `| Z) . x = (- ((arctan . x) / ((sin . x) ^2))) + ((cot . x) / (1 + (x ^2))) ; ::_thesis: verum end; hence ( cot (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((cot (#) arctan) `| Z) . x = (- ((arctan . x) / ((sin . x) ^2))) + ((cot . x) / (1 + (x ^2))) ) ) by A1, A5, A3, FDIFF_1:21; ::_thesis: verum end; theorem :: FDIFF_11:32 for Z being open Subset of REAL st Z c= dom (cot (#) arccot) & Z c= ].(- 1),1.[ holds ( cot (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((cot (#) arccot) `| Z) . x = (- ((arccot . x) / ((sin . x) ^2))) - ((cot . x) / (1 + (x ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (cot (#) arccot) & Z c= ].(- 1),1.[ implies ( cot (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((cot (#) arccot) `| Z) . x = (- ((arccot . x) / ((sin . x) ^2))) - ((cot . x) / (1 + (x ^2))) ) ) ) assume that A1: Z c= dom (cot (#) arccot) and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( cot (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((cot (#) arccot) `| Z) . x = (- ((arccot . x) / ((sin . x) ^2))) - ((cot . x) / (1 + (x ^2))) ) ) A3: arccot is_differentiable_on Z by A2, SIN_COS9:82; Z c= (dom cot) /\ (dom arccot) by A1, VALUED_1:def_4; then A4: Z c= dom cot by XBOOLE_1:18; for x being Real st x in Z holds cot is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies cot is_differentiable_in x ) assume x in Z ; ::_thesis: cot is_differentiable_in x then sin . x <> 0 by A4, FDIFF_8:2; hence cot is_differentiable_in x by FDIFF_7:47; ::_thesis: verum end; then A5: cot is_differentiable_on Z by A4, FDIFF_1:9; for x being Real st x in Z holds ((cot (#) arccot) `| Z) . x = (- ((arccot . x) / ((sin . x) ^2))) - ((cot . x) / (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((cot (#) arccot) `| Z) . x = (- ((arccot . x) / ((sin . x) ^2))) - ((cot . x) / (1 + (x ^2))) ) assume A6: x in Z ; ::_thesis: ((cot (#) arccot) `| Z) . x = (- ((arccot . x) / ((sin . x) ^2))) - ((cot . x) / (1 + (x ^2))) then A7: sin . x <> 0 by A4, FDIFF_8:2; ((cot (#) arccot) `| Z) . x = ((arccot . x) * (diff (cot,x))) + ((cot . x) * (diff (arccot,x))) by A1, A5, A3, A6, FDIFF_1:21 .= ((arccot . x) * (- (1 / ((sin . x) ^2)))) + ((cot . x) * (diff (arccot,x))) by A7, FDIFF_7:47 .= (- ((arccot . x) / ((sin . x) ^2))) + ((cot . x) * ((arccot `| Z) . x)) by A3, A6, FDIFF_1:def_7 .= (- ((arccot . x) / ((sin . x) ^2))) + ((cot . x) * (- (1 / (1 + (x ^2))))) by A2, A6, SIN_COS9:82 .= (- ((arccot . x) / ((sin . x) ^2))) - ((cot . x) / (1 + (x ^2))) ; hence ((cot (#) arccot) `| Z) . x = (- ((arccot . x) / ((sin . x) ^2))) - ((cot . x) / (1 + (x ^2))) ; ::_thesis: verum end; hence ( cot (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((cot (#) arccot) `| Z) . x = (- ((arccot . x) / ((sin . x) ^2))) - ((cot . x) / (1 + (x ^2))) ) ) by A1, A5, A3, FDIFF_1:21; ::_thesis: verum end; theorem :: FDIFF_11:33 for Z being open Subset of REAL st Z c= dom (sec (#) arctan) & Z c= ].(- 1),1.[ holds ( sec (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((sec (#) arctan) `| Z) . x = (((sin . x) * (arctan . x)) / ((cos . x) ^2)) + (1 / ((cos . x) * (1 + (x ^2)))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (sec (#) arctan) & Z c= ].(- 1),1.[ implies ( sec (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((sec (#) arctan) `| Z) . x = (((sin . x) * (arctan . x)) / ((cos . x) ^2)) + (1 / ((cos . x) * (1 + (x ^2)))) ) ) ) assume that A1: Z c= dom (sec (#) arctan) and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( sec (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((sec (#) arctan) `| Z) . x = (((sin . x) * (arctan . x)) / ((cos . x) ^2)) + (1 / ((cos . x) * (1 + (x ^2)))) ) ) A3: arctan is_differentiable_on Z by A2, SIN_COS9:81; Z c= (dom sec) /\ (dom arctan) by A1, VALUED_1:def_4; then A4: Z c= dom sec by XBOOLE_1:18; for x being Real st x in Z holds sec is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies sec is_differentiable_in x ) assume x in Z ; ::_thesis: sec is_differentiable_in x then cos . x <> 0 by A4, RFUNCT_1:3; hence sec is_differentiable_in x by FDIFF_9:1; ::_thesis: verum end; then A5: sec is_differentiable_on Z by A4, FDIFF_1:9; for x being Real st x in Z holds ((sec (#) arctan) `| Z) . x = (((sin . x) * (arctan . x)) / ((cos . x) ^2)) + (1 / ((cos . x) * (1 + (x ^2)))) proof let x be Real; ::_thesis: ( x in Z implies ((sec (#) arctan) `| Z) . x = (((sin . x) * (arctan . x)) / ((cos . x) ^2)) + (1 / ((cos . x) * (1 + (x ^2)))) ) assume A6: x in Z ; ::_thesis: ((sec (#) arctan) `| Z) . x = (((sin . x) * (arctan . x)) / ((cos . x) ^2)) + (1 / ((cos . x) * (1 + (x ^2)))) then A7: cos . x <> 0 by A4, RFUNCT_1:3; ((sec (#) arctan) `| Z) . x = ((arctan . x) * (diff (sec,x))) + ((sec . x) * (diff (arctan,x))) by A1, A5, A3, A6, FDIFF_1:21 .= ((arctan . x) * ((sin . x) / ((cos . x) ^2))) + ((sec . x) * (diff (arctan,x))) by A7, FDIFF_9:1 .= (((sin . x) * (arctan . x)) / ((cos . x) ^2)) + ((sec . x) * ((arctan `| Z) . x)) by A3, A6, FDIFF_1:def_7 .= (((sin . x) * (arctan . x)) / ((cos . x) ^2)) + ((sec . x) * (1 / (1 + (x ^2)))) by A2, A6, SIN_COS9:81 .= (((sin . x) * (arctan . x)) / ((cos . x) ^2)) + ((1 / (cos . x)) * (1 / (1 + (x ^2)))) by A4, A6, RFUNCT_1:def_2 .= (((sin . x) * (arctan . x)) / ((cos . x) ^2)) + (1 / ((cos . x) * (1 + (x ^2)))) by XCMPLX_1:102 ; hence ((sec (#) arctan) `| Z) . x = (((sin . x) * (arctan . x)) / ((cos . x) ^2)) + (1 / ((cos . x) * (1 + (x ^2)))) ; ::_thesis: verum end; hence ( sec (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((sec (#) arctan) `| Z) . x = (((sin . x) * (arctan . x)) / ((cos . x) ^2)) + (1 / ((cos . x) * (1 + (x ^2)))) ) ) by A1, A5, A3, FDIFF_1:21; ::_thesis: verum end; theorem :: FDIFF_11:34 for Z being open Subset of REAL st Z c= dom (sec (#) arccot) & Z c= ].(- 1),1.[ holds ( sec (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((sec (#) arccot) `| Z) . x = (((sin . x) * (arccot . x)) / ((cos . x) ^2)) - (1 / ((cos . x) * (1 + (x ^2)))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (sec (#) arccot) & Z c= ].(- 1),1.[ implies ( sec (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((sec (#) arccot) `| Z) . x = (((sin . x) * (arccot . x)) / ((cos . x) ^2)) - (1 / ((cos . x) * (1 + (x ^2)))) ) ) ) assume that A1: Z c= dom (sec (#) arccot) and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( sec (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((sec (#) arccot) `| Z) . x = (((sin . x) * (arccot . x)) / ((cos . x) ^2)) - (1 / ((cos . x) * (1 + (x ^2)))) ) ) A3: arccot is_differentiable_on Z by A2, SIN_COS9:82; Z c= (dom sec) /\ (dom arccot) by A1, VALUED_1:def_4; then A4: Z c= dom sec by XBOOLE_1:18; for x being Real st x in Z holds sec is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies sec is_differentiable_in x ) assume x in Z ; ::_thesis: sec is_differentiable_in x then cos . x <> 0 by A4, RFUNCT_1:3; hence sec is_differentiable_in x by FDIFF_9:1; ::_thesis: verum end; then A5: sec is_differentiable_on Z by A4, FDIFF_1:9; for x being Real st x in Z holds ((sec (#) arccot) `| Z) . x = (((sin . x) * (arccot . x)) / ((cos . x) ^2)) - (1 / ((cos . x) * (1 + (x ^2)))) proof let x be Real; ::_thesis: ( x in Z implies ((sec (#) arccot) `| Z) . x = (((sin . x) * (arccot . x)) / ((cos . x) ^2)) - (1 / ((cos . x) * (1 + (x ^2)))) ) assume A6: x in Z ; ::_thesis: ((sec (#) arccot) `| Z) . x = (((sin . x) * (arccot . x)) / ((cos . x) ^2)) - (1 / ((cos . x) * (1 + (x ^2)))) then A7: cos . x <> 0 by A4, RFUNCT_1:3; ((sec (#) arccot) `| Z) . x = ((arccot . x) * (diff (sec,x))) + ((sec . x) * (diff (arccot,x))) by A1, A5, A3, A6, FDIFF_1:21 .= ((arccot . x) * ((sin . x) / ((cos . x) ^2))) + ((sec . x) * (diff (arccot,x))) by A7, FDIFF_9:1 .= (((sin . x) * (arccot . x)) / ((cos . x) ^2)) + ((sec . x) * ((arccot `| Z) . x)) by A3, A6, FDIFF_1:def_7 .= (((sin . x) * (arccot . x)) / ((cos . x) ^2)) + ((sec . x) * (- (1 / (1 + (x ^2))))) by A2, A6, SIN_COS9:82 .= (((sin . x) * (arccot . x)) / ((cos . x) ^2)) - ((sec . x) * (1 / (1 + (x ^2)))) .= (((sin . x) * (arccot . x)) / ((cos . x) ^2)) - ((1 / (cos . x)) * (1 / (1 + (x ^2)))) by A4, A6, RFUNCT_1:def_2 .= (((sin . x) * (arccot . x)) / ((cos . x) ^2)) - (1 / ((cos . x) * (1 + (x ^2)))) by XCMPLX_1:102 ; hence ((sec (#) arccot) `| Z) . x = (((sin . x) * (arccot . x)) / ((cos . x) ^2)) - (1 / ((cos . x) * (1 + (x ^2)))) ; ::_thesis: verum end; hence ( sec (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((sec (#) arccot) `| Z) . x = (((sin . x) * (arccot . x)) / ((cos . x) ^2)) - (1 / ((cos . x) * (1 + (x ^2)))) ) ) by A1, A5, A3, FDIFF_1:21; ::_thesis: verum end; theorem :: FDIFF_11:35 for Z being open Subset of REAL st Z c= dom (cosec (#) arctan) & Z c= ].(- 1),1.[ holds ( cosec (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((cosec (#) arctan) `| Z) . x = (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + (1 / ((sin . x) * (1 + (x ^2)))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (cosec (#) arctan) & Z c= ].(- 1),1.[ implies ( cosec (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((cosec (#) arctan) `| Z) . x = (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + (1 / ((sin . x) * (1 + (x ^2)))) ) ) ) assume that A1: Z c= dom (cosec (#) arctan) and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( cosec (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((cosec (#) arctan) `| Z) . x = (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + (1 / ((sin . x) * (1 + (x ^2)))) ) ) A3: arctan is_differentiable_on Z by A2, SIN_COS9:81; Z c= (dom cosec) /\ (dom arctan) by A1, VALUED_1:def_4; then A4: Z c= dom cosec by XBOOLE_1:18; for x being Real st x in Z holds cosec is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies cosec is_differentiable_in x ) assume x in Z ; ::_thesis: cosec is_differentiable_in x then sin . x <> 0 by A4, RFUNCT_1:3; hence cosec is_differentiable_in x by FDIFF_9:2; ::_thesis: verum end; then A5: cosec is_differentiable_on Z by A4, FDIFF_1:9; for x being Real st x in Z holds ((cosec (#) arctan) `| Z) . x = (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + (1 / ((sin . x) * (1 + (x ^2)))) proof let x be Real; ::_thesis: ( x in Z implies ((cosec (#) arctan) `| Z) . x = (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + (1 / ((sin . x) * (1 + (x ^2)))) ) assume A6: x in Z ; ::_thesis: ((cosec (#) arctan) `| Z) . x = (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + (1 / ((sin . x) * (1 + (x ^2)))) then A7: sin . x <> 0 by A4, RFUNCT_1:3; ((cosec (#) arctan) `| Z) . x = ((arctan . x) * (diff (cosec,x))) + ((cosec . x) * (diff (arctan,x))) by A1, A5, A3, A6, FDIFF_1:21 .= ((arctan . x) * (- ((cos . x) / ((sin . x) ^2)))) + ((cosec . x) * (diff (arctan,x))) by A7, FDIFF_9:2 .= (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + ((cosec . x) * ((arctan `| Z) . x)) by A3, A6, FDIFF_1:def_7 .= (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + ((cosec . x) * (1 / (1 + (x ^2)))) by A2, A6, SIN_COS9:81 .= (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + ((1 / (sin . x)) * (1 / (1 + (x ^2)))) by A4, A6, RFUNCT_1:def_2 .= (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + (1 / ((sin . x) * (1 + (x ^2)))) by XCMPLX_1:102 ; hence ((cosec (#) arctan) `| Z) . x = (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + (1 / ((sin . x) * (1 + (x ^2)))) ; ::_thesis: verum end; hence ( cosec (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds ((cosec (#) arctan) `| Z) . x = (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + (1 / ((sin . x) * (1 + (x ^2)))) ) ) by A1, A5, A3, FDIFF_1:21; ::_thesis: verum end; theorem :: FDIFF_11:36 for Z being open Subset of REAL st Z c= dom (cosec (#) arccot) & Z c= ].(- 1),1.[ holds ( cosec (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((cosec (#) arccot) `| Z) . x = (- (((cos . x) * (arccot . x)) / ((sin . x) ^2))) - (1 / ((sin . x) * (1 + (x ^2)))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (cosec (#) arccot) & Z c= ].(- 1),1.[ implies ( cosec (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((cosec (#) arccot) `| Z) . x = (- (((cos . x) * (arccot . x)) / ((sin . x) ^2))) - (1 / ((sin . x) * (1 + (x ^2)))) ) ) ) assume that A1: Z c= dom (cosec (#) arccot) and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( cosec (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((cosec (#) arccot) `| Z) . x = (- (((cos . x) * (arccot . x)) / ((sin . x) ^2))) - (1 / ((sin . x) * (1 + (x ^2)))) ) ) A3: arccot is_differentiable_on Z by A2, SIN_COS9:82; Z c= (dom cosec) /\ (dom arccot) by A1, VALUED_1:def_4; then A4: Z c= dom cosec by XBOOLE_1:18; for x being Real st x in Z holds cosec is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies cosec is_differentiable_in x ) assume x in Z ; ::_thesis: cosec is_differentiable_in x then sin . x <> 0 by A4, RFUNCT_1:3; hence cosec is_differentiable_in x by FDIFF_9:2; ::_thesis: verum end; then A5: cosec is_differentiable_on Z by A4, FDIFF_1:9; for x being Real st x in Z holds ((cosec (#) arccot) `| Z) . x = (- (((cos . x) * (arccot . x)) / ((sin . x) ^2))) - (1 / ((sin . x) * (1 + (x ^2)))) proof let x be Real; ::_thesis: ( x in Z implies ((cosec (#) arccot) `| Z) . x = (- (((cos . x) * (arccot . x)) / ((sin . x) ^2))) - (1 / ((sin . x) * (1 + (x ^2)))) ) assume A6: x in Z ; ::_thesis: ((cosec (#) arccot) `| Z) . x = (- (((cos . x) * (arccot . x)) / ((sin . x) ^2))) - (1 / ((sin . x) * (1 + (x ^2)))) then A7: sin . x <> 0 by A4, RFUNCT_1:3; ((cosec (#) arccot) `| Z) . x = ((arccot . x) * (diff (cosec,x))) + ((cosec . x) * (diff (arccot,x))) by A1, A5, A3, A6, FDIFF_1:21 .= ((arccot . x) * (- ((cos . x) / ((sin . x) ^2)))) + ((cosec . x) * (diff (arccot,x))) by A7, FDIFF_9:2 .= (- (((cos . x) * (arccot . x)) / ((sin . x) ^2))) + ((cosec . x) * ((arccot `| Z) . x)) by A3, A6, FDIFF_1:def_7 .= (- (((cos . x) * (arccot . x)) / ((sin . x) ^2))) + ((cosec . x) * (- (1 / (1 + (x ^2))))) by A2, A6, SIN_COS9:82 .= (- (((cos . x) * (arccot . x)) / ((sin . x) ^2))) - ((cosec . x) * (1 / (1 + (x ^2)))) .= (- (((cos . x) * (arccot . x)) / ((sin . x) ^2))) - ((1 / (sin . x)) * (1 / (1 + (x ^2)))) by A4, A6, RFUNCT_1:def_2 .= (- (((cos . x) * (arccot . x)) / ((sin . x) ^2))) - (1 / ((sin . x) * (1 + (x ^2)))) by XCMPLX_1:102 ; hence ((cosec (#) arccot) `| Z) . x = (- (((cos . x) * (arccot . x)) / ((sin . x) ^2))) - (1 / ((sin . x) * (1 + (x ^2)))) ; ::_thesis: verum end; hence ( cosec (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((cosec (#) arccot) `| Z) . x = (- (((cos . x) * (arccot . x)) / ((sin . x) ^2))) - (1 / ((sin . x) * (1 + (x ^2)))) ) ) by A1, A5, A3, FDIFF_1:21; ::_thesis: verum end; theorem Th37: :: FDIFF_11:37 for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds ( arctan + arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan + arccot) `| Z) . x = 0 ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= ].(- 1),1.[ implies ( arctan + arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan + arccot) `| Z) . x = 0 ) ) ) assume A1: Z c= ].(- 1),1.[ ; ::_thesis: ( arctan + arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan + arccot) `| Z) . x = 0 ) ) then A2: arctan is_differentiable_on Z by SIN_COS9:81; A3: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25; then ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1; then A4: Z c= dom arccot by A1, XBOOLE_1:1; A5: arccot is_differentiable_on Z by A1, SIN_COS9:82; ].(- 1),1.[ c= dom arctan by A3, SIN_COS9:23, XBOOLE_1:1; then Z c= dom arctan by A1, XBOOLE_1:1; then Z c= (dom arctan) /\ (dom arccot) by A4, XBOOLE_1:19; then A6: Z c= dom (arctan + arccot) by VALUED_1:def_1; for x being Real st x in Z holds ((arctan + arccot) `| Z) . x = 0 proof let x be Real; ::_thesis: ( x in Z implies ((arctan + arccot) `| Z) . x = 0 ) assume A7: x in Z ; ::_thesis: ((arctan + arccot) `| Z) . x = 0 then ((arctan + arccot) `| Z) . x = (diff (arctan,x)) + (diff (arccot,x)) by A6, A2, A5, FDIFF_1:18 .= ((arctan `| Z) . x) + (diff (arccot,x)) by A2, A7, FDIFF_1:def_7 .= (1 / (1 + (x ^2))) + (diff (arccot,x)) by A1, A7, SIN_COS9:81 .= (1 / (1 + (x ^2))) + ((arccot `| Z) . x) by A5, A7, FDIFF_1:def_7 .= (1 / (1 + (x ^2))) + (- (1 / (1 + (x ^2)))) by A1, A7, SIN_COS9:82 .= 0 ; hence ((arctan + arccot) `| Z) . x = 0 ; ::_thesis: verum end; hence ( arctan + arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan + arccot) `| Z) . x = 0 ) ) by A6, A2, A5, FDIFF_1:18; ::_thesis: verum end; theorem Th38: :: FDIFF_11:38 for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds ( arctan - arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan - arccot) `| Z) . x = 2 / (1 + (x ^2)) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= ].(- 1),1.[ implies ( arctan - arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan - arccot) `| Z) . x = 2 / (1 + (x ^2)) ) ) ) assume A1: Z c= ].(- 1),1.[ ; ::_thesis: ( arctan - arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan - arccot) `| Z) . x = 2 / (1 + (x ^2)) ) ) then A2: arctan is_differentiable_on Z by SIN_COS9:81; A3: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25; then ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1; then A4: Z c= dom arccot by A1, XBOOLE_1:1; A5: arccot is_differentiable_on Z by A1, SIN_COS9:82; ].(- 1),1.[ c= dom arctan by A3, SIN_COS9:23, XBOOLE_1:1; then Z c= dom arctan by A1, XBOOLE_1:1; then Z c= (dom arctan) /\ (dom arccot) by A4, XBOOLE_1:19; then A6: Z c= dom (arctan - arccot) by VALUED_1:12; for x being Real st x in Z holds ((arctan - arccot) `| Z) . x = 2 / (1 + (x ^2)) proof let x be Real; ::_thesis: ( x in Z implies ((arctan - arccot) `| Z) . x = 2 / (1 + (x ^2)) ) assume A7: x in Z ; ::_thesis: ((arctan - arccot) `| Z) . x = 2 / (1 + (x ^2)) then ((arctan - arccot) `| Z) . x = (diff (arctan,x)) - (diff (arccot,x)) by A6, A2, A5, FDIFF_1:19 .= ((arctan `| Z) . x) - (diff (arccot,x)) by A2, A7, FDIFF_1:def_7 .= (1 / (1 + (x ^2))) - (diff (arccot,x)) by A1, A7, SIN_COS9:81 .= (1 / (1 + (x ^2))) - ((arccot `| Z) . x) by A5, A7, FDIFF_1:def_7 .= (1 / (1 + (x ^2))) - (- (1 / (1 + (x ^2)))) by A1, A7, SIN_COS9:82 .= 2 / (1 + (x ^2)) ; hence ((arctan - arccot) `| Z) . x = 2 / (1 + (x ^2)) ; ::_thesis: verum end; hence ( arctan - arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan - arccot) `| Z) . x = 2 / (1 + (x ^2)) ) ) by A6, A2, A5, FDIFF_1:19; ::_thesis: verum end; theorem :: FDIFF_11:39 for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds ( sin (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((sin (#) (arctan + arccot)) `| Z) . x = (cos . x) * ((arctan . x) + (arccot . x)) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= ].(- 1),1.[ implies ( sin (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((sin (#) (arctan + arccot)) `| Z) . x = (cos . x) * ((arctan . x) + (arccot . x)) ) ) ) for x being Real st x in Z holds sin is_differentiable_in x by SIN_COS:64; then A1: sin is_differentiable_on Z by FDIFF_1:9, SIN_COS:24; assume A2: Z c= ].(- 1),1.[ ; ::_thesis: ( sin (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((sin (#) (arctan + arccot)) `| Z) . x = (cos . x) * ((arctan . x) + (arccot . x)) ) ) then A3: arctan + arccot is_differentiable_on Z by Th37; A4: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25; then ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1; then A5: Z c= dom arccot by A2, XBOOLE_1:1; ].(- 1),1.[ c= dom arctan by A4, SIN_COS9:23, XBOOLE_1:1; then Z c= dom arctan by A2, XBOOLE_1:1; then Z c= (dom arctan) /\ (dom arccot) by A5, XBOOLE_1:19; then A6: Z c= dom (arctan + arccot) by VALUED_1:def_1; then Z c= (dom sin) /\ (dom (arctan + arccot)) by SIN_COS:24, XBOOLE_1:19; then A7: Z c= dom (sin (#) (arctan + arccot)) by VALUED_1:def_4; for x being Real st x in Z holds ((sin (#) (arctan + arccot)) `| Z) . x = (cos . x) * ((arctan . x) + (arccot . x)) proof let x be Real; ::_thesis: ( x in Z implies ((sin (#) (arctan + arccot)) `| Z) . x = (cos . x) * ((arctan . x) + (arccot . x)) ) assume A8: x in Z ; ::_thesis: ((sin (#) (arctan + arccot)) `| Z) . x = (cos . x) * ((arctan . x) + (arccot . x)) then ((sin (#) (arctan + arccot)) `| Z) . x = (((arctan + arccot) . x) * (diff (sin,x))) + ((sin . x) * (diff ((arctan + arccot),x))) by A7, A1, A3, FDIFF_1:21 .= (((arctan . x) + (arccot . x)) * (diff (sin,x))) + ((sin . x) * (diff ((arctan + arccot),x))) by A6, A8, VALUED_1:def_1 .= (((arctan . x) + (arccot . x)) * (cos . x)) + ((sin . x) * (diff ((arctan + arccot),x))) by SIN_COS:64 .= (((arctan . x) + (arccot . x)) * (cos . x)) + ((sin . x) * (((arctan + arccot) `| Z) . x)) by A3, A8, FDIFF_1:def_7 .= (((arctan . x) + (arccot . x)) * (cos . x)) + ((sin . x) * 0) by A2, A8, Th37 .= (cos . x) * ((arctan . x) + (arccot . x)) ; hence ((sin (#) (arctan + arccot)) `| Z) . x = (cos . x) * ((arctan . x) + (arccot . x)) ; ::_thesis: verum end; hence ( sin (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((sin (#) (arctan + arccot)) `| Z) . x = (cos . x) * ((arctan . x) + (arccot . x)) ) ) by A7, A1, A3, FDIFF_1:21; ::_thesis: verum end; theorem :: FDIFF_11:40 for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds ( sin (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((sin (#) (arctan - arccot)) `| Z) . x = ((cos . x) * ((arctan . x) - (arccot . x))) + ((2 * (sin . x)) / (1 + (x ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= ].(- 1),1.[ implies ( sin (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((sin (#) (arctan - arccot)) `| Z) . x = ((cos . x) * ((arctan . x) - (arccot . x))) + ((2 * (sin . x)) / (1 + (x ^2))) ) ) ) for x being Real st x in Z holds sin is_differentiable_in x by SIN_COS:64; then A1: sin is_differentiable_on Z by FDIFF_1:9, SIN_COS:24; assume A2: Z c= ].(- 1),1.[ ; ::_thesis: ( sin (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((sin (#) (arctan - arccot)) `| Z) . x = ((cos . x) * ((arctan . x) - (arccot . x))) + ((2 * (sin . x)) / (1 + (x ^2))) ) ) then A3: arctan - arccot is_differentiable_on Z by Th38; A4: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25; then ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1; then A5: Z c= dom arccot by A2, XBOOLE_1:1; ].(- 1),1.[ c= dom arctan by A4, SIN_COS9:23, XBOOLE_1:1; then Z c= dom arctan by A2, XBOOLE_1:1; then Z c= (dom arctan) /\ (dom arccot) by A5, XBOOLE_1:19; then A6: Z c= dom (arctan - arccot) by VALUED_1:12; then Z c= (dom sin) /\ (dom (arctan - arccot)) by SIN_COS:24, XBOOLE_1:19; then A7: Z c= dom (sin (#) (arctan - arccot)) by VALUED_1:def_4; for x being Real st x in Z holds ((sin (#) (arctan - arccot)) `| Z) . x = ((cos . x) * ((arctan . x) - (arccot . x))) + ((2 * (sin . x)) / (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((sin (#) (arctan - arccot)) `| Z) . x = ((cos . x) * ((arctan . x) - (arccot . x))) + ((2 * (sin . x)) / (1 + (x ^2))) ) assume A8: x in Z ; ::_thesis: ((sin (#) (arctan - arccot)) `| Z) . x = ((cos . x) * ((arctan . x) - (arccot . x))) + ((2 * (sin . x)) / (1 + (x ^2))) then ((sin (#) (arctan - arccot)) `| Z) . x = (((arctan - arccot) . x) * (diff (sin,x))) + ((sin . x) * (diff ((arctan - arccot),x))) by A7, A1, A3, FDIFF_1:21 .= (((arctan . x) - (arccot . x)) * (diff (sin,x))) + ((sin . x) * (diff ((arctan - arccot),x))) by A6, A8, VALUED_1:13 .= (((arctan . x) - (arccot . x)) * (cos . x)) + ((sin . x) * (diff ((arctan - arccot),x))) by SIN_COS:64 .= (((arctan . x) - (arccot . x)) * (cos . x)) + ((sin . x) * (((arctan - arccot) `| Z) . x)) by A3, A8, FDIFF_1:def_7 .= (((arctan . x) - (arccot . x)) * (cos . x)) + ((sin . x) * (2 / (1 + (x ^2)))) by A2, A8, Th38 .= ((cos . x) * ((arctan . x) - (arccot . x))) + ((2 * (sin . x)) / (1 + (x ^2))) ; hence ((sin (#) (arctan - arccot)) `| Z) . x = ((cos . x) * ((arctan . x) - (arccot . x))) + ((2 * (sin . x)) / (1 + (x ^2))) ; ::_thesis: verum end; hence ( sin (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((sin (#) (arctan - arccot)) `| Z) . x = ((cos . x) * ((arctan . x) - (arccot . x))) + ((2 * (sin . x)) / (1 + (x ^2))) ) ) by A7, A1, A3, FDIFF_1:21; ::_thesis: verum end; theorem :: FDIFF_11:41 for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds ( cos (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((cos (#) (arctan + arccot)) `| Z) . x = - ((sin . x) * ((arctan . x) + (arccot . x))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= ].(- 1),1.[ implies ( cos (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((cos (#) (arctan + arccot)) `| Z) . x = - ((sin . x) * ((arctan . x) + (arccot . x))) ) ) ) for x being Real st x in Z holds cos is_differentiable_in x by SIN_COS:63; then A1: cos is_differentiable_on Z by FDIFF_1:9, SIN_COS:24; assume A2: Z c= ].(- 1),1.[ ; ::_thesis: ( cos (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((cos (#) (arctan + arccot)) `| Z) . x = - ((sin . x) * ((arctan . x) + (arccot . x))) ) ) then A3: arctan + arccot is_differentiable_on Z by Th37; A4: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25; then ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1; then A5: Z c= dom arccot by A2, XBOOLE_1:1; ].(- 1),1.[ c= dom arctan by A4, SIN_COS9:23, XBOOLE_1:1; then Z c= dom arctan by A2, XBOOLE_1:1; then Z c= (dom arctan) /\ (dom arccot) by A5, XBOOLE_1:19; then A6: Z c= dom (arctan + arccot) by VALUED_1:def_1; then Z c= (dom cos) /\ (dom (arctan + arccot)) by SIN_COS:24, XBOOLE_1:19; then A7: Z c= dom (cos (#) (arctan + arccot)) by VALUED_1:def_4; for x being Real st x in Z holds ((cos (#) (arctan + arccot)) `| Z) . x = - ((sin . x) * ((arctan . x) + (arccot . x))) proof let x be Real; ::_thesis: ( x in Z implies ((cos (#) (arctan + arccot)) `| Z) . x = - ((sin . x) * ((arctan . x) + (arccot . x))) ) assume A8: x in Z ; ::_thesis: ((cos (#) (arctan + arccot)) `| Z) . x = - ((sin . x) * ((arctan . x) + (arccot . x))) then ((cos (#) (arctan + arccot)) `| Z) . x = (((arctan + arccot) . x) * (diff (cos,x))) + ((cos . x) * (diff ((arctan + arccot),x))) by A7, A1, A3, FDIFF_1:21 .= (((arctan . x) + (arccot . x)) * (diff (cos,x))) + ((cos . x) * (diff ((arctan + arccot),x))) by A6, A8, VALUED_1:def_1 .= (((arctan . x) + (arccot . x)) * (- (sin . x))) + ((cos . x) * (diff ((arctan + arccot),x))) by SIN_COS:63 .= (((arctan . x) + (arccot . x)) * (- (sin . x))) + ((cos . x) * (((arctan + arccot) `| Z) . x)) by A3, A8, FDIFF_1:def_7 .= (- (((arctan . x) + (arccot . x)) * (sin . x))) + ((cos . x) * 0) by A2, A8, Th37 .= - ((sin . x) * ((arctan . x) + (arccot . x))) ; hence ((cos (#) (arctan + arccot)) `| Z) . x = - ((sin . x) * ((arctan . x) + (arccot . x))) ; ::_thesis: verum end; hence ( cos (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((cos (#) (arctan + arccot)) `| Z) . x = - ((sin . x) * ((arctan . x) + (arccot . x))) ) ) by A7, A1, A3, FDIFF_1:21; ::_thesis: verum end; theorem :: FDIFF_11:42 for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds ( cos (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((cos (#) (arctan - arccot)) `| Z) . x = (- ((sin . x) * ((arctan . x) - (arccot . x)))) + ((2 * (cos . x)) / (1 + (x ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= ].(- 1),1.[ implies ( cos (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((cos (#) (arctan - arccot)) `| Z) . x = (- ((sin . x) * ((arctan . x) - (arccot . x)))) + ((2 * (cos . x)) / (1 + (x ^2))) ) ) ) for x being Real st x in Z holds cos is_differentiable_in x by SIN_COS:63; then A1: cos is_differentiable_on Z by FDIFF_1:9, SIN_COS:24; assume A2: Z c= ].(- 1),1.[ ; ::_thesis: ( cos (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((cos (#) (arctan - arccot)) `| Z) . x = (- ((sin . x) * ((arctan . x) - (arccot . x)))) + ((2 * (cos . x)) / (1 + (x ^2))) ) ) then A3: arctan - arccot is_differentiable_on Z by Th38; A4: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25; then ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1; then A5: Z c= dom arccot by A2, XBOOLE_1:1; ].(- 1),1.[ c= dom arctan by A4, SIN_COS9:23, XBOOLE_1:1; then Z c= dom arctan by A2, XBOOLE_1:1; then Z c= (dom arctan) /\ (dom arccot) by A5, XBOOLE_1:19; then A6: Z c= dom (arctan - arccot) by VALUED_1:12; then Z c= (dom cos) /\ (dom (arctan - arccot)) by SIN_COS:24, XBOOLE_1:19; then A7: Z c= dom (cos (#) (arctan - arccot)) by VALUED_1:def_4; for x being Real st x in Z holds ((cos (#) (arctan - arccot)) `| Z) . x = (- ((sin . x) * ((arctan . x) - (arccot . x)))) + ((2 * (cos . x)) / (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((cos (#) (arctan - arccot)) `| Z) . x = (- ((sin . x) * ((arctan . x) - (arccot . x)))) + ((2 * (cos . x)) / (1 + (x ^2))) ) assume A8: x in Z ; ::_thesis: ((cos (#) (arctan - arccot)) `| Z) . x = (- ((sin . x) * ((arctan . x) - (arccot . x)))) + ((2 * (cos . x)) / (1 + (x ^2))) then ((cos (#) (arctan - arccot)) `| Z) . x = (((arctan - arccot) . x) * (diff (cos,x))) + ((cos . x) * (diff ((arctan - arccot),x))) by A7, A1, A3, FDIFF_1:21 .= (((arctan . x) - (arccot . x)) * (diff (cos,x))) + ((cos . x) * (diff ((arctan - arccot),x))) by A6, A8, VALUED_1:13 .= (((arctan . x) - (arccot . x)) * (- (sin . x))) + ((cos . x) * (diff ((arctan - arccot),x))) by SIN_COS:63 .= (((arctan . x) - (arccot . x)) * (- (sin . x))) + ((cos . x) * (((arctan - arccot) `| Z) . x)) by A3, A8, FDIFF_1:def_7 .= (- (((arctan . x) - (arccot . x)) * (sin . x))) + ((cos . x) * (2 / (1 + (x ^2)))) by A2, A8, Th38 .= (- ((sin . x) * ((arctan . x) - (arccot . x)))) + ((2 * (cos . x)) / (1 + (x ^2))) ; hence ((cos (#) (arctan - arccot)) `| Z) . x = (- ((sin . x) * ((arctan . x) - (arccot . x)))) + ((2 * (cos . x)) / (1 + (x ^2))) ; ::_thesis: verum end; hence ( cos (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((cos (#) (arctan - arccot)) `| Z) . x = (- ((sin . x) * ((arctan . x) - (arccot . x)))) + ((2 * (cos . x)) / (1 + (x ^2))) ) ) by A7, A1, A3, FDIFF_1:21; ::_thesis: verum end; theorem :: FDIFF_11:43 for Z being open Subset of REAL st Z c= dom tan & Z c= ].(- 1),1.[ holds ( tan (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((tan (#) (arctan + arccot)) `| Z) . x = ((arctan . x) + (arccot . x)) / ((cos . x) ^2) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom tan & Z c= ].(- 1),1.[ implies ( tan (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((tan (#) (arctan + arccot)) `| Z) . x = ((arctan . x) + (arccot . x)) / ((cos . x) ^2) ) ) ) assume that A1: Z c= dom tan and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( tan (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((tan (#) (arctan + arccot)) `| Z) . x = ((arctan . x) + (arccot . x)) / ((cos . x) ^2) ) ) A3: arctan + arccot is_differentiable_on Z by A2, Th37; for x being Real st x in Z holds tan is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies tan is_differentiable_in x ) assume x in Z ; ::_thesis: tan is_differentiable_in x then cos . x <> 0 by A1, FDIFF_8:1; hence tan is_differentiable_in x by FDIFF_7:46; ::_thesis: verum end; then A4: tan is_differentiable_on Z by A1, FDIFF_1:9; A5: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25; then ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1; then A6: Z c= dom arccot by A2, XBOOLE_1:1; ].(- 1),1.[ c= dom arctan by A5, SIN_COS9:23, XBOOLE_1:1; then Z c= dom arctan by A2, XBOOLE_1:1; then Z c= (dom arctan) /\ (dom arccot) by A6, XBOOLE_1:19; then A7: Z c= dom (arctan + arccot) by VALUED_1:def_1; then Z c= (dom tan) /\ (dom (arctan + arccot)) by A1, XBOOLE_1:19; then A8: Z c= dom (tan (#) (arctan + arccot)) by VALUED_1:def_4; for x being Real st x in Z holds ((tan (#) (arctan + arccot)) `| Z) . x = ((arctan . x) + (arccot . x)) / ((cos . x) ^2) proof let x be Real; ::_thesis: ( x in Z implies ((tan (#) (arctan + arccot)) `| Z) . x = ((arctan . x) + (arccot . x)) / ((cos . x) ^2) ) assume A9: x in Z ; ::_thesis: ((tan (#) (arctan + arccot)) `| Z) . x = ((arctan . x) + (arccot . x)) / ((cos . x) ^2) then A10: cos . x <> 0 by A1, FDIFF_8:1; ((tan (#) (arctan + arccot)) `| Z) . x = (((arctan + arccot) . x) * (diff (tan,x))) + ((tan . x) * (diff ((arctan + arccot),x))) by A8, A4, A3, A9, FDIFF_1:21 .= (((arctan . x) + (arccot . x)) * (diff (tan,x))) + ((tan . x) * (diff ((arctan + arccot),x))) by A7, A9, VALUED_1:def_1 .= (((arctan . x) + (arccot . x)) * (1 / ((cos . x) ^2))) + ((tan . x) * (diff ((arctan + arccot),x))) by A10, FDIFF_7:46 .= (((arctan . x) + (arccot . x)) / ((cos . x) ^2)) + ((tan . x) * (((arctan + arccot) `| Z) . x)) by A3, A9, FDIFF_1:def_7 .= (((arctan . x) + (arccot . x)) / ((cos . x) ^2)) + ((tan . x) * 0) by A2, A9, Th37 .= ((arctan . x) + (arccot . x)) / ((cos . x) ^2) ; hence ((tan (#) (arctan + arccot)) `| Z) . x = ((arctan . x) + (arccot . x)) / ((cos . x) ^2) ; ::_thesis: verum end; hence ( tan (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((tan (#) (arctan + arccot)) `| Z) . x = ((arctan . x) + (arccot . x)) / ((cos . x) ^2) ) ) by A8, A4, A3, FDIFF_1:21; ::_thesis: verum end; theorem :: FDIFF_11:44 for Z being open Subset of REAL st Z c= dom tan & Z c= ].(- 1),1.[ holds ( tan (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((tan (#) (arctan - arccot)) `| Z) . x = (((arctan . x) - (arccot . x)) / ((cos . x) ^2)) + ((2 * (tan . x)) / (1 + (x ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom tan & Z c= ].(- 1),1.[ implies ( tan (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((tan (#) (arctan - arccot)) `| Z) . x = (((arctan . x) - (arccot . x)) / ((cos . x) ^2)) + ((2 * (tan . x)) / (1 + (x ^2))) ) ) ) assume that A1: Z c= dom tan and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( tan (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((tan (#) (arctan - arccot)) `| Z) . x = (((arctan . x) - (arccot . x)) / ((cos . x) ^2)) + ((2 * (tan . x)) / (1 + (x ^2))) ) ) A3: arctan - arccot is_differentiable_on Z by A2, Th38; for x being Real st x in Z holds tan is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies tan is_differentiable_in x ) assume x in Z ; ::_thesis: tan is_differentiable_in x then cos . x <> 0 by A1, FDIFF_8:1; hence tan is_differentiable_in x by FDIFF_7:46; ::_thesis: verum end; then A4: tan is_differentiable_on Z by A1, FDIFF_1:9; A5: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25; then ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1; then A6: Z c= dom arccot by A2, XBOOLE_1:1; ].(- 1),1.[ c= dom arctan by A5, SIN_COS9:23, XBOOLE_1:1; then Z c= dom arctan by A2, XBOOLE_1:1; then Z c= (dom arctan) /\ (dom arccot) by A6, XBOOLE_1:19; then A7: Z c= dom (arctan - arccot) by VALUED_1:12; then Z c= (dom tan) /\ (dom (arctan - arccot)) by A1, XBOOLE_1:19; then A8: Z c= dom (tan (#) (arctan - arccot)) by VALUED_1:def_4; for x being Real st x in Z holds ((tan (#) (arctan - arccot)) `| Z) . x = (((arctan . x) - (arccot . x)) / ((cos . x) ^2)) + ((2 * (tan . x)) / (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((tan (#) (arctan - arccot)) `| Z) . x = (((arctan . x) - (arccot . x)) / ((cos . x) ^2)) + ((2 * (tan . x)) / (1 + (x ^2))) ) assume A9: x in Z ; ::_thesis: ((tan (#) (arctan - arccot)) `| Z) . x = (((arctan . x) - (arccot . x)) / ((cos . x) ^2)) + ((2 * (tan . x)) / (1 + (x ^2))) then A10: cos . x <> 0 by A1, FDIFF_8:1; ((tan (#) (arctan - arccot)) `| Z) . x = (((arctan - arccot) . x) * (diff (tan,x))) + ((tan . x) * (diff ((arctan - arccot),x))) by A8, A4, A3, A9, FDIFF_1:21 .= (((arctan . x) - (arccot . x)) * (diff (tan,x))) + ((tan . x) * (diff ((arctan - arccot),x))) by A7, A9, VALUED_1:13 .= (((arctan . x) - (arccot . x)) * (1 / ((cos . x) ^2))) + ((tan . x) * (diff ((arctan - arccot),x))) by A10, FDIFF_7:46 .= (((arctan . x) - (arccot . x)) / ((cos . x) ^2)) + ((tan . x) * (((arctan - arccot) `| Z) . x)) by A3, A9, FDIFF_1:def_7 .= (((arctan . x) - (arccot . x)) / ((cos . x) ^2)) + ((tan . x) * (2 / (1 + (x ^2)))) by A2, A9, Th38 .= (((arctan . x) - (arccot . x)) / ((cos . x) ^2)) + ((2 * (tan . x)) / (1 + (x ^2))) ; hence ((tan (#) (arctan - arccot)) `| Z) . x = (((arctan . x) - (arccot . x)) / ((cos . x) ^2)) + ((2 * (tan . x)) / (1 + (x ^2))) ; ::_thesis: verum end; hence ( tan (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((tan (#) (arctan - arccot)) `| Z) . x = (((arctan . x) - (arccot . x)) / ((cos . x) ^2)) + ((2 * (tan . x)) / (1 + (x ^2))) ) ) by A8, A4, A3, FDIFF_1:21; ::_thesis: verum end; theorem :: FDIFF_11:45 for Z being open Subset of REAL st Z c= dom cot & Z c= ].(- 1),1.[ holds ( cot (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((cot (#) (arctan + arccot)) `| Z) . x = - (((arctan . x) + (arccot . x)) / ((sin . x) ^2)) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom cot & Z c= ].(- 1),1.[ implies ( cot (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((cot (#) (arctan + arccot)) `| Z) . x = - (((arctan . x) + (arccot . x)) / ((sin . x) ^2)) ) ) ) assume that A1: Z c= dom cot and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( cot (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((cot (#) (arctan + arccot)) `| Z) . x = - (((arctan . x) + (arccot . x)) / ((sin . x) ^2)) ) ) A3: arctan + arccot is_differentiable_on Z by A2, Th37; for x being Real st x in Z holds cot is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies cot is_differentiable_in x ) assume x in Z ; ::_thesis: cot is_differentiable_in x then sin . x <> 0 by A1, FDIFF_8:2; hence cot is_differentiable_in x by FDIFF_7:47; ::_thesis: verum end; then A4: cot is_differentiable_on Z by A1, FDIFF_1:9; A5: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25; then ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1; then A6: Z c= dom arccot by A2, XBOOLE_1:1; ].(- 1),1.[ c= dom arctan by A5, SIN_COS9:23, XBOOLE_1:1; then Z c= dom arctan by A2, XBOOLE_1:1; then Z c= (dom arctan) /\ (dom arccot) by A6, XBOOLE_1:19; then A7: Z c= dom (arctan + arccot) by VALUED_1:def_1; then Z c= (dom cot) /\ (dom (arctan + arccot)) by A1, XBOOLE_1:19; then A8: Z c= dom (cot (#) (arctan + arccot)) by VALUED_1:def_4; for x being Real st x in Z holds ((cot (#) (arctan + arccot)) `| Z) . x = - (((arctan . x) + (arccot . x)) / ((sin . x) ^2)) proof let x be Real; ::_thesis: ( x in Z implies ((cot (#) (arctan + arccot)) `| Z) . x = - (((arctan . x) + (arccot . x)) / ((sin . x) ^2)) ) assume A9: x in Z ; ::_thesis: ((cot (#) (arctan + arccot)) `| Z) . x = - (((arctan . x) + (arccot . x)) / ((sin . x) ^2)) then A10: sin . x <> 0 by A1, FDIFF_8:2; ((cot (#) (arctan + arccot)) `| Z) . x = (((arctan + arccot) . x) * (diff (cot,x))) + ((cot . x) * (diff ((arctan + arccot),x))) by A8, A4, A3, A9, FDIFF_1:21 .= (((arctan . x) + (arccot . x)) * (diff (cot,x))) + ((cot . x) * (diff ((arctan + arccot),x))) by A7, A9, VALUED_1:def_1 .= (((arctan . x) + (arccot . x)) * (- (1 / ((sin . x) ^2)))) + ((cot . x) * (diff ((arctan + arccot),x))) by A10, FDIFF_7:47 .= (- (((arctan . x) + (arccot . x)) / ((sin . x) ^2))) + ((cot . x) * (((arctan + arccot) `| Z) . x)) by A3, A9, FDIFF_1:def_7 .= (- (((arctan . x) + (arccot . x)) / ((sin . x) ^2))) + ((cot . x) * 0) by A2, A9, Th37 .= - (((arctan . x) + (arccot . x)) / ((sin . x) ^2)) ; hence ((cot (#) (arctan + arccot)) `| Z) . x = - (((arctan . x) + (arccot . x)) / ((sin . x) ^2)) ; ::_thesis: verum end; hence ( cot (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((cot (#) (arctan + arccot)) `| Z) . x = - (((arctan . x) + (arccot . x)) / ((sin . x) ^2)) ) ) by A8, A4, A3, FDIFF_1:21; ::_thesis: verum end; theorem :: FDIFF_11:46 for Z being open Subset of REAL st Z c= dom cot & Z c= ].(- 1),1.[ holds ( cot (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((cot (#) (arctan - arccot)) `| Z) . x = (- (((arctan . x) - (arccot . x)) / ((sin . x) ^2))) + ((2 * (cot . x)) / (1 + (x ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom cot & Z c= ].(- 1),1.[ implies ( cot (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((cot (#) (arctan - arccot)) `| Z) . x = (- (((arctan . x) - (arccot . x)) / ((sin . x) ^2))) + ((2 * (cot . x)) / (1 + (x ^2))) ) ) ) assume that A1: Z c= dom cot and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( cot (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((cot (#) (arctan - arccot)) `| Z) . x = (- (((arctan . x) - (arccot . x)) / ((sin . x) ^2))) + ((2 * (cot . x)) / (1 + (x ^2))) ) ) A3: arctan - arccot is_differentiable_on Z by A2, Th38; for x being Real st x in Z holds cot is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies cot is_differentiable_in x ) assume x in Z ; ::_thesis: cot is_differentiable_in x then sin . x <> 0 by A1, FDIFF_8:2; hence cot is_differentiable_in x by FDIFF_7:47; ::_thesis: verum end; then A4: cot is_differentiable_on Z by A1, FDIFF_1:9; A5: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25; then ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1; then A6: Z c= dom arccot by A2, XBOOLE_1:1; ].(- 1),1.[ c= dom arctan by A5, SIN_COS9:23, XBOOLE_1:1; then Z c= dom arctan by A2, XBOOLE_1:1; then Z c= (dom arctan) /\ (dom arccot) by A6, XBOOLE_1:19; then A7: Z c= dom (arctan - arccot) by VALUED_1:12; then Z c= (dom cot) /\ (dom (arctan - arccot)) by A1, XBOOLE_1:19; then A8: Z c= dom (cot (#) (arctan - arccot)) by VALUED_1:def_4; for x being Real st x in Z holds ((cot (#) (arctan - arccot)) `| Z) . x = (- (((arctan . x) - (arccot . x)) / ((sin . x) ^2))) + ((2 * (cot . x)) / (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((cot (#) (arctan - arccot)) `| Z) . x = (- (((arctan . x) - (arccot . x)) / ((sin . x) ^2))) + ((2 * (cot . x)) / (1 + (x ^2))) ) assume A9: x in Z ; ::_thesis: ((cot (#) (arctan - arccot)) `| Z) . x = (- (((arctan . x) - (arccot . x)) / ((sin . x) ^2))) + ((2 * (cot . x)) / (1 + (x ^2))) then A10: sin . x <> 0 by A1, FDIFF_8:2; ((cot (#) (arctan - arccot)) `| Z) . x = (((arctan - arccot) . x) * (diff (cot,x))) + ((cot . x) * (diff ((arctan - arccot),x))) by A8, A4, A3, A9, FDIFF_1:21 .= (((arctan . x) - (arccot . x)) * (diff (cot,x))) + ((cot . x) * (diff ((arctan - arccot),x))) by A7, A9, VALUED_1:13 .= (((arctan . x) - (arccot . x)) * (- (1 / ((sin . x) ^2)))) + ((cot . x) * (diff ((arctan - arccot),x))) by A10, FDIFF_7:47 .= (- (((arctan . x) - (arccot . x)) / ((sin . x) ^2))) + ((cot . x) * (((arctan - arccot) `| Z) . x)) by A3, A9, FDIFF_1:def_7 .= (- (((arctan . x) - (arccot . x)) / ((sin . x) ^2))) + ((cot . x) * (2 / (1 + (x ^2)))) by A2, A9, Th38 .= (- (((arctan . x) - (arccot . x)) / ((sin . x) ^2))) + ((2 * (cot . x)) / (1 + (x ^2))) ; hence ((cot (#) (arctan - arccot)) `| Z) . x = (- (((arctan . x) - (arccot . x)) / ((sin . x) ^2))) + ((2 * (cot . x)) / (1 + (x ^2))) ; ::_thesis: verum end; hence ( cot (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((cot (#) (arctan - arccot)) `| Z) . x = (- (((arctan . x) - (arccot . x)) / ((sin . x) ^2))) + ((2 * (cot . x)) / (1 + (x ^2))) ) ) by A8, A4, A3, FDIFF_1:21; ::_thesis: verum end; theorem :: FDIFF_11:47 for Z being open Subset of REAL st Z c= dom sec & Z c= ].(- 1),1.[ holds ( sec (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((sec (#) (arctan + arccot)) `| Z) . x = (((arctan . x) + (arccot . x)) * (sin . x)) / ((cos . x) ^2) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom sec & Z c= ].(- 1),1.[ implies ( sec (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((sec (#) (arctan + arccot)) `| Z) . x = (((arctan . x) + (arccot . x)) * (sin . x)) / ((cos . x) ^2) ) ) ) assume that A1: Z c= dom sec and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( sec (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((sec (#) (arctan + arccot)) `| Z) . x = (((arctan . x) + (arccot . x)) * (sin . x)) / ((cos . x) ^2) ) ) A3: arctan + arccot is_differentiable_on Z by A2, Th37; for x being Real st x in Z holds sec is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies sec is_differentiable_in x ) assume x in Z ; ::_thesis: sec is_differentiable_in x then cos . x <> 0 by A1, RFUNCT_1:3; hence sec is_differentiable_in x by FDIFF_9:1; ::_thesis: verum end; then A4: sec is_differentiable_on Z by A1, FDIFF_1:9; A5: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25; then ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1; then A6: Z c= dom arccot by A2, XBOOLE_1:1; ].(- 1),1.[ c= dom arctan by A5, SIN_COS9:23, XBOOLE_1:1; then Z c= dom arctan by A2, XBOOLE_1:1; then Z c= (dom arctan) /\ (dom arccot) by A6, XBOOLE_1:19; then A7: Z c= dom (arctan + arccot) by VALUED_1:def_1; then Z c= (dom sec) /\ (dom (arctan + arccot)) by A1, XBOOLE_1:19; then A8: Z c= dom (sec (#) (arctan + arccot)) by VALUED_1:def_4; for x being Real st x in Z holds ((sec (#) (arctan + arccot)) `| Z) . x = (((arctan . x) + (arccot . x)) * (sin . x)) / ((cos . x) ^2) proof let x be Real; ::_thesis: ( x in Z implies ((sec (#) (arctan + arccot)) `| Z) . x = (((arctan . x) + (arccot . x)) * (sin . x)) / ((cos . x) ^2) ) assume A9: x in Z ; ::_thesis: ((sec (#) (arctan + arccot)) `| Z) . x = (((arctan . x) + (arccot . x)) * (sin . x)) / ((cos . x) ^2) then A10: cos . x <> 0 by A1, RFUNCT_1:3; ((sec (#) (arctan + arccot)) `| Z) . x = (((arctan + arccot) . x) * (diff (sec,x))) + ((sec . x) * (diff ((arctan + arccot),x))) by A8, A4, A3, A9, FDIFF_1:21 .= (((arctan . x) + (arccot . x)) * (diff (sec,x))) + ((sec . x) * (diff ((arctan + arccot),x))) by A7, A9, VALUED_1:def_1 .= (((arctan . x) + (arccot . x)) * ((sin . x) / ((cos . x) ^2))) + ((sec . x) * (diff ((arctan + arccot),x))) by A10, FDIFF_9:1 .= ((((arctan . x) + (arccot . x)) * (sin . x)) / ((cos . x) ^2)) + ((sec . x) * (((arctan + arccot) `| Z) . x)) by A3, A9, FDIFF_1:def_7 .= ((((arctan . x) + (arccot . x)) * (sin . x)) / ((cos . x) ^2)) + ((sec . x) * 0) by A2, A9, Th37 .= (((arctan . x) + (arccot . x)) * (sin . x)) / ((cos . x) ^2) ; hence ((sec (#) (arctan + arccot)) `| Z) . x = (((arctan . x) + (arccot . x)) * (sin . x)) / ((cos . x) ^2) ; ::_thesis: verum end; hence ( sec (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((sec (#) (arctan + arccot)) `| Z) . x = (((arctan . x) + (arccot . x)) * (sin . x)) / ((cos . x) ^2) ) ) by A8, A4, A3, FDIFF_1:21; ::_thesis: verum end; theorem :: FDIFF_11:48 for Z being open Subset of REAL st Z c= dom sec & Z c= ].(- 1),1.[ holds ( sec (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((sec (#) (arctan - arccot)) `| Z) . x = ((((arctan . x) - (arccot . x)) * (sin . x)) / ((cos . x) ^2)) + ((2 * (sec . x)) / (1 + (x ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom sec & Z c= ].(- 1),1.[ implies ( sec (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((sec (#) (arctan - arccot)) `| Z) . x = ((((arctan . x) - (arccot . x)) * (sin . x)) / ((cos . x) ^2)) + ((2 * (sec . x)) / (1 + (x ^2))) ) ) ) assume that A1: Z c= dom sec and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( sec (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((sec (#) (arctan - arccot)) `| Z) . x = ((((arctan . x) - (arccot . x)) * (sin . x)) / ((cos . x) ^2)) + ((2 * (sec . x)) / (1 + (x ^2))) ) ) A3: arctan - arccot is_differentiable_on Z by A2, Th38; for x being Real st x in Z holds sec is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies sec is_differentiable_in x ) assume x in Z ; ::_thesis: sec is_differentiable_in x then cos . x <> 0 by A1, RFUNCT_1:3; hence sec is_differentiable_in x by FDIFF_9:1; ::_thesis: verum end; then A4: sec is_differentiable_on Z by A1, FDIFF_1:9; A5: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25; then ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1; then A6: Z c= dom arccot by A2, XBOOLE_1:1; ].(- 1),1.[ c= dom arctan by A5, SIN_COS9:23, XBOOLE_1:1; then Z c= dom arctan by A2, XBOOLE_1:1; then Z c= (dom arctan) /\ (dom arccot) by A6, XBOOLE_1:19; then A7: Z c= dom (arctan - arccot) by VALUED_1:12; then Z c= (dom sec) /\ (dom (arctan - arccot)) by A1, XBOOLE_1:19; then A8: Z c= dom (sec (#) (arctan - arccot)) by VALUED_1:def_4; for x being Real st x in Z holds ((sec (#) (arctan - arccot)) `| Z) . x = ((((arctan . x) - (arccot . x)) * (sin . x)) / ((cos . x) ^2)) + ((2 * (sec . x)) / (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((sec (#) (arctan - arccot)) `| Z) . x = ((((arctan . x) - (arccot . x)) * (sin . x)) / ((cos . x) ^2)) + ((2 * (sec . x)) / (1 + (x ^2))) ) assume A9: x in Z ; ::_thesis: ((sec (#) (arctan - arccot)) `| Z) . x = ((((arctan . x) - (arccot . x)) * (sin . x)) / ((cos . x) ^2)) + ((2 * (sec . x)) / (1 + (x ^2))) then A10: cos . x <> 0 by A1, RFUNCT_1:3; ((sec (#) (arctan - arccot)) `| Z) . x = (((arctan - arccot) . x) * (diff (sec,x))) + ((sec . x) * (diff ((arctan - arccot),x))) by A8, A4, A3, A9, FDIFF_1:21 .= (((arctan . x) - (arccot . x)) * (diff (sec,x))) + ((sec . x) * (diff ((arctan - arccot),x))) by A7, A9, VALUED_1:13 .= (((arctan . x) - (arccot . x)) * ((sin . x) / ((cos . x) ^2))) + ((sec . x) * (diff ((arctan - arccot),x))) by A10, FDIFF_9:1 .= ((((arctan . x) - (arccot . x)) * (sin . x)) / ((cos . x) ^2)) + ((sec . x) * (((arctan - arccot) `| Z) . x)) by A3, A9, FDIFF_1:def_7 .= ((((arctan . x) - (arccot . x)) * (sin . x)) / ((cos . x) ^2)) + ((sec . x) * (2 / (1 + (x ^2)))) by A2, A9, Th38 .= ((((arctan . x) - (arccot . x)) * (sin . x)) / ((cos . x) ^2)) + ((2 * (sec . x)) / (1 + (x ^2))) ; hence ((sec (#) (arctan - arccot)) `| Z) . x = ((((arctan . x) - (arccot . x)) * (sin . x)) / ((cos . x) ^2)) + ((2 * (sec . x)) / (1 + (x ^2))) ; ::_thesis: verum end; hence ( sec (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((sec (#) (arctan - arccot)) `| Z) . x = ((((arctan . x) - (arccot . x)) * (sin . x)) / ((cos . x) ^2)) + ((2 * (sec . x)) / (1 + (x ^2))) ) ) by A8, A4, A3, FDIFF_1:21; ::_thesis: verum end; theorem :: FDIFF_11:49 for Z being open Subset of REAL st Z c= dom cosec & Z c= ].(- 1),1.[ holds ( cosec (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((cosec (#) (arctan + arccot)) `| Z) . x = - ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2)) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom cosec & Z c= ].(- 1),1.[ implies ( cosec (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((cosec (#) (arctan + arccot)) `| Z) . x = - ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2)) ) ) ) assume that A1: Z c= dom cosec and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( cosec (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((cosec (#) (arctan + arccot)) `| Z) . x = - ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2)) ) ) A3: arctan + arccot is_differentiable_on Z by A2, Th37; for x being Real st x in Z holds cosec is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies cosec is_differentiable_in x ) assume x in Z ; ::_thesis: cosec is_differentiable_in x then sin . x <> 0 by A1, RFUNCT_1:3; hence cosec is_differentiable_in x by FDIFF_9:2; ::_thesis: verum end; then A4: cosec is_differentiable_on Z by A1, FDIFF_1:9; A5: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25; then ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1; then A6: Z c= dom arccot by A2, XBOOLE_1:1; ].(- 1),1.[ c= dom arctan by A5, SIN_COS9:23, XBOOLE_1:1; then Z c= dom arctan by A2, XBOOLE_1:1; then Z c= (dom arctan) /\ (dom arccot) by A6, XBOOLE_1:19; then A7: Z c= dom (arctan + arccot) by VALUED_1:def_1; then Z c= (dom cosec) /\ (dom (arctan + arccot)) by A1, XBOOLE_1:19; then A8: Z c= dom (cosec (#) (arctan + arccot)) by VALUED_1:def_4; for x being Real st x in Z holds ((cosec (#) (arctan + arccot)) `| Z) . x = - ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2)) proof let x be Real; ::_thesis: ( x in Z implies ((cosec (#) (arctan + arccot)) `| Z) . x = - ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2)) ) assume A9: x in Z ; ::_thesis: ((cosec (#) (arctan + arccot)) `| Z) . x = - ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2)) then A10: sin . x <> 0 by A1, RFUNCT_1:3; ((cosec (#) (arctan + arccot)) `| Z) . x = (((arctan + arccot) . x) * (diff (cosec,x))) + ((cosec . x) * (diff ((arctan + arccot),x))) by A8, A4, A3, A9, FDIFF_1:21 .= (((arctan . x) + (arccot . x)) * (diff (cosec,x))) + ((cosec . x) * (diff ((arctan + arccot),x))) by A7, A9, VALUED_1:def_1 .= (((arctan . x) + (arccot . x)) * (- ((cos . x) / ((sin . x) ^2)))) + ((cosec . x) * (diff ((arctan + arccot),x))) by A10, FDIFF_9:2 .= (- ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2))) + ((cosec . x) * (((arctan + arccot) `| Z) . x)) by A3, A9, FDIFF_1:def_7 .= (- ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2))) + ((cosec . x) * 0) by A2, A9, Th37 .= - ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2)) ; hence ((cosec (#) (arctan + arccot)) `| Z) . x = - ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2)) ; ::_thesis: verum end; hence ( cosec (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((cosec (#) (arctan + arccot)) `| Z) . x = - ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2)) ) ) by A8, A4, A3, FDIFF_1:21; ::_thesis: verum end; theorem :: FDIFF_11:50 for Z being open Subset of REAL st Z c= dom cosec & Z c= ].(- 1),1.[ holds ( cosec (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((cosec (#) (arctan - arccot)) `| Z) . x = (- ((((arctan . x) - (arccot . x)) * (cos . x)) / ((sin . x) ^2))) + ((2 * (cosec . x)) / (1 + (x ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom cosec & Z c= ].(- 1),1.[ implies ( cosec (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((cosec (#) (arctan - arccot)) `| Z) . x = (- ((((arctan . x) - (arccot . x)) * (cos . x)) / ((sin . x) ^2))) + ((2 * (cosec . x)) / (1 + (x ^2))) ) ) ) assume that A1: Z c= dom cosec and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( cosec (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((cosec (#) (arctan - arccot)) `| Z) . x = (- ((((arctan . x) - (arccot . x)) * (cos . x)) / ((sin . x) ^2))) + ((2 * (cosec . x)) / (1 + (x ^2))) ) ) A3: arctan - arccot is_differentiable_on Z by A2, Th38; for x being Real st x in Z holds cosec is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies cosec is_differentiable_in x ) assume x in Z ; ::_thesis: cosec is_differentiable_in x then sin . x <> 0 by A1, RFUNCT_1:3; hence cosec is_differentiable_in x by FDIFF_9:2; ::_thesis: verum end; then A4: cosec is_differentiable_on Z by A1, FDIFF_1:9; A5: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25; then ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1; then A6: Z c= dom arccot by A2, XBOOLE_1:1; ].(- 1),1.[ c= dom arctan by A5, SIN_COS9:23, XBOOLE_1:1; then Z c= dom arctan by A2, XBOOLE_1:1; then Z c= (dom arctan) /\ (dom arccot) by A6, XBOOLE_1:19; then A7: Z c= dom (arctan - arccot) by VALUED_1:12; then Z c= (dom cosec) /\ (dom (arctan - arccot)) by A1, XBOOLE_1:19; then A8: Z c= dom (cosec (#) (arctan - arccot)) by VALUED_1:def_4; for x being Real st x in Z holds ((cosec (#) (arctan - arccot)) `| Z) . x = (- ((((arctan . x) - (arccot . x)) * (cos . x)) / ((sin . x) ^2))) + ((2 * (cosec . x)) / (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((cosec (#) (arctan - arccot)) `| Z) . x = (- ((((arctan . x) - (arccot . x)) * (cos . x)) / ((sin . x) ^2))) + ((2 * (cosec . x)) / (1 + (x ^2))) ) assume A9: x in Z ; ::_thesis: ((cosec (#) (arctan - arccot)) `| Z) . x = (- ((((arctan . x) - (arccot . x)) * (cos . x)) / ((sin . x) ^2))) + ((2 * (cosec . x)) / (1 + (x ^2))) then A10: sin . x <> 0 by A1, RFUNCT_1:3; ((cosec (#) (arctan - arccot)) `| Z) . x = (((arctan - arccot) . x) * (diff (cosec,x))) + ((cosec . x) * (diff ((arctan - arccot),x))) by A8, A4, A3, A9, FDIFF_1:21 .= (((arctan . x) - (arccot . x)) * (diff (cosec,x))) + ((cosec . x) * (diff ((arctan - arccot),x))) by A7, A9, VALUED_1:13 .= (((arctan . x) - (arccot . x)) * (- ((cos . x) / ((sin . x) ^2)))) + ((cosec . x) * (diff ((arctan - arccot),x))) by A10, FDIFF_9:2 .= (- ((((arctan . x) - (arccot . x)) * (cos . x)) / ((sin . x) ^2))) + ((cosec . x) * (((arctan - arccot) `| Z) . x)) by A3, A9, FDIFF_1:def_7 .= (- ((((arctan . x) - (arccot . x)) * (cos . x)) / ((sin . x) ^2))) + ((cosec . x) * (2 / (1 + (x ^2)))) by A2, A9, Th38 .= (- ((((arctan . x) - (arccot . x)) * (cos . x)) / ((sin . x) ^2))) + ((2 * (cosec . x)) / (1 + (x ^2))) ; hence ((cosec (#) (arctan - arccot)) `| Z) . x = (- ((((arctan . x) - (arccot . x)) * (cos . x)) / ((sin . x) ^2))) + ((2 * (cosec . x)) / (1 + (x ^2))) ; ::_thesis: verum end; hence ( cosec (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((cosec (#) (arctan - arccot)) `| Z) . x = (- ((((arctan . x) - (arccot . x)) * (cos . x)) / ((sin . x) ^2))) + ((2 * (cosec . x)) / (1 + (x ^2))) ) ) by A8, A4, A3, FDIFF_1:21; ::_thesis: verum end; theorem :: FDIFF_11:51 for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds ( exp_R (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R (#) (arctan + arccot)) `| Z) . x = (exp_R . x) * ((arctan . x) + (arccot . x)) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= ].(- 1),1.[ implies ( exp_R (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R (#) (arctan + arccot)) `| Z) . x = (exp_R . x) * ((arctan . x) + (arccot . x)) ) ) ) assume A1: Z c= ].(- 1),1.[ ; ::_thesis: ( exp_R (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R (#) (arctan + arccot)) `| Z) . x = (exp_R . x) * ((arctan . x) + (arccot . x)) ) ) then A2: arctan + arccot is_differentiable_on Z by Th37; A3: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25; then ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1; then A4: Z c= dom arccot by A1, XBOOLE_1:1; ].(- 1),1.[ c= dom arctan by A3, SIN_COS9:23, XBOOLE_1:1; then Z c= dom arctan by A1, XBOOLE_1:1; then Z c= (dom arctan) /\ (dom arccot) by A4, XBOOLE_1:19; then A5: Z c= dom (arctan + arccot) by VALUED_1:def_1; then Z c= (dom exp_R) /\ (dom (arctan + arccot)) by TAYLOR_1:16, XBOOLE_1:19; then A6: Z c= dom (exp_R (#) (arctan + arccot)) by VALUED_1:def_4; A7: exp_R is_differentiable_on Z by FDIFF_1:26, TAYLOR_1:16; for x being Real st x in Z holds ((exp_R (#) (arctan + arccot)) `| Z) . x = (exp_R . x) * ((arctan . x) + (arccot . x)) proof let x be Real; ::_thesis: ( x in Z implies ((exp_R (#) (arctan + arccot)) `| Z) . x = (exp_R . x) * ((arctan . x) + (arccot . x)) ) assume A8: x in Z ; ::_thesis: ((exp_R (#) (arctan + arccot)) `| Z) . x = (exp_R . x) * ((arctan . x) + (arccot . x)) then ((exp_R (#) (arctan + arccot)) `| Z) . x = (((arctan + arccot) . x) * (diff (exp_R,x))) + ((exp_R . x) * (diff ((arctan + arccot),x))) by A6, A7, A2, FDIFF_1:21 .= (((arctan . x) + (arccot . x)) * (diff (exp_R,x))) + ((exp_R . x) * (diff ((arctan + arccot),x))) by A5, A8, VALUED_1:def_1 .= (((arctan . x) + (arccot . x)) * (exp_R . x)) + ((exp_R . x) * (diff ((arctan + arccot),x))) by TAYLOR_1:16 .= (((arctan . x) + (arccot . x)) * (exp_R . x)) + ((exp_R . x) * (((arctan + arccot) `| Z) . x)) by A2, A8, FDIFF_1:def_7 .= (((arctan . x) + (arccot . x)) * (exp_R . x)) + ((exp_R . x) * 0) by A1, A8, Th37 .= (exp_R . x) * ((arctan . x) + (arccot . x)) ; hence ((exp_R (#) (arctan + arccot)) `| Z) . x = (exp_R . x) * ((arctan . x) + (arccot . x)) ; ::_thesis: verum end; hence ( exp_R (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R (#) (arctan + arccot)) `| Z) . x = (exp_R . x) * ((arctan . x) + (arccot . x)) ) ) by A6, A7, A2, FDIFF_1:21; ::_thesis: verum end; theorem :: FDIFF_11:52 for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds ( exp_R (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R (#) (arctan - arccot)) `| Z) . x = ((exp_R . x) * ((arctan . x) - (arccot . x))) + ((2 * (exp_R . x)) / (1 + (x ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= ].(- 1),1.[ implies ( exp_R (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R (#) (arctan - arccot)) `| Z) . x = ((exp_R . x) * ((arctan . x) - (arccot . x))) + ((2 * (exp_R . x)) / (1 + (x ^2))) ) ) ) assume A1: Z c= ].(- 1),1.[ ; ::_thesis: ( exp_R (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R (#) (arctan - arccot)) `| Z) . x = ((exp_R . x) * ((arctan . x) - (arccot . x))) + ((2 * (exp_R . x)) / (1 + (x ^2))) ) ) then A2: arctan - arccot is_differentiable_on Z by Th38; A3: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25; then ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1; then A4: Z c= dom arccot by A1, XBOOLE_1:1; ].(- 1),1.[ c= dom arctan by A3, SIN_COS9:23, XBOOLE_1:1; then Z c= dom arctan by A1, XBOOLE_1:1; then Z c= (dom arctan) /\ (dom arccot) by A4, XBOOLE_1:19; then A5: Z c= dom (arctan - arccot) by VALUED_1:12; then Z c= (dom exp_R) /\ (dom (arctan - arccot)) by TAYLOR_1:16, XBOOLE_1:19; then A6: Z c= dom (exp_R (#) (arctan - arccot)) by VALUED_1:def_4; A7: exp_R is_differentiable_on Z by FDIFF_1:26, TAYLOR_1:16; for x being Real st x in Z holds ((exp_R (#) (arctan - arccot)) `| Z) . x = ((exp_R . x) * ((arctan . x) - (arccot . x))) + ((2 * (exp_R . x)) / (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((exp_R (#) (arctan - arccot)) `| Z) . x = ((exp_R . x) * ((arctan . x) - (arccot . x))) + ((2 * (exp_R . x)) / (1 + (x ^2))) ) assume A8: x in Z ; ::_thesis: ((exp_R (#) (arctan - arccot)) `| Z) . x = ((exp_R . x) * ((arctan . x) - (arccot . x))) + ((2 * (exp_R . x)) / (1 + (x ^2))) then ((exp_R (#) (arctan - arccot)) `| Z) . x = (((arctan - arccot) . x) * (diff (exp_R,x))) + ((exp_R . x) * (diff ((arctan - arccot),x))) by A6, A7, A2, FDIFF_1:21 .= (((arctan . x) - (arccot . x)) * (diff (exp_R,x))) + ((exp_R . x) * (diff ((arctan - arccot),x))) by A5, A8, VALUED_1:13 .= (((arctan . x) - (arccot . x)) * (exp_R . x)) + ((exp_R . x) * (diff ((arctan - arccot),x))) by TAYLOR_1:16 .= (((arctan . x) - (arccot . x)) * (exp_R . x)) + ((exp_R . x) * (((arctan - arccot) `| Z) . x)) by A2, A8, FDIFF_1:def_7 .= (((arctan . x) - (arccot . x)) * (exp_R . x)) + ((exp_R . x) * (2 / (1 + (x ^2)))) by A1, A8, Th38 .= ((exp_R . x) * ((arctan . x) - (arccot . x))) + ((2 * (exp_R . x)) / (1 + (x ^2))) ; hence ((exp_R (#) (arctan - arccot)) `| Z) . x = ((exp_R . x) * ((arctan . x) - (arccot . x))) + ((2 * (exp_R . x)) / (1 + (x ^2))) ; ::_thesis: verum end; hence ( exp_R (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R (#) (arctan - arccot)) `| Z) . x = ((exp_R . x) * ((arctan . x) - (arccot . x))) + ((2 * (exp_R . x)) / (1 + (x ^2))) ) ) by A6, A7, A2, FDIFF_1:21; ::_thesis: verum end; theorem :: FDIFF_11:53 for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds ( (arctan + arccot) / exp_R is_differentiable_on Z & ( for x being Real st x in Z holds (((arctan + arccot) / exp_R) `| Z) . x = - (((arctan . x) + (arccot . x)) / (exp_R . x)) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= ].(- 1),1.[ implies ( (arctan + arccot) / exp_R is_differentiable_on Z & ( for x being Real st x in Z holds (((arctan + arccot) / exp_R) `| Z) . x = - (((arctan . x) + (arccot . x)) / (exp_R . x)) ) ) ) assume A1: Z c= ].(- 1),1.[ ; ::_thesis: ( (arctan + arccot) / exp_R is_differentiable_on Z & ( for x being Real st x in Z holds (((arctan + arccot) / exp_R) `| Z) . x = - (((arctan . x) + (arccot . x)) / (exp_R . x)) ) ) then A2: arctan + arccot is_differentiable_on Z by Th37; A3: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25; then ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1; then A4: Z c= dom arccot by A1, XBOOLE_1:1; ].(- 1),1.[ c= dom arctan by A3, SIN_COS9:23, XBOOLE_1:1; then Z c= dom arctan by A1, XBOOLE_1:1; then Z c= (dom arctan) /\ (dom arccot) by A4, XBOOLE_1:19; then A5: Z c= dom (arctan + arccot) by VALUED_1:def_1; A6: ( exp_R is_differentiable_on Z & ( for x being Real st x in Z holds exp_R . x <> 0 ) ) by FDIFF_1:26, SIN_COS:54, TAYLOR_1:16; then A7: (arctan + arccot) / exp_R is_differentiable_on Z by A2, FDIFF_2:21; for x being Real st x in Z holds (((arctan + arccot) / exp_R) `| Z) . x = - (((arctan . x) + (arccot . x)) / (exp_R . x)) proof let x be Real; ::_thesis: ( x in Z implies (((arctan + arccot) / exp_R) `| Z) . x = - (((arctan . x) + (arccot . x)) / (exp_R . x)) ) A8: exp_R is_differentiable_in x by SIN_COS:65; A9: exp_R . x <> 0 by SIN_COS:54; assume A10: x in Z ; ::_thesis: (((arctan + arccot) / exp_R) `| Z) . x = - (((arctan . x) + (arccot . x)) / (exp_R . x)) then A11: arctan + arccot is_differentiable_in x by A2, FDIFF_1:9; (((arctan + arccot) / exp_R) `| Z) . x = diff (((arctan + arccot) / exp_R),x) by A7, A10, FDIFF_1:def_7 .= (((diff ((arctan + arccot),x)) * (exp_R . x)) - ((diff (exp_R,x)) * ((arctan + arccot) . x))) / ((exp_R . x) ^2) by A11, A8, A9, FDIFF_2:14 .= (((((arctan + arccot) `| Z) . x) * (exp_R . x)) - ((diff (exp_R,x)) * ((arctan + arccot) . x))) / ((exp_R . x) ^2) by A2, A10, FDIFF_1:def_7 .= ((0 * (exp_R . x)) - ((diff (exp_R,x)) * ((arctan + arccot) . x))) / ((exp_R . x) ^2) by A1, A10, Th37 .= - (((diff (exp_R,x)) * ((arctan + arccot) . x)) / ((exp_R . x) ^2)) .= - (((exp_R . x) * ((arctan + arccot) . x)) / ((exp_R . x) ^2)) by SIN_COS:65 .= - ((((arctan . x) + (arccot . x)) * (exp_R . x)) / ((exp_R . x) * (exp_R . x))) by A5, A10, VALUED_1:def_1 .= - (((arctan . x) + (arccot . x)) * ((exp_R . x) / ((exp_R . x) * (exp_R . x)))) .= - (((arctan . x) + (arccot . x)) * (((exp_R . x) / (exp_R . x)) / (exp_R . x))) by XCMPLX_1:78 .= - (((arctan . x) + (arccot . x)) * (1 / (exp_R . x))) by A9, XCMPLX_1:60 .= - (((arctan . x) + (arccot . x)) / (exp_R . x)) ; hence (((arctan + arccot) / exp_R) `| Z) . x = - (((arctan . x) + (arccot . x)) / (exp_R . x)) ; ::_thesis: verum end; hence ( (arctan + arccot) / exp_R is_differentiable_on Z & ( for x being Real st x in Z holds (((arctan + arccot) / exp_R) `| Z) . x = - (((arctan . x) + (arccot . x)) / (exp_R . x)) ) ) by A2, A6, FDIFF_2:21; ::_thesis: verum end; theorem :: FDIFF_11:54 for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds ( (arctan - arccot) / exp_R is_differentiable_on Z & ( for x being Real st x in Z holds (((arctan - arccot) / exp_R) `| Z) . x = (((2 / (1 + (x ^2))) - (arctan . x)) + (arccot . x)) / (exp_R . x) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= ].(- 1),1.[ implies ( (arctan - arccot) / exp_R is_differentiable_on Z & ( for x being Real st x in Z holds (((arctan - arccot) / exp_R) `| Z) . x = (((2 / (1 + (x ^2))) - (arctan . x)) + (arccot . x)) / (exp_R . x) ) ) ) assume A1: Z c= ].(- 1),1.[ ; ::_thesis: ( (arctan - arccot) / exp_R is_differentiable_on Z & ( for x being Real st x in Z holds (((arctan - arccot) / exp_R) `| Z) . x = (((2 / (1 + (x ^2))) - (arctan . x)) + (arccot . x)) / (exp_R . x) ) ) then A2: arctan - arccot is_differentiable_on Z by Th38; A3: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25; then ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1; then A4: Z c= dom arccot by A1, XBOOLE_1:1; ].(- 1),1.[ c= dom arctan by A3, SIN_COS9:23, XBOOLE_1:1; then Z c= dom arctan by A1, XBOOLE_1:1; then Z c= (dom arctan) /\ (dom arccot) by A4, XBOOLE_1:19; then A5: Z c= dom (arctan - arccot) by VALUED_1:12; A6: ( exp_R is_differentiable_on Z & ( for x being Real st x in Z holds exp_R . x <> 0 ) ) by FDIFF_1:26, SIN_COS:54, TAYLOR_1:16; then A7: (arctan - arccot) / exp_R is_differentiable_on Z by A2, FDIFF_2:21; for x being Real st x in Z holds (((arctan - arccot) / exp_R) `| Z) . x = (((2 / (1 + (x ^2))) - (arctan . x)) + (arccot . x)) / (exp_R . x) proof let x be Real; ::_thesis: ( x in Z implies (((arctan - arccot) / exp_R) `| Z) . x = (((2 / (1 + (x ^2))) - (arctan . x)) + (arccot . x)) / (exp_R . x) ) A8: exp_R is_differentiable_in x by SIN_COS:65; A9: exp_R . x <> 0 by SIN_COS:54; assume A10: x in Z ; ::_thesis: (((arctan - arccot) / exp_R) `| Z) . x = (((2 / (1 + (x ^2))) - (arctan . x)) + (arccot . x)) / (exp_R . x) then A11: arctan - arccot is_differentiable_in x by A2, FDIFF_1:9; (((arctan - arccot) / exp_R) `| Z) . x = diff (((arctan - arccot) / exp_R),x) by A7, A10, FDIFF_1:def_7 .= (((diff ((arctan - arccot),x)) * (exp_R . x)) - ((diff (exp_R,x)) * ((arctan - arccot) . x))) / ((exp_R . x) ^2) by A11, A8, A9, FDIFF_2:14 .= (((((arctan - arccot) `| Z) . x) * (exp_R . x)) - ((diff (exp_R,x)) * ((arctan - arccot) . x))) / ((exp_R . x) ^2) by A2, A10, FDIFF_1:def_7 .= (((2 / (1 + (x ^2))) * (exp_R . x)) - ((diff (exp_R,x)) * ((arctan - arccot) . x))) / ((exp_R . x) ^2) by A1, A10, Th38 .= (((2 / (1 + (x ^2))) * (exp_R . x)) - ((exp_R . x) * ((arctan - arccot) . x))) / ((exp_R . x) ^2) by SIN_COS:65 .= (((2 / (1 + (x ^2))) * (exp_R . x)) - ((exp_R . x) * ((arctan . x) - (arccot . x)))) / ((exp_R . x) ^2) by A5, A10, VALUED_1:13 .= ((2 / (1 + (x ^2))) - ((arctan . x) - (arccot . x))) * ((exp_R . x) / ((exp_R . x) * (exp_R . x))) .= ((2 / (1 + (x ^2))) - ((arctan . x) - (arccot . x))) * (((exp_R . x) / (exp_R . x)) / (exp_R . x)) by XCMPLX_1:78 .= ((2 / (1 + (x ^2))) - ((arctan . x) - (arccot . x))) * (1 / (exp_R . x)) by A9, XCMPLX_1:60 .= (((2 / (1 + (x ^2))) - (arctan . x)) + (arccot . x)) / (exp_R . x) ; hence (((arctan - arccot) / exp_R) `| Z) . x = (((2 / (1 + (x ^2))) - (arctan . x)) + (arccot . x)) / (exp_R . x) ; ::_thesis: verum end; hence ( (arctan - arccot) / exp_R is_differentiable_on Z & ( for x being Real st x in Z holds (((arctan - arccot) / exp_R) `| Z) . x = (((2 / (1 + (x ^2))) - (arctan . x)) + (arccot . x)) / (exp_R . x) ) ) by A2, A6, FDIFF_2:21; ::_thesis: verum end; theorem :: FDIFF_11:55 for Z being open Subset of REAL st Z c= dom (exp_R * (arctan + arccot)) & Z c= ].(- 1),1.[ holds ( exp_R * (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R * (arctan + arccot)) `| Z) . x = 0 ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (exp_R * (arctan + arccot)) & Z c= ].(- 1),1.[ implies ( exp_R * (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R * (arctan + arccot)) `| Z) . x = 0 ) ) ) assume that A1: Z c= dom (exp_R * (arctan + arccot)) and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( exp_R * (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R * (arctan + arccot)) `| Z) . x = 0 ) ) A3: arctan + arccot is_differentiable_on Z by A2, Th37; A4: for x being Real st x in Z holds exp_R * (arctan + arccot) is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies exp_R * (arctan + arccot) is_differentiable_in x ) assume x in Z ; ::_thesis: exp_R * (arctan + arccot) is_differentiable_in x then A5: arctan + arccot is_differentiable_in x by A3, FDIFF_1:9; exp_R is_differentiable_in (arctan + arccot) . x by SIN_COS:65; hence exp_R * (arctan + arccot) is_differentiable_in x by A5, FDIFF_2:13; ::_thesis: verum end; then A6: exp_R * (arctan + arccot) is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((exp_R * (arctan + arccot)) `| Z) . x = 0 proof let x be Real; ::_thesis: ( x in Z implies ((exp_R * (arctan + arccot)) `| Z) . x = 0 ) A7: exp_R is_differentiable_in (arctan + arccot) . x by SIN_COS:65; assume A8: x in Z ; ::_thesis: ((exp_R * (arctan + arccot)) `| Z) . x = 0 then A9: arctan + arccot is_differentiable_in x by A3, FDIFF_1:9; ((exp_R * (arctan + arccot)) `| Z) . x = diff ((exp_R * (arctan + arccot)),x) by A6, A8, FDIFF_1:def_7 .= (diff (exp_R,((arctan + arccot) . x))) * (diff ((arctan + arccot),x)) by A9, A7, FDIFF_2:13 .= (exp_R . ((arctan + arccot) . x)) * (diff ((arctan + arccot),x)) by SIN_COS:65 .= (exp_R . ((arctan + arccot) . x)) * (((arctan + arccot) `| Z) . x) by A3, A8, FDIFF_1:def_7 .= (exp_R . ((arctan + arccot) . x)) * 0 by A2, A8, Th37 .= 0 ; hence ((exp_R * (arctan + arccot)) `| Z) . x = 0 ; ::_thesis: verum end; hence ( exp_R * (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R * (arctan + arccot)) `| Z) . x = 0 ) ) by A1, A4, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_11:56 for Z being open Subset of REAL st Z c= dom (exp_R * (arctan - arccot)) & Z c= ].(- 1),1.[ holds ( exp_R * (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R * (arctan - arccot)) `| Z) . x = (2 * (exp_R . ((arctan . x) - (arccot . x)))) / (1 + (x ^2)) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (exp_R * (arctan - arccot)) & Z c= ].(- 1),1.[ implies ( exp_R * (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R * (arctan - arccot)) `| Z) . x = (2 * (exp_R . ((arctan . x) - (arccot . x)))) / (1 + (x ^2)) ) ) ) assume that A1: Z c= dom (exp_R * (arctan - arccot)) and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( exp_R * (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R * (arctan - arccot)) `| Z) . x = (2 * (exp_R . ((arctan . x) - (arccot . x)))) / (1 + (x ^2)) ) ) A3: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25; then ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1; then A4: Z c= dom arccot by A2, XBOOLE_1:1; ].(- 1),1.[ c= dom arctan by A3, SIN_COS9:23, XBOOLE_1:1; then Z c= dom arctan by A2, XBOOLE_1:1; then Z c= (dom arctan) /\ (dom arccot) by A4, XBOOLE_1:19; then A5: Z c= dom (arctan - arccot) by VALUED_1:12; A6: arctan - arccot is_differentiable_on Z by A2, Th38; A7: for x being Real st x in Z holds exp_R * (arctan - arccot) is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies exp_R * (arctan - arccot) is_differentiable_in x ) assume x in Z ; ::_thesis: exp_R * (arctan - arccot) is_differentiable_in x then A8: arctan - arccot is_differentiable_in x by A6, FDIFF_1:9; exp_R is_differentiable_in (arctan - arccot) . x by SIN_COS:65; hence exp_R * (arctan - arccot) is_differentiable_in x by A8, FDIFF_2:13; ::_thesis: verum end; then A9: exp_R * (arctan - arccot) is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((exp_R * (arctan - arccot)) `| Z) . x = (2 * (exp_R . ((arctan . x) - (arccot . x)))) / (1 + (x ^2)) proof let x be Real; ::_thesis: ( x in Z implies ((exp_R * (arctan - arccot)) `| Z) . x = (2 * (exp_R . ((arctan . x) - (arccot . x)))) / (1 + (x ^2)) ) A10: exp_R is_differentiable_in (arctan - arccot) . x by SIN_COS:65; assume A11: x in Z ; ::_thesis: ((exp_R * (arctan - arccot)) `| Z) . x = (2 * (exp_R . ((arctan . x) - (arccot . x)))) / (1 + (x ^2)) then A12: arctan - arccot is_differentiable_in x by A6, FDIFF_1:9; ((exp_R * (arctan - arccot)) `| Z) . x = diff ((exp_R * (arctan - arccot)),x) by A9, A11, FDIFF_1:def_7 .= (diff (exp_R,((arctan - arccot) . x))) * (diff ((arctan - arccot),x)) by A12, A10, FDIFF_2:13 .= (exp_R . ((arctan - arccot) . x)) * (diff ((arctan - arccot),x)) by SIN_COS:65 .= (exp_R . ((arctan - arccot) . x)) * (((arctan - arccot) `| Z) . x) by A6, A11, FDIFF_1:def_7 .= (exp_R . ((arctan - arccot) . x)) * (2 / (1 + (x ^2))) by A2, A11, Th38 .= (exp_R . ((arctan . x) - (arccot . x))) * (2 / (1 + (x ^2))) by A5, A11, VALUED_1:13 .= (2 * (exp_R . ((arctan . x) - (arccot . x)))) / (1 + (x ^2)) ; hence ((exp_R * (arctan - arccot)) `| Z) . x = (2 * (exp_R . ((arctan . x) - (arccot . x)))) / (1 + (x ^2)) ; ::_thesis: verum end; hence ( exp_R * (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R * (arctan - arccot)) `| Z) . x = (2 * (exp_R . ((arctan . x) - (arccot . x)))) / (1 + (x ^2)) ) ) by A1, A7, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_11:57 for Z being open Subset of REAL st Z c= dom (sin * (arctan + arccot)) & Z c= ].(- 1),1.[ holds ( sin * (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((sin * (arctan + arccot)) `| Z) . x = 0 ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (sin * (arctan + arccot)) & Z c= ].(- 1),1.[ implies ( sin * (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((sin * (arctan + arccot)) `| Z) . x = 0 ) ) ) assume that A1: Z c= dom (sin * (arctan + arccot)) and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( sin * (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((sin * (arctan + arccot)) `| Z) . x = 0 ) ) A3: arctan + arccot is_differentiable_on Z by A2, Th37; A4: for x being Real st x in Z holds sin * (arctan + arccot) is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies sin * (arctan + arccot) is_differentiable_in x ) assume x in Z ; ::_thesis: sin * (arctan + arccot) is_differentiable_in x then A5: arctan + arccot is_differentiable_in x by A3, FDIFF_1:9; sin is_differentiable_in (arctan + arccot) . x by SIN_COS:64; hence sin * (arctan + arccot) is_differentiable_in x by A5, FDIFF_2:13; ::_thesis: verum end; then A6: sin * (arctan + arccot) is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((sin * (arctan + arccot)) `| Z) . x = 0 proof let x be Real; ::_thesis: ( x in Z implies ((sin * (arctan + arccot)) `| Z) . x = 0 ) A7: sin is_differentiable_in (arctan + arccot) . x by SIN_COS:64; assume A8: x in Z ; ::_thesis: ((sin * (arctan + arccot)) `| Z) . x = 0 then A9: arctan + arccot is_differentiable_in x by A3, FDIFF_1:9; ((sin * (arctan + arccot)) `| Z) . x = diff ((sin * (arctan + arccot)),x) by A6, A8, FDIFF_1:def_7 .= (diff (sin,((arctan + arccot) . x))) * (diff ((arctan + arccot),x)) by A9, A7, FDIFF_2:13 .= (cos . ((arctan + arccot) . x)) * (diff ((arctan + arccot),x)) by SIN_COS:64 .= (cos . ((arctan + arccot) . x)) * (((arctan + arccot) `| Z) . x) by A3, A8, FDIFF_1:def_7 .= (cos . ((arctan + arccot) . x)) * 0 by A2, A8, Th37 .= 0 ; hence ((sin * (arctan + arccot)) `| Z) . x = 0 ; ::_thesis: verum end; hence ( sin * (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((sin * (arctan + arccot)) `| Z) . x = 0 ) ) by A1, A4, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_11:58 for Z being open Subset of REAL st Z c= dom (sin * (arctan - arccot)) & Z c= ].(- 1),1.[ holds ( sin * (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((sin * (arctan - arccot)) `| Z) . x = (2 * (cos . ((arctan . x) - (arccot . x)))) / (1 + (x ^2)) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (sin * (arctan - arccot)) & Z c= ].(- 1),1.[ implies ( sin * (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((sin * (arctan - arccot)) `| Z) . x = (2 * (cos . ((arctan . x) - (arccot . x)))) / (1 + (x ^2)) ) ) ) assume that A1: Z c= dom (sin * (arctan - arccot)) and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( sin * (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((sin * (arctan - arccot)) `| Z) . x = (2 * (cos . ((arctan . x) - (arccot . x)))) / (1 + (x ^2)) ) ) A3: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25; then ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1; then A4: Z c= dom arccot by A2, XBOOLE_1:1; ].(- 1),1.[ c= dom arctan by A3, SIN_COS9:23, XBOOLE_1:1; then Z c= dom arctan by A2, XBOOLE_1:1; then Z c= (dom arctan) /\ (dom arccot) by A4, XBOOLE_1:19; then A5: Z c= dom (arctan - arccot) by VALUED_1:12; A6: arctan - arccot is_differentiable_on Z by A2, Th38; A7: for x being Real st x in Z holds sin * (arctan - arccot) is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies sin * (arctan - arccot) is_differentiable_in x ) assume x in Z ; ::_thesis: sin * (arctan - arccot) is_differentiable_in x then A8: arctan - arccot is_differentiable_in x by A6, FDIFF_1:9; sin is_differentiable_in (arctan - arccot) . x by SIN_COS:64; hence sin * (arctan - arccot) is_differentiable_in x by A8, FDIFF_2:13; ::_thesis: verum end; then A9: sin * (arctan - arccot) is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((sin * (arctan - arccot)) `| Z) . x = (2 * (cos . ((arctan . x) - (arccot . x)))) / (1 + (x ^2)) proof let x be Real; ::_thesis: ( x in Z implies ((sin * (arctan - arccot)) `| Z) . x = (2 * (cos . ((arctan . x) - (arccot . x)))) / (1 + (x ^2)) ) A10: sin is_differentiable_in (arctan - arccot) . x by SIN_COS:64; assume A11: x in Z ; ::_thesis: ((sin * (arctan - arccot)) `| Z) . x = (2 * (cos . ((arctan . x) - (arccot . x)))) / (1 + (x ^2)) then A12: arctan - arccot is_differentiable_in x by A6, FDIFF_1:9; ((sin * (arctan - arccot)) `| Z) . x = diff ((sin * (arctan - arccot)),x) by A9, A11, FDIFF_1:def_7 .= (diff (sin,((arctan - arccot) . x))) * (diff ((arctan - arccot),x)) by A12, A10, FDIFF_2:13 .= (cos . ((arctan - arccot) . x)) * (diff ((arctan - arccot),x)) by SIN_COS:64 .= (cos . ((arctan - arccot) . x)) * (((arctan - arccot) `| Z) . x) by A6, A11, FDIFF_1:def_7 .= (cos . ((arctan - arccot) . x)) * (2 / (1 + (x ^2))) by A2, A11, Th38 .= (cos . ((arctan . x) - (arccot . x))) * (2 / (1 + (x ^2))) by A5, A11, VALUED_1:13 .= (2 * (cos . ((arctan . x) - (arccot . x)))) / (1 + (x ^2)) ; hence ((sin * (arctan - arccot)) `| Z) . x = (2 * (cos . ((arctan . x) - (arccot . x)))) / (1 + (x ^2)) ; ::_thesis: verum end; hence ( sin * (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((sin * (arctan - arccot)) `| Z) . x = (2 * (cos . ((arctan . x) - (arccot . x)))) / (1 + (x ^2)) ) ) by A1, A7, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_11:59 for Z being open Subset of REAL st Z c= dom (cos * (arctan + arccot)) & Z c= ].(- 1),1.[ holds ( cos * (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((cos * (arctan + arccot)) `| Z) . x = 0 ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (cos * (arctan + arccot)) & Z c= ].(- 1),1.[ implies ( cos * (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((cos * (arctan + arccot)) `| Z) . x = 0 ) ) ) assume that A1: Z c= dom (cos * (arctan + arccot)) and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( cos * (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((cos * (arctan + arccot)) `| Z) . x = 0 ) ) A3: arctan + arccot is_differentiable_on Z by A2, Th37; A4: for x being Real st x in Z holds cos * (arctan + arccot) is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies cos * (arctan + arccot) is_differentiable_in x ) assume x in Z ; ::_thesis: cos * (arctan + arccot) is_differentiable_in x then A5: arctan + arccot is_differentiable_in x by A3, FDIFF_1:9; cos is_differentiable_in (arctan + arccot) . x by SIN_COS:63; hence cos * (arctan + arccot) is_differentiable_in x by A5, FDIFF_2:13; ::_thesis: verum end; then A6: cos * (arctan + arccot) is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((cos * (arctan + arccot)) `| Z) . x = 0 proof let x be Real; ::_thesis: ( x in Z implies ((cos * (arctan + arccot)) `| Z) . x = 0 ) A7: cos is_differentiable_in (arctan + arccot) . x by SIN_COS:63; assume A8: x in Z ; ::_thesis: ((cos * (arctan + arccot)) `| Z) . x = 0 then A9: arctan + arccot is_differentiable_in x by A3, FDIFF_1:9; ((cos * (arctan + arccot)) `| Z) . x = diff ((cos * (arctan + arccot)),x) by A6, A8, FDIFF_1:def_7 .= (diff (cos,((arctan + arccot) . x))) * (diff ((arctan + arccot),x)) by A9, A7, FDIFF_2:13 .= (- (sin . ((arctan + arccot) . x))) * (diff ((arctan + arccot),x)) by SIN_COS:63 .= (- (sin . ((arctan + arccot) . x))) * (((arctan + arccot) `| Z) . x) by A3, A8, FDIFF_1:def_7 .= (- (sin . ((arctan + arccot) . x))) * 0 by A2, A8, Th37 .= 0 ; hence ((cos * (arctan + arccot)) `| Z) . x = 0 ; ::_thesis: verum end; hence ( cos * (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((cos * (arctan + arccot)) `| Z) . x = 0 ) ) by A1, A4, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_11:60 for Z being open Subset of REAL st Z c= dom (cos * (arctan - arccot)) & Z c= ].(- 1),1.[ holds ( cos * (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((cos * (arctan - arccot)) `| Z) . x = - ((2 * (sin . ((arctan . x) - (arccot . x)))) / (1 + (x ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (cos * (arctan - arccot)) & Z c= ].(- 1),1.[ implies ( cos * (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((cos * (arctan - arccot)) `| Z) . x = - ((2 * (sin . ((arctan . x) - (arccot . x)))) / (1 + (x ^2))) ) ) ) assume that A1: Z c= dom (cos * (arctan - arccot)) and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( cos * (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((cos * (arctan - arccot)) `| Z) . x = - ((2 * (sin . ((arctan . x) - (arccot . x)))) / (1 + (x ^2))) ) ) A3: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25; then ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1; then A4: Z c= dom arccot by A2, XBOOLE_1:1; ].(- 1),1.[ c= dom arctan by A3, SIN_COS9:23, XBOOLE_1:1; then Z c= dom arctan by A2, XBOOLE_1:1; then Z c= (dom arctan) /\ (dom arccot) by A4, XBOOLE_1:19; then A5: Z c= dom (arctan - arccot) by VALUED_1:12; A6: arctan - arccot is_differentiable_on Z by A2, Th38; A7: for x being Real st x in Z holds cos * (arctan - arccot) is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies cos * (arctan - arccot) is_differentiable_in x ) assume x in Z ; ::_thesis: cos * (arctan - arccot) is_differentiable_in x then A8: arctan - arccot is_differentiable_in x by A6, FDIFF_1:9; cos is_differentiable_in (arctan - arccot) . x by SIN_COS:63; hence cos * (arctan - arccot) is_differentiable_in x by A8, FDIFF_2:13; ::_thesis: verum end; then A9: cos * (arctan - arccot) is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((cos * (arctan - arccot)) `| Z) . x = - ((2 * (sin . ((arctan . x) - (arccot . x)))) / (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((cos * (arctan - arccot)) `| Z) . x = - ((2 * (sin . ((arctan . x) - (arccot . x)))) / (1 + (x ^2))) ) A10: cos is_differentiable_in (arctan - arccot) . x by SIN_COS:63; assume A11: x in Z ; ::_thesis: ((cos * (arctan - arccot)) `| Z) . x = - ((2 * (sin . ((arctan . x) - (arccot . x)))) / (1 + (x ^2))) then A12: arctan - arccot is_differentiable_in x by A6, FDIFF_1:9; ((cos * (arctan - arccot)) `| Z) . x = diff ((cos * (arctan - arccot)),x) by A9, A11, FDIFF_1:def_7 .= (diff (cos,((arctan - arccot) . x))) * (diff ((arctan - arccot),x)) by A12, A10, FDIFF_2:13 .= (- (sin . ((arctan - arccot) . x))) * (diff ((arctan - arccot),x)) by SIN_COS:63 .= (- (sin . ((arctan - arccot) . x))) * (((arctan - arccot) `| Z) . x) by A6, A11, FDIFF_1:def_7 .= (- (sin . ((arctan - arccot) . x))) * (2 / (1 + (x ^2))) by A2, A11, Th38 .= (- (sin . ((arctan . x) - (arccot . x)))) * (2 / (1 + (x ^2))) by A5, A11, VALUED_1:13 .= - ((2 * (sin . ((arctan . x) - (arccot . x)))) / (1 + (x ^2))) ; hence ((cos * (arctan - arccot)) `| Z) . x = - ((2 * (sin . ((arctan . x) - (arccot . x)))) / (1 + (x ^2))) ; ::_thesis: verum end; hence ( cos * (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((cos * (arctan - arccot)) `| Z) . x = - ((2 * (sin . ((arctan . x) - (arccot . x)))) / (1 + (x ^2))) ) ) by A1, A7, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_11:61 for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds ( arctan (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan (#) arccot) `| Z) . x = ((arccot . x) - (arctan . x)) / (1 + (x ^2)) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= ].(- 1),1.[ implies ( arctan (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan (#) arccot) `| Z) . x = ((arccot . x) - (arctan . x)) / (1 + (x ^2)) ) ) ) assume A1: Z c= ].(- 1),1.[ ; ::_thesis: ( arctan (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan (#) arccot) `| Z) . x = ((arccot . x) - (arctan . x)) / (1 + (x ^2)) ) ) then A2: arctan is_differentiable_on Z by SIN_COS9:81; A3: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25; then ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1; then A4: Z c= dom arccot by A1, XBOOLE_1:1; A5: arccot is_differentiable_on Z by A1, SIN_COS9:82; ].(- 1),1.[ c= dom arctan by A3, SIN_COS9:23, XBOOLE_1:1; then Z c= dom arctan by A1, XBOOLE_1:1; then Z c= (dom arctan) /\ (dom arccot) by A4, XBOOLE_1:19; then A6: Z c= dom (arctan (#) arccot) by VALUED_1:def_4; for x being Real st x in Z holds ((arctan (#) arccot) `| Z) . x = ((arccot . x) - (arctan . x)) / (1 + (x ^2)) proof let x be Real; ::_thesis: ( x in Z implies ((arctan (#) arccot) `| Z) . x = ((arccot . x) - (arctan . x)) / (1 + (x ^2)) ) assume A7: x in Z ; ::_thesis: ((arctan (#) arccot) `| Z) . x = ((arccot . x) - (arctan . x)) / (1 + (x ^2)) then ((arctan (#) arccot) `| Z) . x = ((arccot . x) * (diff (arctan,x))) + ((arctan . x) * (diff (arccot,x))) by A6, A2, A5, FDIFF_1:21 .= ((arccot . x) * ((arctan `| Z) . x)) + ((arctan . x) * (diff (arccot,x))) by A2, A7, FDIFF_1:def_7 .= ((arccot . x) * (1 / (1 + (x ^2)))) + ((arctan . x) * (diff (arccot,x))) by A1, A7, SIN_COS9:81 .= ((arccot . x) * (1 / (1 + (x ^2)))) + ((arctan . x) * ((arccot `| Z) . x)) by A5, A7, FDIFF_1:def_7 .= ((arccot . x) * (1 / (1 + (x ^2)))) + ((arctan . x) * (- (1 / (1 + (x ^2))))) by A1, A7, SIN_COS9:82 .= ((arccot . x) - (arctan . x)) / (1 + (x ^2)) ; hence ((arctan (#) arccot) `| Z) . x = ((arccot . x) - (arctan . x)) / (1 + (x ^2)) ; ::_thesis: verum end; hence ( arctan (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan (#) arccot) `| Z) . x = ((arccot . x) - (arctan . x)) / (1 + (x ^2)) ) ) by A6, A2, A5, FDIFF_1:21; ::_thesis: verum end; theorem :: FDIFF_11:62 for Z being open Subset of REAL st not 0 in Z & Z c= dom ((arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^))) & ( for x being Real st x in Z holds ( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ) holds ( (arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds (((arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^))) `| Z) . x = ((arctan . (1 / x)) - (arccot . (1 / x))) / (1 + (x ^2)) ) ) proof let Z be open Subset of REAL; ::_thesis: ( not 0 in Z & Z c= dom ((arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^))) & ( for x being Real st x in Z holds ( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ) implies ( (arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds (((arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^))) `| Z) . x = ((arctan . (1 / x)) - (arccot . (1 / x))) / (1 + (x ^2)) ) ) ) set f = id Z; assume that A1: not 0 in Z and A2: Z c= dom ((arctan * ((id Z) ^)) (#) (arccot * ((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) ^)) (#) (arccot * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds (((arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^))) `| Z) . x = ((arctan . (1 / x)) - (arccot . (1 / x))) / (1 + (x ^2)) ) ) A4: Z c= (dom (arctan * ((id Z) ^))) /\ (dom (arccot * ((id Z) ^))) by A2, VALUED_1:def_4; then A5: Z c= dom (arctan * ((id Z) ^)) by XBOOLE_1:18; then A6: arctan * ((id Z) ^) is_differentiable_on Z by A1, A3, SIN_COS9:111; A7: Z c= dom (arccot * ((id Z) ^)) by A4, XBOOLE_1:18; then A8: arccot * ((id Z) ^) is_differentiable_on Z by A1, A3, SIN_COS9:112; for y being set st y in Z holds y in dom ((id Z) ^) by A5, FUNCT_1:11; then A9: Z c= dom ((id Z) ^) by TARSKI:def_3; for x being Real st x in Z holds (((arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^))) `| Z) . x = ((arctan . (1 / x)) - (arccot . (1 / x))) / (1 + (x ^2)) proof let x be Real; ::_thesis: ( x in Z implies (((arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^))) `| Z) . x = ((arctan . (1 / x)) - (arccot . (1 / x))) / (1 + (x ^2)) ) assume A10: x in Z ; ::_thesis: (((arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^))) `| Z) . x = ((arctan . (1 / x)) - (arccot . (1 / x))) / (1 + (x ^2)) then (((arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^))) `| Z) . x = (((arccot * ((id Z) ^)) . x) * (diff ((arctan * ((id Z) ^)),x))) + (((arctan * ((id Z) ^)) . x) * (diff ((arccot * ((id Z) ^)),x))) by A2, A6, A8, FDIFF_1:21 .= (((arccot * ((id Z) ^)) . x) * (((arctan * ((id Z) ^)) `| Z) . x)) + (((arctan * ((id Z) ^)) . x) * (diff ((arccot * ((id Z) ^)),x))) by A6, A10, FDIFF_1:def_7 .= (((arccot * ((id Z) ^)) . x) * (- (1 / (1 + (x ^2))))) + (((arctan * ((id Z) ^)) . x) * (diff ((arccot * ((id Z) ^)),x))) by A1, A3, A5, A10, SIN_COS9:111 .= (((arccot * ((id Z) ^)) . x) * (- (1 / (1 + (x ^2))))) + (((arctan * ((id Z) ^)) . x) * (((arccot * ((id Z) ^)) `| Z) . x)) by A8, A10, FDIFF_1:def_7 .= (((arccot * ((id Z) ^)) . x) * (- (1 / (1 + (x ^2))))) + (((arctan * ((id Z) ^)) . x) * (1 / (1 + (x ^2)))) by A1, A3, A7, A10, SIN_COS9:112 .= ((arccot . (((id Z) ^) . x)) * (- (1 / (1 + (x ^2))))) + (((arctan * ((id Z) ^)) . x) * (1 / (1 + (x ^2)))) by A7, A10, FUNCT_1:12 .= ((arccot . (((id Z) . x) ")) * (- (1 / (1 + (x ^2))))) + (((arctan * ((id Z) ^)) . x) * (1 / (1 + (x ^2)))) by A9, A10, RFUNCT_1:def_2 .= ((arccot . (1 / x)) * (- (1 / (1 + (x ^2))))) + (((arctan * ((id Z) ^)) . x) * (1 / (1 + (x ^2)))) by A10, FUNCT_1:18 .= ((arccot . (1 / x)) * (- (1 / (1 + (x ^2))))) + ((arctan . (((id Z) ^) . x)) * (1 / (1 + (x ^2)))) by A5, A10, FUNCT_1:12 .= ((arccot . (1 / x)) * (- (1 / (1 + (x ^2))))) + ((arctan . (((id Z) . x) ")) * (1 / (1 + (x ^2)))) by A9, A10, RFUNCT_1:def_2 .= (- ((arccot . (1 / x)) * (1 / (1 + (x ^2))))) + ((arctan . (1 / x)) * (1 / (1 + (x ^2)))) by A10, FUNCT_1:18 .= ((arctan . (1 / x)) - (arccot . (1 / x))) / (1 + (x ^2)) ; hence (((arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^))) `| Z) . x = ((arctan . (1 / x)) - (arccot . (1 / x))) / (1 + (x ^2)) ; ::_thesis: verum end; hence ( (arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds (((arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^))) `| Z) . x = ((arctan . (1 / x)) - (arccot . (1 / x))) / (1 + (x ^2)) ) ) by A2, A6, A8, FDIFF_1:21; ::_thesis: verum end; theorem :: FDIFF_11:63 for Z being open Subset of REAL st not 0 in Z & Z c= dom ((id Z) (#) (arctan * ((id Z) ^))) & ( for x being Real st x in Z holds ( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ) holds ( (id Z) (#) (arctan * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds (((id Z) (#) (arctan * ((id Z) ^))) `| Z) . x = (arctan . (1 / x)) - (x / (1 + (x ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( not 0 in Z & Z c= dom ((id Z) (#) (arctan * ((id Z) ^))) & ( for x being Real st x in Z holds ( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ) implies ( (id Z) (#) (arctan * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds (((id Z) (#) (arctan * ((id Z) ^))) `| Z) . x = (arctan . (1 / x)) - (x / (1 + (x ^2))) ) ) ) set f = id Z; assume that A1: not 0 in Z and A2: Z c= dom ((id Z) (#) (arctan * ((id Z) ^))) and A3: for x being Real st x in Z holds ( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ; ::_thesis: ( (id Z) (#) (arctan * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds (((id Z) (#) (arctan * ((id Z) ^))) `| Z) . x = (arctan . (1 / x)) - (x / (1 + (x ^2))) ) ) A4: Z c= (dom (id Z)) /\ (dom (arctan * ((id Z) ^))) by A2, VALUED_1:def_4; then A5: Z c= dom (arctan * ((id Z) ^)) by XBOOLE_1:18; then A6: arctan * ((id Z) ^) is_differentiable_on Z by A1, A3, SIN_COS9:111; A7: for x being Real st x in Z holds (id Z) . x = (1 * x) + 0 by FUNCT_1:18; A8: Z c= dom (id Z) by A4, XBOOLE_1:18; then A9: id Z is_differentiable_on Z by A7, FDIFF_1:23; for y being set st y in Z holds y in dom ((id Z) ^) by A5, FUNCT_1:11; then A10: Z c= dom ((id Z) ^) by TARSKI:def_3; for x being Real st x in Z holds (((id Z) (#) (arctan * ((id Z) ^))) `| Z) . x = (arctan . (1 / x)) - (x / (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies (((id Z) (#) (arctan * ((id Z) ^))) `| Z) . x = (arctan . (1 / x)) - (x / (1 + (x ^2))) ) assume A11: x in Z ; ::_thesis: (((id Z) (#) (arctan * ((id Z) ^))) `| Z) . x = (arctan . (1 / x)) - (x / (1 + (x ^2))) then (((id Z) (#) (arctan * ((id Z) ^))) `| Z) . x = (((arctan * ((id Z) ^)) . x) * (diff ((id Z),x))) + (((id Z) . x) * (diff ((arctan * ((id Z) ^)),x))) by A2, A6, A9, FDIFF_1:21 .= (((arctan * ((id Z) ^)) . x) * (((id Z) `| Z) . x)) + (((id Z) . x) * (diff ((arctan * ((id Z) ^)),x))) by A9, A11, FDIFF_1:def_7 .= (((arctan * ((id Z) ^)) . x) * 1) + (((id Z) . x) * (diff ((arctan * ((id Z) ^)),x))) by A8, A7, A11, FDIFF_1:23 .= (((arctan * ((id Z) ^)) . x) * 1) + (x * (diff ((arctan * ((id Z) ^)),x))) by A11, FUNCT_1:18 .= ((arctan * ((id Z) ^)) . x) + (x * (((arctan * ((id Z) ^)) `| Z) . x)) by A6, A11, FDIFF_1:def_7 .= ((arctan * ((id Z) ^)) . x) + (x * (- (1 / (1 + (x ^2))))) by A1, A3, A5, A11, SIN_COS9:111 .= (arctan . (((id Z) ^) . x)) - (x / (1 + (x ^2))) by A5, A11, FUNCT_1:12 .= (arctan . (((id Z) . x) ")) - (x / (1 + (x ^2))) by A10, A11, RFUNCT_1:def_2 .= (arctan . (1 / x)) - (x / (1 + (x ^2))) by A11, FUNCT_1:18 ; hence (((id Z) (#) (arctan * ((id Z) ^))) `| Z) . x = (arctan . (1 / x)) - (x / (1 + (x ^2))) ; ::_thesis: verum end; hence ( (id Z) (#) (arctan * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds (((id Z) (#) (arctan * ((id Z) ^))) `| Z) . x = (arctan . (1 / x)) - (x / (1 + (x ^2))) ) ) by A2, A6, A9, FDIFF_1:21; ::_thesis: verum end; theorem :: FDIFF_11:64 for Z being open Subset of REAL st not 0 in Z & Z c= dom ((id Z) (#) (arccot * ((id Z) ^))) & ( for x being Real st x in Z holds ( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ) holds ( (id Z) (#) (arccot * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds (((id Z) (#) (arccot * ((id Z) ^))) `| Z) . x = (arccot . (1 / x)) + (x / (1 + (x ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( not 0 in Z & Z c= dom ((id Z) (#) (arccot * ((id Z) ^))) & ( for x being Real st x in Z holds ( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ) implies ( (id Z) (#) (arccot * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds (((id Z) (#) (arccot * ((id Z) ^))) `| Z) . x = (arccot . (1 / x)) + (x / (1 + (x ^2))) ) ) ) set f = id Z; assume that A1: not 0 in Z and A2: Z c= dom ((id Z) (#) (arccot * ((id Z) ^))) and A3: for x being Real st x in Z holds ( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ; ::_thesis: ( (id Z) (#) (arccot * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds (((id Z) (#) (arccot * ((id Z) ^))) `| Z) . x = (arccot . (1 / x)) + (x / (1 + (x ^2))) ) ) A4: Z c= (dom (id Z)) /\ (dom (arccot * ((id Z) ^))) by A2, VALUED_1:def_4; then A5: Z c= dom (arccot * ((id Z) ^)) by XBOOLE_1:18; then A6: arccot * ((id Z) ^) is_differentiable_on Z by A1, A3, SIN_COS9:112; A7: for x being Real st x in Z holds (id Z) . x = (1 * x) + 0 by FUNCT_1:18; A8: Z c= dom (id Z) by A4, XBOOLE_1:18; then A9: id Z is_differentiable_on Z by A7, FDIFF_1:23; for y being set st y in Z holds y in dom ((id Z) ^) by A5, FUNCT_1:11; then A10: Z c= dom ((id Z) ^) by TARSKI:def_3; for x being Real st x in Z holds (((id Z) (#) (arccot * ((id Z) ^))) `| Z) . x = (arccot . (1 / x)) + (x / (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies (((id Z) (#) (arccot * ((id Z) ^))) `| Z) . x = (arccot . (1 / x)) + (x / (1 + (x ^2))) ) assume A11: x in Z ; ::_thesis: (((id Z) (#) (arccot * ((id Z) ^))) `| Z) . x = (arccot . (1 / x)) + (x / (1 + (x ^2))) then (((id Z) (#) (arccot * ((id Z) ^))) `| Z) . x = (((arccot * ((id Z) ^)) . x) * (diff ((id Z),x))) + (((id Z) . x) * (diff ((arccot * ((id Z) ^)),x))) by A2, A6, A9, FDIFF_1:21 .= (((arccot * ((id Z) ^)) . x) * (((id Z) `| Z) . x)) + (((id Z) . x) * (diff ((arccot * ((id Z) ^)),x))) by A9, A11, FDIFF_1:def_7 .= (((arccot * ((id Z) ^)) . x) * 1) + (((id Z) . x) * (diff ((arccot * ((id Z) ^)),x))) by A8, A7, A11, FDIFF_1:23 .= (((arccot * ((id Z) ^)) . x) * 1) + (x * (diff ((arccot * ((id Z) ^)),x))) by A11, FUNCT_1:18 .= ((arccot * ((id Z) ^)) . x) + (x * (((arccot * ((id Z) ^)) `| Z) . x)) by A6, A11, FDIFF_1:def_7 .= ((arccot * ((id Z) ^)) . x) + (x * (1 / (1 + (x ^2)))) by A1, A3, A5, A11, SIN_COS9:112 .= (arccot . (((id Z) ^) . x)) + (x / (1 + (x ^2))) by A5, A11, FUNCT_1:12 .= (arccot . (((id Z) . x) ")) + (x / (1 + (x ^2))) by A10, A11, RFUNCT_1:def_2 .= (arccot . (1 / x)) + (x / (1 + (x ^2))) by A11, FUNCT_1:18 ; hence (((id Z) (#) (arccot * ((id Z) ^))) `| Z) . x = (arccot . (1 / x)) + (x / (1 + (x ^2))) ; ::_thesis: verum end; hence ( (id Z) (#) (arccot * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds (((id Z) (#) (arccot * ((id Z) ^))) `| Z) . x = (arccot . (1 / x)) + (x / (1 + (x ^2))) ) ) by A2, A6, A9, FDIFF_1:21; ::_thesis: verum end; theorem :: FDIFF_11:65 for Z being open Subset of REAL for g being PartFunc of REAL,REAL st not 0 in Z & Z c= dom (g (#) (arctan * ((id Z) ^))) & g = #Z 2 & ( for x being Real st x in Z holds ( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ) holds ( g (#) (arctan * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds ((g (#) (arctan * ((id Z) ^))) `| Z) . x = ((2 * x) * (arctan . (1 / x))) - ((x ^2) / (1 + (x ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: for g being PartFunc of REAL,REAL st not 0 in Z & Z c= dom (g (#) (arctan * ((id Z) ^))) & g = #Z 2 & ( for x being Real st x in Z holds ( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ) holds ( g (#) (arctan * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds ((g (#) (arctan * ((id Z) ^))) `| Z) . x = ((2 * x) * (arctan . (1 / x))) - ((x ^2) / (1 + (x ^2))) ) ) let g be PartFunc of REAL,REAL; ::_thesis: ( not 0 in Z & Z c= dom (g (#) (arctan * ((id Z) ^))) & g = #Z 2 & ( for x being Real st x in Z holds ( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ) implies ( g (#) (arctan * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds ((g (#) (arctan * ((id Z) ^))) `| Z) . x = ((2 * x) * (arctan . (1 / x))) - ((x ^2) / (1 + (x ^2))) ) ) ) set f = id Z; assume that A1: not 0 in Z and A2: Z c= dom (g (#) (arctan * ((id Z) ^))) and A3: g = #Z 2 and A4: for x being Real st x in Z holds ( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ; ::_thesis: ( g (#) (arctan * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds ((g (#) (arctan * ((id Z) ^))) `| Z) . x = ((2 * x) * (arctan . (1 / x))) - ((x ^2) / (1 + (x ^2))) ) ) A5: for x being Real st x in Z holds g is_differentiable_in x by A3, TAYLOR_1:2; A6: Z c= (dom g) /\ (dom (arctan * ((id Z) ^))) by A2, VALUED_1:def_4; then A7: Z c= dom (arctan * ((id Z) ^)) by XBOOLE_1:18; then A8: arctan * ((id Z) ^) is_differentiable_on Z by A1, A4, SIN_COS9:111; Z c= dom g by A6, XBOOLE_1:18; then A9: g is_differentiable_on Z by A5, FDIFF_1:9; A10: for x being Real st x in Z holds (g `| Z) . x = 2 * x proof let x be Real; ::_thesis: ( x in Z implies (g `| Z) . x = 2 * x ) assume x in Z ; ::_thesis: (g `| Z) . x = 2 * x then (g `| Z) . x = diff (g,x) by A9, FDIFF_1:def_7 .= 2 * (x #Z (2 - 1)) by A3, TAYLOR_1:2 .= 2 * x by PREPOWER:35 ; hence (g `| Z) . x = 2 * x ; ::_thesis: verum end; for y being set st y in Z holds y in dom ((id Z) ^) by A7, FUNCT_1:11; then A11: Z c= dom ((id Z) ^) by TARSKI:def_3; for x being Real st x in Z holds ((g (#) (arctan * ((id Z) ^))) `| Z) . x = ((2 * x) * (arctan . (1 / x))) - ((x ^2) / (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((g (#) (arctan * ((id Z) ^))) `| Z) . x = ((2 * x) * (arctan . (1 / x))) - ((x ^2) / (1 + (x ^2))) ) assume A12: x in Z ; ::_thesis: ((g (#) (arctan * ((id Z) ^))) `| Z) . x = ((2 * x) * (arctan . (1 / x))) - ((x ^2) / (1 + (x ^2))) then ((g (#) (arctan * ((id Z) ^))) `| Z) . x = (((arctan * ((id Z) ^)) . x) * (diff (g,x))) + ((g . x) * (diff ((arctan * ((id Z) ^)),x))) by A2, A8, A9, FDIFF_1:21 .= (((arctan * ((id Z) ^)) . x) * ((g `| Z) . x)) + ((g . x) * (diff ((arctan * ((id Z) ^)),x))) by A9, A12, FDIFF_1:def_7 .= (((arctan * ((id Z) ^)) . x) * (2 * x)) + ((g . x) * (diff ((arctan * ((id Z) ^)),x))) by A10, A12 .= (((arctan * ((id Z) ^)) . x) * (2 * x)) + ((x #Z 2) * (diff ((arctan * ((id Z) ^)),x))) by A3, TAYLOR_1:def_1 .= (((arctan * ((id Z) ^)) . x) * (2 * x)) + ((x #Z 2) * (((arctan * ((id Z) ^)) `| Z) . x)) by A8, A12, FDIFF_1:def_7 .= (((arctan * ((id Z) ^)) . x) * (2 * x)) + ((x #Z (1 + 1)) * (- (1 / (1 + (x ^2))))) by A1, A4, A7, A12, SIN_COS9:111 .= (((arctan * ((id Z) ^)) . x) * (2 * x)) + (((x #Z 1) * (x #Z 1)) * (- (1 / (1 + (x ^2))))) by TAYLOR_1:1 .= (((arctan * ((id Z) ^)) . x) * (2 * x)) + ((x * (x #Z 1)) * (- (1 / (1 + (x ^2))))) by PREPOWER:35 .= (((arctan * ((id Z) ^)) . x) * (2 * x)) + ((x ^2) * (- (1 / (1 + (x ^2))))) by PREPOWER:35 .= ((arctan . (((id Z) ^) . x)) * (2 * x)) - ((x ^2) / (1 + (x ^2))) by A7, A12, FUNCT_1:12 .= ((arctan . (((id Z) . x) ")) * (2 * x)) - ((x ^2) / (1 + (x ^2))) by A11, A12, RFUNCT_1:def_2 .= ((2 * x) * (arctan . (1 / x))) - ((x ^2) / (1 + (x ^2))) by A12, FUNCT_1:18 ; hence ((g (#) (arctan * ((id Z) ^))) `| Z) . x = ((2 * x) * (arctan . (1 / x))) - ((x ^2) / (1 + (x ^2))) ; ::_thesis: verum end; hence ( g (#) (arctan * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds ((g (#) (arctan * ((id Z) ^))) `| Z) . x = ((2 * x) * (arctan . (1 / x))) - ((x ^2) / (1 + (x ^2))) ) ) by A2, A8, A9, FDIFF_1:21; ::_thesis: verum end; theorem :: FDIFF_11:66 for Z being open Subset of REAL for g being PartFunc of REAL,REAL st not 0 in Z & Z c= dom (g (#) (arccot * ((id Z) ^))) & g = #Z 2 & ( for x being Real st x in Z holds ( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ) holds ( g (#) (arccot * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds ((g (#) (arccot * ((id Z) ^))) `| Z) . x = ((2 * x) * (arccot . (1 / x))) + ((x ^2) / (1 + (x ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: for g being PartFunc of REAL,REAL st not 0 in Z & Z c= dom (g (#) (arccot * ((id Z) ^))) & g = #Z 2 & ( for x being Real st x in Z holds ( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ) holds ( g (#) (arccot * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds ((g (#) (arccot * ((id Z) ^))) `| Z) . x = ((2 * x) * (arccot . (1 / x))) + ((x ^2) / (1 + (x ^2))) ) ) let g be PartFunc of REAL,REAL; ::_thesis: ( not 0 in Z & Z c= dom (g (#) (arccot * ((id Z) ^))) & g = #Z 2 & ( for x being Real st x in Z holds ( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ) implies ( g (#) (arccot * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds ((g (#) (arccot * ((id Z) ^))) `| Z) . x = ((2 * x) * (arccot . (1 / x))) + ((x ^2) / (1 + (x ^2))) ) ) ) set f = id Z; assume that A1: not 0 in Z and A2: Z c= dom (g (#) (arccot * ((id Z) ^))) and A3: g = #Z 2 and A4: for x being Real st x in Z holds ( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ; ::_thesis: ( g (#) (arccot * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds ((g (#) (arccot * ((id Z) ^))) `| Z) . x = ((2 * x) * (arccot . (1 / x))) + ((x ^2) / (1 + (x ^2))) ) ) A5: for x being Real st x in Z holds g is_differentiable_in x by A3, TAYLOR_1:2; A6: Z c= (dom g) /\ (dom (arccot * ((id Z) ^))) by A2, VALUED_1:def_4; then A7: Z c= dom (arccot * ((id Z) ^)) by XBOOLE_1:18; then A8: arccot * ((id Z) ^) is_differentiable_on Z by A1, A4, SIN_COS9:112; Z c= dom g by A6, XBOOLE_1:18; then A9: g is_differentiable_on Z by A5, FDIFF_1:9; A10: for x being Real st x in Z holds (g `| Z) . x = 2 * x proof let x be Real; ::_thesis: ( x in Z implies (g `| Z) . x = 2 * x ) assume x in Z ; ::_thesis: (g `| Z) . x = 2 * x then (g `| Z) . x = diff (g,x) by A9, FDIFF_1:def_7 .= 2 * (x #Z (2 - 1)) by A3, TAYLOR_1:2 .= 2 * x by PREPOWER:35 ; hence (g `| Z) . x = 2 * x ; ::_thesis: verum end; for y being set st y in Z holds y in dom ((id Z) ^) by A7, FUNCT_1:11; then A11: Z c= dom ((id Z) ^) by TARSKI:def_3; for x being Real st x in Z holds ((g (#) (arccot * ((id Z) ^))) `| Z) . x = ((2 * x) * (arccot . (1 / x))) + ((x ^2) / (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((g (#) (arccot * ((id Z) ^))) `| Z) . x = ((2 * x) * (arccot . (1 / x))) + ((x ^2) / (1 + (x ^2))) ) assume A12: x in Z ; ::_thesis: ((g (#) (arccot * ((id Z) ^))) `| Z) . x = ((2 * x) * (arccot . (1 / x))) + ((x ^2) / (1 + (x ^2))) then ((g (#) (arccot * ((id Z) ^))) `| Z) . x = (((arccot * ((id Z) ^)) . x) * (diff (g,x))) + ((g . x) * (diff ((arccot * ((id Z) ^)),x))) by A2, A8, A9, FDIFF_1:21 .= (((arccot * ((id Z) ^)) . x) * ((g `| Z) . x)) + ((g . x) * (diff ((arccot * ((id Z) ^)),x))) by A9, A12, FDIFF_1:def_7 .= (((arccot * ((id Z) ^)) . x) * (2 * x)) + ((g . x) * (diff ((arccot * ((id Z) ^)),x))) by A10, A12 .= (((arccot * ((id Z) ^)) . x) * (2 * x)) + ((x #Z 2) * (diff ((arccot * ((id Z) ^)),x))) by A3, TAYLOR_1:def_1 .= (((arccot * ((id Z) ^)) . x) * (2 * x)) + ((x #Z 2) * (((arccot * ((id Z) ^)) `| Z) . x)) by A8, A12, FDIFF_1:def_7 .= (((arccot * ((id Z) ^)) . x) * (2 * x)) + ((x #Z (1 + 1)) * (1 / (1 + (x ^2)))) by A1, A4, A7, A12, SIN_COS9:112 .= (((arccot * ((id Z) ^)) . x) * (2 * x)) + (((x #Z 1) * (x #Z 1)) * (1 / (1 + (x ^2)))) by TAYLOR_1:1 .= (((arccot * ((id Z) ^)) . x) * (2 * x)) + ((x * (x #Z 1)) * (1 / (1 + (x ^2)))) by PREPOWER:35 .= (((arccot * ((id Z) ^)) . x) * (2 * x)) + ((x ^2) / (1 + (x ^2))) by PREPOWER:35 .= ((arccot . (((id Z) ^) . x)) * (2 * x)) + ((x ^2) / (1 + (x ^2))) by A7, A12, FUNCT_1:12 .= ((arccot . (((id Z) . x) ")) * (2 * x)) + ((x ^2) / (1 + (x ^2))) by A11, A12, RFUNCT_1:def_2 .= ((2 * x) * (arccot . (1 / x))) + ((x ^2) / (1 + (x ^2))) by A12, FUNCT_1:18 ; hence ((g (#) (arccot * ((id Z) ^))) `| Z) . x = ((2 * x) * (arccot . (1 / x))) + ((x ^2) / (1 + (x ^2))) ; ::_thesis: verum end; hence ( g (#) (arccot * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds ((g (#) (arccot * ((id Z) ^))) `| Z) . x = ((2 * x) * (arccot . (1 / x))) + ((x ^2) / (1 + (x ^2))) ) ) by A2, A8, A9, FDIFF_1:21; ::_thesis: verum end; theorem Th67: :: FDIFF_11:67 for Z being open Subset of REAL st Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds arctan . x <> 0 ) holds ( arctan ^ is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan ^) `| Z) . x = - (1 / (((arctan . x) ^2) * (1 + (x ^2)))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds arctan . x <> 0 ) implies ( arctan ^ is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan ^) `| Z) . x = - (1 / (((arctan . x) ^2) * (1 + (x ^2)))) ) ) ) assume that A1: Z c= ].(- 1),1.[ and A2: for x being Real st x in Z holds arctan . x <> 0 ; ::_thesis: ( arctan ^ is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan ^) `| Z) . x = - (1 / (((arctan . x) ^2) * (1 + (x ^2)))) ) ) A3: arctan is_differentiable_on Z by A1, SIN_COS9:81; then A4: arctan ^ is_differentiable_on Z by A2, FDIFF_2:22; for x being Real st x in Z holds ((arctan ^) `| Z) . x = - (1 / (((arctan . x) ^2) * (1 + (x ^2)))) proof let x be Real; ::_thesis: ( x in Z implies ((arctan ^) `| Z) . x = - (1 / (((arctan . x) ^2) * (1 + (x ^2)))) ) assume A5: x in Z ; ::_thesis: ((arctan ^) `| Z) . x = - (1 / (((arctan . x) ^2) * (1 + (x ^2)))) then A6: ( arctan . x <> 0 & arctan is_differentiable_in x ) by A2, A3, FDIFF_1:9; ((arctan ^) `| Z) . x = diff ((arctan ^),x) by A4, A5, FDIFF_1:def_7 .= - ((diff (arctan,x)) / ((arctan . x) ^2)) by A6, FDIFF_2:15 .= - (((arctan `| Z) . x) / ((arctan . x) ^2)) by A3, A5, FDIFF_1:def_7 .= - ((1 / (1 + (x ^2))) / ((arctan . x) ^2)) by A1, A5, SIN_COS9:81 .= - (1 / (((arctan . x) ^2) * (1 + (x ^2)))) by XCMPLX_1:78 ; hence ((arctan ^) `| Z) . x = - (1 / (((arctan . x) ^2) * (1 + (x ^2)))) ; ::_thesis: verum end; hence ( arctan ^ is_differentiable_on Z & ( for x being Real st x in Z holds ((arctan ^) `| Z) . x = - (1 / (((arctan . x) ^2) * (1 + (x ^2)))) ) ) by A2, A3, FDIFF_2:22; ::_thesis: verum end; theorem Th68: :: FDIFF_11:68 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 / (((arccot . x) ^2) * (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 / (((arccot . x) ^2) * (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 / (((arccot . x) ^2) * (1 + (x ^2))) ) ) then A2: arccot is_differentiable_on Z by SIN_COS9:82; A3: for x being Real st x in Z holds arccot . x <> 0 proof PI in ].0,4.[ by SIN_COS:def_28; then PI > 0 by XXREAL_1:4; then A4: PI / 4 > 0 / 4 by XREAL_1:74; let x be Real; ::_thesis: ( x in Z implies arccot . x <> 0 ) assume A5: x in Z ; ::_thesis: arccot . x <> 0 assume A6: arccot . x = 0 ; ::_thesis: contradiction ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25; then Z c= [.(- 1),1.] by A1, XBOOLE_1:1; then x in [.(- 1),1.] by A5; then 0 in arccot .: [.(- 1),1.] by A6, FUNCT_1:def_6, SIN_COS9:24; then 0 in [.(PI / 4),((3 / 4) * PI).] by RELAT_1:115, SIN_COS9:56; hence contradiction by A4, XXREAL_1:1; ::_thesis: verum end; then A7: arccot ^ is_differentiable_on Z by A2, FDIFF_2:22; for x being Real st x in Z holds ((arccot ^) `| Z) . x = 1 / (((arccot . x) ^2) * (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((arccot ^) `| Z) . x = 1 / (((arccot . x) ^2) * (1 + (x ^2))) ) assume A8: x in Z ; ::_thesis: ((arccot ^) `| Z) . x = 1 / (((arccot . x) ^2) * (1 + (x ^2))) then A9: ( arccot . x <> 0 & arccot is_differentiable_in x ) by A3, A2, FDIFF_1:9; ((arccot ^) `| Z) . x = diff ((arccot ^),x) by A7, A8, FDIFF_1:def_7 .= - ((diff (arccot,x)) / ((arccot . x) ^2)) by A9, FDIFF_2:15 .= - (((arccot `| Z) . x) / ((arccot . x) ^2)) by A2, A8, FDIFF_1:def_7 .= - ((- (1 / (1 + (x ^2)))) / ((arccot . x) ^2)) by A1, A8, SIN_COS9:82 .= (1 / (1 + (x ^2))) / ((arccot . x) ^2) .= 1 / (((arccot . x) ^2) * (1 + (x ^2))) by XCMPLX_1:78 ; hence ((arccot ^) `| Z) . x = 1 / (((arccot . x) ^2) * (1 + (x ^2))) ; ::_thesis: verum end; hence ( arccot ^ is_differentiable_on Z & ( for x being Real st x in Z holds ((arccot ^) `| Z) . x = 1 / (((arccot . x) ^2) * (1 + (x ^2))) ) ) by A3, A2, FDIFF_2:22; ::_thesis: verum end; theorem :: FDIFF_11:69 for n being Element of NAT for Z being open Subset of REAL st Z c= dom ((1 / n) (#) ((#Z n) * (arctan ^))) & Z c= ].(- 1),1.[ & n > 0 & ( for x being Real st x in Z holds arctan . x <> 0 ) holds ( (1 / n) (#) ((#Z n) * (arctan ^)) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / n) (#) ((#Z n) * (arctan ^))) `| Z) . x = - (1 / (((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 ((1 / n) (#) ((#Z n) * (arctan ^))) & Z c= ].(- 1),1.[ & n > 0 & ( for x being Real st x in Z holds arctan . x <> 0 ) holds ( (1 / n) (#) ((#Z n) * (arctan ^)) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / n) (#) ((#Z n) * (arctan ^))) `| Z) . x = - (1 / (((arctan . x) #Z (n + 1)) * (1 + (x ^2)))) ) ) let Z be open Subset of REAL; ::_thesis: ( Z c= dom ((1 / n) (#) ((#Z n) * (arctan ^))) & Z c= ].(- 1),1.[ & n > 0 & ( for x being Real st x in Z holds arctan . x <> 0 ) implies ( (1 / n) (#) ((#Z n) * (arctan ^)) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / n) (#) ((#Z n) * (arctan ^))) `| Z) . x = - (1 / (((arctan . x) #Z (n + 1)) * (1 + (x ^2)))) ) ) ) assume that A1: Z c= dom ((1 / n) (#) ((#Z n) * (arctan ^))) and A2: Z c= ].(- 1),1.[ and A3: n > 0 and A4: for x being Real st x in Z holds arctan . x <> 0 ; ::_thesis: ( (1 / n) (#) ((#Z n) * (arctan ^)) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / n) (#) ((#Z n) * (arctan ^))) `| Z) . x = - (1 / (((arctan . x) #Z (n + 1)) * (1 + (x ^2)))) ) ) A5: Z c= dom ((#Z n) * (arctan ^)) by A1, VALUED_1:def_5; A6: arctan ^ is_differentiable_on Z by A2, A4, Th67; 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 x in Z ; ::_thesis: (#Z n) * (arctan ^) is_differentiable_in x then arctan ^ is_differentiable_in x by A6, FDIFF_1:9; hence (#Z n) * (arctan ^) is_differentiable_in x by TAYLOR_1:3; ::_thesis: verum end; then A7: (#Z n) * (arctan ^) is_differentiable_on Z by A5, FDIFF_1:9; for y being set st y in Z holds y in dom (arctan ^) by A5, FUNCT_1:11; then A8: Z c= dom (arctan ^) by TARSKI:def_3; for x being Real st x in Z holds (((1 / n) (#) ((#Z n) * (arctan ^))) `| Z) . x = - (1 / (((arctan . x) #Z (n + 1)) * (1 + (x ^2)))) proof let x be Real; ::_thesis: ( x in Z implies (((1 / n) (#) ((#Z n) * (arctan ^))) `| Z) . x = - (1 / (((arctan . x) #Z (n + 1)) * (1 + (x ^2)))) ) assume A9: x in Z ; ::_thesis: (((1 / n) (#) ((#Z n) * (arctan ^))) `| Z) . x = - (1 / (((arctan . x) #Z (n + 1)) * (1 + (x ^2)))) then A10: arctan ^ is_differentiable_in x by A6, FDIFF_1:9; A11: (arctan ^) . x = 1 / (arctan . x) by A8, A9, RFUNCT_1:def_2; (((1 / n) (#) ((#Z n) * (arctan ^))) `| Z) . x = (1 / n) * (diff (((#Z n) * (arctan ^)),x)) by A1, A7, A9, FDIFF_1:20 .= (1 / n) * ((n * (((arctan ^) . x) #Z (n - 1))) * (diff ((arctan ^),x))) by A10, TAYLOR_1:3 .= (1 / n) * ((n * (((arctan ^) . x) #Z (n - 1))) * (((arctan ^) `| Z) . x)) by A6, A9, FDIFF_1:def_7 .= (1 / n) * ((n * (((arctan ^) . x) #Z (n - 1))) * (- (1 / (((arctan . x) ^2) * (1 + (x ^2)))))) by A2, A4, A9, Th67 .= - ((((1 / n) * n) * (((arctan ^) . x) #Z (n - 1))) * (1 / (((arctan . x) ^2) * (1 + (x ^2))))) .= - ((1 * (((arctan ^) . x) #Z (n - 1))) * (1 / (((arctan . x) ^2) * (1 + (x ^2))))) by A3, XCMPLX_1:106 .= - (((1 / (arctan . x)) #Z (n - 1)) * (1 / (((arctan . x) #Z 2) * (1 + (x ^2))))) by A11, FDIFF_7:1 .= - ((1 / ((arctan . x) #Z (n - 1))) / (((arctan . x) #Z 2) * (1 + (x ^2)))) by PREPOWER:42 .= - (1 / (((arctan . x) #Z (n - 1)) * (((arctan . x) #Z 2) * (1 + (x ^2))))) by XCMPLX_1:78 .= - (1 / ((((arctan . x) #Z (n - 1)) * ((arctan . x) #Z 2)) * (1 + (x ^2)))) .= - (1 / (((arctan . x) #Z ((n - 1) + 2)) * (1 + (x ^2)))) by A4, A9, PREPOWER:44 .= - (1 / (((arctan . x) #Z (n + 1)) * (1 + (x ^2)))) ; hence (((1 / n) (#) ((#Z n) * (arctan ^))) `| Z) . x = - (1 / (((arctan . x) #Z (n + 1)) * (1 + (x ^2)))) ; ::_thesis: verum end; hence ( (1 / n) (#) ((#Z n) * (arctan ^)) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / n) (#) ((#Z n) * (arctan ^))) `| Z) . x = - (1 / (((arctan . x) #Z (n + 1)) * (1 + (x ^2)))) ) ) by A1, A7, FDIFF_1:20; ::_thesis: verum end; theorem :: FDIFF_11:70 for n being Element of NAT for Z being open Subset of REAL st Z c= dom ((1 / n) (#) ((#Z n) * (arccot ^))) & Z c= ].(- 1),1.[ & n > 0 holds ( (1 / n) (#) ((#Z n) * (arccot ^)) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / n) (#) ((#Z n) * (arccot ^))) `| Z) . x = 1 / (((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 ((1 / n) (#) ((#Z n) * (arccot ^))) & Z c= ].(- 1),1.[ & n > 0 holds ( (1 / n) (#) ((#Z n) * (arccot ^)) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / n) (#) ((#Z n) * (arccot ^))) `| Z) . x = 1 / (((arccot . x) #Z (n + 1)) * (1 + (x ^2))) ) ) let Z be open Subset of REAL; ::_thesis: ( Z c= dom ((1 / n) (#) ((#Z n) * (arccot ^))) & Z c= ].(- 1),1.[ & n > 0 implies ( (1 / n) (#) ((#Z n) * (arccot ^)) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / n) (#) ((#Z n) * (arccot ^))) `| Z) . x = 1 / (((arccot . x) #Z (n + 1)) * (1 + (x ^2))) ) ) ) assume that A1: Z c= dom ((1 / n) (#) ((#Z n) * (arccot ^))) and A2: Z c= ].(- 1),1.[ and A3: n > 0 ; ::_thesis: ( (1 / n) (#) ((#Z n) * (arccot ^)) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / n) (#) ((#Z n) * (arccot ^))) `| Z) . x = 1 / (((arccot . x) #Z (n + 1)) * (1 + (x ^2))) ) ) A4: Z c= dom ((#Z n) * (arccot ^)) by A1, VALUED_1:def_5; A5: for x being Real st x in Z holds arccot . x <> 0 proof PI in ].0,4.[ by SIN_COS:def_28; then PI > 0 by XXREAL_1:4; then A6: PI / 4 > 0 / 4 by XREAL_1:74; let x be Real; ::_thesis: ( x in Z implies arccot . x <> 0 ) assume A7: x in Z ; ::_thesis: arccot . x <> 0 assume A8: arccot . x = 0 ; ::_thesis: contradiction ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25; then Z c= [.(- 1),1.] by A2, XBOOLE_1:1; then x in [.(- 1),1.] by A7; then 0 in arccot .: [.(- 1),1.] by A8, FUNCT_1:def_6, SIN_COS9:24; then 0 in [.(PI / 4),((3 / 4) * PI).] by RELAT_1:115, SIN_COS9:56; hence contradiction by A6, XXREAL_1:1; ::_thesis: verum end; A9: arccot ^ is_differentiable_on Z by A2, Th68; 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 x in Z ; ::_thesis: (#Z n) * (arccot ^) is_differentiable_in x then arccot ^ is_differentiable_in x by A9, FDIFF_1:9; hence (#Z n) * (arccot ^) is_differentiable_in x by TAYLOR_1:3; ::_thesis: verum end; then A10: (#Z n) * (arccot ^) is_differentiable_on Z by A4, FDIFF_1:9; for y being set st y in Z holds y in dom (arccot ^) by A4, FUNCT_1:11; then A11: Z c= dom (arccot ^) by TARSKI:def_3; for x being Real st x in Z holds (((1 / n) (#) ((#Z n) * (arccot ^))) `| Z) . x = 1 / (((arccot . x) #Z (n + 1)) * (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies (((1 / n) (#) ((#Z n) * (arccot ^))) `| Z) . x = 1 / (((arccot . x) #Z (n + 1)) * (1 + (x ^2))) ) assume A12: x in Z ; ::_thesis: (((1 / n) (#) ((#Z n) * (arccot ^))) `| Z) . x = 1 / (((arccot . x) #Z (n + 1)) * (1 + (x ^2))) then A13: arccot ^ is_differentiable_in x by A9, FDIFF_1:9; A14: (arccot ^) . x = 1 / (arccot . x) by A11, A12, RFUNCT_1:def_2; (((1 / n) (#) ((#Z n) * (arccot ^))) `| Z) . x = (1 / n) * (diff (((#Z n) * (arccot ^)),x)) by A1, A10, A12, FDIFF_1:20 .= (1 / n) * ((n * (((arccot ^) . x) #Z (n - 1))) * (diff ((arccot ^),x))) by A13, TAYLOR_1:3 .= (1 / n) * ((n * (((arccot ^) . x) #Z (n - 1))) * (((arccot ^) `| Z) . x)) by A9, A12, FDIFF_1:def_7 .= (1 / n) * ((n * (((arccot ^) . x) #Z (n - 1))) * (1 / (((arccot . x) ^2) * (1 + (x ^2))))) by A2, A12, Th68 .= (((1 / n) * n) * (((arccot ^) . x) #Z (n - 1))) * (1 / (((arccot . x) ^2) * (1 + (x ^2)))) .= (1 * (((arccot ^) . x) #Z (n - 1))) * (1 / (((arccot . x) ^2) * (1 + (x ^2)))) by A3, XCMPLX_1:106 .= ((1 / (arccot . x)) #Z (n - 1)) * (1 / (((arccot . x) #Z 2) * (1 + (x ^2)))) by A14, FDIFF_7:1 .= (1 / ((arccot . x) #Z (n - 1))) / (((arccot . x) #Z 2) * (1 + (x ^2))) by PREPOWER:42 .= 1 / (((arccot . x) #Z (n - 1)) * (((arccot . x) #Z 2) * (1 + (x ^2)))) by XCMPLX_1:78 .= 1 / ((((arccot . x) #Z (n - 1)) * ((arccot . x) #Z 2)) * (1 + (x ^2))) .= 1 / (((arccot . x) #Z ((n - 1) + 2)) * (1 + (x ^2))) by A5, A12, PREPOWER:44 .= 1 / (((arccot . x) #Z (n + 1)) * (1 + (x ^2))) ; hence (((1 / n) (#) ((#Z n) * (arccot ^))) `| Z) . x = 1 / (((arccot . x) #Z (n + 1)) * (1 + (x ^2))) ; ::_thesis: verum end; hence ( (1 / n) (#) ((#Z n) * (arccot ^)) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / n) (#) ((#Z n) * (arccot ^))) `| Z) . x = 1 / (((arccot . x) #Z (n + 1)) * (1 + (x ^2))) ) ) by A1, A10, FDIFF_1:20; ::_thesis: verum end; theorem :: FDIFF_11:71 for Z being open Subset of REAL st Z c= dom (2 (#) ((#R (1 / 2)) * arctan)) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds arctan . x > 0 ) holds ( 2 (#) ((#R (1 / 2)) * arctan) is_differentiable_on Z & ( for x being Real st x in Z holds ((2 (#) ((#R (1 / 2)) * arctan)) `| Z) . x = ((arctan . x) #R (- (1 / 2))) / (1 + (x ^2)) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (2 (#) ((#R (1 / 2)) * arctan)) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds arctan . x > 0 ) implies ( 2 (#) ((#R (1 / 2)) * arctan) is_differentiable_on Z & ( for x being Real st x in Z holds ((2 (#) ((#R (1 / 2)) * arctan)) `| Z) . x = ((arctan . x) #R (- (1 / 2))) / (1 + (x ^2)) ) ) ) assume that A1: Z c= dom (2 (#) ((#R (1 / 2)) * arctan)) and A2: Z c= ].(- 1),1.[ and A3: for x being Real st x in Z holds arctan . x > 0 ; ::_thesis: ( 2 (#) ((#R (1 / 2)) * arctan) is_differentiable_on Z & ( for x being Real st x in Z holds ((2 (#) ((#R (1 / 2)) * arctan)) `| Z) . x = ((arctan . x) #R (- (1 / 2))) / (1 + (x ^2)) ) ) A4: for x being Real st x in Z holds (#R (1 / 2)) * arctan is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies (#R (1 / 2)) * arctan is_differentiable_in x ) assume A5: x in Z ; ::_thesis: (#R (1 / 2)) * arctan is_differentiable_in x then A6: arctan . x > 0 by A3; arctan is_differentiable_on Z by A2, SIN_COS9:81; then arctan is_differentiable_in x by A5, FDIFF_1:9; hence (#R (1 / 2)) * arctan is_differentiable_in x by A6, TAYLOR_1:22; ::_thesis: verum end; Z c= dom ((#R (1 / 2)) * arctan) by A1, VALUED_1:def_5; then A7: (#R (1 / 2)) * arctan is_differentiable_on Z by A4, FDIFF_1:9; for x being Real st x in Z holds ((2 (#) ((#R (1 / 2)) * arctan)) `| Z) . x = ((arctan . x) #R (- (1 / 2))) / (1 + (x ^2)) proof let x be Real; ::_thesis: ( x in Z implies ((2 (#) ((#R (1 / 2)) * arctan)) `| Z) . x = ((arctan . x) #R (- (1 / 2))) / (1 + (x ^2)) ) assume A8: x in Z ; ::_thesis: ((2 (#) ((#R (1 / 2)) * arctan)) `| Z) . x = ((arctan . x) #R (- (1 / 2))) / (1 + (x ^2)) then A9: arctan . x > 0 by A3; A10: arctan is_differentiable_on Z by A2, SIN_COS9:81; then A11: arctan is_differentiable_in x by A8, FDIFF_1:9; ((2 (#) ((#R (1 / 2)) * arctan)) `| Z) . x = 2 * (diff (((#R (1 / 2)) * arctan),x)) by A1, A7, A8, FDIFF_1:20 .= 2 * (((1 / 2) * ((arctan . x) #R ((1 / 2) - 1))) * (diff (arctan,x))) by A11, A9, TAYLOR_1:22 .= 2 * (((1 / 2) * ((arctan . x) #R ((1 / 2) - 1))) * ((arctan `| Z) . x)) by A8, A10, FDIFF_1:def_7 .= 2 * (((1 / 2) * ((arctan . x) #R ((1 / 2) - 1))) * (1 / (1 + (x ^2)))) by A2, A8, SIN_COS9:81 .= ((arctan . x) #R (- (1 / 2))) / (1 + (x ^2)) ; hence ((2 (#) ((#R (1 / 2)) * arctan)) `| Z) . x = ((arctan . x) #R (- (1 / 2))) / (1 + (x ^2)) ; ::_thesis: verum end; hence ( 2 (#) ((#R (1 / 2)) * arctan) is_differentiable_on Z & ( for x being Real st x in Z holds ((2 (#) ((#R (1 / 2)) * arctan)) `| Z) . x = ((arctan . x) #R (- (1 / 2))) / (1 + (x ^2)) ) ) by A1, A7, FDIFF_1:20; ::_thesis: verum end; theorem :: FDIFF_11:72 for Z being open Subset of REAL st Z c= dom (2 (#) ((#R (1 / 2)) * arccot)) & Z c= ].(- 1),1.[ holds ( 2 (#) ((#R (1 / 2)) * arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((2 (#) ((#R (1 / 2)) * arccot)) `| Z) . x = - (((arccot . x) #R (- (1 / 2))) / (1 + (x ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (2 (#) ((#R (1 / 2)) * arccot)) & Z c= ].(- 1),1.[ implies ( 2 (#) ((#R (1 / 2)) * arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((2 (#) ((#R (1 / 2)) * arccot)) `| Z) . x = - (((arccot . x) #R (- (1 / 2))) / (1 + (x ^2))) ) ) ) assume that A1: Z c= dom (2 (#) ((#R (1 / 2)) * arccot)) and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( 2 (#) ((#R (1 / 2)) * arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((2 (#) ((#R (1 / 2)) * arccot)) `| Z) . x = - (((arccot . x) #R (- (1 / 2))) / (1 + (x ^2))) ) ) A3: for x being Real st x in Z holds arccot . x > 0 proof let x be Real; ::_thesis: ( x in Z implies arccot . x > 0 ) ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25; then A4: Z c= [.(- 1),1.] by A2, XBOOLE_1:1; assume x in Z ; ::_thesis: arccot . x > 0 then x in [.(- 1),1.] by A4; then arccot . x in arccot .: [.(- 1),1.] by FUNCT_1:def_6, SIN_COS9:24; then arccot . x in [.(PI / 4),((3 / 4) * PI).] by RELAT_1:115, SIN_COS9:56; then arccot . x >= PI / 4 by XXREAL_1:1; then A5: (arccot . x) + 0 > (PI / 4) + (- (PI / 4)) by XREAL_1:8; assume arccot . x <= 0 ; ::_thesis: contradiction hence contradiction by A5; ::_thesis: verum end; A6: for x being Real st x in Z holds (#R (1 / 2)) * arccot is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies (#R (1 / 2)) * arccot is_differentiable_in x ) assume A7: x in Z ; ::_thesis: (#R (1 / 2)) * arccot is_differentiable_in x then A8: arccot . x > 0 by A3; arccot is_differentiable_on Z by A2, SIN_COS9:82; then arccot is_differentiable_in x by A7, FDIFF_1:9; hence (#R (1 / 2)) * arccot is_differentiable_in x by A8, TAYLOR_1:22; ::_thesis: verum end; Z c= dom ((#R (1 / 2)) * arccot) by A1, VALUED_1:def_5; then A9: (#R (1 / 2)) * arccot is_differentiable_on Z by A6, FDIFF_1:9; for x being Real st x in Z holds ((2 (#) ((#R (1 / 2)) * arccot)) `| Z) . x = - (((arccot . x) #R (- (1 / 2))) / (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies ((2 (#) ((#R (1 / 2)) * arccot)) `| Z) . x = - (((arccot . x) #R (- (1 / 2))) / (1 + (x ^2))) ) assume A10: x in Z ; ::_thesis: ((2 (#) ((#R (1 / 2)) * arccot)) `| Z) . x = - (((arccot . x) #R (- (1 / 2))) / (1 + (x ^2))) then A11: arccot . x > 0 by A3; A12: arccot is_differentiable_on Z by A2, SIN_COS9:82; then A13: arccot is_differentiable_in x by A10, FDIFF_1:9; ((2 (#) ((#R (1 / 2)) * arccot)) `| Z) . x = 2 * (diff (((#R (1 / 2)) * arccot),x)) by A1, A9, A10, FDIFF_1:20 .= 2 * (((1 / 2) * ((arccot . x) #R ((1 / 2) - 1))) * (diff (arccot,x))) by A13, A11, TAYLOR_1:22 .= 2 * (((1 / 2) * ((arccot . x) #R ((1 / 2) - 1))) * ((arccot `| Z) . x)) by A10, A12, FDIFF_1:def_7 .= 2 * (((1 / 2) * ((arccot . x) #R ((1 / 2) - 1))) * (- (1 / (1 + (x ^2))))) by A2, A10, SIN_COS9:82 .= - (((arccot . x) #R (- (1 / 2))) / (1 + (x ^2))) ; hence ((2 (#) ((#R (1 / 2)) * arccot)) `| Z) . x = - (((arccot . x) #R (- (1 / 2))) / (1 + (x ^2))) ; ::_thesis: verum end; hence ( 2 (#) ((#R (1 / 2)) * arccot) is_differentiable_on Z & ( for x being Real st x in Z holds ((2 (#) ((#R (1 / 2)) * arccot)) `| Z) . x = - (((arccot . x) #R (- (1 / 2))) / (1 + (x ^2))) ) ) by A1, A9, FDIFF_1:20; ::_thesis: verum end; theorem :: FDIFF_11:73 for Z being open Subset of REAL st Z c= dom ((2 / 3) (#) ((#R (3 / 2)) * arctan)) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds arctan . x > 0 ) holds ( (2 / 3) (#) ((#R (3 / 2)) * arctan) is_differentiable_on Z & ( for x being Real st x in Z holds (((2 / 3) (#) ((#R (3 / 2)) * arctan)) `| Z) . x = ((arctan . x) #R (1 / 2)) / (1 + (x ^2)) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom ((2 / 3) (#) ((#R (3 / 2)) * arctan)) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds arctan . x > 0 ) implies ( (2 / 3) (#) ((#R (3 / 2)) * arctan) is_differentiable_on Z & ( for x being Real st x in Z holds (((2 / 3) (#) ((#R (3 / 2)) * arctan)) `| Z) . x = ((arctan . x) #R (1 / 2)) / (1 + (x ^2)) ) ) ) assume that A1: Z c= dom ((2 / 3) (#) ((#R (3 / 2)) * arctan)) and A2: Z c= ].(- 1),1.[ and A3: for x being Real st x in Z holds arctan . x > 0 ; ::_thesis: ( (2 / 3) (#) ((#R (3 / 2)) * arctan) is_differentiable_on Z & ( for x being Real st x in Z holds (((2 / 3) (#) ((#R (3 / 2)) * arctan)) `| Z) . x = ((arctan . x) #R (1 / 2)) / (1 + (x ^2)) ) ) A4: for x being Real st x in Z holds (#R (3 / 2)) * arctan is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies (#R (3 / 2)) * arctan is_differentiable_in x ) assume A5: x in Z ; ::_thesis: (#R (3 / 2)) * arctan is_differentiable_in x then A6: arctan . x > 0 by A3; arctan is_differentiable_on Z by A2, SIN_COS9:81; then arctan is_differentiable_in x by A5, FDIFF_1:9; hence (#R (3 / 2)) * arctan is_differentiable_in x by A6, TAYLOR_1:22; ::_thesis: verum end; Z c= dom ((#R (3 / 2)) * arctan) by A1, VALUED_1:def_5; then A7: (#R (3 / 2)) * arctan is_differentiable_on Z by A4, FDIFF_1:9; for x being Real st x in Z holds (((2 / 3) (#) ((#R (3 / 2)) * arctan)) `| Z) . x = ((arctan . x) #R (1 / 2)) / (1 + (x ^2)) proof let x be Real; ::_thesis: ( x in Z implies (((2 / 3) (#) ((#R (3 / 2)) * arctan)) `| Z) . x = ((arctan . x) #R (1 / 2)) / (1 + (x ^2)) ) assume A8: x in Z ; ::_thesis: (((2 / 3) (#) ((#R (3 / 2)) * arctan)) `| Z) . x = ((arctan . x) #R (1 / 2)) / (1 + (x ^2)) then A9: arctan . x > 0 by A3; A10: arctan is_differentiable_on Z by A2, SIN_COS9:81; then A11: arctan is_differentiable_in x by A8, FDIFF_1:9; (((2 / 3) (#) ((#R (3 / 2)) * arctan)) `| Z) . x = (2 / 3) * (diff (((#R (3 / 2)) * arctan),x)) by A1, A7, A8, FDIFF_1:20 .= (2 / 3) * (((3 / 2) * ((arctan . x) #R ((3 / 2) - 1))) * (diff (arctan,x))) by A11, A9, TAYLOR_1:22 .= (2 / 3) * (((3 / 2) * ((arctan . x) #R ((3 / 2) - 1))) * ((arctan `| Z) . x)) by A8, A10, FDIFF_1:def_7 .= (2 / 3) * (((3 / 2) * ((arctan . x) #R ((3 / 2) - 1))) * (1 / (1 + (x ^2)))) by A2, A8, SIN_COS9:81 .= ((arctan . x) #R (1 / 2)) / (1 + (x ^2)) ; hence (((2 / 3) (#) ((#R (3 / 2)) * arctan)) `| Z) . x = ((arctan . x) #R (1 / 2)) / (1 + (x ^2)) ; ::_thesis: verum end; hence ( (2 / 3) (#) ((#R (3 / 2)) * arctan) is_differentiable_on Z & ( for x being Real st x in Z holds (((2 / 3) (#) ((#R (3 / 2)) * arctan)) `| Z) . x = ((arctan . x) #R (1 / 2)) / (1 + (x ^2)) ) ) by A1, A7, FDIFF_1:20; ::_thesis: verum end; theorem :: FDIFF_11:74 for Z being open Subset of REAL st Z c= dom ((2 / 3) (#) ((#R (3 / 2)) * arccot)) & Z c= ].(- 1),1.[ holds ( (2 / 3) (#) ((#R (3 / 2)) * arccot) is_differentiable_on Z & ( for x being Real st x in Z holds (((2 / 3) (#) ((#R (3 / 2)) * arccot)) `| Z) . x = - (((arccot . x) #R (1 / 2)) / (1 + (x ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom ((2 / 3) (#) ((#R (3 / 2)) * arccot)) & Z c= ].(- 1),1.[ implies ( (2 / 3) (#) ((#R (3 / 2)) * arccot) is_differentiable_on Z & ( for x being Real st x in Z holds (((2 / 3) (#) ((#R (3 / 2)) * arccot)) `| Z) . x = - (((arccot . x) #R (1 / 2)) / (1 + (x ^2))) ) ) ) assume that A1: Z c= dom ((2 / 3) (#) ((#R (3 / 2)) * arccot)) and A2: Z c= ].(- 1),1.[ ; ::_thesis: ( (2 / 3) (#) ((#R (3 / 2)) * arccot) is_differentiable_on Z & ( for x being Real st x in Z holds (((2 / 3) (#) ((#R (3 / 2)) * arccot)) `| Z) . x = - (((arccot . x) #R (1 / 2)) / (1 + (x ^2))) ) ) A3: for x being Real st x in Z holds arccot . x > 0 proof let x be Real; ::_thesis: ( x in Z implies arccot . x > 0 ) ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25; then A4: Z c= [.(- 1),1.] by A2, XBOOLE_1:1; assume x in Z ; ::_thesis: arccot . x > 0 then x in [.(- 1),1.] by A4; then arccot . x in arccot .: [.(- 1),1.] by FUNCT_1:def_6, SIN_COS9:24; then arccot . x in [.(PI / 4),((3 / 4) * PI).] by RELAT_1:115, SIN_COS9:56; then arccot . x >= PI / 4 by XXREAL_1:1; then A5: (arccot . x) + 0 > (PI / 4) + (- (PI / 4)) by XREAL_1:8; assume arccot . x <= 0 ; ::_thesis: contradiction hence contradiction by A5; ::_thesis: verum end; A6: for x being Real st x in Z holds (#R (3 / 2)) * arccot is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies (#R (3 / 2)) * arccot is_differentiable_in x ) assume A7: x in Z ; ::_thesis: (#R (3 / 2)) * arccot is_differentiable_in x then A8: arccot . x > 0 by A3; arccot is_differentiable_on Z by A2, SIN_COS9:82; then arccot is_differentiable_in x by A7, FDIFF_1:9; hence (#R (3 / 2)) * arccot is_differentiable_in x by A8, TAYLOR_1:22; ::_thesis: verum end; Z c= dom ((#R (3 / 2)) * arccot) by A1, VALUED_1:def_5; then A9: (#R (3 / 2)) * arccot is_differentiable_on Z by A6, FDIFF_1:9; for x being Real st x in Z holds (((2 / 3) (#) ((#R (3 / 2)) * arccot)) `| Z) . x = - (((arccot . x) #R (1 / 2)) / (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies (((2 / 3) (#) ((#R (3 / 2)) * arccot)) `| Z) . x = - (((arccot . x) #R (1 / 2)) / (1 + (x ^2))) ) assume A10: x in Z ; ::_thesis: (((2 / 3) (#) ((#R (3 / 2)) * arccot)) `| Z) . x = - (((arccot . x) #R (1 / 2)) / (1 + (x ^2))) then A11: arccot . x > 0 by A3; A12: arccot is_differentiable_on Z by A2, SIN_COS9:82; then A13: arccot is_differentiable_in x by A10, FDIFF_1:9; (((2 / 3) (#) ((#R (3 / 2)) * arccot)) `| Z) . x = (2 / 3) * (diff (((#R (3 / 2)) * arccot),x)) by A1, A9, A10, FDIFF_1:20 .= (2 / 3) * (((3 / 2) * ((arccot . x) #R ((3 / 2) - 1))) * (diff (arccot,x))) by A13, A11, TAYLOR_1:22 .= (2 / 3) * (((3 / 2) * ((arccot . x) #R ((3 / 2) - 1))) * ((arccot `| Z) . x)) by A10, A12, FDIFF_1:def_7 .= (2 / 3) * (((3 / 2) * ((arccot . x) #R ((3 / 2) - 1))) * (- (1 / (1 + (x ^2))))) by A2, A10, SIN_COS9:82 .= - (((arccot . x) #R (1 / 2)) / (1 + (x ^2))) ; hence (((2 / 3) (#) ((#R (3 / 2)) * arccot)) `| Z) . x = - (((arccot . x) #R (1 / 2)) / (1 + (x ^2))) ; ::_thesis: verum end; hence ( (2 / 3) (#) ((#R (3 / 2)) * arccot) is_differentiable_on Z & ( for x being Real st x in Z holds (((2 / 3) (#) ((#R (3 / 2)) * arccot)) `| Z) . x = - (((arccot . x) #R (1 / 2)) / (1 + (x ^2))) ) ) by A1, A9, FDIFF_1:20; ::_thesis: verum end;