:: FDIFF_6 semantic presentation begin Lm1: for x being Real holds 1 - (cos (2 * x)) = 2 * ((sin x) ^2) proof let x be Real; ::_thesis: 1 - (cos (2 * x)) = 2 * ((sin x) ^2) 1 - (cos (2 * x)) = 1 - (cos (x + x)) .= 1 - (((cos x) * (cos x)) - ((sin x) * (sin x))) by SIN_COS:75 .= (((cos x) ^2) + ((sin x) ^2)) - (((cos x) ^2) - ((sin x) ^2)) by SIN_COS:29 .= 2 * ((sin x) ^2) ; hence 1 - (cos (2 * x)) = 2 * ((sin x) ^2) ; ::_thesis: verum end; Lm2: for x being Real holds 1 + (cos (2 * x)) = 2 * ((cos x) ^2) proof let x be Real; ::_thesis: 1 + (cos (2 * x)) = 2 * ((cos x) ^2) 1 + (cos (2 * x)) = 1 + (cos (x + x)) .= 1 + (((cos x) ^2) - ((sin x) ^2)) by SIN_COS:75 .= (((cos x) ^2) + ((sin x) ^2)) + (((cos x) ^2) - ((sin x) ^2)) by SIN_COS:29 .= 2 * ((cos x) ^2) ; hence 1 + (cos (2 * x)) = 2 * ((cos x) ^2) ; ::_thesis: verum end; Lm3: for x being Real st sin . x > 0 holds sin . x = ((1 - (cos . x)) * (1 + (cos . x))) #R (1 / 2) proof let x be Real; ::_thesis: ( sin . x > 0 implies sin . x = ((1 - (cos . x)) * (1 + (cos . x))) #R (1 / 2) ) assume A1: sin . x > 0 ; ::_thesis: sin . x = ((1 - (cos . x)) * (1 + (cos . x))) #R (1 / 2) then sin . x = (sin . x) #R (2 * (1 / 2)) by PREPOWER:72 .= ((sin . x) #R (1 + 1)) #R (1 / 2) by A1, PREPOWER:91 .= (((sin . x) #R 1) * ((sin . x) #R 1)) #R (1 / 2) by A1, PREPOWER:75 .= ((sin . x) * ((sin . x) #R 1)) #R (1 / 2) by A1, PREPOWER:72 .= ((((sin . x) ^2) + ((cos . x) ^2)) - ((cos . x) ^2)) #R (1 / 2) by A1, PREPOWER:72 .= ((1 ^2) - ((cos . x) ^2)) #R (1 / 2) by SIN_COS:28 .= ((1 - (cos . x)) * (1 + (cos . x))) #R (1 / 2) ; hence sin . x = ((1 - (cos . x)) * (1 + (cos . x))) #R (1 / 2) ; ::_thesis: verum end; Lm4: for x being Real st sin . x > 0 & cos . x < 1 & cos . x > - 1 holds (sin . x) / ((1 - (cos . x)) #R (1 / 2)) = (1 + (cos . x)) #R (1 / 2) proof let x be Real; ::_thesis: ( sin . x > 0 & cos . x < 1 & cos . x > - 1 implies (sin . x) / ((1 - (cos . x)) #R (1 / 2)) = (1 + (cos . x)) #R (1 / 2) ) assume that A1: sin . x > 0 and A2: cos . x < 1 and A3: cos . x > - 1 ; ::_thesis: (sin . x) / ((1 - (cos . x)) #R (1 / 2)) = (1 + (cos . x)) #R (1 / 2) A4: 1 - (cos . x) > 1 - 1 by A2, XREAL_1:15; 1 + (cos . x) > 1 + (- 1) by A3, XREAL_1:8; then A5: (1 - (cos . x)) * (1 + (cos . x)) > 0 by A4, XREAL_1:129; (sin . x) / ((1 - (cos . x)) #R (1 / 2)) = (((1 - (cos . x)) * (1 + (cos . x))) #R (1 / 2)) / ((1 - (cos . x)) #R (1 / 2)) by A1, Lm3 .= (((1 - (cos . x)) * (1 + (cos . x))) / (1 - (cos . x))) #R (1 / 2) by A4, A5, PREPOWER:80 .= (1 + (cos . x)) #R (1 / 2) by A4, XCMPLX_1:89 ; hence (sin . x) / ((1 - (cos . x)) #R (1 / 2)) = (1 + (cos . x)) #R (1 / 2) ; ::_thesis: verum end; theorem Th1: :: FDIFF_6:1 for a, x being Real st a > 0 holds exp_R . (x * (log (number_e,a))) = a #R x proof let a, x be Real; ::_thesis: ( a > 0 implies exp_R . (x * (log (number_e,a))) = a #R x ) assume A1: a > 0 ; ::_thesis: exp_R . (x * (log (number_e,a))) = a #R x number_e <> 1 by TAYLOR_1:11; then exp_R . (x * (log (number_e,a))) = exp_R . (log (number_e,(a to_power x))) by A1, POWER:55, TAYLOR_1:11 .= exp_R . (log (number_e,(a #R x))) by A1, POWER:def_2 .= a #R x by A1, PREPOWER:81, TAYLOR_1:15 ; hence exp_R . (x * (log (number_e,a))) = a #R x ; ::_thesis: verum end; theorem Th2: :: FDIFF_6:2 for a, x being Real st a > 0 holds exp_R . (- (x * (log (number_e,a)))) = a #R (- x) proof let a, x be Real; ::_thesis: ( a > 0 implies exp_R . (- (x * (log (number_e,a)))) = a #R (- x) ) A1: number_e <> 1 by TAYLOR_1:11; assume A2: a > 0 ; ::_thesis: exp_R . (- (x * (log (number_e,a)))) = a #R (- x) exp_R . (- (x * (log (number_e,a)))) = exp_R . ((- x) * (log (number_e,a))) .= exp_R . (log (number_e,(a to_power (- x)))) by A2, A1, POWER:55, TAYLOR_1:11 .= exp_R . (log (number_e,(a #R (- x)))) by A2, POWER:def_2 .= a #R (- x) by A2, PREPOWER:81, TAYLOR_1:15 ; hence exp_R . (- (x * (log (number_e,a)))) = a #R (- x) ; ::_thesis: verum end; Lm5: for x being Real st sin . x > 0 & cos . x < 1 & cos . x > - 1 holds (sin . x) / ((1 + (cos . x)) #R (1 / 2)) = (1 - (cos . x)) #R (1 / 2) proof let x be Real; ::_thesis: ( sin . x > 0 & cos . x < 1 & cos . x > - 1 implies (sin . x) / ((1 + (cos . x)) #R (1 / 2)) = (1 - (cos . x)) #R (1 / 2) ) assume that A1: sin . x > 0 and A2: cos . x < 1 and A3: cos . x > - 1 ; ::_thesis: (sin . x) / ((1 + (cos . x)) #R (1 / 2)) = (1 - (cos . x)) #R (1 / 2) A4: 1 + (cos . x) > 1 + (- 1) by A3, XREAL_1:8; 1 - (cos . x) > 1 - 1 by A2, XREAL_1:15; then A5: (1 - (cos . x)) * (1 + (cos . x)) > 0 by A4, XREAL_1:129; (sin . x) / ((1 + (cos . x)) #R (1 / 2)) = (((1 - (cos . x)) * (1 + (cos . x))) #R (1 / 2)) / ((1 + (cos . x)) #R (1 / 2)) by A1, Lm3 .= (((1 - (cos . x)) * (1 + (cos . x))) / (1 + (cos . x))) #R (1 / 2) by A4, A5, PREPOWER:80 .= (1 - (cos . x)) #R (1 / 2) by A4, XCMPLX_1:89 ; hence (sin . x) / ((1 + (cos . x)) #R (1 / 2)) = (1 - (cos . x)) #R (1 / 2) ; ::_thesis: verum end; theorem Th3: :: FDIFF_6:3 for a being Real for Z being open Subset of REAL for f1, f2 being PartFunc of REAL,REAL st Z c= dom (f1 - f2) & ( for x being Real st x in Z holds f1 . x = a ^2 ) & f2 = #Z 2 holds ( f1 - f2 is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 - f2) `| Z) . x = - (2 * x) ) ) proof let a be Real; ::_thesis: for Z being open Subset of REAL for f1, f2 being PartFunc of REAL,REAL st Z c= dom (f1 - f2) & ( for x being Real st x in Z holds f1 . x = a ^2 ) & f2 = #Z 2 holds ( f1 - f2 is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 - f2) `| Z) . x = - (2 * x) ) ) let Z be open Subset of REAL; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st Z c= dom (f1 - f2) & ( for x being Real st x in Z holds f1 . x = a ^2 ) & f2 = #Z 2 holds ( f1 - f2 is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 - f2) `| Z) . x = - (2 * x) ) ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (f1 - f2) & ( for x being Real st x in Z holds f1 . x = a ^2 ) & f2 = #Z 2 implies ( f1 - f2 is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 - f2) `| Z) . x = - (2 * x) ) ) ) assume that A1: Z c= dom (f1 - f2) and A2: for x being Real st x in Z holds f1 . x = a ^2 and A3: f2 = #Z 2 ; ::_thesis: ( f1 - f2 is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 - f2) `| Z) . x = - (2 * x) ) ) A4: f1 + ((- 1) (#) f2) = f1 - f2 ; A5: for x being Real st x in Z holds f1 . x = (a ^2) + (0 * x) by A2; for x being Real st x in Z holds ((f1 - f2) `| Z) . x = - (2 * x) proof let x be Real; ::_thesis: ( x in Z implies ((f1 - f2) `| Z) . x = - (2 * x) ) assume x in Z ; ::_thesis: ((f1 - f2) `| Z) . x = - (2 * x) hence ((f1 - f2) `| Z) . x = 0 + ((2 * (- 1)) * x) by A1, A3, A5, FDIFF_4:12 .= - (2 * x) ; ::_thesis: verum end; hence ( f1 - f2 is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 - f2) `| Z) . x = - (2 * x) ) ) by A1, A3, A4, A5, FDIFF_4:12; ::_thesis: verum end; theorem Th4: :: FDIFF_6:4 for a being Real for Z being open Subset of REAL for f1, f2 being PartFunc of REAL,REAL st Z c= dom ((f1 + f2) / (f1 - f2)) & f2 = #Z 2 & ( for x being Real st x in Z holds ( f1 . x = a ^2 & (f1 - f2) . x <> 0 ) ) holds ( (f1 + f2) / (f1 - f2) is_differentiable_on Z & ( for x being Real st x in Z holds (((f1 + f2) / (f1 - f2)) `| Z) . x = ((4 * (a ^2)) * x) / (((a ^2) - (x |^ 2)) ^2) ) ) proof let a be Real; ::_thesis: for Z being open Subset of REAL for f1, f2 being PartFunc of REAL,REAL st Z c= dom ((f1 + f2) / (f1 - f2)) & f2 = #Z 2 & ( for x being Real st x in Z holds ( f1 . x = a ^2 & (f1 - f2) . x <> 0 ) ) holds ( (f1 + f2) / (f1 - f2) is_differentiable_on Z & ( for x being Real st x in Z holds (((f1 + f2) / (f1 - f2)) `| Z) . x = ((4 * (a ^2)) * x) / (((a ^2) - (x |^ 2)) ^2) ) ) let Z be open Subset of REAL; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st Z c= dom ((f1 + f2) / (f1 - f2)) & f2 = #Z 2 & ( for x being Real st x in Z holds ( f1 . x = a ^2 & (f1 - f2) . x <> 0 ) ) holds ( (f1 + f2) / (f1 - f2) is_differentiable_on Z & ( for x being Real st x in Z holds (((f1 + f2) / (f1 - f2)) `| Z) . x = ((4 * (a ^2)) * x) / (((a ^2) - (x |^ 2)) ^2) ) ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom ((f1 + f2) / (f1 - f2)) & f2 = #Z 2 & ( for x being Real st x in Z holds ( f1 . x = a ^2 & (f1 - f2) . x <> 0 ) ) implies ( (f1 + f2) / (f1 - f2) is_differentiable_on Z & ( for x being Real st x in Z holds (((f1 + f2) / (f1 - f2)) `| Z) . x = ((4 * (a ^2)) * x) / (((a ^2) - (x |^ 2)) ^2) ) ) ) assume that A1: Z c= dom ((f1 + f2) / (f1 - f2)) and A2: f2 = #Z 2 and A3: for x being Real st x in Z holds ( f1 . x = a ^2 & (f1 - f2) . x <> 0 ) ; ::_thesis: ( (f1 + f2) / (f1 - f2) is_differentiable_on Z & ( for x being Real st x in Z holds (((f1 + f2) / (f1 - f2)) `| Z) . x = ((4 * (a ^2)) * x) / (((a ^2) - (x |^ 2)) ^2) ) ) A4: for x being Real st x in Z holds f1 . x = a ^2 by A3; A5: Z c= (dom (f1 + f2)) /\ ((dom (f1 - f2)) \ ((f1 - f2) " {0})) by A1, RFUNCT_1:def_1; then A6: Z c= dom (f1 + f2) by XBOOLE_1:18; then A7: f1 + f2 is_differentiable_on Z by A2, A4, FDIFF_4:17; A8: Z c= dom (f1 - f2) by A5, XBOOLE_1:1; then A9: f1 - f2 is_differentiable_on Z by A2, A4, Th3; A10: for x being Real st x in Z holds (f1 - f2) . x <> 0 by A3; then A11: (f1 + f2) / (f1 - f2) is_differentiable_on Z by A7, A9, FDIFF_2:21; for x being Real st x in Z holds (((f1 + f2) / (f1 - f2)) `| Z) . x = ((4 * (a ^2)) * x) / (((a ^2) - (x |^ 2)) ^2) proof let x be Real; ::_thesis: ( x in Z implies (((f1 + f2) / (f1 - f2)) `| Z) . x = ((4 * (a ^2)) * x) / (((a ^2) - (x |^ 2)) ^2) ) A12: f2 . x = x #Z 2 by A2, TAYLOR_1:def_1 .= x |^ 2 by PREPOWER:36 ; assume A13: x in Z ; ::_thesis: (((f1 + f2) / (f1 - f2)) `| Z) . x = ((4 * (a ^2)) * x) / (((a ^2) - (x |^ 2)) ^2) then A14: (f1 - f2) . x <> 0 by A3; A15: (f1 - f2) . x = (f1 . x) - (f2 . x) by A8, A13, VALUED_1:13 .= (a ^2) - (x |^ 2) by A3, A13, A12 ; A16: (f1 + f2) . x = (f1 . x) + (f2 . x) by A6, A13, VALUED_1:def_1 .= (a ^2) + (x |^ 2) by A3, A13, A12 ; ( f1 + f2 is_differentiable_in x & f1 - f2 is_differentiable_in x ) by A7, A9, A13, FDIFF_1:9; then diff (((f1 + f2) / (f1 - f2)),x) = (((diff ((f1 + f2),x)) * ((f1 - f2) . x)) - ((diff ((f1 - f2),x)) * ((f1 + f2) . x))) / (((f1 - f2) . x) ^2) by A14, FDIFF_2:14 .= (((((f1 + f2) `| Z) . x) * ((f1 - f2) . x)) - ((diff ((f1 - f2),x)) * ((f1 + f2) . x))) / (((f1 - f2) . x) ^2) by A7, A13, FDIFF_1:def_7 .= (((((f1 + f2) `| Z) . x) * ((f1 - f2) . x)) - ((((f1 - f2) `| Z) . x) * ((f1 + f2) . x))) / (((f1 - f2) . x) ^2) by A9, A13, FDIFF_1:def_7 .= (((2 * x) * ((f1 - f2) . x)) - ((((f1 - f2) `| Z) . x) * ((f1 + f2) . x))) / (((f1 - f2) . x) ^2) by A2, A6, A4, A13, FDIFF_4:17 .= (((2 * x) * ((f1 - f2) . x)) - ((- (2 * x)) * ((f1 + f2) . x))) / (((f1 - f2) . x) ^2) by A2, A8, A4, A13, Th3 .= ((4 * (a ^2)) * x) / (((a ^2) - (x |^ 2)) ^2) by A16, A15 ; hence (((f1 + f2) / (f1 - f2)) `| Z) . x = ((4 * (a ^2)) * x) / (((a ^2) - (x |^ 2)) ^2) by A11, A13, FDIFF_1:def_7; ::_thesis: verum end; hence ( (f1 + f2) / (f1 - f2) is_differentiable_on Z & ( for x being Real st x in Z holds (((f1 + f2) / (f1 - f2)) `| Z) . x = ((4 * (a ^2)) * x) / (((a ^2) - (x |^ 2)) ^2) ) ) by A7, A9, A10, FDIFF_2:21; ::_thesis: verum end; theorem Th5: :: FDIFF_6:5 for a being Real for Z being open Subset of REAL for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom f & f = ln * ((f1 + f2) / (f1 - f2)) & f2 = #Z 2 & ( for x being Real st x in Z holds ( f1 . x = a ^2 & (f1 - f2) . x > 0 & a <> 0 ) ) holds ( f is_differentiable_on Z & ( for x being Real st x in Z holds (f `| Z) . x = ((4 * (a ^2)) * x) / ((a |^ 4) - (x |^ 4)) ) ) proof let a be Real; ::_thesis: for Z being open Subset of REAL for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom f & f = ln * ((f1 + f2) / (f1 - f2)) & f2 = #Z 2 & ( for x being Real st x in Z holds ( f1 . x = a ^2 & (f1 - f2) . x > 0 & a <> 0 ) ) holds ( f is_differentiable_on Z & ( for x being Real st x in Z holds (f `| Z) . x = ((4 * (a ^2)) * x) / ((a |^ 4) - (x |^ 4)) ) ) let Z be open Subset of REAL; ::_thesis: for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom f & f = ln * ((f1 + f2) / (f1 - f2)) & f2 = #Z 2 & ( for x being Real st x in Z holds ( f1 . x = a ^2 & (f1 - f2) . x > 0 & a <> 0 ) ) holds ( f is_differentiable_on Z & ( for x being Real st x in Z holds (f `| Z) . x = ((4 * (a ^2)) * x) / ((a |^ 4) - (x |^ 4)) ) ) let f, f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom f & f = ln * ((f1 + f2) / (f1 - f2)) & f2 = #Z 2 & ( for x being Real st x in Z holds ( f1 . x = a ^2 & (f1 - f2) . x > 0 & a <> 0 ) ) implies ( f is_differentiable_on Z & ( for x being Real st x in Z holds (f `| Z) . x = ((4 * (a ^2)) * x) / ((a |^ 4) - (x |^ 4)) ) ) ) assume that A1: Z c= dom f and A2: f = ln * ((f1 + f2) / (f1 - f2)) and A3: f2 = #Z 2 and A4: for x being Real st x in Z holds ( f1 . x = a ^2 & (f1 - f2) . x > 0 & a <> 0 ) ; ::_thesis: ( f is_differentiable_on Z & ( for x being Real st x in Z holds (f `| Z) . x = ((4 * (a ^2)) * x) / ((a |^ 4) - (x |^ 4)) ) ) for y being set st y in Z holds y in dom ((f1 + f2) / (f1 - f2)) by A1, A2, FUNCT_1:11; then A5: Z c= dom ((f1 + f2) / (f1 - f2)) by TARSKI:def_3; then A6: Z c= (dom (f1 + f2)) /\ ((dom (f1 - f2)) \ ((f1 - f2) " {0})) by RFUNCT_1:def_1; then A7: Z c= dom (f1 - f2) by XBOOLE_1:1; A8: for x being Real st x in Z holds ( f1 . x = a ^2 & (f1 - f2) . x <> 0 ) by A4; then A9: (f1 + f2) / (f1 - f2) is_differentiable_on Z by A3, A5, Th4; A10: Z c= dom (f1 + f2) by A6, XBOOLE_1:18; A11: for x being Real st x in Z holds ((f1 + f2) / (f1 - f2)) . x > 0 proof let x be Real; ::_thesis: ( x in Z implies ((f1 + f2) / (f1 - f2)) . x > 0 ) A12: f2 . x = x #Z 2 by A3, TAYLOR_1:def_1 .= x |^ 2 by PREPOWER:36 ; assume A13: x in Z ; ::_thesis: ((f1 + f2) / (f1 - f2)) . x > 0 then A14: (f1 - f2) . x > 0 by A4; a <> 0 by A4, A13; then A15: a ^2 > 0 by SQUARE_1:12; x |^ 2 = x ^2 by NEWTON:81; then A16: (a ^2) + (x |^ 2) > 0 + 0 by A15, XREAL_1:8, XREAL_1:63; A17: ((f1 + f2) / (f1 - f2)) . x = ((f1 + f2) . x) * (((f1 - f2) . x) ") by A5, A13, RFUNCT_1:def_1 .= ((f1 + f2) . x) / ((f1 - f2) . x) by XCMPLX_0:def_9 ; (f1 + f2) . x = (f1 . x) + (f2 . x) by A10, A13, VALUED_1:def_1 .= (a ^2) + (x |^ 2) by A4, A13, A12 ; hence ((f1 + f2) / (f1 - f2)) . x > 0 by A14, A16, A17, XREAL_1:139; ::_thesis: verum end; A18: for x being Real st x in Z holds ln * ((f1 + f2) / (f1 - f2)) is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies ln * ((f1 + f2) / (f1 - f2)) is_differentiable_in x ) assume x in Z ; ::_thesis: ln * ((f1 + f2) / (f1 - f2)) is_differentiable_in x then ( (f1 + f2) / (f1 - f2) is_differentiable_in x & ((f1 + f2) / (f1 - f2)) . x > 0 ) by A9, A11, FDIFF_1:9; hence ln * ((f1 + f2) / (f1 - f2)) is_differentiable_in x by TAYLOR_1:20; ::_thesis: verum end; then A19: f is_differentiable_on Z by A1, A2, FDIFF_1:9; for x being Real st x in Z holds (f `| Z) . x = ((4 * (a ^2)) * x) / ((a |^ 4) - (x |^ 4)) proof let x be Real; ::_thesis: ( x in Z implies (f `| Z) . x = ((4 * (a ^2)) * x) / ((a |^ 4) - (x |^ 4)) ) A20: (a ^2) ^2 = (a |^ 2) * (a ^2) by NEWTON:81 .= (a |^ 2) * (a |^ 2) by NEWTON:81 .= (a |^ 2) |^ 2 by WSIERP_1:1 .= a |^ (2 * 2) by NEWTON:9 .= a |^ 4 ; A21: (x |^ 2) ^2 = (x |^ 2) |^ 2 by WSIERP_1:1 .= x |^ (2 * 2) by NEWTON:9 .= x |^ 4 ; A22: f2 . x = x #Z 2 by A3, TAYLOR_1:def_1 .= x |^ 2 by PREPOWER:36 ; assume A23: x in Z ; ::_thesis: (f `| Z) . x = ((4 * (a ^2)) * x) / ((a |^ 4) - (x |^ 4)) then A24: (f1 + f2) . x = (f1 . x) + (f2 . x) by A10, VALUED_1:def_1 .= (a ^2) + (x |^ 2) by A4, A23, A22 ; A25: (f1 - f2) . x = (f1 . x) - (f2 . x) by A7, A23, VALUED_1:13 .= (a ^2) - (x |^ 2) by A4, A23, A22 ; then A26: (a ^2) - (x |^ 2) > 0 by A4, A23; A27: ((f1 + f2) / (f1 - f2)) . x = ((f1 + f2) . x) * (((f1 - f2) . x) ") by A5, A23, RFUNCT_1:def_1 .= ((a ^2) + (x |^ 2)) / ((a ^2) - (x |^ 2)) by A24, A25, XCMPLX_0:def_9 ; ( (f1 + f2) / (f1 - f2) is_differentiable_in x & ((f1 + f2) / (f1 - f2)) . x > 0 ) by A9, A11, A23, FDIFF_1:9; then diff ((ln * ((f1 + f2) / (f1 - f2))),x) = (diff (((f1 + f2) / (f1 - f2)),x)) / (((f1 + f2) / (f1 - f2)) . x) by TAYLOR_1:20 .= ((((f1 + f2) / (f1 - f2)) `| Z) . x) / (((f1 + f2) / (f1 - f2)) . x) by A9, A23, FDIFF_1:def_7 .= (((4 * (a ^2)) * x) / (((a ^2) - (x |^ 2)) ^2)) / (((a ^2) + (x |^ 2)) / ((a ^2) - (x |^ 2))) by A3, A5, A8, A23, A27, Th4 .= ((((4 * (a ^2)) * x) / ((a ^2) - (x |^ 2))) / ((a ^2) - (x |^ 2))) / (((a ^2) + (x |^ 2)) / ((a ^2) - (x |^ 2))) by XCMPLX_1:78 .= ((((4 * (a ^2)) * x) / ((a ^2) - (x |^ 2))) / (((a ^2) + (x |^ 2)) / ((a ^2) - (x |^ 2)))) / ((a ^2) - (x |^ 2)) by XCMPLX_1:48 .= (((4 * (a ^2)) * x) / ((a ^2) + (x |^ 2))) / ((a ^2) - (x |^ 2)) by A26, XCMPLX_1:55 .= ((4 * (a ^2)) * x) / (((a ^2) + (x |^ 2)) * ((a ^2) - (x |^ 2))) by XCMPLX_1:78 .= ((4 * (a ^2)) * x) / ((a |^ 4) - (x |^ 4)) by A20, A21 ; hence (f `| Z) . x = ((4 * (a ^2)) * x) / ((a |^ 4) - (x |^ 4)) by A2, A19, A23, FDIFF_1:def_7; ::_thesis: verum end; hence ( f is_differentiable_on Z & ( for x being Real st x in Z holds (f `| Z) . x = ((4 * (a ^2)) * x) / ((a |^ 4) - (x |^ 4)) ) ) by A1, A2, A18, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_6:6 for a being Real for Z being open Subset of REAL for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom ((1 / (4 * (a ^2))) (#) f) & f = ln * ((f1 + f2) / (f1 - f2)) & f2 = #Z 2 & ( for x being Real st x in Z holds ( f1 . x = a ^2 & (f1 - f2) . x > 0 & a <> 0 ) ) holds ( (1 / (4 * (a ^2))) (#) f is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / (4 * (a ^2))) (#) f) `| Z) . x = x / ((a |^ 4) - (x |^ 4)) ) ) proof let a be Real; ::_thesis: for Z being open Subset of REAL for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom ((1 / (4 * (a ^2))) (#) f) & f = ln * ((f1 + f2) / (f1 - f2)) & f2 = #Z 2 & ( for x being Real st x in Z holds ( f1 . x = a ^2 & (f1 - f2) . x > 0 & a <> 0 ) ) holds ( (1 / (4 * (a ^2))) (#) f is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / (4 * (a ^2))) (#) f) `| Z) . x = x / ((a |^ 4) - (x |^ 4)) ) ) let Z be open Subset of REAL; ::_thesis: for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom ((1 / (4 * (a ^2))) (#) f) & f = ln * ((f1 + f2) / (f1 - f2)) & f2 = #Z 2 & ( for x being Real st x in Z holds ( f1 . x = a ^2 & (f1 - f2) . x > 0 & a <> 0 ) ) holds ( (1 / (4 * (a ^2))) (#) f is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / (4 * (a ^2))) (#) f) `| Z) . x = x / ((a |^ 4) - (x |^ 4)) ) ) let f, f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom ((1 / (4 * (a ^2))) (#) f) & f = ln * ((f1 + f2) / (f1 - f2)) & f2 = #Z 2 & ( for x being Real st x in Z holds ( f1 . x = a ^2 & (f1 - f2) . x > 0 & a <> 0 ) ) implies ( (1 / (4 * (a ^2))) (#) f is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / (4 * (a ^2))) (#) f) `| Z) . x = x / ((a |^ 4) - (x |^ 4)) ) ) ) assume that A1: Z c= dom ((1 / (4 * (a ^2))) (#) f) and A2: ( f = ln * ((f1 + f2) / (f1 - f2)) & f2 = #Z 2 ) and A3: for x being Real st x in Z holds ( f1 . x = a ^2 & (f1 - f2) . x > 0 & a <> 0 ) ; ::_thesis: ( (1 / (4 * (a ^2))) (#) f is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / (4 * (a ^2))) (#) f) `| Z) . x = x / ((a |^ 4) - (x |^ 4)) ) ) A4: Z c= dom f by A1, VALUED_1:def_5; then A5: f is_differentiable_on Z by A2, A3, Th5; for x being Real st x in Z holds (((1 / (4 * (a ^2))) (#) f) `| Z) . x = x / ((a |^ 4) - (x |^ 4)) proof let x be Real; ::_thesis: ( x in Z implies (((1 / (4 * (a ^2))) (#) f) `| Z) . x = x / ((a |^ 4) - (x |^ 4)) ) assume A6: x in Z ; ::_thesis: (((1 / (4 * (a ^2))) (#) f) `| Z) . x = x / ((a |^ 4) - (x |^ 4)) then a <> 0 by A3; then a ^2 > 0 by SQUARE_1:12; then A7: 4 * (a ^2) > 4 * 0 by XREAL_1:68; (((1 / (4 * (a ^2))) (#) f) `| Z) . x = (1 / (4 * (a ^2))) * (diff (f,x)) by A1, A5, A6, FDIFF_1:20 .= (1 / (4 * (a ^2))) * ((f `| Z) . x) by A5, A6, FDIFF_1:def_7 .= (1 / (4 * (a ^2))) * (((4 * (a ^2)) * x) / ((a |^ 4) - (x |^ 4))) by A2, A3, A4, A6, Th5 .= (1 / (4 * (a ^2))) * ((4 * (a ^2)) * (x / ((a |^ 4) - (x |^ 4)))) by XCMPLX_1:74 .= (x / ((a |^ 4) - (x |^ 4))) * ((1 / (4 * (a ^2))) * (4 * (a ^2))) .= x / ((a |^ 4) - (x |^ 4)) by A7, XCMPLX_1:108 ; hence (((1 / (4 * (a ^2))) (#) f) `| Z) . x = x / ((a |^ 4) - (x |^ 4)) ; ::_thesis: verum end; hence ( (1 / (4 * (a ^2))) (#) f is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / (4 * (a ^2))) (#) f) `| Z) . x = x / ((a |^ 4) - (x |^ 4)) ) ) by A1, A5, FDIFF_1:20; ::_thesis: verum end; theorem Th7: :: FDIFF_6:7 for Z being open Subset of REAL for f1, f2 being PartFunc of REAL,REAL st Z c= dom (f1 / (f2 + f1)) & f1 = #Z 2 & ( for x being Real st x in Z holds ( f2 . x = 1 & x <> 0 ) ) holds ( f1 / (f2 + f1) is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 / (f2 + f1)) `| Z) . x = (2 * x) / ((1 + (x ^2)) ^2) ) ) proof let Z be open Subset of REAL; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st Z c= dom (f1 / (f2 + f1)) & f1 = #Z 2 & ( for x being Real st x in Z holds ( f2 . x = 1 & x <> 0 ) ) holds ( f1 / (f2 + f1) is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 / (f2 + f1)) `| Z) . x = (2 * x) / ((1 + (x ^2)) ^2) ) ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (f1 / (f2 + f1)) & f1 = #Z 2 & ( for x being Real st x in Z holds ( f2 . x = 1 & x <> 0 ) ) implies ( f1 / (f2 + f1) is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 / (f2 + f1)) `| Z) . x = (2 * x) / ((1 + (x ^2)) ^2) ) ) ) assume that A1: Z c= dom (f1 / (f2 + f1)) and A2: f1 = #Z 2 and A3: for x being Real st x in Z holds ( f2 . x = 1 & x <> 0 ) ; ::_thesis: ( f1 / (f2 + f1) is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 / (f2 + f1)) `| Z) . x = (2 * x) / ((1 + (x ^2)) ^2) ) ) A4: Z c= (dom f1) /\ ((dom (f2 + f1)) \ ((f2 + f1) " {0})) by A1, RFUNCT_1:def_1; then A5: Z c= dom (f2 + f1) by XBOOLE_1:1; A6: for x being Real st x in Z holds f1 is_differentiable_in x by A2, TAYLOR_1:2; Z c= dom f1 by A4, XBOOLE_1:18; then A7: f1 is_differentiable_on Z by A6, FDIFF_1:9; A8: for x being Real st x in Z holds (f1 `| Z) . x = 2 * x proof let x be Real; ::_thesis: ( x in Z implies (f1 `| Z) . x = 2 * x ) 2 * (x #Z (2 - 1)) = 2 * x by PREPOWER:35; then A9: diff (f1,x) = 2 * x by A2, TAYLOR_1:2; assume x in Z ; ::_thesis: (f1 `| Z) . x = 2 * x hence (f1 `| Z) . x = 2 * x by A7, A9, FDIFF_1:def_7; ::_thesis: verum end; A10: for x being Real st x in Z holds f2 . x = 1 ^2 by A3; then A11: f2 + f1 is_differentiable_on Z by A2, A5, FDIFF_4:17; A12: for x being Real st x in Z holds (f2 + f1) . x <> 0 proof let x be Real; ::_thesis: ( x in Z implies (f2 + f1) . x <> 0 ) A13: 1 + (x ^2) > 0 + 0 by XREAL_1:8, XREAL_1:63; assume A14: x in Z ; ::_thesis: (f2 + f1) . x <> 0 then (f2 + f1) . x = (f2 . x) + (f1 . x) by A5, VALUED_1:def_1 .= 1 + (f1 . x) by A3, A14 .= 1 + (x #Z 2) by A2, TAYLOR_1:def_1 .= 1 + (x |^ 2) by PREPOWER:36 .= 1 + (x ^2) by NEWTON:81 ; hence (f2 + f1) . x <> 0 by A13; ::_thesis: verum end; then A15: f1 / (f2 + f1) is_differentiable_on Z by A11, A7, FDIFF_2:21; for x being Real st x in Z holds ((f1 / (f2 + f1)) `| Z) . x = (2 * x) / ((1 + (x ^2)) ^2) proof let x be Real; ::_thesis: ( x in Z implies ((f1 / (f2 + f1)) `| Z) . x = (2 * x) / ((1 + (x ^2)) ^2) ) A16: f1 is_differentiable_in x by A2, TAYLOR_1:2; A17: f1 . x = x #Z 2 by A2, TAYLOR_1:def_1 .= x |^ 2 by PREPOWER:36 .= x ^2 by NEWTON:81 ; assume A18: x in Z ; ::_thesis: ((f1 / (f2 + f1)) `| Z) . x = (2 * x) / ((1 + (x ^2)) ^2) then A19: (f2 + f1) . x = (f2 . x) + (f1 . x) by A5, VALUED_1:def_1 .= 1 + (f1 . x) by A3, A18 .= 1 + (x #Z 2) by A2, TAYLOR_1:def_1 .= 1 + (x |^ 2) by PREPOWER:36 .= 1 + (x ^2) by NEWTON:81 ; ( f2 + f1 is_differentiable_in x & (f2 + f1) . x <> 0 ) by A11, A12, A18, FDIFF_1:9; then diff ((f1 / (f2 + f1)),x) = (((diff (f1,x)) * ((f2 + f1) . x)) - ((diff ((f2 + f1),x)) * (f1 . x))) / (((f2 + f1) . x) ^2) by A16, FDIFF_2:14 .= ((((f1 `| Z) . x) * ((f2 + f1) . x)) - ((diff ((f2 + f1),x)) * (f1 . x))) / (((f2 + f1) . x) ^2) by A7, A18, FDIFF_1:def_7 .= ((((f1 `| Z) . x) * ((f2 + f1) . x)) - ((((f2 + f1) `| Z) . x) * (f1 . x))) / (((f2 + f1) . x) ^2) by A11, A18, FDIFF_1:def_7 .= (((2 * x) * ((f2 + f1) . x)) - ((((f2 + f1) `| Z) . x) * (f1 . x))) / (((f2 + f1) . x) ^2) by A8, A18 .= (((2 * x) * (1 + (x ^2))) - ((2 * x) * (x ^2))) / ((1 + (x ^2)) ^2) by A2, A10, A5, A18, A17, A19, FDIFF_4:17 .= (2 * x) / ((1 + (x ^2)) ^2) ; hence ((f1 / (f2 + f1)) `| Z) . x = (2 * x) / ((1 + (x ^2)) ^2) by A15, A18, FDIFF_1:def_7; ::_thesis: verum end; hence ( f1 / (f2 + f1) is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 / (f2 + f1)) `| Z) . x = (2 * x) / ((1 + (x ^2)) ^2) ) ) by A11, A7, A12, FDIFF_2:21; ::_thesis: verum end; theorem :: FDIFF_6:8 for Z being open Subset of REAL for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom ((1 / 2) (#) f) & f = ln * (f1 / (f2 + f1)) & f1 = #Z 2 & ( for x being Real st x in Z holds ( f2 . x = 1 & x <> 0 ) ) holds ( (1 / 2) (#) f is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) f) `| Z) . x = 1 / (x * (1 + (x ^2))) ) ) proof let Z be open Subset of REAL; ::_thesis: for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom ((1 / 2) (#) f) & f = ln * (f1 / (f2 + f1)) & f1 = #Z 2 & ( for x being Real st x in Z holds ( f2 . x = 1 & x <> 0 ) ) holds ( (1 / 2) (#) f is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) f) `| Z) . x = 1 / (x * (1 + (x ^2))) ) ) let f, f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom ((1 / 2) (#) f) & f = ln * (f1 / (f2 + f1)) & f1 = #Z 2 & ( for x being Real st x in Z holds ( f2 . x = 1 & x <> 0 ) ) implies ( (1 / 2) (#) f is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) f) `| Z) . x = 1 / (x * (1 + (x ^2))) ) ) ) assume that A1: Z c= dom ((1 / 2) (#) f) and A2: f = ln * (f1 / (f2 + f1)) and A3: f1 = #Z 2 and A4: for x being Real st x in Z holds ( f2 . x = 1 & x <> 0 ) ; ::_thesis: ( (1 / 2) (#) f is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) f) `| Z) . x = 1 / (x * (1 + (x ^2))) ) ) A5: Z c= dom f by A1, VALUED_1:def_5; then for y being set st y in Z holds y in dom (f1 / (f2 + f1)) by A2, FUNCT_1:11; then A6: Z c= dom (f1 / (f2 + f1)) by TARSKI:def_3; then A7: f1 / (f2 + f1) is_differentiable_on Z by A3, A4, Th7; Z c= (dom f1) /\ ((dom (f2 + f1)) \ ((f2 + f1) " {0})) by A6, RFUNCT_1:def_1; then A8: Z c= dom (f2 + f1) by XBOOLE_1:1; A9: for x being Real st x in Z holds (f1 / (f1 + f2)) . x > 0 proof let x be Real; ::_thesis: ( x in Z implies (f1 / (f1 + f2)) . x > 0 ) assume A10: x in Z ; ::_thesis: (f1 / (f1 + f2)) . x > 0 then A11: (f1 / (f2 + f1)) . x = (f1 . x) * (((f2 + f1) . x) ") by A6, RFUNCT_1:def_1 .= (f1 . x) / ((f2 + f1) . x) by XCMPLX_0:def_9 ; A12: x <> 0 by A4, A10; then A13: 1 + (x ^2) > 0 + 0 by SQUARE_1:12, XREAL_1:8; A14: f1 . x = x #Z 2 by A3, TAYLOR_1:def_1 .= x |^ 2 by PREPOWER:36 .= x ^2 by NEWTON:81 ; then A15: f1 . x > 0 by A12, SQUARE_1:12; (f2 + f1) . x = (f2 . x) + (f1 . x) by A8, A10, VALUED_1:def_1 .= 1 + (x ^2) by A4, A10, A14 ; hence (f1 / (f1 + f2)) . x > 0 by A15, A13, A11, XREAL_1:139; ::_thesis: verum end; for x being Real st x in Z holds ln * (f1 / (f2 + f1)) is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies ln * (f1 / (f2 + f1)) is_differentiable_in x ) assume x in Z ; ::_thesis: ln * (f1 / (f2 + f1)) is_differentiable_in x then ( f1 / (f2 + f1) is_differentiable_in x & (f1 / (f1 + f2)) . x > 0 ) by A7, A9, FDIFF_1:9; hence ln * (f1 / (f2 + f1)) is_differentiable_in x by TAYLOR_1:20; ::_thesis: verum end; then A16: f is_differentiable_on Z by A2, A5, FDIFF_1:9; for x being Real st x in Z holds (((1 / 2) (#) f) `| Z) . x = 1 / (x * (1 + (x ^2))) proof let x be Real; ::_thesis: ( x in Z implies (((1 / 2) (#) f) `| Z) . x = 1 / (x * (1 + (x ^2))) ) A17: f1 . x = x #Z 2 by A3, TAYLOR_1:def_1 .= x |^ 2 by PREPOWER:36 .= x ^2 by NEWTON:81 ; assume A18: x in Z ; ::_thesis: (((1 / 2) (#) f) `| Z) . x = 1 / (x * (1 + (x ^2))) then A19: ( f1 / (f2 + f1) is_differentiable_in x & (f1 / (f1 + f2)) . x > 0 ) by A7, A9, FDIFF_1:9; x <> 0 by A4, A18; then A20: 1 + (x ^2) > 0 + 0 by SQUARE_1:12, XREAL_1:8; A21: (f2 + f1) . x = (f2 . x) + (f1 . x) by A8, A18, VALUED_1:def_1 .= 1 + (x ^2) by A4, A18, A17 ; A22: (f1 / (f2 + f1)) . x = (f1 . x) * (((f2 + f1) . x) ") by A6, A18, RFUNCT_1:def_1 .= (x ^2) / (1 + (x ^2)) by A17, A21, XCMPLX_0:def_9 ; (((1 / 2) (#) f) `| Z) . x = (1 / 2) * (diff ((ln * (f1 / (f2 + f1))),x)) by A1, A2, A16, A18, FDIFF_1:20 .= (1 / 2) * ((diff ((f1 / (f2 + f1)),x)) / ((f1 / (f2 + f1)) . x)) by A19, TAYLOR_1:20 .= (1 / 2) * ((((f1 / (f2 + f1)) `| Z) . x) / ((f1 / (f2 + f1)) . x)) by A7, A18, FDIFF_1:def_7 .= (1 / 2) * (((2 * x) / ((1 + (x ^2)) ^2)) / ((x ^2) / (1 + (x ^2)))) by A3, A4, A6, A18, A22, Th7 .= ((1 / 2) * ((2 * x) / ((1 + (x ^2)) ^2))) / ((x ^2) / (1 + (x ^2))) by XCMPLX_1:74 .= (((1 / 2) * (2 * x)) / ((1 + (x ^2)) ^2)) / ((x ^2) / (1 + (x ^2))) by XCMPLX_1:74 .= ((x / (1 + (x ^2))) / (1 + (x ^2))) / ((x ^2) / (1 + (x ^2))) by XCMPLX_1:78 .= ((x / (1 + (x ^2))) / ((x ^2) / (1 + (x ^2)))) / (1 + (x ^2)) by XCMPLX_1:48 .= (x / (x ^2)) / (1 + (x ^2)) by A20, XCMPLX_1:55 .= ((x / x) / x) / (1 + (x ^2)) by XCMPLX_1:78 .= (1 / x) / (1 + (x ^2)) by A4, A18, XCMPLX_1:60 .= 1 / (x * (1 + (x ^2))) by XCMPLX_1:78 ; hence (((1 / 2) (#) f) `| Z) . x = 1 / (x * (1 + (x ^2))) ; ::_thesis: verum end; hence ( (1 / 2) (#) f is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) f) `| Z) . x = 1 / (x * (1 + (x ^2))) ) ) by A1, A16, FDIFF_1:20; ::_thesis: verum end; theorem :: FDIFF_6:9 for n being Element of NAT for Z being open Subset of REAL st Z c= dom (ln * (#Z n)) & ( for x being Real st x in Z holds x > 0 ) holds ( ln * (#Z n) is_differentiable_on Z & ( for x being Real st x in Z holds ((ln * (#Z n)) `| Z) . x = n / x ) ) proof let n be Element of NAT ; ::_thesis: for Z being open Subset of REAL st Z c= dom (ln * (#Z n)) & ( for x being Real st x in Z holds x > 0 ) holds ( ln * (#Z n) is_differentiable_on Z & ( for x being Real st x in Z holds ((ln * (#Z n)) `| Z) . x = n / x ) ) let Z be open Subset of REAL; ::_thesis: ( Z c= dom (ln * (#Z n)) & ( for x being Real st x in Z holds x > 0 ) implies ( ln * (#Z n) is_differentiable_on Z & ( for x being Real st x in Z holds ((ln * (#Z n)) `| Z) . x = n / x ) ) ) assume that A1: Z c= dom (ln * (#Z n)) and A2: for x being Real st x in Z holds x > 0 ; ::_thesis: ( ln * (#Z n) is_differentiable_on Z & ( for x being Real st x in Z holds ((ln * (#Z n)) `| Z) . x = n / x ) ) A3: for x being Real st x in Z holds ln * (#Z n) is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies ln * (#Z n) is_differentiable_in x ) A4: (#Z n) . x = x #Z n by TAYLOR_1:def_1; assume x in Z ; ::_thesis: ln * (#Z n) is_differentiable_in x then ( #Z n is_differentiable_in x & (#Z n) . x > 0 ) by A2, A4, PREPOWER:39, TAYLOR_1:2; hence ln * (#Z n) is_differentiable_in x by TAYLOR_1:20; ::_thesis: verum end; then A5: ln * (#Z n) is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((ln * (#Z n)) `| Z) . x = n / x proof let x be Real; ::_thesis: ( x in Z implies ((ln * (#Z n)) `| Z) . x = n / x ) A6: ( #Z n is_differentiable_in x & diff ((#Z n),x) = n * (x #Z (n - 1)) ) by TAYLOR_1:2; assume A7: x in Z ; ::_thesis: ((ln * (#Z n)) `| Z) . x = n / x then A8: x > 0 by A2; A9: x |^ n > 0 by A2, A7, NEWTON:83; A10: (#Z n) . x = x #Z n by TAYLOR_1:def_1; then (#Z n) . x > 0 by A2, A7, PREPOWER:39; then diff ((ln * (#Z n)),x) = (n * (x #Z (n - 1))) / (x #Z n) by A6, A10, TAYLOR_1:20 .= (n * (x #Z (n - 1))) / (x |^ n) by PREPOWER:36 .= (n * ((x |^ n) / (x |^ 1))) / (x |^ n) by A8, PREPOWER:43 .= n * (((x |^ n) / (x |^ 1)) / (x |^ n)) by XCMPLX_1:74 .= n * (((x |^ n) / (x |^ n)) / (x |^ 1)) by XCMPLX_1:48 .= n * (1 / (x |^ 1)) by A9, XCMPLX_1:60 .= n * (1 / x) by NEWTON:5 .= (n * 1) / x by XCMPLX_1:74 .= n / x ; hence ((ln * (#Z n)) `| Z) . x = n / x by A5, A7, FDIFF_1:def_7; ::_thesis: verum end; hence ( ln * (#Z n) is_differentiable_on Z & ( for x being Real st x in Z holds ((ln * (#Z n)) `| Z) . x = n / x ) ) by A1, A3, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_6:10 for Z being open Subset of REAL for f2, f1 being PartFunc of REAL,REAL st Z c= dom ((f2 ^) + (ln * (f1 / f2))) & ( for x being Real st x in Z holds ( f2 . x = x & f2 . x > 0 & f1 . x = x - 1 & f1 . x > 0 ) ) holds ( (f2 ^) + (ln * (f1 / f2)) is_differentiable_on Z & ( for x being Real st x in Z holds (((f2 ^) + (ln * (f1 / f2))) `| Z) . x = 1 / ((x ^2) * (x - 1)) ) ) proof let Z be open Subset of REAL; ::_thesis: for f2, f1 being PartFunc of REAL,REAL st Z c= dom ((f2 ^) + (ln * (f1 / f2))) & ( for x being Real st x in Z holds ( f2 . x = x & f2 . x > 0 & f1 . x = x - 1 & f1 . x > 0 ) ) holds ( (f2 ^) + (ln * (f1 / f2)) is_differentiable_on Z & ( for x being Real st x in Z holds (((f2 ^) + (ln * (f1 / f2))) `| Z) . x = 1 / ((x ^2) * (x - 1)) ) ) let f2, f1 be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom ((f2 ^) + (ln * (f1 / f2))) & ( for x being Real st x in Z holds ( f2 . x = x & f2 . x > 0 & f1 . x = x - 1 & f1 . x > 0 ) ) implies ( (f2 ^) + (ln * (f1 / f2)) is_differentiable_on Z & ( for x being Real st x in Z holds (((f2 ^) + (ln * (f1 / f2))) `| Z) . x = 1 / ((x ^2) * (x - 1)) ) ) ) assume that A1: Z c= dom ((f2 ^) + (ln * (f1 / f2))) and A2: for x being Real st x in Z holds ( f2 . x = x & f2 . x > 0 & f1 . x = x - 1 & f1 . x > 0 ) ; ::_thesis: ( (f2 ^) + (ln * (f1 / f2)) is_differentiable_on Z & ( for x being Real st x in Z holds (((f2 ^) + (ln * (f1 / f2))) `| Z) . x = 1 / ((x ^2) * (x - 1)) ) ) A3: Z c= (dom (f2 ^)) /\ (dom (ln * (f1 / f2))) by A1, VALUED_1:def_1; then A4: Z c= dom (ln * (f1 / f2)) by XBOOLE_1:18; A5: dom (f2 ^) c= dom f2 by RFUNCT_1:1; Z c= dom (f2 ^) by A3, XBOOLE_1:18; then A6: Z c= dom f2 by A5, XBOOLE_1:1; A7: for x being Real st x in Z holds ( f1 . x = x - 1 & f1 . x > 0 & f2 . x = x - 0 & f2 . x > 0 ) by A2; then A8: ln * (f1 / f2) is_differentiable_on Z by A4, FDIFF_4:24; A9: for x being Real st x in Z holds ( f2 . x = 0 + x & f2 . x <> 0 ) by A2; then A10: f2 ^ is_differentiable_on Z by A6, FDIFF_4:14; for x being Real st x in Z holds (((f2 ^) + (ln * (f1 / f2))) `| Z) . x = 1 / ((x ^2) * (x - 1)) proof let x be Real; ::_thesis: ( x in Z implies (((f2 ^) + (ln * (f1 / f2))) `| Z) . x = 1 / ((x ^2) * (x - 1)) ) assume A11: x in Z ; ::_thesis: (((f2 ^) + (ln * (f1 / f2))) `| Z) . x = 1 / ((x ^2) * (x - 1)) then A12: ( f2 . x = x & f2 . x > 0 ) by A2; A13: ( f1 . x = x - 1 & f1 . x > 0 ) by A2, A11; (((f2 ^) + (ln * (f1 / f2))) `| Z) . x = (diff ((f2 ^),x)) + (diff ((ln * (f1 / f2)),x)) by A1, A10, A8, A11, FDIFF_1:18 .= (((f2 ^) `| Z) . x) + (diff ((ln * (f1 / f2)),x)) by A10, A11, FDIFF_1:def_7 .= (((f2 ^) `| Z) . x) + (((ln * (f1 / f2)) `| Z) . x) by A8, A11, FDIFF_1:def_7 .= (- (1 / ((0 + x) ^2))) + (((ln * (f1 / f2)) `| Z) . x) by A6, A9, A11, FDIFF_4:14 .= (- (1 / ((0 + x) ^2))) + ((1 - 0) / ((x - 1) * (x - 0))) by A4, A7, A11, FDIFF_4:24 .= (- ((1 * (x - 1)) / ((x ^2) * (x - 1)))) + (1 / ((x - 1) * x)) by A13, XCMPLX_1:91 .= (- ((1 * (x - 1)) / ((x ^2) * (x - 1)))) + ((1 * x) / (((x - 1) * x) * x)) by A12, XCMPLX_1:91 .= ((- (x - 1)) / ((x ^2) * (x - 1))) + (x / ((x ^2) * (x - 1))) by XCMPLX_1:187 .= (((- x) + 1) + x) / ((x ^2) * (x - 1)) by XCMPLX_1:62 .= 1 / ((x ^2) * (x - 1)) ; hence (((f2 ^) + (ln * (f1 / f2))) `| Z) . x = 1 / ((x ^2) * (x - 1)) ; ::_thesis: verum end; hence ( (f2 ^) + (ln * (f1 / f2)) is_differentiable_on Z & ( for x being Real st x in Z holds (((f2 ^) + (ln * (f1 / f2))) `| Z) . x = 1 / ((x ^2) * (x - 1)) ) ) by A1, A10, A8, FDIFF_1:18; ::_thesis: verum end; theorem Th11: :: FDIFF_6:11 for a being Real for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom (exp_R * f) & ( for x being Real st x in Z holds f . x = x * (log (number_e,a)) ) & a > 0 holds ( exp_R * f is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R * f) `| Z) . x = (a #R x) * (log (number_e,a)) ) ) proof let a be Real; ::_thesis: for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom (exp_R * f) & ( for x being Real st x in Z holds f . x = x * (log (number_e,a)) ) & a > 0 holds ( exp_R * f is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R * f) `| Z) . x = (a #R x) * (log (number_e,a)) ) ) let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st Z c= dom (exp_R * f) & ( for x being Real st x in Z holds f . x = x * (log (number_e,a)) ) & a > 0 holds ( exp_R * f is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R * f) `| Z) . x = (a #R x) * (log (number_e,a)) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (exp_R * f) & ( for x being Real st x in Z holds f . x = x * (log (number_e,a)) ) & a > 0 implies ( exp_R * f is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R * f) `| Z) . x = (a #R x) * (log (number_e,a)) ) ) ) assume that A1: Z c= dom (exp_R * f) and A2: for x being Real st x in Z holds f . x = x * (log (number_e,a)) and A3: a > 0 ; ::_thesis: ( exp_R * f is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R * f) `| Z) . x = (a #R x) * (log (number_e,a)) ) ) for y being set st y in Z holds y in dom f by A1, FUNCT_1:11; then A4: Z c= dom f by TARSKI:def_3; A5: for x being Real st x in Z holds f . x = ((log (number_e,a)) * x) + 0 by A2; then A6: f is_differentiable_on Z by A4, FDIFF_1:23; A7: for x being Real st x in Z holds exp_R * f is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies exp_R * f is_differentiable_in x ) assume x in Z ; ::_thesis: exp_R * f is_differentiable_in x then f is_differentiable_in x by A6, FDIFF_1:9; hence exp_R * f is_differentiable_in x by TAYLOR_1:19; ::_thesis: verum end; then A8: exp_R * f is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((exp_R * f) `| Z) . x = (a #R x) * (log (number_e,a)) proof let x be Real; ::_thesis: ( x in Z implies ((exp_R * f) `| Z) . x = (a #R x) * (log (number_e,a)) ) assume A9: x in Z ; ::_thesis: ((exp_R * f) `| Z) . x = (a #R x) * (log (number_e,a)) then f is_differentiable_in x by A6, FDIFF_1:9; then diff ((exp_R * f),x) = (exp_R . (f . x)) * (diff (f,x)) by TAYLOR_1:19 .= (exp_R . (f . x)) * ((f `| Z) . x) by A6, A9, FDIFF_1:def_7 .= (exp_R . (f . x)) * (log (number_e,a)) by A4, A5, A9, FDIFF_1:23 .= (exp_R . (x * (log (number_e,a)))) * (log (number_e,a)) by A2, A9 .= (a #R x) * (log (number_e,a)) by A3, Th1 ; hence ((exp_R * f) `| Z) . x = (a #R x) * (log (number_e,a)) by A8, A9, FDIFF_1:def_7; ::_thesis: verum end; hence ( exp_R * f is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R * f) `| Z) . x = (a #R x) * (log (number_e,a)) ) ) by A1, A7, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_6:12 for a being Real for Z being open Subset of REAL for f1, f2 being PartFunc of REAL,REAL st Z c= dom ((1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2)) & ( for x being Real st x in Z holds ( f1 . x = x * (log (number_e,a)) & f2 . x = x - (1 / (log (number_e,a))) ) ) & a > 0 & a <> 1 holds ( (1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2)) `| Z) . x = x * (a #R x) ) ) proof let a be Real; ::_thesis: for Z being open Subset of REAL for f1, f2 being PartFunc of REAL,REAL st Z c= dom ((1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2)) & ( for x being Real st x in Z holds ( f1 . x = x * (log (number_e,a)) & f2 . x = x - (1 / (log (number_e,a))) ) ) & a > 0 & a <> 1 holds ( (1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2)) `| Z) . x = x * (a #R x) ) ) let Z be open Subset of REAL; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st Z c= dom ((1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2)) & ( for x being Real st x in Z holds ( f1 . x = x * (log (number_e,a)) & f2 . x = x - (1 / (log (number_e,a))) ) ) & a > 0 & a <> 1 holds ( (1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2)) `| Z) . x = x * (a #R x) ) ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom ((1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2)) & ( for x being Real st x in Z holds ( f1 . x = x * (log (number_e,a)) & f2 . x = x - (1 / (log (number_e,a))) ) ) & a > 0 & a <> 1 implies ( (1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2)) `| Z) . x = x * (a #R x) ) ) ) assume that A1: Z c= dom ((1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2)) and A2: for x being Real st x in Z holds ( f1 . x = x * (log (number_e,a)) & f2 . x = x - (1 / (log (number_e,a))) ) and A3: a > 0 and A4: a <> 1 ; ::_thesis: ( (1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2)) `| Z) . x = x * (a #R x) ) ) A5: Z c= dom ((exp_R * f1) (#) f2) by A1, VALUED_1:def_5; then A6: Z c= (dom (exp_R * f1)) /\ (dom f2) by VALUED_1:def_4; then A7: Z c= dom (exp_R * f1) by XBOOLE_1:18; A8: for x being Real st x in Z holds f2 . x = (1 * x) + (- (1 / (log (number_e,a)))) proof let x be Real; ::_thesis: ( x in Z implies f2 . x = (1 * x) + (- (1 / (log (number_e,a)))) ) A9: (1 * x) + (- (1 / (log (number_e,a)))) = (1 * x) - (1 / (log (number_e,a))) ; assume x in Z ; ::_thesis: f2 . x = (1 * x) + (- (1 / (log (number_e,a)))) hence f2 . x = (1 * x) + (- (1 / (log (number_e,a)))) by A2, A9; ::_thesis: verum end; A10: for x being Real st x in Z holds f1 . x = x * (log (number_e,a)) by A2; then A11: exp_R * f1 is_differentiable_on Z by A3, A7, Th11; A12: Z c= dom f2 by A6, XBOOLE_1:18; then A13: f2 is_differentiable_on Z by A8, FDIFF_1:23; then A14: (exp_R * f1) (#) f2 is_differentiable_on Z by A5, A11, FDIFF_1:21; A15: log (number_e,a) <> 0 proof A16: number_e <> 1 by TAYLOR_1:11; assume log (number_e,a) = 0 ; ::_thesis: contradiction then log (number_e,a) = log (number_e,1) by SIN_COS2:13, TAYLOR_1:13; then a = number_e to_power (log (number_e,1)) by A3, A16, POWER:def_3, TAYLOR_1:11 .= 1 by A16, POWER:def_3, TAYLOR_1:11 ; hence contradiction by A4; ::_thesis: verum end; for x being Real st x in Z holds (((1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2)) `| Z) . x = x * (a #R x) proof let x be Real; ::_thesis: ( x in Z implies (((1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2)) `| Z) . x = x * (a #R x) ) assume A17: x in Z ; ::_thesis: (((1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2)) `| Z) . x = x * (a #R x) then A18: (exp_R * f1) . x = exp_R . (f1 . x) by A7, FUNCT_1:12 .= exp_R . (x * (log (number_e,a))) by A2, A17 .= a #R x by A3, Th1 ; (((1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2)) `| Z) . x = (1 / (log (number_e,a))) * (diff (((exp_R * f1) (#) f2),x)) by A1, A14, A17, FDIFF_1:20 .= (1 / (log (number_e,a))) * ((((exp_R * f1) (#) f2) `| Z) . x) by A14, A17, FDIFF_1:def_7 .= (1 / (log (number_e,a))) * (((f2 . x) * (diff ((exp_R * f1),x))) + (((exp_R * f1) . x) * (diff (f2,x)))) by A5, A11, A13, A17, FDIFF_1:21 .= (1 / (log (number_e,a))) * (((f2 . x) * (((exp_R * f1) `| Z) . x)) + (((exp_R * f1) . x) * (diff (f2,x)))) by A11, A17, FDIFF_1:def_7 .= (1 / (log (number_e,a))) * (((f2 . x) * (((exp_R * f1) `| Z) . x)) + (((exp_R * f1) . x) * ((f2 `| Z) . x))) by A13, A17, FDIFF_1:def_7 .= (1 / (log (number_e,a))) * (((f2 . x) * ((a #R x) * (log (number_e,a)))) + (((exp_R * f1) . x) * ((f2 `| Z) . x))) by A3, A10, A7, A17, Th11 .= (1 / (log (number_e,a))) * (((f2 . x) * ((a #R x) * (log (number_e,a)))) + (((exp_R * f1) . x) * 1)) by A12, A8, A17, FDIFF_1:23 .= (1 / (log (number_e,a))) * ((((f2 . x) * (log (number_e,a))) + 1) * (a #R x)) by A18 .= (1 / (log (number_e,a))) * ((((x - (1 / (log (number_e,a)))) * (log (number_e,a))) + 1) * (a #R x)) by A2, A17 .= ((1 / (log (number_e,a))) * (((x * (log (number_e,a))) - ((1 / (log (number_e,a))) * (log (number_e,a)))) + 1)) * (a #R x) .= ((1 / (log (number_e,a))) * (((x * (log (number_e,a))) - 1) + 1)) * (a #R x) by A15, XCMPLX_1:106 .= (((1 / (log (number_e,a))) * (log (number_e,a))) * x) * (a #R x) .= (1 * x) * (a #R x) by A15, XCMPLX_1:106 .= x * (a #R x) ; hence (((1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2)) `| Z) . x = x * (a #R x) ; ::_thesis: verum end; hence ( (1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2)) `| Z) . x = x * (a #R x) ) ) by A1, A14, FDIFF_1:20; ::_thesis: verum end; theorem :: FDIFF_6:13 for a being Real for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom ((1 / (1 + (log (number_e,a)))) (#) ((exp_R * f) (#) exp_R)) & ( for x being Real st x in Z holds f . x = x * (log (number_e,a)) ) & a > 0 & a <> 1 / number_e holds ( (1 / (1 + (log (number_e,a)))) (#) ((exp_R * f) (#) exp_R) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / (1 + (log (number_e,a)))) (#) ((exp_R * f) (#) exp_R)) `| Z) . x = (a #R x) * (exp_R . x) ) ) proof let a be Real; ::_thesis: for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom ((1 / (1 + (log (number_e,a)))) (#) ((exp_R * f) (#) exp_R)) & ( for x being Real st x in Z holds f . x = x * (log (number_e,a)) ) & a > 0 & a <> 1 / number_e holds ( (1 / (1 + (log (number_e,a)))) (#) ((exp_R * f) (#) exp_R) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / (1 + (log (number_e,a)))) (#) ((exp_R * f) (#) exp_R)) `| Z) . x = (a #R x) * (exp_R . x) ) ) let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st Z c= dom ((1 / (1 + (log (number_e,a)))) (#) ((exp_R * f) (#) exp_R)) & ( for x being Real st x in Z holds f . x = x * (log (number_e,a)) ) & a > 0 & a <> 1 / number_e holds ( (1 / (1 + (log (number_e,a)))) (#) ((exp_R * f) (#) exp_R) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / (1 + (log (number_e,a)))) (#) ((exp_R * f) (#) exp_R)) `| Z) . x = (a #R x) * (exp_R . x) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom ((1 / (1 + (log (number_e,a)))) (#) ((exp_R * f) (#) exp_R)) & ( for x being Real st x in Z holds f . x = x * (log (number_e,a)) ) & a > 0 & a <> 1 / number_e implies ( (1 / (1 + (log (number_e,a)))) (#) ((exp_R * f) (#) exp_R) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / (1 + (log (number_e,a)))) (#) ((exp_R * f) (#) exp_R)) `| Z) . x = (a #R x) * (exp_R . x) ) ) ) assume that A1: Z c= dom ((1 / (1 + (log (number_e,a)))) (#) ((exp_R * f) (#) exp_R)) and A2: for x being Real st x in Z holds f . x = x * (log (number_e,a)) and A3: a > 0 and A4: a <> 1 / number_e ; ::_thesis: ( (1 / (1 + (log (number_e,a)))) (#) ((exp_R * f) (#) exp_R) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / (1 + (log (number_e,a)))) (#) ((exp_R * f) (#) exp_R)) `| Z) . x = (a #R x) * (exp_R . x) ) ) A5: Z c= dom ((exp_R * f) (#) exp_R) by A1, VALUED_1:def_5; then Z c= (dom (exp_R * f)) /\ (dom exp_R) by VALUED_1:def_4; then A6: Z c= dom (exp_R * f) by XBOOLE_1:18; then A7: exp_R * f is_differentiable_on Z by A2, A3, Th11; A8: exp_R is_differentiable_on Z by FDIFF_1:26, TAYLOR_1:16; then A9: (exp_R * f) (#) exp_R is_differentiable_on Z by A5, A7, FDIFF_1:21; A10: 1 + (log (number_e,a)) <> 0 proof A11: number_e * a > 0 * a by A3, TAYLOR_1:11, XREAL_1:68; assume A12: 1 + (log (number_e,a)) = 0 ; ::_thesis: contradiction A13: number_e <> 1 by TAYLOR_1:11; log (number_e,1) = 0 by SIN_COS2:13, TAYLOR_1:13 .= (log (number_e,number_e)) + (log (number_e,a)) by A12, A13, POWER:52, TAYLOR_1:11 .= log (number_e,(number_e * a)) by A3, A13, POWER:53, TAYLOR_1:11 ; then number_e * a = number_e to_power (log (number_e,1)) by A13, A11, POWER:def_3, TAYLOR_1:11 .= 1 by A13, POWER:def_3, TAYLOR_1:11 ; hence contradiction by A4, XCMPLX_1:73; ::_thesis: verum end; for x being Real st x in Z holds (((1 / (1 + (log (number_e,a)))) (#) ((exp_R * f) (#) exp_R)) `| Z) . x = (a #R x) * (exp_R . x) proof let x be Real; ::_thesis: ( x in Z implies (((1 / (1 + (log (number_e,a)))) (#) ((exp_R * f) (#) exp_R)) `| Z) . x = (a #R x) * (exp_R . x) ) assume A14: x in Z ; ::_thesis: (((1 / (1 + (log (number_e,a)))) (#) ((exp_R * f) (#) exp_R)) `| Z) . x = (a #R x) * (exp_R . x) then A15: (exp_R * f) . x = exp_R . (f . x) by A6, FUNCT_1:12 .= exp_R . (x * (log (number_e,a))) by A2, A14 .= a #R x by A3, Th1 ; (((1 / (1 + (log (number_e,a)))) (#) ((exp_R * f) (#) exp_R)) `| Z) . x = (1 / (1 + (log (number_e,a)))) * (diff (((exp_R * f) (#) exp_R),x)) by A1, A9, A14, FDIFF_1:20 .= (1 / (1 + (log (number_e,a)))) * ((((exp_R * f) (#) exp_R) `| Z) . x) by A9, A14, FDIFF_1:def_7 .= (1 / (1 + (log (number_e,a)))) * (((exp_R . x) * (diff ((exp_R * f),x))) + (((exp_R * f) . x) * (diff (exp_R,x)))) by A5, A7, A8, A14, FDIFF_1:21 .= (1 / (1 + (log (number_e,a)))) * (((exp_R . x) * (((exp_R * f) `| Z) . x)) + (((exp_R * f) . x) * (diff (exp_R,x)))) by A7, A14, FDIFF_1:def_7 .= (1 / (1 + (log (number_e,a)))) * (((exp_R . x) * (((exp_R * f) `| Z) . x)) + (((exp_R * f) . x) * (exp_R . x))) by TAYLOR_1:16 .= (1 / (1 + (log (number_e,a)))) * (((((exp_R * f) `| Z) . x) + ((exp_R * f) . x)) * (exp_R . x)) .= (1 / (1 + (log (number_e,a)))) * ((((a #R x) * (log (number_e,a))) + ((exp_R * f) . x)) * (exp_R . x)) by A2, A3, A6, A14, Th11 .= (((1 / (1 + (log (number_e,a)))) * ((log (number_e,a)) + 1)) * (a #R x)) * (exp_R . x) by A15 .= (1 * (a #R x)) * (exp_R . x) by A10, XCMPLX_1:106 .= (a #R x) * (exp_R . x) ; hence (((1 / (1 + (log (number_e,a)))) (#) ((exp_R * f) (#) exp_R)) `| Z) . x = (a #R x) * (exp_R . x) ; ::_thesis: verum end; hence ( (1 / (1 + (log (number_e,a)))) (#) ((exp_R * f) (#) exp_R) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / (1 + (log (number_e,a)))) (#) ((exp_R * f) (#) exp_R)) `| Z) . x = (a #R x) * (exp_R . x) ) ) by A1, A9, FDIFF_1:20; ::_thesis: verum end; theorem Th14: :: FDIFF_6:14 for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom (exp_R * f) & ( for x being Real st x in Z holds f . x = - x ) holds ( exp_R * f is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R * f) `| Z) . x = - (exp_R (- x)) ) ) proof let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st Z c= dom (exp_R * f) & ( for x being Real st x in Z holds f . x = - x ) holds ( exp_R * f is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R * f) `| Z) . x = - (exp_R (- x)) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (exp_R * f) & ( for x being Real st x in Z holds f . x = - x ) implies ( exp_R * f is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R * f) `| Z) . x = - (exp_R (- x)) ) ) ) assume that A1: Z c= dom (exp_R * f) and A2: for x being Real st x in Z holds f . x = - x ; ::_thesis: ( exp_R * f is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R * f) `| Z) . x = - (exp_R (- x)) ) ) A3: for x being Real st x in Z holds f . x = ((- 1) * x) + 0 proof let x be Real; ::_thesis: ( x in Z implies f . x = ((- 1) * x) + 0 ) assume x in Z ; ::_thesis: f . x = ((- 1) * x) + 0 then f . x = - x by A2 .= ((- 1) * x) + 0 ; hence f . x = ((- 1) * x) + 0 ; ::_thesis: verum end; for y being set st y in Z holds y in dom f by A1, FUNCT_1:11; then A4: Z c= dom f by TARSKI:def_3; then A5: f is_differentiable_on Z by A3, FDIFF_1:23; A6: for x being Real st x in Z holds exp_R * f is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies exp_R * f is_differentiable_in x ) assume x in Z ; ::_thesis: exp_R * f is_differentiable_in x then f is_differentiable_in x by A5, FDIFF_1:9; hence exp_R * f is_differentiable_in x by TAYLOR_1:19; ::_thesis: verum end; then A7: exp_R * f is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((exp_R * f) `| Z) . x = - (exp_R (- x)) proof let x be Real; ::_thesis: ( x in Z implies ((exp_R * f) `| Z) . x = - (exp_R (- x)) ) assume A8: x in Z ; ::_thesis: ((exp_R * f) `| Z) . x = - (exp_R (- x)) then f is_differentiable_in x by A5, FDIFF_1:9; then diff ((exp_R * f),x) = (exp_R . (f . x)) * (diff (f,x)) by TAYLOR_1:19 .= (exp_R . (f . x)) * ((f `| Z) . x) by A5, A8, FDIFF_1:def_7 .= (exp_R . (f . x)) * (- 1) by A4, A3, A8, FDIFF_1:23 .= (exp_R . (- x)) * (- 1) by A2, A8 .= - (exp_R . (- x)) .= - (exp_R (- x)) by SIN_COS:def_23 ; hence ((exp_R * f) `| Z) . x = - (exp_R (- x)) by A7, A8, FDIFF_1:def_7; ::_thesis: verum end; hence ( exp_R * f is_differentiable_on Z & ( for x being Real st x in Z holds ((exp_R * f) `| Z) . x = - (exp_R (- x)) ) ) by A1, A6, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_6:15 for Z being open Subset of REAL for f1, f2 being PartFunc of REAL,REAL st Z c= dom (f1 (#) (exp_R * f2)) & ( for x being Real st x in Z holds ( f1 . x = (- x) - 1 & f2 . x = - x ) ) holds ( f1 (#) (exp_R * f2) is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 (#) (exp_R * f2)) `| Z) . x = x / (exp_R x) ) ) proof let Z be open Subset of REAL; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st Z c= dom (f1 (#) (exp_R * f2)) & ( for x being Real st x in Z holds ( f1 . x = (- x) - 1 & f2 . x = - x ) ) holds ( f1 (#) (exp_R * f2) is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 (#) (exp_R * f2)) `| Z) . x = x / (exp_R x) ) ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (f1 (#) (exp_R * f2)) & ( for x being Real st x in Z holds ( f1 . x = (- x) - 1 & f2 . x = - x ) ) implies ( f1 (#) (exp_R * f2) is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 (#) (exp_R * f2)) `| Z) . x = x / (exp_R x) ) ) ) assume that A1: Z c= dom (f1 (#) (exp_R * f2)) and A2: for x being Real st x in Z holds ( f1 . x = (- x) - 1 & f2 . x = - x ) ; ::_thesis: ( f1 (#) (exp_R * f2) is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 (#) (exp_R * f2)) `| Z) . x = x / (exp_R x) ) ) A3: Z c= (dom f1) /\ (dom (exp_R * f2)) by A1, VALUED_1:def_4; then A4: Z c= dom f1 by XBOOLE_1:18; A5: Z c= dom (exp_R * f2) by A3, XBOOLE_1:18; A6: for x being Real st x in Z holds f1 . x = ((- 1) * x) + (- 1) proof let x be Real; ::_thesis: ( x in Z implies f1 . x = ((- 1) * x) + (- 1) ) assume x in Z ; ::_thesis: f1 . x = ((- 1) * x) + (- 1) then f1 . x = (- x) - 1 by A2 .= ((- 1) * x) + (- 1) ; hence f1 . x = ((- 1) * x) + (- 1) ; ::_thesis: verum end; then A7: f1 is_differentiable_on Z by A4, FDIFF_1:23; A8: for x being Real st x in Z holds f2 . x = - x by A2; then A9: exp_R * f2 is_differentiable_on Z by A5, Th14; for x being Real st x in Z holds ((f1 (#) (exp_R * f2)) `| Z) . x = x / (exp_R x) proof let x be Real; ::_thesis: ( x in Z implies ((f1 (#) (exp_R * f2)) `| Z) . x = x / (exp_R x) ) assume A10: x in Z ; ::_thesis: ((f1 (#) (exp_R * f2)) `| Z) . x = x / (exp_R x) then A11: (exp_R * f2) . x = exp_R . (f2 . x) by A5, FUNCT_1:12 .= exp_R . (- x) by A2, A10 .= exp_R (- x) by SIN_COS:def_23 ; ((f1 (#) (exp_R * f2)) `| Z) . x = (((exp_R * f2) . x) * (diff (f1,x))) + ((f1 . x) * (diff ((exp_R * f2),x))) by A1, A7, A9, A10, FDIFF_1:21 .= ((exp_R (- x)) * (diff (f1,x))) + (((- x) - 1) * (diff ((exp_R * f2),x))) by A2, A10, A11 .= ((exp_R (- x)) * ((f1 `| Z) . x)) + (((- x) - 1) * (diff ((exp_R * f2),x))) by A7, A10, FDIFF_1:def_7 .= ((exp_R (- x)) * ((f1 `| Z) . x)) + (((- x) - 1) * (((exp_R * f2) `| Z) . x)) by A9, A10, FDIFF_1:def_7 .= ((exp_R (- x)) * (- 1)) + (((- x) - 1) * (((exp_R * f2) `| Z) . x)) by A4, A6, A10, FDIFF_1:23 .= ((exp_R (- x)) * (- 1)) + (((- x) - 1) * (- (exp_R (- x)))) by A8, A5, A10, Th14 .= x * (exp_R (- x)) .= x * (1 / (exp_R x)) by TAYLOR_1:4 .= x / (exp_R x) by XCMPLX_1:99 ; hence ((f1 (#) (exp_R * f2)) `| Z) . x = x / (exp_R x) ; ::_thesis: verum end; hence ( f1 (#) (exp_R * f2) is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 (#) (exp_R * f2)) `| Z) . x = x / (exp_R x) ) ) by A1, A7, A9, FDIFF_1:21; ::_thesis: verum end; theorem Th16: :: FDIFF_6:16 for a being Real for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom (- (exp_R * f)) & ( for x being Real st x in Z holds f . x = - (x * (log (number_e,a))) ) & a > 0 holds ( - (exp_R * f) is_differentiable_on Z & ( for x being Real st x in Z holds ((- (exp_R * f)) `| Z) . x = (a #R (- x)) * (log (number_e,a)) ) ) proof let a be Real; ::_thesis: for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom (- (exp_R * f)) & ( for x being Real st x in Z holds f . x = - (x * (log (number_e,a))) ) & a > 0 holds ( - (exp_R * f) is_differentiable_on Z & ( for x being Real st x in Z holds ((- (exp_R * f)) `| Z) . x = (a #R (- x)) * (log (number_e,a)) ) ) let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st Z c= dom (- (exp_R * f)) & ( for x being Real st x in Z holds f . x = - (x * (log (number_e,a))) ) & a > 0 holds ( - (exp_R * f) is_differentiable_on Z & ( for x being Real st x in Z holds ((- (exp_R * f)) `| Z) . x = (a #R (- x)) * (log (number_e,a)) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (- (exp_R * f)) & ( for x being Real st x in Z holds f . x = - (x * (log (number_e,a))) ) & a > 0 implies ( - (exp_R * f) is_differentiable_on Z & ( for x being Real st x in Z holds ((- (exp_R * f)) `| Z) . x = (a #R (- x)) * (log (number_e,a)) ) ) ) assume that A1: Z c= dom (- (exp_R * f)) and A2: for x being Real st x in Z holds f . x = - (x * (log (number_e,a))) and A3: a > 0 ; ::_thesis: ( - (exp_R * f) is_differentiable_on Z & ( for x being Real st x in Z holds ((- (exp_R * f)) `| Z) . x = (a #R (- x)) * (log (number_e,a)) ) ) A4: Z c= dom (exp_R * f) by A1, VALUED_1:8; then for y being set st y in Z holds y in dom f by FUNCT_1:11; then A5: Z c= dom f by TARSKI:def_3; A6: for x being Real st x in Z holds f . x = ((- (log (number_e,a))) * x) + 0 proof let x be Real; ::_thesis: ( x in Z implies f . x = ((- (log (number_e,a))) * x) + 0 ) assume x in Z ; ::_thesis: f . x = ((- (log (number_e,a))) * x) + 0 then f . x = - ((log (number_e,a)) * x) by A2 .= ((- (log (number_e,a))) * x) + 0 ; hence f . x = ((- (log (number_e,a))) * x) + 0 ; ::_thesis: verum end; then A7: f is_differentiable_on Z by A5, FDIFF_1:23; for x being Real st x in Z holds exp_R * f is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies exp_R * f is_differentiable_in x ) assume x in Z ; ::_thesis: exp_R * f is_differentiable_in x then f is_differentiable_in x by A7, FDIFF_1:9; hence exp_R * f is_differentiable_in x by TAYLOR_1:19; ::_thesis: verum end; then A8: exp_R * f is_differentiable_on Z by A4, FDIFF_1:9; A9: for x being Real st x in Z holds ((- (exp_R * f)) `| Z) . x = (a #R (- x)) * (log (number_e,a)) proof let x be Real; ::_thesis: ( x in Z implies ((- (exp_R * f)) `| Z) . x = (a #R (- x)) * (log (number_e,a)) ) assume A10: x in Z ; ::_thesis: ((- (exp_R * f)) `| Z) . x = (a #R (- x)) * (log (number_e,a)) then A11: f is_differentiable_in x by A7, FDIFF_1:9; ((- (exp_R * f)) `| Z) . x = (- 1) * (diff ((exp_R * f),x)) by A1, A8, A10, FDIFF_1:20 .= (- 1) * ((exp_R . (f . x)) * (diff (f,x))) by A11, TAYLOR_1:19 .= (- 1) * ((exp_R . (f . x)) * ((f `| Z) . x)) by A7, A10, FDIFF_1:def_7 .= (- 1) * ((exp_R . (f . x)) * (- (log (number_e,a)))) by A5, A6, A10, FDIFF_1:23 .= (- 1) * ((exp_R . (- (x * (log (number_e,a))))) * (- (log (number_e,a)))) by A2, A10 .= (- 1) * ((a #R (- x)) * (- (log (number_e,a)))) by A3, Th2 .= (a #R (- x)) * (log (number_e,a)) ; hence ((- (exp_R * f)) `| Z) . x = (a #R (- x)) * (log (number_e,a)) ; ::_thesis: verum end; Z c= dom ((- 1) (#) (exp_R * f)) by A1; hence ( - (exp_R * f) is_differentiable_on Z & ( for x being Real st x in Z holds ((- (exp_R * f)) `| Z) . x = (a #R (- x)) * (log (number_e,a)) ) ) by A8, A9, FDIFF_1:20; ::_thesis: verum end; theorem :: FDIFF_6:17 for a being Real for Z being open Subset of REAL for f1, f2 being PartFunc of REAL,REAL st Z c= dom ((1 / (log (number_e,a))) (#) ((- (exp_R * f1)) (#) f2)) & ( for x being Real st x in Z holds ( f1 . x = - (x * (log (number_e,a))) & f2 . x = x + (1 / (log (number_e,a))) ) ) & a > 0 & a <> 1 holds ( (1 / (log (number_e,a))) (#) ((- (exp_R * f1)) (#) f2) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / (log (number_e,a))) (#) ((- (exp_R * f1)) (#) f2)) `| Z) . x = x / (a #R x) ) ) proof let a be Real; ::_thesis: for Z being open Subset of REAL for f1, f2 being PartFunc of REAL,REAL st Z c= dom ((1 / (log (number_e,a))) (#) ((- (exp_R * f1)) (#) f2)) & ( for x being Real st x in Z holds ( f1 . x = - (x * (log (number_e,a))) & f2 . x = x + (1 / (log (number_e,a))) ) ) & a > 0 & a <> 1 holds ( (1 / (log (number_e,a))) (#) ((- (exp_R * f1)) (#) f2) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / (log (number_e,a))) (#) ((- (exp_R * f1)) (#) f2)) `| Z) . x = x / (a #R x) ) ) let Z be open Subset of REAL; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st Z c= dom ((1 / (log (number_e,a))) (#) ((- (exp_R * f1)) (#) f2)) & ( for x being Real st x in Z holds ( f1 . x = - (x * (log (number_e,a))) & f2 . x = x + (1 / (log (number_e,a))) ) ) & a > 0 & a <> 1 holds ( (1 / (log (number_e,a))) (#) ((- (exp_R * f1)) (#) f2) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / (log (number_e,a))) (#) ((- (exp_R * f1)) (#) f2)) `| Z) . x = x / (a #R x) ) ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom ((1 / (log (number_e,a))) (#) ((- (exp_R * f1)) (#) f2)) & ( for x being Real st x in Z holds ( f1 . x = - (x * (log (number_e,a))) & f2 . x = x + (1 / (log (number_e,a))) ) ) & a > 0 & a <> 1 implies ( (1 / (log (number_e,a))) (#) ((- (exp_R * f1)) (#) f2) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / (log (number_e,a))) (#) ((- (exp_R * f1)) (#) f2)) `| Z) . x = x / (a #R x) ) ) ) assume that A1: Z c= dom ((1 / (log (number_e,a))) (#) ((- (exp_R * f1)) (#) f2)) and A2: for x being Real st x in Z holds ( f1 . x = - (x * (log (number_e,a))) & f2 . x = x + (1 / (log (number_e,a))) ) and A3: a > 0 and A4: a <> 1 ; ::_thesis: ( (1 / (log (number_e,a))) (#) ((- (exp_R * f1)) (#) f2) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / (log (number_e,a))) (#) ((- (exp_R * f1)) (#) f2)) `| Z) . x = x / (a #R x) ) ) A5: for x being Real st x in Z holds f2 . x = (1 * x) + (1 / (log (number_e,a))) by A2; A6: Z c= dom ((- (exp_R * f1)) (#) f2) by A1, VALUED_1:def_5; then A7: Z c= (dom (- (exp_R * f1))) /\ (dom f2) by VALUED_1:def_4; then A8: Z c= dom (- (exp_R * f1)) by XBOOLE_1:18; A9: for x being Real st x in Z holds f1 . x = - (x * (log (number_e,a))) by A2; then A10: - (exp_R * f1) is_differentiable_on Z by A3, A8, Th16; A11: Z c= dom f2 by A7, XBOOLE_1:18; then A12: f2 is_differentiable_on Z by A5, FDIFF_1:23; then A13: (- (exp_R * f1)) (#) f2 is_differentiable_on Z by A6, A10, FDIFF_1:21; A14: log (number_e,a) <> 0 proof A15: number_e <> 1 by TAYLOR_1:11; assume log (number_e,a) = 0 ; ::_thesis: contradiction then log (number_e,a) = log (number_e,1) by SIN_COS2:13, TAYLOR_1:13; then a = number_e to_power (log (number_e,1)) by A3, A15, POWER:def_3, TAYLOR_1:11 .= 1 by A15, POWER:def_3, TAYLOR_1:11 ; hence contradiction by A4; ::_thesis: verum end; for x being Real st x in Z holds (((1 / (log (number_e,a))) (#) ((- (exp_R * f1)) (#) f2)) `| Z) . x = x / (a #R x) proof let x be Real; ::_thesis: ( x in Z implies (((1 / (log (number_e,a))) (#) ((- (exp_R * f1)) (#) f2)) `| Z) . x = x / (a #R x) ) assume A16: x in Z ; ::_thesis: (((1 / (log (number_e,a))) (#) ((- (exp_R * f1)) (#) f2)) `| Z) . x = x / (a #R x) then x in dom (- (exp_R * f1)) by A8; then A17: x in dom (exp_R * f1) by VALUED_1:8; A18: (- (exp_R * f1)) . x = - ((exp_R * f1) . x) by VALUED_1:8 .= - (exp_R . (f1 . x)) by A17, FUNCT_1:12 .= - (exp_R . (- (x * (log (number_e,a))))) by A2, A16 .= - (a #R (- x)) by A3, Th2 ; (((1 / (log (number_e,a))) (#) ((- (exp_R * f1)) (#) f2)) `| Z) . x = (1 / (log (number_e,a))) * (diff (((- (exp_R * f1)) (#) f2),x)) by A1, A13, A16, FDIFF_1:20 .= (1 / (log (number_e,a))) * ((((- (exp_R * f1)) (#) f2) `| Z) . x) by A13, A16, FDIFF_1:def_7 .= (1 / (log (number_e,a))) * (((f2 . x) * (diff ((- (exp_R * f1)),x))) + (((- (exp_R * f1)) . x) * (diff (f2,x)))) by A6, A10, A12, A16, FDIFF_1:21 .= (1 / (log (number_e,a))) * (((f2 . x) * (((- (exp_R * f1)) `| Z) . x)) + (((- (exp_R * f1)) . x) * (diff (f2,x)))) by A10, A16, FDIFF_1:def_7 .= (1 / (log (number_e,a))) * (((f2 . x) * (((- (exp_R * f1)) `| Z) . x)) + (((- (exp_R * f1)) . x) * ((f2 `| Z) . x))) by A12, A16, FDIFF_1:def_7 .= (1 / (log (number_e,a))) * (((f2 . x) * ((a #R (- x)) * (log (number_e,a)))) + (((- (exp_R * f1)) . x) * ((f2 `| Z) . x))) by A3, A9, A8, A16, Th16 .= (1 / (log (number_e,a))) * (((f2 . x) * ((a #R (- x)) * (log (number_e,a)))) + (((- (exp_R * f1)) . x) * 1)) by A11, A5, A16, FDIFF_1:23 .= (1 / (log (number_e,a))) * ((((f2 . x) * (log (number_e,a))) - 1) * (a #R (- x))) by A18 .= (1 / (log (number_e,a))) * ((((x + (1 / (log (number_e,a)))) * (log (number_e,a))) - 1) * (a #R (- x))) by A2, A16 .= ((1 / (log (number_e,a))) * (((x * (log (number_e,a))) + ((1 / (log (number_e,a))) * (log (number_e,a)))) - 1)) * (a #R (- x)) .= ((1 / (log (number_e,a))) * (((x * (log (number_e,a))) + 1) - 1)) * (a #R (- x)) by A14, XCMPLX_1:106 .= (((1 / (log (number_e,a))) * (log (number_e,a))) * x) * (a #R (- x)) .= (1 * x) * (a #R (- x)) by A14, XCMPLX_1:106 .= x * (1 / (a #R x)) by A3, PREPOWER:76 .= x / (a #R x) by XCMPLX_1:99 ; hence (((1 / (log (number_e,a))) (#) ((- (exp_R * f1)) (#) f2)) `| Z) . x = x / (a #R x) ; ::_thesis: verum end; hence ( (1 / (log (number_e,a))) (#) ((- (exp_R * f1)) (#) f2) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / (log (number_e,a))) (#) ((- (exp_R * f1)) (#) f2)) `| Z) . x = x / (a #R x) ) ) by A1, A13, FDIFF_1:20; ::_thesis: verum end; theorem :: FDIFF_6:18 for a being Real for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom ((1 / ((log (number_e,a)) - 1)) (#) ((exp_R * f) / exp_R)) & ( for x being Real st x in Z holds f . x = x * (log (number_e,a)) ) & a > 0 & a <> number_e holds ( (1 / ((log (number_e,a)) - 1)) (#) ((exp_R * f) / exp_R) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / ((log (number_e,a)) - 1)) (#) ((exp_R * f) / exp_R)) `| Z) . x = (a #R x) / (exp_R . x) ) ) proof let a be Real; ::_thesis: for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom ((1 / ((log (number_e,a)) - 1)) (#) ((exp_R * f) / exp_R)) & ( for x being Real st x in Z holds f . x = x * (log (number_e,a)) ) & a > 0 & a <> number_e holds ( (1 / ((log (number_e,a)) - 1)) (#) ((exp_R * f) / exp_R) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / ((log (number_e,a)) - 1)) (#) ((exp_R * f) / exp_R)) `| Z) . x = (a #R x) / (exp_R . x) ) ) let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st Z c= dom ((1 / ((log (number_e,a)) - 1)) (#) ((exp_R * f) / exp_R)) & ( for x being Real st x in Z holds f . x = x * (log (number_e,a)) ) & a > 0 & a <> number_e holds ( (1 / ((log (number_e,a)) - 1)) (#) ((exp_R * f) / exp_R) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / ((log (number_e,a)) - 1)) (#) ((exp_R * f) / exp_R)) `| Z) . x = (a #R x) / (exp_R . x) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom ((1 / ((log (number_e,a)) - 1)) (#) ((exp_R * f) / exp_R)) & ( for x being Real st x in Z holds f . x = x * (log (number_e,a)) ) & a > 0 & a <> number_e implies ( (1 / ((log (number_e,a)) - 1)) (#) ((exp_R * f) / exp_R) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / ((log (number_e,a)) - 1)) (#) ((exp_R * f) / exp_R)) `| Z) . x = (a #R x) / (exp_R . x) ) ) ) assume that A1: Z c= dom ((1 / ((log (number_e,a)) - 1)) (#) ((exp_R * f) / exp_R)) and A2: for x being Real st x in Z holds f . x = x * (log (number_e,a)) and A3: a > 0 and A4: a <> number_e ; ::_thesis: ( (1 / ((log (number_e,a)) - 1)) (#) ((exp_R * f) / exp_R) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / ((log (number_e,a)) - 1)) (#) ((exp_R * f) / exp_R)) `| Z) . x = (a #R x) / (exp_R . x) ) ) Z c= dom ((exp_R * f) / exp_R) by A1, VALUED_1:def_5; then Z c= (dom (exp_R * f)) /\ ((dom exp_R) \ (exp_R " {0})) by RFUNCT_1:def_1; then A5: Z c= dom (exp_R * f) by XBOOLE_1:18; then A6: exp_R * f is_differentiable_on Z by A2, A3, Th11; ( 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: (exp_R * f) / exp_R is_differentiable_on Z by A6, FDIFF_2:21; A8: (log (number_e,a)) - 1 <> 0 proof A9: number_e <> 1 by TAYLOR_1:11; assume (log (number_e,a)) - 1 = 0 ; ::_thesis: contradiction then log (number_e,a) = log (number_e,number_e) by A9, POWER:52, TAYLOR_1:11; then a = number_e to_power (log (number_e,number_e)) by A3, A9, POWER:def_3, TAYLOR_1:11 .= number_e by A9, POWER:def_3, TAYLOR_1:11 ; hence contradiction by A4; ::_thesis: verum end; for x being Real st x in Z holds (((1 / ((log (number_e,a)) - 1)) (#) ((exp_R * f) / exp_R)) `| Z) . x = (a #R x) / (exp_R . x) proof let x be Real; ::_thesis: ( x in Z implies (((1 / ((log (number_e,a)) - 1)) (#) ((exp_R * f) / exp_R)) `| Z) . x = (a #R x) / (exp_R . x) ) A10: exp_R . x <> 0 by SIN_COS:54; assume A11: x in Z ; ::_thesis: (((1 / ((log (number_e,a)) - 1)) (#) ((exp_R * f) / exp_R)) `| Z) . x = (a #R x) / (exp_R . x) then A12: (exp_R * f) . x = exp_R . (f . x) by A5, FUNCT_1:12 .= exp_R . (x * (log (number_e,a))) by A2, A11 .= a #R x by A3, Th1 ; A13: ( exp_R is_differentiable_in x & exp_R * f is_differentiable_in x ) by A6, A11, FDIFF_1:9, SIN_COS:65; (((1 / ((log (number_e,a)) - 1)) (#) ((exp_R * f) / exp_R)) `| Z) . x = (1 / ((log (number_e,a)) - 1)) * (diff (((exp_R * f) / exp_R),x)) by A1, A7, A11, FDIFF_1:20 .= (1 / ((log (number_e,a)) - 1)) * ((((diff ((exp_R * f),x)) * (exp_R . x)) - ((diff (exp_R,x)) * ((exp_R * f) . x))) / ((exp_R . x) ^2)) by A13, A10, FDIFF_2:14 .= (1 / ((log (number_e,a)) - 1)) * ((((diff ((exp_R * f),x)) * (exp_R . x)) - ((exp_R . x) * (a #R x))) / ((exp_R . x) ^2)) by A12, SIN_COS:65 .= (1 / ((log (number_e,a)) - 1)) * ((((diff ((exp_R * f),x)) - (a #R x)) * (exp_R . x)) / ((exp_R . x) * (exp_R . x))) .= (1 / ((log (number_e,a)) - 1)) * (((diff ((exp_R * f),x)) - (a #R x)) / (exp_R . x)) by A10, XCMPLX_1:91 .= ((1 / ((log (number_e,a)) - 1)) * ((diff ((exp_R * f),x)) - (a #R x))) / (exp_R . x) by XCMPLX_1:74 .= ((1 / ((log (number_e,a)) - 1)) * ((((exp_R * f) `| Z) . x) - (a #R x))) / (exp_R . x) by A6, A11, FDIFF_1:def_7 .= ((1 / ((log (number_e,a)) - 1)) * (((a #R x) * (log (number_e,a))) - (a #R x))) / (exp_R . x) by A2, A3, A5, A11, Th11 .= (((1 / ((log (number_e,a)) - 1)) * ((log (number_e,a)) - 1)) * (a #R x)) / (exp_R . x) .= (1 * (a #R x)) / (exp_R . x) by A8, XCMPLX_1:106 .= (a #R x) / (exp_R . x) ; hence (((1 / ((log (number_e,a)) - 1)) (#) ((exp_R * f) / exp_R)) `| Z) . x = (a #R x) / (exp_R . x) ; ::_thesis: verum end; hence ( (1 / ((log (number_e,a)) - 1)) (#) ((exp_R * f) / exp_R) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / ((log (number_e,a)) - 1)) (#) ((exp_R * f) / exp_R)) `| Z) . x = (a #R x) / (exp_R . x) ) ) by A1, A7, FDIFF_1:20; ::_thesis: verum end; theorem :: FDIFF_6:19 for a being Real for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom ((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) & ( for x being Real st x in Z holds f . x = x * (log (number_e,a)) ) & a > 0 & a <> number_e holds ( (1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f)) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) `| Z) . x = (exp_R . x) / (a #R x) ) ) proof let a be Real; ::_thesis: for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom ((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) & ( for x being Real st x in Z holds f . x = x * (log (number_e,a)) ) & a > 0 & a <> number_e holds ( (1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f)) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) `| Z) . x = (exp_R . x) / (a #R x) ) ) let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st Z c= dom ((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) & ( for x being Real st x in Z holds f . x = x * (log (number_e,a)) ) & a > 0 & a <> number_e holds ( (1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f)) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) `| Z) . x = (exp_R . x) / (a #R x) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom ((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) & ( for x being Real st x in Z holds f . x = x * (log (number_e,a)) ) & a > 0 & a <> number_e implies ( (1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f)) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) `| Z) . x = (exp_R . x) / (a #R x) ) ) ) assume that A1: Z c= dom ((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) and A2: for x being Real st x in Z holds f . x = x * (log (number_e,a)) and A3: a > 0 and A4: a <> number_e ; ::_thesis: ( (1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f)) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) `| Z) . x = (exp_R . x) / (a #R x) ) ) Z c= dom (exp_R / (exp_R * f)) by A1, VALUED_1:def_5; then Z c= (dom exp_R) /\ ((dom (exp_R * f)) \ ((exp_R * f) " {0})) by RFUNCT_1:def_1; then A5: Z c= dom (exp_R * f) by XBOOLE_1:1; then A6: exp_R * f is_differentiable_on Z by A2, A3, Th11; A7: for x being Real st x in Z holds (exp_R * f) . x <> 0 proof let x be Real; ::_thesis: ( x in Z implies (exp_R * f) . x <> 0 ) assume x in Z ; ::_thesis: (exp_R * f) . x <> 0 then (exp_R * f) . x = exp_R . (f . x) by A5, FUNCT_1:12; hence (exp_R * f) . x <> 0 by SIN_COS:54; ::_thesis: verum end; exp_R is_differentiable_on Z by FDIFF_1:26, TAYLOR_1:16; then A8: exp_R / (exp_R * f) is_differentiable_on Z by A6, A7, FDIFF_2:21; A9: 1 - (log (number_e,a)) <> 0 proof A10: number_e <> 1 by TAYLOR_1:11; assume 1 - (log (number_e,a)) = 0 ; ::_thesis: contradiction then log (number_e,a) = log (number_e,number_e) by A10, POWER:52, TAYLOR_1:11; then a = number_e to_power (log (number_e,number_e)) by A3, A10, POWER:def_3, TAYLOR_1:11 .= number_e by A10, POWER:def_3, TAYLOR_1:11 ; hence contradiction by A4; ::_thesis: verum end; for x being Real st x in Z holds (((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) `| Z) . x = (exp_R . x) / (a #R x) proof let x be Real; ::_thesis: ( x in Z implies (((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) `| Z) . x = (exp_R . x) / (a #R x) ) A11: exp_R is_differentiable_in x by SIN_COS:65; A12: a #R x > 0 by A3, PREPOWER:81; assume A13: x in Z ; ::_thesis: (((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) `| Z) . x = (exp_R . x) / (a #R x) then A14: (exp_R * f) . x = exp_R . (f . x) by A5, FUNCT_1:12 .= exp_R . (x * (log (number_e,a))) by A2, A13 .= a #R x by A3, Th1 ; A15: ( exp_R * f is_differentiable_in x & (exp_R * f) . x <> 0 ) by A6, A7, A13, FDIFF_1:9; (((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) `| Z) . x = (1 / (1 - (log (number_e,a)))) * (diff ((exp_R / (exp_R * f)),x)) by A1, A8, A13, FDIFF_1:20 .= (1 / (1 - (log (number_e,a)))) * ((((diff (exp_R,x)) * ((exp_R * f) . x)) - ((diff ((exp_R * f),x)) * (exp_R . x))) / (((exp_R * f) . x) ^2)) by A11, A15, FDIFF_2:14 .= (1 / (1 - (log (number_e,a)))) * ((((exp_R . x) * (a #R x)) - ((diff ((exp_R * f),x)) * (exp_R . x))) / ((a #R x) ^2)) by A14, SIN_COS:65 .= (1 / (1 - (log (number_e,a)))) * (((exp_R . x) * ((a #R x) - (diff ((exp_R * f),x)))) / ((a #R x) ^2)) .= (1 / (1 - (log (number_e,a)))) * (((exp_R . x) * ((a #R x) - (((exp_R * f) `| Z) . x))) / ((a #R x) ^2)) by A6, A13, FDIFF_1:def_7 .= (1 / (1 - (log (number_e,a)))) * (((exp_R . x) * ((a #R x) - ((a #R x) * (log (number_e,a))))) / ((a #R x) ^2)) by A2, A3, A5, A13, Th11 .= ((1 / (1 - (log (number_e,a)))) * (((1 - (log (number_e,a))) * (exp_R . x)) * (a #R x))) / ((a #R x) ^2) by XCMPLX_1:74 .= ((((1 / (1 - (log (number_e,a)))) * (1 - (log (number_e,a)))) * (exp_R . x)) * (a #R x)) / ((a #R x) ^2) .= ((1 * (exp_R . x)) * (a #R x)) / ((a #R x) ^2) by A9, XCMPLX_1:106 .= (exp_R . x) / (a #R x) by A12, XCMPLX_1:91 ; hence (((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) `| Z) . x = (exp_R . x) / (a #R x) ; ::_thesis: verum end; hence ( (1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f)) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) `| Z) . x = (exp_R . x) / (a #R x) ) ) by A1, A8, FDIFF_1:20; ::_thesis: verum end; theorem :: FDIFF_6:20 for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom (ln * (exp_R + f)) & ( for x being Real st x in Z holds f . x = 1 ) holds ( ln * (exp_R + f) is_differentiable_on Z & ( for x being Real st x in Z holds ((ln * (exp_R + f)) `| Z) . x = (exp_R . x) / ((exp_R . x) + 1) ) ) proof let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st Z c= dom (ln * (exp_R + f)) & ( for x being Real st x in Z holds f . x = 1 ) holds ( ln * (exp_R + f) is_differentiable_on Z & ( for x being Real st x in Z holds ((ln * (exp_R + f)) `| Z) . x = (exp_R . x) / ((exp_R . x) + 1) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (ln * (exp_R + f)) & ( for x being Real st x in Z holds f . x = 1 ) implies ( ln * (exp_R + f) is_differentiable_on Z & ( for x being Real st x in Z holds ((ln * (exp_R + f)) `| Z) . x = (exp_R . x) / ((exp_R . x) + 1) ) ) ) assume that A1: Z c= dom (ln * (exp_R + f)) and A2: for x being Real st x in Z holds f . x = 1 ; ::_thesis: ( ln * (exp_R + f) is_differentiable_on Z & ( for x being Real st x in Z holds ((ln * (exp_R + f)) `| Z) . x = (exp_R . x) / ((exp_R . x) + 1) ) ) A3: for x being Real st x in Z holds f . x = (0 * x) + 1 by A2; for y being set st y in Z holds y in dom (exp_R + f) by A1, FUNCT_1:11; then A4: Z c= dom (exp_R + f) by TARSKI:def_3; then Z c= (dom exp_R) /\ (dom f) by VALUED_1:def_1; then A5: Z c= dom f by XBOOLE_1:18; then A6: f is_differentiable_on Z by A3, FDIFF_1:23; A7: exp_R is_differentiable_on Z by FDIFF_1:26, TAYLOR_1:16; then A8: exp_R + f is_differentiable_on Z by A4, A6, FDIFF_1:18; A9: for x being Real st x in Z holds ((exp_R + f) `| Z) . x = exp_R . x proof let x be Real; ::_thesis: ( x in Z implies ((exp_R + f) `| Z) . x = exp_R . x ) assume A10: x in Z ; ::_thesis: ((exp_R + f) `| Z) . x = exp_R . x hence ((exp_R + f) `| Z) . x = (diff (exp_R,x)) + (diff (f,x)) by A4, A6, A7, FDIFF_1:18 .= (exp_R . x) + (diff (f,x)) by SIN_COS:65 .= (exp_R . x) + ((f `| Z) . x) by A6, A10, FDIFF_1:def_7 .= (exp_R . x) + 0 by A5, A3, A10, FDIFF_1:23 .= exp_R . x ; ::_thesis: verum end; A11: for x being Real st x in Z holds (exp_R + f) . x > 0 proof let x be Real; ::_thesis: ( x in Z implies (exp_R + f) . x > 0 ) assume A12: x in Z ; ::_thesis: (exp_R + f) . x > 0 then (exp_R + f) . x = (exp_R . x) + (f . x) by A4, VALUED_1:def_1 .= (exp_R . x) + 1 by A2, A12 ; hence (exp_R + f) . x > 0 by SIN_COS:54, XREAL_1:34; ::_thesis: verum end; A13: for x being Real st x in Z holds ln * (exp_R + f) is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies ln * (exp_R + f) is_differentiable_in x ) assume x in Z ; ::_thesis: ln * (exp_R + f) is_differentiable_in x then ( exp_R + f is_differentiable_in x & (exp_R + f) . x > 0 ) by A8, A11, FDIFF_1:9; hence ln * (exp_R + f) is_differentiable_in x by TAYLOR_1:20; ::_thesis: verum end; then A14: ln * (exp_R + f) is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((ln * (exp_R + f)) `| Z) . x = (exp_R . x) / ((exp_R . x) + 1) proof let x be Real; ::_thesis: ( x in Z implies ((ln * (exp_R + f)) `| Z) . x = (exp_R . x) / ((exp_R . x) + 1) ) assume A15: x in Z ; ::_thesis: ((ln * (exp_R + f)) `| Z) . x = (exp_R . x) / ((exp_R . x) + 1) then A16: (exp_R + f) . x = (exp_R . x) + (f . x) by A4, VALUED_1:def_1 .= (exp_R . x) + 1 by A2, A15 ; ( exp_R + f is_differentiable_in x & (exp_R + f) . x > 0 ) by A8, A11, A15, FDIFF_1:9; then diff ((ln * (exp_R + f)),x) = (diff ((exp_R + f),x)) / ((exp_R + f) . x) by TAYLOR_1:20 .= (((exp_R + f) `| Z) . x) / ((exp_R + f) . x) by A8, A15, FDIFF_1:def_7 .= (exp_R . x) / ((exp_R . x) + 1) by A9, A15, A16 ; hence ((ln * (exp_R + f)) `| Z) . x = (exp_R . x) / ((exp_R . x) + 1) by A14, A15, FDIFF_1:def_7; ::_thesis: verum end; hence ( ln * (exp_R + f) is_differentiable_on Z & ( for x being Real st x in Z holds ((ln * (exp_R + f)) `| Z) . x = (exp_R . x) / ((exp_R . x) + 1) ) ) by A1, A13, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_6:21 for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom (ln * (exp_R - f)) & ( for x being Real st x in Z holds ( f . x = 1 & (exp_R - f) . x > 0 ) ) holds ( ln * (exp_R - f) is_differentiable_on Z & ( for x being Real st x in Z holds ((ln * (exp_R - f)) `| Z) . x = (exp_R . x) / ((exp_R . x) - 1) ) ) proof let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st Z c= dom (ln * (exp_R - f)) & ( for x being Real st x in Z holds ( f . x = 1 & (exp_R - f) . x > 0 ) ) holds ( ln * (exp_R - f) is_differentiable_on Z & ( for x being Real st x in Z holds ((ln * (exp_R - f)) `| Z) . x = (exp_R . x) / ((exp_R . x) - 1) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (ln * (exp_R - f)) & ( for x being Real st x in Z holds ( f . x = 1 & (exp_R - f) . x > 0 ) ) implies ( ln * (exp_R - f) is_differentiable_on Z & ( for x being Real st x in Z holds ((ln * (exp_R - f)) `| Z) . x = (exp_R . x) / ((exp_R . x) - 1) ) ) ) assume that A1: Z c= dom (ln * (exp_R - f)) and A2: for x being Real st x in Z holds ( f . x = 1 & (exp_R - f) . x > 0 ) ; ::_thesis: ( ln * (exp_R - f) is_differentiable_on Z & ( for x being Real st x in Z holds ((ln * (exp_R - f)) `| Z) . x = (exp_R . x) / ((exp_R . x) - 1) ) ) A3: for x being Real st x in Z holds f . x = (0 * x) + 1 by A2; for y being set st y in Z holds y in dom (exp_R - f) by A1, FUNCT_1:11; then A4: Z c= dom (exp_R - f) by TARSKI:def_3; then Z c= (dom exp_R) /\ (dom f) by VALUED_1:12; then A5: Z c= dom f by XBOOLE_1:18; then A6: f is_differentiable_on Z by A3, FDIFF_1:23; A7: exp_R is_differentiable_on Z by FDIFF_1:26, TAYLOR_1:16; then A8: exp_R - f is_differentiable_on Z by A4, A6, FDIFF_1:19; A9: for x being Real st x in Z holds ((exp_R - f) `| Z) . x = exp_R . x proof let x be Real; ::_thesis: ( x in Z implies ((exp_R - f) `| Z) . x = exp_R . x ) assume A10: x in Z ; ::_thesis: ((exp_R - f) `| Z) . x = exp_R . x hence ((exp_R - f) `| Z) . x = (diff (exp_R,x)) - (diff (f,x)) by A4, A6, A7, FDIFF_1:19 .= (exp_R . x) - (diff (f,x)) by SIN_COS:65 .= (exp_R . x) - ((f `| Z) . x) by A6, A10, FDIFF_1:def_7 .= (exp_R . x) - 0 by A5, A3, A10, FDIFF_1:23 .= exp_R . x ; ::_thesis: verum end; A11: for x being Real st x in Z holds ln * (exp_R - f) is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies ln * (exp_R - f) is_differentiable_in x ) assume x in Z ; ::_thesis: ln * (exp_R - f) is_differentiable_in x then ( exp_R - f is_differentiable_in x & (exp_R - f) . x > 0 ) by A2, A8, FDIFF_1:9; hence ln * (exp_R - f) is_differentiable_in x by TAYLOR_1:20; ::_thesis: verum end; then A12: ln * (exp_R - f) is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((ln * (exp_R - f)) `| Z) . x = (exp_R . x) / ((exp_R . x) - 1) proof let x be Real; ::_thesis: ( x in Z implies ((ln * (exp_R - f)) `| Z) . x = (exp_R . x) / ((exp_R . x) - 1) ) assume A13: x in Z ; ::_thesis: ((ln * (exp_R - f)) `| Z) . x = (exp_R . x) / ((exp_R . x) - 1) then A14: (exp_R - f) . x = (exp_R . x) - (f . x) by A4, VALUED_1:13 .= (exp_R . x) - 1 by A2, A13 ; ( exp_R - f is_differentiable_in x & (exp_R - f) . x > 0 ) by A2, A8, A13, FDIFF_1:9; then diff ((ln * (exp_R - f)),x) = (diff ((exp_R - f),x)) / ((exp_R - f) . x) by TAYLOR_1:20 .= (((exp_R - f) `| Z) . x) / ((exp_R - f) . x) by A8, A13, FDIFF_1:def_7 .= (exp_R . x) / ((exp_R . x) - 1) by A9, A13, A14 ; hence ((ln * (exp_R - f)) `| Z) . x = (exp_R . x) / ((exp_R . x) - 1) by A12, A13, FDIFF_1:def_7; ::_thesis: verum end; hence ( ln * (exp_R - f) is_differentiable_on Z & ( for x being Real st x in Z holds ((ln * (exp_R - f)) `| Z) . x = (exp_R . x) / ((exp_R . x) - 1) ) ) by A1, A11, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_6:22 for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom (- (ln * (f - exp_R))) & ( for x being Real st x in Z holds ( f . x = 1 & (f - exp_R) . x > 0 ) ) holds ( - (ln * (f - exp_R)) is_differentiable_on Z & ( for x being Real st x in Z holds ((- (ln * (f - exp_R))) `| Z) . x = (exp_R . x) / (1 - (exp_R . x)) ) ) proof let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st Z c= dom (- (ln * (f - exp_R))) & ( for x being Real st x in Z holds ( f . x = 1 & (f - exp_R) . x > 0 ) ) holds ( - (ln * (f - exp_R)) is_differentiable_on Z & ( for x being Real st x in Z holds ((- (ln * (f - exp_R))) `| Z) . x = (exp_R . x) / (1 - (exp_R . x)) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (- (ln * (f - exp_R))) & ( for x being Real st x in Z holds ( f . x = 1 & (f - exp_R) . x > 0 ) ) implies ( - (ln * (f - exp_R)) is_differentiable_on Z & ( for x being Real st x in Z holds ((- (ln * (f - exp_R))) `| Z) . x = (exp_R . x) / (1 - (exp_R . x)) ) ) ) assume that A1: Z c= dom (- (ln * (f - exp_R))) and A2: for x being Real st x in Z holds ( f . x = 1 & (f - exp_R) . x > 0 ) ; ::_thesis: ( - (ln * (f - exp_R)) is_differentiable_on Z & ( for x being Real st x in Z holds ((- (ln * (f - exp_R))) `| Z) . x = (exp_R . x) / (1 - (exp_R . x)) ) ) A3: for x being Real st x in Z holds f . x = (0 * x) + 1 by A2; A4: Z c= dom (ln * (f - exp_R)) by A1, VALUED_1:8; then for y being set st y in Z holds y in dom (f - exp_R) by FUNCT_1:11; then A5: Z c= dom (f - exp_R) by TARSKI:def_3; then Z c= (dom exp_R) /\ (dom f) by VALUED_1:12; then A6: Z c= dom f by XBOOLE_1:18; then A7: f is_differentiable_on Z by A3, FDIFF_1:23; A8: exp_R is_differentiable_on Z by FDIFF_1:26, TAYLOR_1:16; then A9: f - exp_R is_differentiable_on Z by A5, A7, FDIFF_1:19; for x being Real st x in Z holds ln * (f - exp_R) is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies ln * (f - exp_R) is_differentiable_in x ) assume x in Z ; ::_thesis: ln * (f - exp_R) is_differentiable_in x then ( f - exp_R is_differentiable_in x & (f - exp_R) . x > 0 ) by A2, A9, FDIFF_1:9; hence ln * (f - exp_R) is_differentiable_in x by TAYLOR_1:20; ::_thesis: verum end; then A10: ln * (f - exp_R) is_differentiable_on Z by A4, FDIFF_1:9; A11: for x being Real st x in Z holds ((f - exp_R) `| Z) . x = - (exp_R . x) proof let x be Real; ::_thesis: ( x in Z implies ((f - exp_R) `| Z) . x = - (exp_R . x) ) assume A12: x in Z ; ::_thesis: ((f - exp_R) `| Z) . x = - (exp_R . x) hence ((f - exp_R) `| Z) . x = (diff (f,x)) - (diff (exp_R,x)) by A5, A7, A8, FDIFF_1:19 .= (diff (f,x)) - (exp_R . x) by SIN_COS:65 .= ((f `| Z) . x) - (exp_R . x) by A7, A12, FDIFF_1:def_7 .= 0 - (exp_R . x) by A6, A3, A12, FDIFF_1:23 .= - (exp_R . x) ; ::_thesis: verum end; A13: for x being Real st x in Z holds ((- (ln * (f - exp_R))) `| Z) . x = (exp_R . x) / (1 - (exp_R . x)) proof let x be Real; ::_thesis: ( x in Z implies ((- (ln * (f - exp_R))) `| Z) . x = (exp_R . x) / (1 - (exp_R . x)) ) assume A14: x in Z ; ::_thesis: ((- (ln * (f - exp_R))) `| Z) . x = (exp_R . x) / (1 - (exp_R . x)) then A15: (f - exp_R) . x = (f . x) - (exp_R . x) by A5, VALUED_1:13 .= 1 - (exp_R . x) by A2, A14 ; A16: ( f - exp_R is_differentiable_in x & (f - exp_R) . x > 0 ) by A2, A9, A14, FDIFF_1:9; (((- 1) (#) (ln * (f - exp_R))) `| Z) . x = (- 1) * (diff ((ln * (f - exp_R)),x)) by A1, A10, A14, FDIFF_1:20 .= (- 1) * ((diff ((f - exp_R),x)) / ((f - exp_R) . x)) by A16, TAYLOR_1:20 .= (- 1) * ((((f - exp_R) `| Z) . x) / ((f - exp_R) . x)) by A9, A14, FDIFF_1:def_7 .= (- 1) * ((- (exp_R . x)) / (1 - (exp_R . x))) by A11, A14, A15 .= ((- 1) * (- (exp_R . x))) / (1 - (exp_R . x)) by XCMPLX_1:74 .= (exp_R . x) / (1 - (exp_R . x)) ; hence ((- (ln * (f - exp_R))) `| Z) . x = (exp_R . x) / (1 - (exp_R . x)) ; ::_thesis: verum end; Z c= dom ((- 1) (#) (ln * (f - exp_R))) by A1; hence ( - (ln * (f - exp_R)) is_differentiable_on Z & ( for x being Real st x in Z holds ((- (ln * (f - exp_R))) `| Z) . x = (exp_R . x) / (1 - (exp_R . x)) ) ) by A10, A13, FDIFF_1:20; ::_thesis: verum end; theorem Th23: :: FDIFF_6:23 for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom (((#Z 2) * exp_R) + f) & ( for x being Real st x in Z holds f . x = 1 ) holds ( ((#Z 2) * exp_R) + f is_differentiable_on Z & ( for x being Real st x in Z holds ((((#Z 2) * exp_R) + f) `| Z) . x = 2 * (exp_R (2 * x)) ) ) proof let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st Z c= dom (((#Z 2) * exp_R) + f) & ( for x being Real st x in Z holds f . x = 1 ) holds ( ((#Z 2) * exp_R) + f is_differentiable_on Z & ( for x being Real st x in Z holds ((((#Z 2) * exp_R) + f) `| Z) . x = 2 * (exp_R (2 * x)) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (((#Z 2) * exp_R) + f) & ( for x being Real st x in Z holds f . x = 1 ) implies ( ((#Z 2) * exp_R) + f is_differentiable_on Z & ( for x being Real st x in Z holds ((((#Z 2) * exp_R) + f) `| Z) . x = 2 * (exp_R (2 * x)) ) ) ) assume that A1: Z c= dom (((#Z 2) * exp_R) + f) and A2: for x being Real st x in Z holds f . x = 1 ; ::_thesis: ( ((#Z 2) * exp_R) + f is_differentiable_on Z & ( for x being Real st x in Z holds ((((#Z 2) * exp_R) + f) `| Z) . x = 2 * (exp_R (2 * x)) ) ) A3: Z c= (dom ((#Z 2) * exp_R)) /\ (dom f) by A1, VALUED_1:def_1; then A4: Z c= dom f by XBOOLE_1:18; A5: now__::_thesis:_for_x_being_Real_st_x_in_Z_holds_ (#Z_2)_*_exp_R_is_differentiable_in_x let x be Real; ::_thesis: ( x in Z implies (#Z 2) * exp_R is_differentiable_in x ) assume x in Z ; ::_thesis: (#Z 2) * exp_R is_differentiable_in x exp_R is_differentiable_in x by SIN_COS:65; hence (#Z 2) * exp_R is_differentiable_in x by TAYLOR_1:3; ::_thesis: verum end; A6: for x being Real st x in Z holds f . x = (0 * x) + 1 by A2; then A7: f is_differentiable_on Z by A4, FDIFF_1:23; Z c= dom ((#Z 2) * exp_R) by A3, XBOOLE_1:18; then A8: (#Z 2) * exp_R is_differentiable_on Z by A5, FDIFF_1:9; for x being Real st x in Z holds ((((#Z 2) * exp_R) + f) `| Z) . x = 2 * (exp_R (2 * x)) proof let x be Real; ::_thesis: ( x in Z implies ((((#Z 2) * exp_R) + f) `| Z) . x = 2 * (exp_R (2 * x)) ) A9: exp_R is_differentiable_in x by SIN_COS:65; assume A10: x in Z ; ::_thesis: ((((#Z 2) * exp_R) + f) `| Z) . x = 2 * (exp_R (2 * x)) then ((((#Z 2) * exp_R) + f) `| Z) . x = (diff (((#Z 2) * exp_R),x)) + (diff (f,x)) by A1, A7, A8, FDIFF_1:18 .= ((2 * ((exp_R . x) #Z (2 - 1))) * (diff (exp_R,x))) + (diff (f,x)) by A9, TAYLOR_1:3 .= ((2 * ((exp_R . x) #Z (2 - 1))) * (exp_R . x)) + (diff (f,x)) by SIN_COS:65 .= ((2 * (exp_R . x)) * (exp_R . x)) + (diff (f,x)) by PREPOWER:35 .= (2 * ((exp_R . x) * (exp_R . x))) + (diff (f,x)) .= (2 * ((exp_R x) * (exp_R . x))) + (diff (f,x)) by SIN_COS:def_23 .= (2 * ((exp_R x) * (exp_R x))) + (diff (f,x)) by SIN_COS:def_23 .= (2 * (exp_R (x + x))) + (diff (f,x)) by SIN_COS:50 .= (2 * (exp_R (2 * x))) + ((f `| Z) . x) by A7, A10, FDIFF_1:def_7 .= (2 * (exp_R (2 * x))) + 0 by A4, A6, A10, FDIFF_1:23 .= 2 * (exp_R (2 * x)) ; hence ((((#Z 2) * exp_R) + f) `| Z) . x = 2 * (exp_R (2 * x)) ; ::_thesis: verum end; hence ( ((#Z 2) * exp_R) + f is_differentiable_on Z & ( for x being Real st x in Z holds ((((#Z 2) * exp_R) + f) `| Z) . x = 2 * (exp_R (2 * x)) ) ) by A1, A7, A8, FDIFF_1:18; ::_thesis: verum end; theorem :: FDIFF_6:24 for Z being open Subset of REAL for f, f1 being PartFunc of REAL,REAL st Z c= dom ((1 / 2) (#) (ln * f)) & f = ((#Z 2) * exp_R) + f1 & ( for x being Real st x in Z holds f1 . x = 1 ) holds ( (1 / 2) (#) (ln * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) (ln * f)) `| Z) . x = (exp_R x) / ((exp_R x) + (exp_R (- x))) ) ) proof let Z be open Subset of REAL; ::_thesis: for f, f1 being PartFunc of REAL,REAL st Z c= dom ((1 / 2) (#) (ln * f)) & f = ((#Z 2) * exp_R) + f1 & ( for x being Real st x in Z holds f1 . x = 1 ) holds ( (1 / 2) (#) (ln * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) (ln * f)) `| Z) . x = (exp_R x) / ((exp_R x) + (exp_R (- x))) ) ) let f, f1 be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom ((1 / 2) (#) (ln * f)) & f = ((#Z 2) * exp_R) + f1 & ( for x being Real st x in Z holds f1 . x = 1 ) implies ( (1 / 2) (#) (ln * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) (ln * f)) `| Z) . x = (exp_R x) / ((exp_R x) + (exp_R (- x))) ) ) ) assume that A1: Z c= dom ((1 / 2) (#) (ln * f)) and A2: f = ((#Z 2) * exp_R) + f1 and A3: for x being Real st x in Z holds f1 . x = 1 ; ::_thesis: ( (1 / 2) (#) (ln * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) (ln * f)) `| Z) . x = (exp_R x) / ((exp_R x) + (exp_R (- x))) ) ) A4: Z c= dom (ln * f) by A1, VALUED_1:def_5; then for y being set st y in Z holds y in dom f by FUNCT_1:11; then A5: Z c= dom (((#Z 2) * exp_R) + f1) by A2, TARSKI:def_3; then A6: ((#Z 2) * exp_R) + f1 is_differentiable_on Z by A3, Th23; Z c= (dom ((#Z 2) * exp_R)) /\ (dom f1) by A5, VALUED_1:def_1; then A7: Z c= dom ((#Z 2) * exp_R) by XBOOLE_1:18; A8: for x being Real st x in Z holds f . x > 0 proof let x be Real; ::_thesis: ( x in Z implies f . x > 0 ) A9: (exp_R . x) #Z 2 > 0 by PREPOWER:39, SIN_COS:54; assume A10: x in Z ; ::_thesis: f . x > 0 then (((#Z 2) * exp_R) + f1) . x = (((#Z 2) * exp_R) . x) + (f1 . x) by A5, VALUED_1:def_1 .= ((#Z 2) . (exp_R . x)) + (f1 . x) by A7, A10, FUNCT_1:12 .= ((exp_R . x) #Z 2) + (f1 . x) by TAYLOR_1:def_1 .= ((exp_R . x) #Z 2) + 1 by A3, A10 ; hence f . x > 0 by A2, A9, XREAL_1:34; ::_thesis: verum end; for x being Real st x in Z holds ln * f is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies ln * f is_differentiable_in x ) assume x in Z ; ::_thesis: ln * f is_differentiable_in x then ( f is_differentiable_in x & f . x > 0 ) by A2, A6, A8, FDIFF_1:9; hence ln * f is_differentiable_in x by TAYLOR_1:20; ::_thesis: verum end; then A11: ln * f is_differentiable_on Z by A4, FDIFF_1:9; for x being Real st x in Z holds (((1 / 2) (#) (ln * f)) `| Z) . x = (exp_R x) / ((exp_R x) + (exp_R (- x))) proof let x be Real; ::_thesis: ( x in Z implies (((1 / 2) (#) (ln * f)) `| Z) . x = (exp_R x) / ((exp_R x) + (exp_R (- x))) ) A12: exp_R x > 0 by SIN_COS:55; assume A13: x in Z ; ::_thesis: (((1 / 2) (#) (ln * f)) `| Z) . x = (exp_R x) / ((exp_R x) + (exp_R (- x))) then A14: ( f is_differentiable_in x & f . x > 0 ) by A2, A6, A8, FDIFF_1:9; A15: (((#Z 2) * exp_R) + f1) . x = (((#Z 2) * exp_R) . x) + (f1 . x) by A5, A13, VALUED_1:def_1 .= ((#Z 2) . (exp_R . x)) + (f1 . x) by A7, A13, FUNCT_1:12 .= ((exp_R . x) #Z 2) + (f1 . x) by TAYLOR_1:def_1 .= ((exp_R . x) #Z 2) + 1 by A3, A13 .= ((exp_R x) #Z (1 + 1)) + 1 by SIN_COS:def_23 .= (((exp_R x) #Z 1) * ((exp_R x) #Z 1)) + 1 by A12, PREPOWER:44 .= ((exp_R x) * ((exp_R x) #Z 1)) + 1 by PREPOWER:35 .= ((exp_R x) * (exp_R x)) + 1 by PREPOWER:35 ; (((1 / 2) (#) (ln * f)) `| Z) . x = (1 / 2) * (diff ((ln * f),x)) by A1, A11, A13, FDIFF_1:20 .= (1 / 2) * ((diff (f,x)) / (f . x)) by A14, TAYLOR_1:20 .= (1 / 2) * ((((((#Z 2) * exp_R) + f1) `| Z) . x) / ((((#Z 2) * exp_R) + f1) . x)) by A2, A6, A13, FDIFF_1:def_7 .= (1 / 2) * ((2 * (exp_R (2 * x))) / (((exp_R x) * (exp_R x)) + 1)) by A3, A5, A13, A15, Th23 .= ((1 / 2) * (2 * (exp_R (2 * x)))) / (((exp_R x) * (exp_R x)) + 1) by XCMPLX_1:74 .= ((exp_R (x + x)) / (exp_R x)) / ((((exp_R x) * (exp_R x)) + 1) / (exp_R x)) by A12, XCMPLX_1:55 .= (((exp_R x) * (exp_R x)) / (exp_R x)) / ((((exp_R x) * (exp_R x)) + 1) / (exp_R x)) by SIN_COS:50 .= (((exp_R x) * (exp_R x)) / (exp_R x)) / ((((exp_R x) * (exp_R x)) / (exp_R x)) + (1 / (exp_R x))) by XCMPLX_1:62 .= (exp_R x) / ((((exp_R x) * (exp_R x)) / (exp_R x)) + (1 / (exp_R x))) by A12, XCMPLX_1:89 .= (exp_R x) / ((exp_R x) + (1 / (exp_R x))) by A12, XCMPLX_1:89 .= (exp_R x) / ((exp_R x) + (exp_R (- x))) by TAYLOR_1:4 ; hence (((1 / 2) (#) (ln * f)) `| Z) . x = (exp_R x) / ((exp_R x) + (exp_R (- x))) ; ::_thesis: verum end; hence ( (1 / 2) (#) (ln * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) (ln * f)) `| Z) . x = (exp_R x) / ((exp_R x) + (exp_R (- x))) ) ) by A1, A11, FDIFF_1:20; ::_thesis: verum end; theorem Th25: :: FDIFF_6:25 for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom (((#Z 2) * exp_R) - f) & ( for x being Real st x in Z holds f . x = 1 ) holds ( ((#Z 2) * exp_R) - f is_differentiable_on Z & ( for x being Real st x in Z holds ((((#Z 2) * exp_R) - f) `| Z) . x = 2 * (exp_R (2 * x)) ) ) proof let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st Z c= dom (((#Z 2) * exp_R) - f) & ( for x being Real st x in Z holds f . x = 1 ) holds ( ((#Z 2) * exp_R) - f is_differentiable_on Z & ( for x being Real st x in Z holds ((((#Z 2) * exp_R) - f) `| Z) . x = 2 * (exp_R (2 * x)) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (((#Z 2) * exp_R) - f) & ( for x being Real st x in Z holds f . x = 1 ) implies ( ((#Z 2) * exp_R) - f is_differentiable_on Z & ( for x being Real st x in Z holds ((((#Z 2) * exp_R) - f) `| Z) . x = 2 * (exp_R (2 * x)) ) ) ) assume that A1: Z c= dom (((#Z 2) * exp_R) - f) and A2: for x being Real st x in Z holds f . x = 1 ; ::_thesis: ( ((#Z 2) * exp_R) - f is_differentiable_on Z & ( for x being Real st x in Z holds ((((#Z 2) * exp_R) - f) `| Z) . x = 2 * (exp_R (2 * x)) ) ) A3: Z c= (dom ((#Z 2) * exp_R)) /\ (dom f) by A1, VALUED_1:12; then A4: Z c= dom f by XBOOLE_1:18; A5: now__::_thesis:_for_x_being_Real_st_x_in_Z_holds_ (#Z_2)_*_exp_R_is_differentiable_in_x let x be Real; ::_thesis: ( x in Z implies (#Z 2) * exp_R is_differentiable_in x ) assume x in Z ; ::_thesis: (#Z 2) * exp_R is_differentiable_in x exp_R is_differentiable_in x by SIN_COS:65; hence (#Z 2) * exp_R is_differentiable_in x by TAYLOR_1:3; ::_thesis: verum end; A6: for x being Real st x in Z holds f . x = (0 * x) + 1 by A2; then A7: f is_differentiable_on Z by A4, FDIFF_1:23; Z c= dom ((#Z 2) * exp_R) by A3, XBOOLE_1:18; then A8: (#Z 2) * exp_R is_differentiable_on Z by A5, FDIFF_1:9; for x being Real st x in Z holds ((((#Z 2) * exp_R) - f) `| Z) . x = 2 * (exp_R (2 * x)) proof let x be Real; ::_thesis: ( x in Z implies ((((#Z 2) * exp_R) - f) `| Z) . x = 2 * (exp_R (2 * x)) ) A9: exp_R is_differentiable_in x by SIN_COS:65; assume A10: x in Z ; ::_thesis: ((((#Z 2) * exp_R) - f) `| Z) . x = 2 * (exp_R (2 * x)) then ((((#Z 2) * exp_R) - f) `| Z) . x = (diff (((#Z 2) * exp_R),x)) - (diff (f,x)) by A1, A7, A8, FDIFF_1:19 .= ((2 * ((exp_R . x) #Z (2 - 1))) * (diff (exp_R,x))) - (diff (f,x)) by A9, TAYLOR_1:3 .= ((2 * ((exp_R . x) #Z 1)) * (exp_R . x)) - (diff (f,x)) by SIN_COS:65 .= ((2 * (exp_R . x)) * (exp_R . x)) - (diff (f,x)) by PREPOWER:35 .= (2 * ((exp_R . x) * (exp_R . x))) - (diff (f,x)) .= (2 * ((exp_R x) * (exp_R . x))) - (diff (f,x)) by SIN_COS:def_23 .= (2 * ((exp_R x) * (exp_R x))) - (diff (f,x)) by SIN_COS:def_23 .= (2 * (exp_R (x + x))) - (diff (f,x)) by SIN_COS:50 .= (2 * (exp_R (2 * x))) - ((f `| Z) . x) by A7, A10, FDIFF_1:def_7 .= (2 * (exp_R (2 * x))) - 0 by A4, A6, A10, FDIFF_1:23 .= 2 * (exp_R (2 * x)) ; hence ((((#Z 2) * exp_R) - f) `| Z) . x = 2 * (exp_R (2 * x)) ; ::_thesis: verum end; hence ( ((#Z 2) * exp_R) - f is_differentiable_on Z & ( for x being Real st x in Z holds ((((#Z 2) * exp_R) - f) `| Z) . x = 2 * (exp_R (2 * x)) ) ) by A1, A7, A8, FDIFF_1:19; ::_thesis: verum end; theorem :: FDIFF_6:26 for Z being open Subset of REAL for f, f1 being PartFunc of REAL,REAL st Z c= dom ((1 / 2) (#) (ln * f)) & f = ((#Z 2) * exp_R) - f1 & ( for x being Real st x in Z holds ( f1 . x = 1 & f . x > 0 ) ) holds ( (1 / 2) (#) (ln * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) (ln * f)) `| Z) . x = (exp_R x) / ((exp_R x) - (exp_R (- x))) ) ) proof let Z be open Subset of REAL; ::_thesis: for f, f1 being PartFunc of REAL,REAL st Z c= dom ((1 / 2) (#) (ln * f)) & f = ((#Z 2) * exp_R) - f1 & ( for x being Real st x in Z holds ( f1 . x = 1 & f . x > 0 ) ) holds ( (1 / 2) (#) (ln * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) (ln * f)) `| Z) . x = (exp_R x) / ((exp_R x) - (exp_R (- x))) ) ) let f, f1 be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom ((1 / 2) (#) (ln * f)) & f = ((#Z 2) * exp_R) - f1 & ( for x being Real st x in Z holds ( f1 . x = 1 & f . x > 0 ) ) implies ( (1 / 2) (#) (ln * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) (ln * f)) `| Z) . x = (exp_R x) / ((exp_R x) - (exp_R (- x))) ) ) ) assume that A1: Z c= dom ((1 / 2) (#) (ln * f)) and A2: f = ((#Z 2) * exp_R) - f1 and A3: for x being Real st x in Z holds ( f1 . x = 1 & f . x > 0 ) ; ::_thesis: ( (1 / 2) (#) (ln * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) (ln * f)) `| Z) . x = (exp_R x) / ((exp_R x) - (exp_R (- x))) ) ) A4: Z c= dom (ln * f) by A1, VALUED_1:def_5; then for y being set st y in Z holds y in dom f by FUNCT_1:11; then A5: Z c= dom (((#Z 2) * exp_R) - f1) by A2, TARSKI:def_3; A6: for x being Real st x in Z holds f1 . x = 1 by A3; then A7: ((#Z 2) * exp_R) - f1 is_differentiable_on Z by A5, Th25; for x being Real st x in Z holds ln * f is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies ln * f is_differentiable_in x ) assume x in Z ; ::_thesis: ln * f is_differentiable_in x then ( f is_differentiable_in x & f . x > 0 ) by A2, A3, A7, FDIFF_1:9; hence ln * f is_differentiable_in x by TAYLOR_1:20; ::_thesis: verum end; then A8: ln * f is_differentiable_on Z by A4, FDIFF_1:9; Z c= (dom ((#Z 2) * exp_R)) /\ (dom f1) by A5, VALUED_1:12; then A9: Z c= dom ((#Z 2) * exp_R) by XBOOLE_1:18; for x being Real st x in Z holds (((1 / 2) (#) (ln * f)) `| Z) . x = (exp_R x) / ((exp_R x) - (exp_R (- x))) proof let x be Real; ::_thesis: ( x in Z implies (((1 / 2) (#) (ln * f)) `| Z) . x = (exp_R x) / ((exp_R x) - (exp_R (- x))) ) A10: exp_R x > 0 by SIN_COS:55; assume A11: x in Z ; ::_thesis: (((1 / 2) (#) (ln * f)) `| Z) . x = (exp_R x) / ((exp_R x) - (exp_R (- x))) then A12: ( f is_differentiable_in x & f . x > 0 ) by A2, A3, A7, FDIFF_1:9; A13: (((#Z 2) * exp_R) - f1) . x = (((#Z 2) * exp_R) . x) - (f1 . x) by A5, A11, VALUED_1:13 .= ((#Z 2) . (exp_R . x)) - (f1 . x) by A9, A11, FUNCT_1:12 .= ((exp_R . x) #Z 2) - (f1 . x) by TAYLOR_1:def_1 .= ((exp_R . x) #Z 2) - 1 by A3, A11 .= ((exp_R x) #Z (1 + 1)) - 1 by SIN_COS:def_23 .= (((exp_R x) #Z 1) * ((exp_R x) #Z 1)) - 1 by A10, PREPOWER:44 .= ((exp_R x) * ((exp_R x) #Z 1)) - 1 by PREPOWER:35 .= ((exp_R x) * (exp_R x)) - 1 by PREPOWER:35 ; (((1 / 2) (#) (ln * f)) `| Z) . x = (1 / 2) * (diff ((ln * f),x)) by A1, A8, A11, FDIFF_1:20 .= (1 / 2) * ((diff (f,x)) / (f . x)) by A12, TAYLOR_1:20 .= (1 / 2) * ((((((#Z 2) * exp_R) - f1) `| Z) . x) / ((((#Z 2) * exp_R) - f1) . x)) by A2, A7, A11, FDIFF_1:def_7 .= (1 / 2) * ((2 * (exp_R (2 * x))) / (((exp_R x) * (exp_R x)) - 1)) by A6, A5, A11, A13, Th25 .= ((1 / 2) * (2 * (exp_R (2 * x)))) / (((exp_R x) * (exp_R x)) - 1) by XCMPLX_1:74 .= ((exp_R (x + x)) / (exp_R x)) / ((((exp_R x) * (exp_R x)) - 1) / (exp_R x)) by A10, XCMPLX_1:55 .= (((exp_R x) * (exp_R x)) / (exp_R x)) / ((((exp_R x) * (exp_R x)) - 1) / (exp_R x)) by SIN_COS:50 .= (((exp_R x) * (exp_R x)) / (exp_R x)) / ((((exp_R x) * (exp_R x)) / (exp_R x)) - (1 / (exp_R x))) by XCMPLX_1:120 .= (exp_R x) / ((((exp_R x) * (exp_R x)) / (exp_R x)) - (1 / (exp_R x))) by A10, XCMPLX_1:89 .= (exp_R x) / ((exp_R x) - (1 / (exp_R x))) by A10, XCMPLX_1:89 .= (exp_R x) / ((exp_R x) - (exp_R (- x))) by TAYLOR_1:4 ; hence (((1 / 2) (#) (ln * f)) `| Z) . x = (exp_R x) / ((exp_R x) - (exp_R (- x))) ; ::_thesis: verum end; hence ( (1 / 2) (#) (ln * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) (ln * f)) `| Z) . x = (exp_R x) / ((exp_R x) - (exp_R (- x))) ) ) by A1, A8, FDIFF_1:20; ::_thesis: verum end; theorem Th27: :: FDIFF_6:27 for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom ((#Z 2) * (exp_R - f)) & ( for x being Real st x in Z holds f . x = 1 ) holds ( (#Z 2) * (exp_R - f) is_differentiable_on Z & ( for x being Real st x in Z holds (((#Z 2) * (exp_R - f)) `| Z) . x = (2 * (exp_R . x)) * ((exp_R . x) - 1) ) ) proof let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st Z c= dom ((#Z 2) * (exp_R - f)) & ( for x being Real st x in Z holds f . x = 1 ) holds ( (#Z 2) * (exp_R - f) is_differentiable_on Z & ( for x being Real st x in Z holds (((#Z 2) * (exp_R - f)) `| Z) . x = (2 * (exp_R . x)) * ((exp_R . x) - 1) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom ((#Z 2) * (exp_R - f)) & ( for x being Real st x in Z holds f . x = 1 ) implies ( (#Z 2) * (exp_R - f) is_differentiable_on Z & ( for x being Real st x in Z holds (((#Z 2) * (exp_R - f)) `| Z) . x = (2 * (exp_R . x)) * ((exp_R . x) - 1) ) ) ) assume that A1: Z c= dom ((#Z 2) * (exp_R - f)) and A2: for x being Real st x in Z holds f . x = 1 ; ::_thesis: ( (#Z 2) * (exp_R - f) is_differentiable_on Z & ( for x being Real st x in Z holds (((#Z 2) * (exp_R - f)) `| Z) . x = (2 * (exp_R . x)) * ((exp_R . x) - 1) ) ) for y being set st y in Z holds y in dom (exp_R - f) by A1, FUNCT_1:11; then A3: Z c= dom (exp_R - f) by TARSKI:def_3; then Z c= (dom exp_R) /\ (dom f) by VALUED_1:12; then A4: Z c= dom f by XBOOLE_1:18; A5: for x being Real st x in Z holds f . x = (0 * x) + 1 by A2; then A6: f is_differentiable_on Z by A4, FDIFF_1:23; A7: for x being Real st x in Z holds (#Z 2) * (exp_R - f) is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies (#Z 2) * (exp_R - f) is_differentiable_in x ) assume x in Z ; ::_thesis: (#Z 2) * (exp_R - f) is_differentiable_in x then A8: f is_differentiable_in x by A6, FDIFF_1:9; exp_R is_differentiable_in x by SIN_COS:65; then exp_R - f is_differentiable_in x by A8, FDIFF_1:14; hence (#Z 2) * (exp_R - f) is_differentiable_in x by TAYLOR_1:3; ::_thesis: verum end; then A9: (#Z 2) * (exp_R - f) is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds (((#Z 2) * (exp_R - f)) `| Z) . x = (2 * (exp_R . x)) * ((exp_R . x) - 1) proof let x be Real; ::_thesis: ( x in Z implies (((#Z 2) * (exp_R - f)) `| Z) . x = (2 * (exp_R . x)) * ((exp_R . x) - 1) ) A10: exp_R is_differentiable_in x by SIN_COS:65; assume A11: x in Z ; ::_thesis: (((#Z 2) * (exp_R - f)) `| Z) . x = (2 * (exp_R . x)) * ((exp_R . x) - 1) then A12: (exp_R - f) . x = (exp_R . x) - (f . x) by A3, VALUED_1:13 .= (exp_R . x) - 1 by A2, A11 ; A13: f is_differentiable_in x by A6, A11, FDIFF_1:9; then A14: diff ((exp_R - f),x) = (diff (exp_R,x)) - (diff (f,x)) by A10, FDIFF_1:14 .= (diff (exp_R,x)) - ((f `| Z) . x) by A6, A11, FDIFF_1:def_7 .= (exp_R . x) - ((f `| Z) . x) by SIN_COS:65 .= (exp_R . x) - 0 by A4, A5, A11, FDIFF_1:23 .= exp_R . x ; A15: exp_R - f is_differentiable_in x by A13, A10, FDIFF_1:14; (((#Z 2) * (exp_R - f)) `| Z) . x = diff (((#Z 2) * (exp_R - f)),x) by A9, A11, FDIFF_1:def_7 .= (2 * (((exp_R - f) . x) #Z (2 - 1))) * (diff ((exp_R - f),x)) by A15, TAYLOR_1:3 .= (2 * ((exp_R . x) - 1)) * (exp_R . x) by A14, A12, PREPOWER:35 .= (2 * (exp_R . x)) * ((exp_R . x) - 1) ; hence (((#Z 2) * (exp_R - f)) `| Z) . x = (2 * (exp_R . x)) * ((exp_R . x) - 1) ; ::_thesis: verum end; hence ( (#Z 2) * (exp_R - f) is_differentiable_on Z & ( for x being Real st x in Z holds (((#Z 2) * (exp_R - f)) `| Z) . x = (2 * (exp_R . x)) * ((exp_R . x) - 1) ) ) by A1, A7, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_6:28 for Z being open Subset of REAL for f, f1 being PartFunc of REAL,REAL st Z c= dom f & f = ln * (((#Z 2) * (exp_R - f1)) / exp_R) & ( for x being Real st x in Z holds ( f1 . x = 1 & (exp_R - f1) . x > 0 ) ) holds ( f is_differentiable_on Z & ( for x being Real st x in Z holds (f `| Z) . x = ((exp_R . x) + 1) / ((exp_R . x) - 1) ) ) proof let Z be open Subset of REAL; ::_thesis: for f, f1 being PartFunc of REAL,REAL st Z c= dom f & f = ln * (((#Z 2) * (exp_R - f1)) / exp_R) & ( for x being Real st x in Z holds ( f1 . x = 1 & (exp_R - f1) . x > 0 ) ) holds ( f is_differentiable_on Z & ( for x being Real st x in Z holds (f `| Z) . x = ((exp_R . x) + 1) / ((exp_R . x) - 1) ) ) let f, f1 be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom f & f = ln * (((#Z 2) * (exp_R - f1)) / exp_R) & ( for x being Real st x in Z holds ( f1 . x = 1 & (exp_R - f1) . x > 0 ) ) implies ( f is_differentiable_on Z & ( for x being Real st x in Z holds (f `| Z) . x = ((exp_R . x) + 1) / ((exp_R . x) - 1) ) ) ) assume that A1: Z c= dom f and A2: f = ln * (((#Z 2) * (exp_R - f1)) / exp_R) and A3: for x being Real st x in Z holds ( f1 . x = 1 & (exp_R - f1) . x > 0 ) ; ::_thesis: ( f is_differentiable_on Z & ( for x being Real st x in Z holds (f `| Z) . x = ((exp_R . x) + 1) / ((exp_R . x) - 1) ) ) for y being set st y in Z holds y in dom (((#Z 2) * (exp_R - f1)) / exp_R) by A1, A2, FUNCT_1:11; then A4: Z c= dom (((#Z 2) * (exp_R - f1)) / exp_R) by TARSKI:def_3; then Z c= (dom ((#Z 2) * (exp_R - f1))) /\ ((dom exp_R) \ (exp_R " {0})) by RFUNCT_1:def_1; then A5: Z c= dom ((#Z 2) * (exp_R - f1)) by XBOOLE_1:18; then for y being set st y in Z holds y in dom (exp_R - f1) by FUNCT_1:11; then A6: Z c= dom (exp_R - f1) by TARSKI:def_3; A7: for x being Real st x in Z holds (((#Z 2) * (exp_R - f1)) / exp_R) . x > 0 proof let x be Real; ::_thesis: ( x in Z implies (((#Z 2) * (exp_R - f1)) / exp_R) . x > 0 ) A8: exp_R . x > 0 by SIN_COS:54; assume A9: x in Z ; ::_thesis: (((#Z 2) * (exp_R - f1)) / exp_R) . x > 0 then A10: ((exp_R - f1) . x) #Z 2 > 0 by A3, PREPOWER:39; (((#Z 2) * (exp_R - f1)) / exp_R) . x = (((#Z 2) * (exp_R - f1)) . x) * ((exp_R . x) ") by A4, A9, RFUNCT_1:def_1 .= (((#Z 2) * (exp_R - f1)) . x) * (1 / (exp_R . x)) by XCMPLX_1:215 .= (((#Z 2) * (exp_R - f1)) . x) / (exp_R . x) by XCMPLX_1:99 .= ((#Z 2) . ((exp_R - f1) . x)) / (exp_R . x) by A5, A9, FUNCT_1:12 .= (((exp_R - f1) . x) #Z 2) / (exp_R . x) by TAYLOR_1:def_1 ; hence (((#Z 2) * (exp_R - f1)) / exp_R) . x > 0 by A10, A8, XREAL_1:139; ::_thesis: verum end; A11: for x being Real st x in Z holds f1 . x = 1 by A3; then A12: (#Z 2) * (exp_R - f1) is_differentiable_on Z by A5, Th27; ( 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 A13: ((#Z 2) * (exp_R - f1)) / exp_R is_differentiable_on Z by A12, FDIFF_2:21; A14: for x being Real st x in Z holds ln * (((#Z 2) * (exp_R - f1)) / exp_R) is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies ln * (((#Z 2) * (exp_R - f1)) / exp_R) is_differentiable_in x ) assume x in Z ; ::_thesis: ln * (((#Z 2) * (exp_R - f1)) / exp_R) is_differentiable_in x then ( ((#Z 2) * (exp_R - f1)) / exp_R is_differentiable_in x & (((#Z 2) * (exp_R - f1)) / exp_R) . x > 0 ) by A13, A7, FDIFF_1:9; hence ln * (((#Z 2) * (exp_R - f1)) / exp_R) is_differentiable_in x by TAYLOR_1:20; ::_thesis: verum end; then A15: f is_differentiable_on Z by A1, A2, FDIFF_1:9; for x being Real st x in Z holds (f `| Z) . x = ((exp_R . x) + 1) / ((exp_R . x) - 1) proof let x be Real; ::_thesis: ( x in Z implies (f `| Z) . x = ((exp_R . x) + 1) / ((exp_R . x) - 1) ) A16: exp_R . x > 0 by SIN_COS:54; A17: exp_R is_differentiable_in x by SIN_COS:65; assume A18: x in Z ; ::_thesis: (f `| Z) . x = ((exp_R . x) + 1) / ((exp_R . x) - 1) then A19: (exp_R - f1) . x = (exp_R . x) - (f1 . x) by A6, VALUED_1:13 .= (exp_R . x) - 1 by A3, A18 ; then A20: (exp_R . x) - 1 > 0 by A3, A18; A21: (((#Z 2) * (exp_R - f1)) / exp_R) . x = (((#Z 2) * (exp_R - f1)) . x) * ((exp_R . x) ") by A4, A18, RFUNCT_1:def_1 .= (((#Z 2) * (exp_R - f1)) . x) * (1 / (exp_R . x)) by XCMPLX_1:215 .= (((#Z 2) * (exp_R - f1)) . x) / (exp_R . x) by XCMPLX_1:99 .= ((#Z 2) . ((exp_R - f1) . x)) / (exp_R . x) by A5, A18, FUNCT_1:12 .= (((exp_R . x) - 1) #Z (1 + 1)) / (exp_R . x) by A19, TAYLOR_1:def_1 .= ((((exp_R . x) - 1) #Z 1) * (((exp_R . x) - 1) #Z 1)) / (exp_R . x) by A20, PREPOWER:44 .= (((exp_R . x) - 1) * (((exp_R . x) - 1) #Z 1)) / (exp_R . x) by PREPOWER:35 .= (((exp_R . x) - 1) * ((exp_R . x) - 1)) / (exp_R . x) by PREPOWER:35 ; A22: ( ((#Z 2) * (exp_R - f1)) / exp_R is_differentiable_in x & (((#Z 2) * (exp_R - f1)) / exp_R) . x > 0 ) by A13, A7, A18, FDIFF_1:9; (#Z 2) * (exp_R - f1) is_differentiable_in x by A12, A18, FDIFF_1:9; then A23: diff ((((#Z 2) * (exp_R - f1)) / exp_R),x) = (((diff (((#Z 2) * (exp_R - f1)),x)) * (exp_R . x)) - ((diff (exp_R,x)) * (((#Z 2) * (exp_R - f1)) . x))) / ((exp_R . x) ^2) by A16, A17, FDIFF_2:14 .= ((((((#Z 2) * (exp_R - f1)) `| Z) . x) * (exp_R . x)) - ((diff (exp_R,x)) * (((#Z 2) * (exp_R - f1)) . x))) / ((exp_R . x) ^2) by A12, A18, FDIFF_1:def_7 .= ((((2 * (exp_R . x)) * ((exp_R . x) - 1)) * (exp_R . x)) - ((diff (exp_R,x)) * (((#Z 2) * (exp_R - f1)) . x))) / ((exp_R . x) ^2) by A11, A5, A18, Th27 .= ((((2 * (exp_R . x)) * ((exp_R . x) - 1)) * (exp_R . x)) - ((exp_R . x) * (((#Z 2) * (exp_R - f1)) . x))) / ((exp_R . x) ^2) by SIN_COS:65 .= ((((2 * (exp_R . x)) * ((exp_R . x) - 1)) - (((#Z 2) * (exp_R - f1)) . x)) * (exp_R . x)) / ((exp_R . x) * (exp_R . x)) .= (((2 * (exp_R . x)) * ((exp_R . x) - 1)) - (((#Z 2) * (exp_R - f1)) . x)) / (exp_R . x) by A16, XCMPLX_1:91 .= (((2 * (exp_R . x)) * ((exp_R . x) - 1)) - ((#Z 2) . ((exp_R - f1) . x))) / (exp_R . x) by A5, A18, FUNCT_1:12 .= (((2 * (exp_R . x)) * ((exp_R . x) - 1)) - (((exp_R - f1) . x) #Z 2)) / (exp_R . x) by TAYLOR_1:def_1 .= (((2 * (exp_R . x)) * ((exp_R . x) - 1)) - (((exp_R . x) - (f1 . x)) #Z 2)) / (exp_R . x) by A6, A18, VALUED_1:13 .= (((2 * (exp_R . x)) * ((exp_R . x) - 1)) - (((exp_R . x) - 1) #Z (1 + 1))) / (exp_R . x) by A3, A18 .= (((2 * (exp_R . x)) * ((exp_R . x) - 1)) - ((((exp_R . x) - 1) #Z 1) * (((exp_R . x) - 1) #Z 1))) / (exp_R . x) by A20, PREPOWER:44 .= (((2 * (exp_R . x)) * ((exp_R . x) - 1)) - (((exp_R . x) - 1) * (((exp_R . x) - 1) #Z 1))) / (exp_R . x) by PREPOWER:35 .= (((2 * (exp_R . x)) * ((exp_R . x) - 1)) - (((exp_R . x) - 1) * ((exp_R . x) - 1))) / (exp_R . x) by PREPOWER:35 .= (((exp_R . x) + 1) * ((exp_R . x) - 1)) / (exp_R . x) ; (f `| Z) . x = diff ((ln * (((#Z 2) * (exp_R - f1)) / exp_R)),x) by A2, A15, A18, FDIFF_1:def_7 .= ((((exp_R . x) + 1) * ((exp_R . x) - 1)) / (exp_R . x)) / ((((exp_R . x) - 1) * ((exp_R . x) - 1)) / (exp_R . x)) by A22, A23, A21, TAYLOR_1:20 .= (((exp_R . x) + 1) * ((exp_R . x) - 1)) / (((exp_R . x) - 1) * ((exp_R . x) - 1)) by A16, XCMPLX_1:55 .= ((exp_R . x) + 1) / ((exp_R . x) - 1) by A20, XCMPLX_1:91 ; hence (f `| Z) . x = ((exp_R . x) + 1) / ((exp_R . x) - 1) ; ::_thesis: verum end; hence ( f is_differentiable_on Z & ( for x being Real st x in Z holds (f `| Z) . x = ((exp_R . x) + 1) / ((exp_R . x) - 1) ) ) by A1, A2, A14, FDIFF_1:9; ::_thesis: verum end; theorem Th29: :: FDIFF_6:29 for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom ((#Z 2) * (exp_R + f)) & ( for x being Real st x in Z holds f . x = 1 ) holds ( (#Z 2) * (exp_R + f) is_differentiable_on Z & ( for x being Real st x in Z holds (((#Z 2) * (exp_R + f)) `| Z) . x = (2 * (exp_R . x)) * ((exp_R . x) + 1) ) ) proof let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st Z c= dom ((#Z 2) * (exp_R + f)) & ( for x being Real st x in Z holds f . x = 1 ) holds ( (#Z 2) * (exp_R + f) is_differentiable_on Z & ( for x being Real st x in Z holds (((#Z 2) * (exp_R + f)) `| Z) . x = (2 * (exp_R . x)) * ((exp_R . x) + 1) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom ((#Z 2) * (exp_R + f)) & ( for x being Real st x in Z holds f . x = 1 ) implies ( (#Z 2) * (exp_R + f) is_differentiable_on Z & ( for x being Real st x in Z holds (((#Z 2) * (exp_R + f)) `| Z) . x = (2 * (exp_R . x)) * ((exp_R . x) + 1) ) ) ) assume that A1: Z c= dom ((#Z 2) * (exp_R + f)) and A2: for x being Real st x in Z holds f . x = 1 ; ::_thesis: ( (#Z 2) * (exp_R + f) is_differentiable_on Z & ( for x being Real st x in Z holds (((#Z 2) * (exp_R + f)) `| Z) . x = (2 * (exp_R . x)) * ((exp_R . x) + 1) ) ) for y being set st y in Z holds y in dom (exp_R + f) by A1, FUNCT_1:11; then A3: Z c= dom (exp_R + f) by TARSKI:def_3; then Z c= (dom exp_R) /\ (dom f) by VALUED_1:def_1; then A4: Z c= dom f by XBOOLE_1:18; A5: for x being Real st x in Z holds f . x = (0 * x) + 1 by A2; then A6: f is_differentiable_on Z by A4, FDIFF_1:23; A7: for x being Real st x in Z holds (#Z 2) * (exp_R + f) is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies (#Z 2) * (exp_R + f) is_differentiable_in x ) assume x in Z ; ::_thesis: (#Z 2) * (exp_R + f) is_differentiable_in x then A8: f is_differentiable_in x by A6, FDIFF_1:9; exp_R is_differentiable_in x by SIN_COS:65; then exp_R + f is_differentiable_in x by A8, FDIFF_1:13; hence (#Z 2) * (exp_R + f) is_differentiable_in x by TAYLOR_1:3; ::_thesis: verum end; then A9: (#Z 2) * (exp_R + f) is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds (((#Z 2) * (exp_R + f)) `| Z) . x = (2 * (exp_R . x)) * ((exp_R . x) + 1) proof let x be Real; ::_thesis: ( x in Z implies (((#Z 2) * (exp_R + f)) `| Z) . x = (2 * (exp_R . x)) * ((exp_R . x) + 1) ) A10: exp_R is_differentiable_in x by SIN_COS:65; assume A11: x in Z ; ::_thesis: (((#Z 2) * (exp_R + f)) `| Z) . x = (2 * (exp_R . x)) * ((exp_R . x) + 1) then A12: (exp_R + f) . x = (exp_R . x) + (f . x) by A3, VALUED_1:def_1 .= (exp_R . x) + 1 by A2, A11 ; A13: f is_differentiable_in x by A6, A11, FDIFF_1:9; then A14: diff ((exp_R + f),x) = (diff (exp_R,x)) + (diff (f,x)) by A10, FDIFF_1:13 .= (diff (exp_R,x)) + ((f `| Z) . x) by A6, A11, FDIFF_1:def_7 .= (exp_R . x) + ((f `| Z) . x) by SIN_COS:65 .= (exp_R . x) + 0 by A4, A5, A11, FDIFF_1:23 .= exp_R . x ; A15: exp_R + f is_differentiable_in x by A13, A10, FDIFF_1:13; (((#Z 2) * (exp_R + f)) `| Z) . x = diff (((#Z 2) * (exp_R + f)),x) by A9, A11, FDIFF_1:def_7 .= (2 * (((exp_R + f) . x) #Z (2 - 1))) * (diff ((exp_R + f),x)) by A15, TAYLOR_1:3 .= (2 * ((exp_R . x) + 1)) * (exp_R . x) by A14, A12, PREPOWER:35 .= (2 * (exp_R . x)) * ((exp_R . x) + 1) ; hence (((#Z 2) * (exp_R + f)) `| Z) . x = (2 * (exp_R . x)) * ((exp_R . x) + 1) ; ::_thesis: verum end; hence ( (#Z 2) * (exp_R + f) is_differentiable_on Z & ( for x being Real st x in Z holds (((#Z 2) * (exp_R + f)) `| Z) . x = (2 * (exp_R . x)) * ((exp_R . x) + 1) ) ) by A1, A7, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_6:30 for Z being open Subset of REAL for f, f1 being PartFunc of REAL,REAL st Z c= dom f & f = ln * (((#Z 2) * (exp_R + f1)) / exp_R) & ( for x being Real st x in Z holds f1 . x = 1 ) holds ( f is_differentiable_on Z & ( for x being Real st x in Z holds (f `| Z) . x = ((exp_R . x) - 1) / ((exp_R . x) + 1) ) ) proof let Z be open Subset of REAL; ::_thesis: for f, f1 being PartFunc of REAL,REAL st Z c= dom f & f = ln * (((#Z 2) * (exp_R + f1)) / exp_R) & ( for x being Real st x in Z holds f1 . x = 1 ) holds ( f is_differentiable_on Z & ( for x being Real st x in Z holds (f `| Z) . x = ((exp_R . x) - 1) / ((exp_R . x) + 1) ) ) let f, f1 be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom f & f = ln * (((#Z 2) * (exp_R + f1)) / exp_R) & ( for x being Real st x in Z holds f1 . x = 1 ) implies ( f is_differentiable_on Z & ( for x being Real st x in Z holds (f `| Z) . x = ((exp_R . x) - 1) / ((exp_R . x) + 1) ) ) ) assume that A1: Z c= dom f and A2: f = ln * (((#Z 2) * (exp_R + f1)) / exp_R) and A3: for x being Real st x in Z holds f1 . x = 1 ; ::_thesis: ( f is_differentiable_on Z & ( for x being Real st x in Z holds (f `| Z) . x = ((exp_R . x) - 1) / ((exp_R . x) + 1) ) ) for y being set st y in Z holds y in dom (((#Z 2) * (exp_R + f1)) / exp_R) by A1, A2, FUNCT_1:11; then A4: Z c= dom (((#Z 2) * (exp_R + f1)) / exp_R) by TARSKI:def_3; then Z c= (dom ((#Z 2) * (exp_R + f1))) /\ ((dom exp_R) \ (exp_R " {0})) by RFUNCT_1:def_1; then A5: Z c= dom ((#Z 2) * (exp_R + f1)) by XBOOLE_1:18; then A6: (#Z 2) * (exp_R + f1) is_differentiable_on Z by A3, Th29; for y being set st y in Z holds y in dom (exp_R + f1) by A5, FUNCT_1:11; then A7: Z c= dom (exp_R + f1) by TARSKI:def_3; A8: for x being Real st x in Z holds (((#Z 2) * (exp_R + f1)) / exp_R) . x > 0 proof let x be Real; ::_thesis: ( x in Z implies (((#Z 2) * (exp_R + f1)) / exp_R) . x > 0 ) A9: exp_R . x > 0 by SIN_COS:54; assume A10: x in Z ; ::_thesis: (((#Z 2) * (exp_R + f1)) / exp_R) . x > 0 then (exp_R + f1) . x = (exp_R . x) + (f1 . x) by A7, VALUED_1:def_1 .= (exp_R . x) + 1 by A3, A10 ; then (exp_R + f1) . x > 0 by SIN_COS:54, XREAL_1:34; then A11: ((exp_R + f1) . x) #Z 2 > 0 by PREPOWER:39; (((#Z 2) * (exp_R + f1)) / exp_R) . x = (((#Z 2) * (exp_R + f1)) . x) * ((exp_R . x) ") by A4, A10, RFUNCT_1:def_1 .= (((#Z 2) * (exp_R + f1)) . x) * (1 / (exp_R . x)) by XCMPLX_1:215 .= (((#Z 2) * (exp_R + f1)) . x) / (exp_R . x) by XCMPLX_1:99 .= ((#Z 2) . ((exp_R + f1) . x)) / (exp_R . x) by A5, A10, FUNCT_1:12 .= (((exp_R + f1) . x) #Z 2) / (exp_R . x) by TAYLOR_1:def_1 ; hence (((#Z 2) * (exp_R + f1)) / exp_R) . x > 0 by A11, A9, XREAL_1:139; ::_thesis: verum end; ( 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 A12: ((#Z 2) * (exp_R + f1)) / exp_R is_differentiable_on Z by A6, FDIFF_2:21; A13: for x being Real st x in Z holds ln * (((#Z 2) * (exp_R + f1)) / exp_R) is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies ln * (((#Z 2) * (exp_R + f1)) / exp_R) is_differentiable_in x ) assume x in Z ; ::_thesis: ln * (((#Z 2) * (exp_R + f1)) / exp_R) is_differentiable_in x then ( ((#Z 2) * (exp_R + f1)) / exp_R is_differentiable_in x & (((#Z 2) * (exp_R + f1)) / exp_R) . x > 0 ) by A12, A8, FDIFF_1:9; hence ln * (((#Z 2) * (exp_R + f1)) / exp_R) is_differentiable_in x by TAYLOR_1:20; ::_thesis: verum end; then A14: f is_differentiable_on Z by A1, A2, FDIFF_1:9; for x being Real st x in Z holds (f `| Z) . x = ((exp_R . x) - 1) / ((exp_R . x) + 1) proof let x be Real; ::_thesis: ( x in Z implies (f `| Z) . x = ((exp_R . x) - 1) / ((exp_R . x) + 1) ) A15: exp_R . x > 0 by SIN_COS:54; A16: exp_R is_differentiable_in x by SIN_COS:65; A17: (exp_R . x) + 1 > 0 by SIN_COS:54, XREAL_1:34; assume A18: x in Z ; ::_thesis: (f `| Z) . x = ((exp_R . x) - 1) / ((exp_R . x) + 1) then A19: (exp_R + f1) . x = (exp_R . x) + (f1 . x) by A7, VALUED_1:def_1 .= (exp_R . x) + 1 by A3, A18 ; A20: ((#Z 2) * (exp_R + f1)) . x = (#Z 2) . ((exp_R + f1) . x) by A5, A18, FUNCT_1:12 .= ((exp_R . x) + 1) #Z (1 + 1) by A19, TAYLOR_1:def_1 .= (((exp_R . x) + 1) #Z 1) * (((exp_R . x) + 1) #Z 1) by A17, PREPOWER:44 .= ((exp_R . x) + 1) * (((exp_R . x) + 1) #Z 1) by PREPOWER:35 .= ((exp_R . x) + 1) * ((exp_R . x) + 1) by PREPOWER:35 ; A21: ( ((#Z 2) * (exp_R + f1)) / exp_R is_differentiable_in x & (((#Z 2) * (exp_R + f1)) / exp_R) . x > 0 ) by A12, A8, A18, FDIFF_1:9; (#Z 2) * (exp_R + f1) is_differentiable_in x by A6, A18, FDIFF_1:9; then A22: diff ((((#Z 2) * (exp_R + f1)) / exp_R),x) = (((diff (((#Z 2) * (exp_R + f1)),x)) * (exp_R . x)) - ((diff (exp_R,x)) * (((#Z 2) * (exp_R + f1)) . x))) / ((exp_R . x) ^2) by A15, A16, FDIFF_2:14 .= ((((((#Z 2) * (exp_R + f1)) `| Z) . x) * (exp_R . x)) - ((diff (exp_R,x)) * (((#Z 2) * (exp_R + f1)) . x))) / ((exp_R . x) ^2) by A6, A18, FDIFF_1:def_7 .= ((((2 * (exp_R . x)) * ((exp_R . x) + 1)) * (exp_R . x)) - ((diff (exp_R,x)) * (((#Z 2) * (exp_R + f1)) . x))) / ((exp_R . x) ^2) by A3, A5, A18, Th29 .= ((((2 * (exp_R . x)) * ((exp_R . x) + 1)) * (exp_R . x)) - ((exp_R . x) * (((#Z 2) * (exp_R + f1)) . x))) / ((exp_R . x) ^2) by SIN_COS:65 .= ((((2 * (exp_R . x)) * ((exp_R . x) + 1)) - (((#Z 2) * (exp_R + f1)) . x)) * (exp_R . x)) / ((exp_R . x) * (exp_R . x)) .= (((exp_R . x) - 1) * ((exp_R . x) + 1)) / (exp_R . x) by A15, A20, XCMPLX_1:91 ; A23: (((#Z 2) * (exp_R + f1)) / exp_R) . x = (((#Z 2) * (exp_R + f1)) . x) * ((exp_R . x) ") by A4, A18, RFUNCT_1:def_1 .= (((#Z 2) * (exp_R + f1)) . x) * (1 / (exp_R . x)) by XCMPLX_1:215 .= (((exp_R . x) + 1) * ((exp_R . x) + 1)) / (exp_R . x) by A20, XCMPLX_1:99 ; (f `| Z) . x = diff ((ln * (((#Z 2) * (exp_R + f1)) / exp_R)),x) by A2, A14, A18, FDIFF_1:def_7 .= ((((exp_R . x) + 1) * ((exp_R . x) - 1)) / (exp_R . x)) / ((((exp_R . x) + 1) * ((exp_R . x) + 1)) / (exp_R . x)) by A21, A22, A23, TAYLOR_1:20 .= (((exp_R . x) + 1) * ((exp_R . x) - 1)) / (((exp_R . x) + 1) * ((exp_R . x) + 1)) by A15, XCMPLX_1:55 .= ((exp_R . x) - 1) / ((exp_R . x) + 1) by A17, XCMPLX_1:91 ; hence (f `| Z) . x = ((exp_R . x) - 1) / ((exp_R . x) + 1) ; ::_thesis: verum end; hence ( f is_differentiable_on Z & ( for x being Real st x in Z holds (f `| Z) . x = ((exp_R . x) - 1) / ((exp_R . x) + 1) ) ) by A1, A2, A13, FDIFF_1:9; ::_thesis: verum end; theorem Th31: :: FDIFF_6:31 for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom ((#Z 2) * (f - exp_R)) & ( for x being Real st x in Z holds f . x = 1 ) holds ( (#Z 2) * (f - exp_R) is_differentiable_on Z & ( for x being Real st x in Z holds (((#Z 2) * (f - exp_R)) `| Z) . x = - ((2 * (exp_R . x)) * (1 - (exp_R . x))) ) ) proof let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st Z c= dom ((#Z 2) * (f - exp_R)) & ( for x being Real st x in Z holds f . x = 1 ) holds ( (#Z 2) * (f - exp_R) is_differentiable_on Z & ( for x being Real st x in Z holds (((#Z 2) * (f - exp_R)) `| Z) . x = - ((2 * (exp_R . x)) * (1 - (exp_R . x))) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom ((#Z 2) * (f - exp_R)) & ( for x being Real st x in Z holds f . x = 1 ) implies ( (#Z 2) * (f - exp_R) is_differentiable_on Z & ( for x being Real st x in Z holds (((#Z 2) * (f - exp_R)) `| Z) . x = - ((2 * (exp_R . x)) * (1 - (exp_R . x))) ) ) ) assume that A1: Z c= dom ((#Z 2) * (f - exp_R)) and A2: for x being Real st x in Z holds f . x = 1 ; ::_thesis: ( (#Z 2) * (f - exp_R) is_differentiable_on Z & ( for x being Real st x in Z holds (((#Z 2) * (f - exp_R)) `| Z) . x = - ((2 * (exp_R . x)) * (1 - (exp_R . x))) ) ) for y being set st y in Z holds y in dom (f - exp_R) by A1, FUNCT_1:11; then A3: Z c= dom (f - exp_R) by TARSKI:def_3; then Z c= (dom exp_R) /\ (dom f) by VALUED_1:12; then A4: Z c= dom f by XBOOLE_1:18; A5: for x being Real st x in Z holds f . x = (0 * x) + 1 by A2; then A6: f is_differentiable_on Z by A4, FDIFF_1:23; A7: for x being Real st x in Z holds (#Z 2) * (f - exp_R) is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies (#Z 2) * (f - exp_R) is_differentiable_in x ) assume x in Z ; ::_thesis: (#Z 2) * (f - exp_R) is_differentiable_in x then A8: f is_differentiable_in x by A6, FDIFF_1:9; exp_R is_differentiable_in x by SIN_COS:65; then f - exp_R is_differentiable_in x by A8, FDIFF_1:14; hence (#Z 2) * (f - exp_R) is_differentiable_in x by TAYLOR_1:3; ::_thesis: verum end; then A9: (#Z 2) * (f - exp_R) is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds (((#Z 2) * (f - exp_R)) `| Z) . x = - ((2 * (exp_R . x)) * (1 - (exp_R . x))) proof let x be Real; ::_thesis: ( x in Z implies (((#Z 2) * (f - exp_R)) `| Z) . x = - ((2 * (exp_R . x)) * (1 - (exp_R . x))) ) A10: exp_R is_differentiable_in x by SIN_COS:65; assume A11: x in Z ; ::_thesis: (((#Z 2) * (f - exp_R)) `| Z) . x = - ((2 * (exp_R . x)) * (1 - (exp_R . x))) then A12: (f - exp_R) . x = (f . x) - (exp_R . x) by A3, VALUED_1:13 .= 1 - (exp_R . x) by A2, A11 ; A13: f is_differentiable_in x by A6, A11, FDIFF_1:9; then A14: diff ((f - exp_R),x) = (diff (f,x)) - (diff (exp_R,x)) by A10, FDIFF_1:14 .= ((f `| Z) . x) - (diff (exp_R,x)) by A6, A11, FDIFF_1:def_7 .= ((f `| Z) . x) - (exp_R . x) by SIN_COS:65 .= 0 - (exp_R . x) by A4, A5, A11, FDIFF_1:23 .= - (exp_R . x) ; A15: f - exp_R is_differentiable_in x by A13, A10, FDIFF_1:14; (((#Z 2) * (f - exp_R)) `| Z) . x = diff (((#Z 2) * (f - exp_R)),x) by A9, A11, FDIFF_1:def_7 .= (2 * (((f - exp_R) . x) #Z (2 - 1))) * (diff ((f - exp_R),x)) by A15, TAYLOR_1:3 .= (2 * (1 - (exp_R . x))) * (- (exp_R . x)) by A14, A12, PREPOWER:35 .= - ((2 * (exp_R . x)) * (1 - (exp_R . x))) ; hence (((#Z 2) * (f - exp_R)) `| Z) . x = - ((2 * (exp_R . x)) * (1 - (exp_R . x))) ; ::_thesis: verum end; hence ( (#Z 2) * (f - exp_R) is_differentiable_on Z & ( for x being Real st x in Z holds (((#Z 2) * (f - exp_R)) `| Z) . x = - ((2 * (exp_R . x)) * (1 - (exp_R . x))) ) ) by A1, A7, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_6:32 for Z being open Subset of REAL for f, f1 being PartFunc of REAL,REAL st Z c= dom f & f = ln * (exp_R / ((#Z 2) * (f1 - exp_R))) & ( for x being Real st x in Z holds ( f1 . x = 1 & (f1 - exp_R) . x > 0 ) ) holds ( f is_differentiable_on Z & ( for x being Real st x in Z holds (f `| Z) . x = (1 + (exp_R . x)) / (1 - (exp_R . x)) ) ) proof let Z be open Subset of REAL; ::_thesis: for f, f1 being PartFunc of REAL,REAL st Z c= dom f & f = ln * (exp_R / ((#Z 2) * (f1 - exp_R))) & ( for x being Real st x in Z holds ( f1 . x = 1 & (f1 - exp_R) . x > 0 ) ) holds ( f is_differentiable_on Z & ( for x being Real st x in Z holds (f `| Z) . x = (1 + (exp_R . x)) / (1 - (exp_R . x)) ) ) let f, f1 be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom f & f = ln * (exp_R / ((#Z 2) * (f1 - exp_R))) & ( for x being Real st x in Z holds ( f1 . x = 1 & (f1 - exp_R) . x > 0 ) ) implies ( f is_differentiable_on Z & ( for x being Real st x in Z holds (f `| Z) . x = (1 + (exp_R . x)) / (1 - (exp_R . x)) ) ) ) assume that A1: Z c= dom f and A2: f = ln * (exp_R / ((#Z 2) * (f1 - exp_R))) and A3: for x being Real st x in Z holds ( f1 . x = 1 & (f1 - exp_R) . x > 0 ) ; ::_thesis: ( f is_differentiable_on Z & ( for x being Real st x in Z holds (f `| Z) . x = (1 + (exp_R . x)) / (1 - (exp_R . x)) ) ) for y being set st y in Z holds y in dom (exp_R / ((#Z 2) * (f1 - exp_R))) by A1, A2, FUNCT_1:11; then A4: Z c= dom (exp_R / ((#Z 2) * (f1 - exp_R))) by TARSKI:def_3; then Z c= (dom exp_R) /\ ((dom ((#Z 2) * (f1 - exp_R))) \ (((#Z 2) * (f1 - exp_R)) " {0})) by RFUNCT_1:def_1; then A5: Z c= dom ((#Z 2) * (f1 - exp_R)) by XBOOLE_1:1; then for y being set st y in Z holds y in dom (f1 - exp_R) by FUNCT_1:11; then A6: Z c= dom (f1 - exp_R) by TARSKI:def_3; A7: for x being Real st x in Z holds ((#Z 2) * (f1 - exp_R)) . x > 0 proof let x be Real; ::_thesis: ( x in Z implies ((#Z 2) * (f1 - exp_R)) . x > 0 ) assume A8: x in Z ; ::_thesis: ((#Z 2) * (f1 - exp_R)) . x > 0 then ((#Z 2) * (f1 - exp_R)) . x = (#Z 2) . ((f1 - exp_R) . x) by A5, FUNCT_1:12 .= ((f1 - exp_R) . x) #Z 2 by TAYLOR_1:def_1 ; hence ((#Z 2) * (f1 - exp_R)) . x > 0 by A3, A8, PREPOWER:39; ::_thesis: verum end; A9: for x being Real st x in Z holds (exp_R / ((#Z 2) * (f1 - exp_R))) . x > 0 proof let x be Real; ::_thesis: ( x in Z implies (exp_R / ((#Z 2) * (f1 - exp_R))) . x > 0 ) A10: exp_R . x > 0 by SIN_COS:54; assume A11: x in Z ; ::_thesis: (exp_R / ((#Z 2) * (f1 - exp_R))) . x > 0 then A12: ((#Z 2) * (f1 - exp_R)) . x > 0 by A7; (exp_R / ((#Z 2) * (f1 - exp_R))) . x = (exp_R . x) * ((((#Z 2) * (f1 - exp_R)) . x) ") by A4, A11, RFUNCT_1:def_1 .= (exp_R . x) * (1 / (((#Z 2) * (f1 - exp_R)) . x)) by XCMPLX_1:215 .= (exp_R . x) / (((#Z 2) * (f1 - exp_R)) . x) by XCMPLX_1:99 ; hence (exp_R / ((#Z 2) * (f1 - exp_R))) . x > 0 by A12, A10, XREAL_1:139; ::_thesis: verum end; A13: for x being Real st x in Z holds f1 . x = 1 by A3; then A14: (#Z 2) * (f1 - exp_R) is_differentiable_on Z by A5, Th31; ( exp_R is_differentiable_on Z & ( for x being Real st x in Z holds ((#Z 2) * (f1 - exp_R)) . x <> 0 ) ) by A7, FDIFF_1:26, TAYLOR_1:16; then A15: exp_R / ((#Z 2) * (f1 - exp_R)) is_differentiable_on Z by A14, FDIFF_2:21; A16: for x being Real st x in Z holds ln * (exp_R / ((#Z 2) * (f1 - exp_R))) is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies ln * (exp_R / ((#Z 2) * (f1 - exp_R))) is_differentiable_in x ) assume x in Z ; ::_thesis: ln * (exp_R / ((#Z 2) * (f1 - exp_R))) is_differentiable_in x then ( exp_R / ((#Z 2) * (f1 - exp_R)) is_differentiable_in x & (exp_R / ((#Z 2) * (f1 - exp_R))) . x > 0 ) by A15, A9, FDIFF_1:9; hence ln * (exp_R / ((#Z 2) * (f1 - exp_R))) is_differentiable_in x by TAYLOR_1:20; ::_thesis: verum end; then A17: f is_differentiable_on Z by A1, A2, FDIFF_1:9; for x being Real st x in Z holds (f `| Z) . x = (1 + (exp_R . x)) / (1 - (exp_R . x)) proof let x be Real; ::_thesis: ( x in Z implies (f `| Z) . x = (1 + (exp_R . x)) / (1 - (exp_R . x)) ) A18: exp_R is_differentiable_in x by SIN_COS:65; assume A19: x in Z ; ::_thesis: (f `| Z) . x = (1 + (exp_R . x)) / (1 - (exp_R . x)) then A20: ((#Z 2) * (f1 - exp_R)) . x = (#Z 2) . ((f1 - exp_R) . x) by A5, FUNCT_1:12 .= ((f1 - exp_R) . x) #Z 2 by TAYLOR_1:def_1 .= ((f1 . x) - (exp_R . x)) #Z 2 by A6, A19, VALUED_1:13 .= (1 - (exp_R . x)) #Z 2 by A3, A19 ; A21: (exp_R / ((#Z 2) * (f1 - exp_R))) . x = (exp_R . x) * ((((#Z 2) * (f1 - exp_R)) . x) ") by A4, A19, RFUNCT_1:def_1 .= (exp_R . x) * (1 / (((#Z 2) * (f1 - exp_R)) . x)) by XCMPLX_1:215 .= (exp_R . x) / ((1 - (exp_R . x)) #Z 2) by A20, XCMPLX_1:99 ; A22: ( exp_R / ((#Z 2) * (f1 - exp_R)) is_differentiable_in x & (exp_R / ((#Z 2) * (f1 - exp_R))) . x > 0 ) by A15, A9, A19, FDIFF_1:9; A23: (f1 - exp_R) . x > 0 by A3, A19; ( ((#Z 2) * (f1 - exp_R)) . x <> 0 & (#Z 2) * (f1 - exp_R) is_differentiable_in x ) by A14, A7, A19, FDIFF_1:9; then A24: diff ((exp_R / ((#Z 2) * (f1 - exp_R))),x) = (((diff (exp_R,x)) * (((#Z 2) * (f1 - exp_R)) . x)) - ((diff (((#Z 2) * (f1 - exp_R)),x)) * (exp_R . x))) / ((((#Z 2) * (f1 - exp_R)) . x) ^2) by A18, FDIFF_2:14 .= (((exp_R . x) * (((#Z 2) * (f1 - exp_R)) . x)) - ((diff (((#Z 2) * (f1 - exp_R)),x)) * (exp_R . x))) / ((((#Z 2) * (f1 - exp_R)) . x) ^2) by SIN_COS:65 .= (((exp_R . x) * (((#Z 2) * (f1 - exp_R)) . x)) - (((((#Z 2) * (f1 - exp_R)) `| Z) . x) * (exp_R . x))) / ((((#Z 2) * (f1 - exp_R)) . x) ^2) by A14, A19, FDIFF_1:def_7 .= (((exp_R . x) * ((1 - (exp_R . x)) #Z 2)) - ((- ((2 * (exp_R . x)) * (1 - (exp_R . x)))) * (exp_R . x))) / (((1 - (exp_R . x)) #Z 2) ^2) by A13, A5, A19, A20, Th31 .= ((exp_R . x) * (((1 - (exp_R . x)) #Z 2) + ((2 * (1 - (exp_R . x))) * (exp_R . x)))) / (((1 - (exp_R . x)) #Z 2) * ((1 - (exp_R . x)) #Z 2)) .= (((exp_R . x) / ((1 - (exp_R . x)) #Z 2)) * (((1 - (exp_R . x)) #Z 2) + ((2 * (1 - (exp_R . x))) * (exp_R . x)))) / ((1 - (exp_R . x)) #Z 2) by XCMPLX_1:83 ; A25: exp_R . x > 0 by SIN_COS:54; A26: (f1 - exp_R) . x = (f1 . x) - (exp_R . x) by A6, A19, VALUED_1:13 .= 1 - (exp_R . x) by A3, A19 ; then (1 - (exp_R . x)) #Z 2 > 0 by A3, A19, PREPOWER:39; then A27: (exp_R . x) / ((1 - (exp_R . x)) #Z 2) <> 0 by A25, XREAL_1:139; A28: (1 - (exp_R . x)) #Z 2 = (1 - (exp_R . x)) #Z (1 + 1) .= ((1 - (exp_R . x)) #Z 1) * ((1 - (exp_R . x)) #Z 1) by A23, A26, PREPOWER:44 .= (1 - (exp_R . x)) * ((1 - (exp_R . x)) #Z 1) by PREPOWER:35 .= (1 - (exp_R . x)) * (1 - (exp_R . x)) by PREPOWER:35 ; (f `| Z) . x = diff ((ln * (exp_R / ((#Z 2) * (f1 - exp_R)))),x) by A2, A17, A19, FDIFF_1:def_7 .= ((((exp_R . x) / ((1 - (exp_R . x)) #Z 2)) * (((1 - (exp_R . x)) #Z 2) + ((2 * (1 - (exp_R . x))) * (exp_R . x)))) / ((1 - (exp_R . x)) #Z 2)) / ((exp_R . x) / ((1 - (exp_R . x)) #Z 2)) by A22, A24, A21, TAYLOR_1:20 .= (((exp_R . x) / ((1 - (exp_R . x)) #Z 2)) * (((1 - (exp_R . x)) #Z 2) + ((2 * (1 - (exp_R . x))) * (exp_R . x)))) / (((exp_R . x) / ((1 - (exp_R . x)) #Z 2)) * ((1 - (exp_R . x)) #Z 2)) by XCMPLX_1:78 .= ((1 - (exp_R . x)) * (1 + (exp_R . x))) / ((1 - (exp_R . x)) * (1 - (exp_R . x))) by A27, A28, XCMPLX_1:91 .= (1 + (exp_R . x)) / (1 - (exp_R . x)) by A23, A26, XCMPLX_1:91 ; hence (f `| Z) . x = (1 + (exp_R . x)) / (1 - (exp_R . x)) ; ::_thesis: verum end; hence ( f is_differentiable_on Z & ( for x being Real st x in Z holds (f `| Z) . x = (1 + (exp_R . x)) / (1 - (exp_R . x)) ) ) by A1, A2, A16, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_6:33 for Z being open Subset of REAL for f, f1 being PartFunc of REAL,REAL st Z c= dom f & f = ln * (exp_R / ((#Z 2) * (f1 + exp_R))) & ( for x being Real st x in Z holds f1 . x = 1 ) holds ( f is_differentiable_on Z & ( for x being Real st x in Z holds (f `| Z) . x = (1 - (exp_R . x)) / (1 + (exp_R . x)) ) ) proof let Z be open Subset of REAL; ::_thesis: for f, f1 being PartFunc of REAL,REAL st Z c= dom f & f = ln * (exp_R / ((#Z 2) * (f1 + exp_R))) & ( for x being Real st x in Z holds f1 . x = 1 ) holds ( f is_differentiable_on Z & ( for x being Real st x in Z holds (f `| Z) . x = (1 - (exp_R . x)) / (1 + (exp_R . x)) ) ) let f, f1 be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom f & f = ln * (exp_R / ((#Z 2) * (f1 + exp_R))) & ( for x being Real st x in Z holds f1 . x = 1 ) implies ( f is_differentiable_on Z & ( for x being Real st x in Z holds (f `| Z) . x = (1 - (exp_R . x)) / (1 + (exp_R . x)) ) ) ) assume that A1: Z c= dom f and A2: f = ln * (exp_R / ((#Z 2) * (f1 + exp_R))) and A3: for x being Real st x in Z holds f1 . x = 1 ; ::_thesis: ( f is_differentiable_on Z & ( for x being Real st x in Z holds (f `| Z) . x = (1 - (exp_R . x)) / (1 + (exp_R . x)) ) ) for y being set st y in Z holds y in dom (exp_R / ((#Z 2) * (f1 + exp_R))) by A1, A2, FUNCT_1:11; then A4: Z c= dom (exp_R / ((#Z 2) * (f1 + exp_R))) by TARSKI:def_3; then Z c= (dom exp_R) /\ ((dom ((#Z 2) * (f1 + exp_R))) \ (((#Z 2) * (f1 + exp_R)) " {0})) by RFUNCT_1:def_1; then A5: Z c= dom ((#Z 2) * (f1 + exp_R)) by XBOOLE_1:1; then A6: (#Z 2) * (f1 + exp_R) is_differentiable_on Z by A3, Th29; for y being set st y in Z holds y in dom (f1 + exp_R) by A5, FUNCT_1:11; then A7: Z c= dom (f1 + exp_R) by TARSKI:def_3; A8: for x being Real st x in Z holds ((#Z 2) * (f1 + exp_R)) . x > 0 proof let x be Real; ::_thesis: ( x in Z implies ((#Z 2) * (f1 + exp_R)) . x > 0 ) assume A9: x in Z ; ::_thesis: ((#Z 2) * (f1 + exp_R)) . x > 0 then (f1 + exp_R) . x = (f1 . x) + (exp_R . x) by A7, VALUED_1:def_1 .= 1 + (exp_R . x) by A3, A9 ; then A10: (f1 + exp_R) . x > 0 by SIN_COS:54, XREAL_1:34; ((#Z 2) * (f1 + exp_R)) . x = (#Z 2) . ((f1 + exp_R) . x) by A5, A9, FUNCT_1:12 .= ((f1 + exp_R) . x) #Z 2 by TAYLOR_1:def_1 ; hence ((#Z 2) * (f1 + exp_R)) . x > 0 by A10, PREPOWER:39; ::_thesis: verum end; A11: for x being Real st x in Z holds (exp_R / ((#Z 2) * (f1 + exp_R))) . x > 0 proof let x be Real; ::_thesis: ( x in Z implies (exp_R / ((#Z 2) * (f1 + exp_R))) . x > 0 ) A12: exp_R . x > 0 by SIN_COS:54; assume A13: x in Z ; ::_thesis: (exp_R / ((#Z 2) * (f1 + exp_R))) . x > 0 then A14: ((#Z 2) * (f1 + exp_R)) . x > 0 by A8; (exp_R / ((#Z 2) * (f1 + exp_R))) . x = (exp_R . x) * ((((#Z 2) * (f1 + exp_R)) . x) ") by A4, A13, RFUNCT_1:def_1 .= (exp_R . x) * (1 / (((#Z 2) * (f1 + exp_R)) . x)) by XCMPLX_1:215 .= (exp_R . x) / (((#Z 2) * (f1 + exp_R)) . x) by XCMPLX_1:99 ; hence (exp_R / ((#Z 2) * (f1 + exp_R))) . x > 0 by A14, A12, XREAL_1:139; ::_thesis: verum end; ( exp_R is_differentiable_on Z & ( for x being Real st x in Z holds ((#Z 2) * (f1 + exp_R)) . x <> 0 ) ) by A8, FDIFF_1:26, TAYLOR_1:16; then A15: exp_R / ((#Z 2) * (f1 + exp_R)) is_differentiable_on Z by A6, FDIFF_2:21; A16: for x being Real st x in Z holds ln * (exp_R / ((#Z 2) * (f1 + exp_R))) is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies ln * (exp_R / ((#Z 2) * (f1 + exp_R))) is_differentiable_in x ) assume x in Z ; ::_thesis: ln * (exp_R / ((#Z 2) * (f1 + exp_R))) is_differentiable_in x then ( exp_R / ((#Z 2) * (f1 + exp_R)) is_differentiable_in x & (exp_R / ((#Z 2) * (f1 + exp_R))) . x > 0 ) by A15, A11, FDIFF_1:9; hence ln * (exp_R / ((#Z 2) * (f1 + exp_R))) is_differentiable_in x by TAYLOR_1:20; ::_thesis: verum end; then A17: f is_differentiable_on Z by A1, A2, FDIFF_1:9; for x being Real st x in Z holds (f `| Z) . x = (1 - (exp_R . x)) / (1 + (exp_R . x)) proof let x be Real; ::_thesis: ( x in Z implies (f `| Z) . x = (1 - (exp_R . x)) / (1 + (exp_R . x)) ) A18: exp_R is_differentiable_in x by SIN_COS:65; assume A19: x in Z ; ::_thesis: (f `| Z) . x = (1 - (exp_R . x)) / (1 + (exp_R . x)) then A20: ((#Z 2) * (f1 + exp_R)) . x = (#Z 2) . ((f1 + exp_R) . x) by A5, FUNCT_1:12 .= ((f1 + exp_R) . x) #Z 2 by TAYLOR_1:def_1 .= ((f1 . x) + (exp_R . x)) #Z 2 by A7, A19, VALUED_1:def_1 .= (1 + (exp_R . x)) #Z 2 by A3, A19 ; ( ((#Z 2) * (f1 + exp_R)) . x <> 0 & (#Z 2) * (f1 + exp_R) is_differentiable_in x ) by A6, A8, A19, FDIFF_1:9; then A21: diff ((exp_R / ((#Z 2) * (f1 + exp_R))),x) = (((diff (exp_R,x)) * (((#Z 2) * (f1 + exp_R)) . x)) - ((diff (((#Z 2) * (f1 + exp_R)),x)) * (exp_R . x))) / ((((#Z 2) * (f1 + exp_R)) . x) ^2) by A18, FDIFF_2:14 .= (((exp_R . x) * (((#Z 2) * (f1 + exp_R)) . x)) - ((diff (((#Z 2) * (f1 + exp_R)),x)) * (exp_R . x))) / ((((#Z 2) * (f1 + exp_R)) . x) ^2) by SIN_COS:65 .= (((exp_R . x) * (((#Z 2) * (f1 + exp_R)) . x)) - (((((#Z 2) * (f1 + exp_R)) `| Z) . x) * (exp_R . x))) / ((((#Z 2) * (f1 + exp_R)) . x) ^2) by A6, A19, FDIFF_1:def_7 .= (((exp_R . x) * ((1 + (exp_R . x)) #Z 2)) - (((2 * (exp_R . x)) * (1 + (exp_R . x))) * (exp_R . x))) / (((1 + (exp_R . x)) #Z 2) ^2) by A3, A5, A19, A20, Th29 .= ((exp_R . x) * (((1 + (exp_R . x)) #Z 2) - ((2 * (1 + (exp_R . x))) * (exp_R . x)))) / (((1 + (exp_R . x)) #Z 2) * ((1 + (exp_R . x)) #Z 2)) .= (((exp_R . x) / ((1 + (exp_R . x)) #Z 2)) * (((1 + (exp_R . x)) #Z 2) - ((2 * (1 + (exp_R . x))) * (exp_R . x)))) / ((1 + (exp_R . x)) #Z 2) by XCMPLX_1:83 ; A22: 1 + (exp_R . x) > 0 by SIN_COS:54, XREAL_1:34; then ( exp_R . x > 0 & (1 + (exp_R . x)) #Z 2 > 0 ) by PREPOWER:39, SIN_COS:54; then A23: (exp_R . x) / ((1 + (exp_R . x)) #Z 2) <> 0 by XREAL_1:139; A24: (exp_R / ((#Z 2) * (f1 + exp_R))) . x = (exp_R . x) * ((((#Z 2) * (f1 + exp_R)) . x) ") by A4, A19, RFUNCT_1:def_1 .= (exp_R . x) * (1 / (((#Z 2) * (f1 + exp_R)) . x)) by XCMPLX_1:215 .= (exp_R . x) / ((1 + (exp_R . x)) #Z 2) by A20, XCMPLX_1:99 ; A25: ( exp_R / ((#Z 2) * (f1 + exp_R)) is_differentiable_in x & (exp_R / ((#Z 2) * (f1 + exp_R))) . x > 0 ) by A15, A11, A19, FDIFF_1:9; A26: (1 + (exp_R . x)) #Z 2 = (1 + (exp_R . x)) #Z (1 + 1) .= ((1 + (exp_R . x)) #Z 1) * ((1 + (exp_R . x)) #Z 1) by A22, PREPOWER:44 .= (1 + (exp_R . x)) * ((1 + (exp_R . x)) #Z 1) by PREPOWER:35 .= (1 + (exp_R . x)) * (1 + (exp_R . x)) by PREPOWER:35 ; (f `| Z) . x = diff ((ln * (exp_R / ((#Z 2) * (f1 + exp_R)))),x) by A2, A17, A19, FDIFF_1:def_7 .= ((((exp_R . x) / ((1 + (exp_R . x)) #Z 2)) * (((1 + (exp_R . x)) #Z 2) - ((2 * (1 + (exp_R . x))) * (exp_R . x)))) / ((1 + (exp_R . x)) #Z 2)) / ((exp_R . x) / ((1 + (exp_R . x)) #Z 2)) by A25, A21, A24, TAYLOR_1:20 .= (((exp_R . x) / ((1 + (exp_R . x)) #Z 2)) * (((1 + (exp_R . x)) #Z 2) - ((2 * (1 + (exp_R . x))) * (exp_R . x)))) / (((exp_R . x) / ((1 + (exp_R . x)) #Z 2)) * ((1 + (exp_R . x)) #Z 2)) by XCMPLX_1:78 .= ((1 + (exp_R . x)) * (1 - (exp_R . x))) / ((1 + (exp_R . x)) * (1 + (exp_R . x))) by A23, A26, XCMPLX_1:91 .= (1 - (exp_R . x)) / (1 + (exp_R . x)) by A22, XCMPLX_1:91 ; hence (f `| Z) . x = (1 - (exp_R . x)) / (1 + (exp_R . x)) ; ::_thesis: verum end; hence ( f is_differentiable_on Z & ( for x being Real st x in Z holds (f `| Z) . x = (1 - (exp_R . x)) / (1 + (exp_R . x)) ) ) by A1, A2, A16, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_6:34 for Z being open Subset of REAL for f, f1 being PartFunc of REAL,REAL st Z c= dom (ln * f) & f = exp_R + (exp_R * f1) & ( for x being Real st x in Z holds f1 . x = - x ) holds ( ln * f is_differentiable_on Z & ( for x being Real st x in Z holds ((ln * f) `| Z) . x = ((exp_R x) - (exp_R (- x))) / ((exp_R x) + (exp_R (- x))) ) ) proof let Z be open Subset of REAL; ::_thesis: for f, f1 being PartFunc of REAL,REAL st Z c= dom (ln * f) & f = exp_R + (exp_R * f1) & ( for x being Real st x in Z holds f1 . x = - x ) holds ( ln * f is_differentiable_on Z & ( for x being Real st x in Z holds ((ln * f) `| Z) . x = ((exp_R x) - (exp_R (- x))) / ((exp_R x) + (exp_R (- x))) ) ) let f, f1 be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (ln * f) & f = exp_R + (exp_R * f1) & ( for x being Real st x in Z holds f1 . x = - x ) implies ( ln * f is_differentiable_on Z & ( for x being Real st x in Z holds ((ln * f) `| Z) . x = ((exp_R x) - (exp_R (- x))) / ((exp_R x) + (exp_R (- x))) ) ) ) assume that A1: Z c= dom (ln * f) and A2: f = exp_R + (exp_R * f1) and A3: for x being Real st x in Z holds f1 . x = - x ; ::_thesis: ( ln * f is_differentiable_on Z & ( for x being Real st x in Z holds ((ln * f) `| Z) . x = ((exp_R x) - (exp_R (- x))) / ((exp_R x) + (exp_R (- x))) ) ) for y being set st y in Z holds y in dom f by A1, FUNCT_1:11; then A4: Z c= dom (exp_R + (exp_R * f1)) by A2, TARSKI:def_3; then Z c= (dom exp_R) /\ (dom (exp_R * f1)) by VALUED_1:def_1; then A5: Z c= dom (exp_R * f1) by XBOOLE_1:18; then A6: exp_R * f1 is_differentiable_on Z by A3, Th14; A7: exp_R is_differentiable_on Z by FDIFF_1:26, TAYLOR_1:16; then A8: f is_differentiable_on Z by A2, A4, A6, FDIFF_1:18; A9: for x being Real st x in Z holds ((exp_R + (exp_R * f1)) `| Z) . x = (exp_R x) - (exp_R (- x)) proof let x be Real; ::_thesis: ( x in Z implies ((exp_R + (exp_R * f1)) `| Z) . x = (exp_R x) - (exp_R (- x)) ) assume A10: x in Z ; ::_thesis: ((exp_R + (exp_R * f1)) `| Z) . x = (exp_R x) - (exp_R (- x)) hence ((exp_R + (exp_R * f1)) `| Z) . x = (diff (exp_R,x)) + (diff ((exp_R * f1),x)) by A4, A6, A7, FDIFF_1:18 .= (exp_R . x) + (diff ((exp_R * f1),x)) by SIN_COS:65 .= (exp_R . x) + (((exp_R * f1) `| Z) . x) by A6, A10, FDIFF_1:def_7 .= (exp_R . x) + (- (exp_R (- x))) by A3, A5, A10, Th14 .= (exp_R x) + (- (exp_R (- x))) by SIN_COS:def_23 .= (exp_R x) - (exp_R (- x)) ; ::_thesis: verum end; A11: for x being Real st x in Z holds (exp_R + (exp_R * f1)) . x > 0 proof let x be Real; ::_thesis: ( x in Z implies (exp_R + (exp_R * f1)) . x > 0 ) A12: exp_R x > 0 by SIN_COS:55; assume A13: x in Z ; ::_thesis: (exp_R + (exp_R * f1)) . x > 0 then (exp_R + (exp_R * f1)) . x = (exp_R . x) + ((exp_R * f1) . x) by A4, VALUED_1:def_1 .= (exp_R . x) + (exp_R . (f1 . x)) by A5, A13, FUNCT_1:12 .= (exp_R . x) + (exp_R . (- x)) by A3, A13 .= (exp_R x) + (exp_R . (- x)) by SIN_COS:def_23 .= (exp_R x) + (exp_R (- x)) by SIN_COS:def_23 ; hence (exp_R + (exp_R * f1)) . x > 0 by A12, SIN_COS:55, XREAL_1:34; ::_thesis: verum end; A14: for x being Real st x in Z holds ln * f is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies ln * f is_differentiable_in x ) assume x in Z ; ::_thesis: ln * f is_differentiable_in x then ( f is_differentiable_in x & f . x > 0 ) by A2, A8, A11, FDIFF_1:9; hence ln * f is_differentiable_in x by TAYLOR_1:20; ::_thesis: verum end; then A15: ln * f is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((ln * f) `| Z) . x = ((exp_R x) - (exp_R (- x))) / ((exp_R x) + (exp_R (- x))) proof let x be Real; ::_thesis: ( x in Z implies ((ln * f) `| Z) . x = ((exp_R x) - (exp_R (- x))) / ((exp_R x) + (exp_R (- x))) ) assume A16: x in Z ; ::_thesis: ((ln * f) `| Z) . x = ((exp_R x) - (exp_R (- x))) / ((exp_R x) + (exp_R (- x))) then A17: f . x = (exp_R . x) + ((exp_R * f1) . x) by A2, A4, VALUED_1:def_1 .= (exp_R . x) + (exp_R . (f1 . x)) by A5, A16, FUNCT_1:12 .= (exp_R . x) + (exp_R . (- x)) by A3, A16 .= (exp_R x) + (exp_R . (- x)) by SIN_COS:def_23 .= (exp_R x) + (exp_R (- x)) by SIN_COS:def_23 ; ( f is_differentiable_in x & f . x > 0 ) by A2, A8, A11, A16, FDIFF_1:9; then diff ((ln * f),x) = (diff (f,x)) / (f . x) by TAYLOR_1:20 .= ((f `| Z) . x) / (f . x) by A8, A16, FDIFF_1:def_7 .= ((exp_R x) - (exp_R (- x))) / ((exp_R x) + (exp_R (- x))) by A2, A9, A16, A17 ; hence ((ln * f) `| Z) . x = ((exp_R x) - (exp_R (- x))) / ((exp_R x) + (exp_R (- x))) by A15, A16, FDIFF_1:def_7; ::_thesis: verum end; hence ( ln * f is_differentiable_on Z & ( for x being Real st x in Z holds ((ln * f) `| Z) . x = ((exp_R x) - (exp_R (- x))) / ((exp_R x) + (exp_R (- x))) ) ) by A1, A14, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_6:35 for Z being open Subset of REAL for f, f1 being PartFunc of REAL,REAL st Z c= dom (ln * f) & f = exp_R - (exp_R * f1) & ( for x being Real st x in Z holds ( f1 . x = - x & f . x > 0 ) ) holds ( ln * f is_differentiable_on Z & ( for x being Real st x in Z holds ((ln * f) `| Z) . x = ((exp_R x) + (exp_R (- x))) / ((exp_R x) - (exp_R (- x))) ) ) proof let Z be open Subset of REAL; ::_thesis: for f, f1 being PartFunc of REAL,REAL st Z c= dom (ln * f) & f = exp_R - (exp_R * f1) & ( for x being Real st x in Z holds ( f1 . x = - x & f . x > 0 ) ) holds ( ln * f is_differentiable_on Z & ( for x being Real st x in Z holds ((ln * f) `| Z) . x = ((exp_R x) + (exp_R (- x))) / ((exp_R x) - (exp_R (- x))) ) ) let f, f1 be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (ln * f) & f = exp_R - (exp_R * f1) & ( for x being Real st x in Z holds ( f1 . x = - x & f . x > 0 ) ) implies ( ln * f is_differentiable_on Z & ( for x being Real st x in Z holds ((ln * f) `| Z) . x = ((exp_R x) + (exp_R (- x))) / ((exp_R x) - (exp_R (- x))) ) ) ) assume that A1: Z c= dom (ln * f) and A2: f = exp_R - (exp_R * f1) and A3: for x being Real st x in Z holds ( f1 . x = - x & f . x > 0 ) ; ::_thesis: ( ln * f is_differentiable_on Z & ( for x being Real st x in Z holds ((ln * f) `| Z) . x = ((exp_R x) + (exp_R (- x))) / ((exp_R x) - (exp_R (- x))) ) ) for y being set st y in Z holds y in dom f by A1, FUNCT_1:11; then A4: Z c= dom (exp_R - (exp_R * f1)) by A2, TARSKI:def_3; then Z c= (dom exp_R) /\ (dom (exp_R * f1)) by VALUED_1:12; then A5: Z c= dom (exp_R * f1) by XBOOLE_1:18; A6: for x being Real st x in Z holds f1 . x = - x by A3; then A7: exp_R * f1 is_differentiable_on Z by A5, Th14; A8: exp_R is_differentiable_on Z by FDIFF_1:26, TAYLOR_1:16; then A9: f is_differentiable_on Z by A2, A4, A7, FDIFF_1:19; A10: for x being Real st x in Z holds ((exp_R - (exp_R * f1)) `| Z) . x = (exp_R x) + (exp_R (- x)) proof let x be Real; ::_thesis: ( x in Z implies ((exp_R - (exp_R * f1)) `| Z) . x = (exp_R x) + (exp_R (- x)) ) assume A11: x in Z ; ::_thesis: ((exp_R - (exp_R * f1)) `| Z) . x = (exp_R x) + (exp_R (- x)) hence ((exp_R - (exp_R * f1)) `| Z) . x = (diff (exp_R,x)) - (diff ((exp_R * f1),x)) by A4, A7, A8, FDIFF_1:19 .= (exp_R . x) - (diff ((exp_R * f1),x)) by SIN_COS:65 .= (exp_R . x) - (((exp_R * f1) `| Z) . x) by A7, A11, FDIFF_1:def_7 .= (exp_R . x) - (- (exp_R (- x))) by A6, A5, A11, Th14 .= (exp_R . x) + (exp_R (- x)) .= (exp_R x) + (exp_R (- x)) by SIN_COS:def_23 ; ::_thesis: verum end; A12: for x being Real st x in Z holds ln * f is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies ln * f is_differentiable_in x ) assume x in Z ; ::_thesis: ln * f is_differentiable_in x then ( f is_differentiable_in x & f . x > 0 ) by A3, A9, FDIFF_1:9; hence ln * f is_differentiable_in x by TAYLOR_1:20; ::_thesis: verum end; then A13: ln * f is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((ln * f) `| Z) . x = ((exp_R x) + (exp_R (- x))) / ((exp_R x) - (exp_R (- x))) proof let x be Real; ::_thesis: ( x in Z implies ((ln * f) `| Z) . x = ((exp_R x) + (exp_R (- x))) / ((exp_R x) - (exp_R (- x))) ) assume A14: x in Z ; ::_thesis: ((ln * f) `| Z) . x = ((exp_R x) + (exp_R (- x))) / ((exp_R x) - (exp_R (- x))) then A15: f . x = (exp_R . x) - ((exp_R * f1) . x) by A2, A4, VALUED_1:13 .= (exp_R . x) - (exp_R . (f1 . x)) by A5, A14, FUNCT_1:12 .= (exp_R . x) - (exp_R . (- x)) by A3, A14 .= (exp_R x) - (exp_R . (- x)) by SIN_COS:def_23 .= (exp_R x) - (exp_R (- x)) by SIN_COS:def_23 ; ( f is_differentiable_in x & f . x > 0 ) by A3, A9, A14, FDIFF_1:9; then diff ((ln * f),x) = (diff (f,x)) / (f . x) by TAYLOR_1:20 .= ((f `| Z) . x) / (f . x) by A9, A14, FDIFF_1:def_7 .= ((exp_R x) + (exp_R (- x))) / ((exp_R x) - (exp_R (- x))) by A2, A10, A14, A15 ; hence ((ln * f) `| Z) . x = ((exp_R x) + (exp_R (- x))) / ((exp_R x) - (exp_R (- x))) by A13, A14, FDIFF_1:def_7; ::_thesis: verum end; hence ( ln * f is_differentiable_on Z & ( for x being Real st x in Z holds ((ln * f) `| Z) . x = ((exp_R x) + (exp_R (- x))) / ((exp_R x) - (exp_R (- x))) ) ) by A1, A12, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_6:36 for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom ((2 / 3) (#) ((#R (3 / 2)) * (f + exp_R))) & ( for x being Real st x in Z holds f . x = 1 ) holds ( (2 / 3) (#) ((#R (3 / 2)) * (f + exp_R)) is_differentiable_on Z & ( for x being Real st x in Z holds (((2 / 3) (#) ((#R (3 / 2)) * (f + exp_R))) `| Z) . x = (exp_R . x) * ((1 + (exp_R . x)) #R (1 / 2)) ) ) proof let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st Z c= dom ((2 / 3) (#) ((#R (3 / 2)) * (f + exp_R))) & ( for x being Real st x in Z holds f . x = 1 ) holds ( (2 / 3) (#) ((#R (3 / 2)) * (f + exp_R)) is_differentiable_on Z & ( for x being Real st x in Z holds (((2 / 3) (#) ((#R (3 / 2)) * (f + exp_R))) `| Z) . x = (exp_R . x) * ((1 + (exp_R . x)) #R (1 / 2)) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom ((2 / 3) (#) ((#R (3 / 2)) * (f + exp_R))) & ( for x being Real st x in Z holds f . x = 1 ) implies ( (2 / 3) (#) ((#R (3 / 2)) * (f + exp_R)) is_differentiable_on Z & ( for x being Real st x in Z holds (((2 / 3) (#) ((#R (3 / 2)) * (f + exp_R))) `| Z) . x = (exp_R . x) * ((1 + (exp_R . x)) #R (1 / 2)) ) ) ) assume that A1: Z c= dom ((2 / 3) (#) ((#R (3 / 2)) * (f + exp_R))) and A2: for x being Real st x in Z holds f . x = 1 ; ::_thesis: ( (2 / 3) (#) ((#R (3 / 2)) * (f + exp_R)) is_differentiable_on Z & ( for x being Real st x in Z holds (((2 / 3) (#) ((#R (3 / 2)) * (f + exp_R))) `| Z) . x = (exp_R . x) * ((1 + (exp_R . x)) #R (1 / 2)) ) ) A3: for x being Real st x in Z holds f . x = (0 * x) + 1 by A2; A4: Z c= dom ((#R (3 / 2)) * (f + exp_R)) by A1, VALUED_1:def_5; then for y being set st y in Z holds y in dom (f + exp_R) by FUNCT_1:11; then A5: Z c= dom (f + exp_R) by TARSKI:def_3; then Z c= (dom exp_R) /\ (dom f) by VALUED_1:def_1; then A6: Z c= dom f by XBOOLE_1:18; then A7: f is_differentiable_on Z by A3, FDIFF_1:23; A8: exp_R is_differentiable_on Z by FDIFF_1:26, TAYLOR_1:16; then A9: f + exp_R is_differentiable_on Z by A5, A7, FDIFF_1:18; A10: for x being Real st x in Z holds (f + exp_R) . x > 0 proof let x be Real; ::_thesis: ( x in Z implies (f + exp_R) . x > 0 ) assume A11: x in Z ; ::_thesis: (f + exp_R) . x > 0 then (f + exp_R) . x = (f . x) + (exp_R . x) by A5, VALUED_1:def_1 .= 1 + (exp_R . x) by A2, A11 ; hence (f + exp_R) . x > 0 by SIN_COS:54, XREAL_1:34; ::_thesis: verum end; now__::_thesis:_for_x_being_Real_st_x_in_Z_holds_ (#R_(3_/_2))_*_(f_+_exp_R)_is_differentiable_in_x let x be Real; ::_thesis: ( x in Z implies (#R (3 / 2)) * (f + exp_R) is_differentiable_in x ) assume x in Z ; ::_thesis: (#R (3 / 2)) * (f + exp_R) is_differentiable_in x then ( f + exp_R is_differentiable_in x & (f + exp_R) . x > 0 ) by A9, A10, FDIFF_1:9; hence (#R (3 / 2)) * (f + exp_R) is_differentiable_in x by TAYLOR_1:22; ::_thesis: verum end; then A12: (#R (3 / 2)) * (f + exp_R) is_differentiable_on Z by A4, FDIFF_1:9; for x being Real st x in Z holds (((2 / 3) (#) ((#R (3 / 2)) * (f + exp_R))) `| Z) . x = (exp_R . x) * ((1 + (exp_R . x)) #R (1 / 2)) proof let x be Real; ::_thesis: ( x in Z implies (((2 / 3) (#) ((#R (3 / 2)) * (f + exp_R))) `| Z) . x = (exp_R . x) * ((1 + (exp_R . x)) #R (1 / 2)) ) assume A13: x in Z ; ::_thesis: (((2 / 3) (#) ((#R (3 / 2)) * (f + exp_R))) `| Z) . x = (exp_R . x) * ((1 + (exp_R . x)) #R (1 / 2)) then A14: ((f + exp_R) `| Z) . x = (diff (f,x)) + (diff (exp_R,x)) by A5, A7, A8, FDIFF_1:18 .= (diff (f,x)) + (exp_R . x) by SIN_COS:65 .= ((f `| Z) . x) + (exp_R . x) by A7, A13, FDIFF_1:def_7 .= 0 + (exp_R . x) by A6, A3, A13, FDIFF_1:23 .= exp_R . x ; A15: ( f + exp_R is_differentiable_in x & (f + exp_R) . x > 0 ) by A9, A10, A13, FDIFF_1:9; A16: (f + exp_R) . x = (f . x) + (exp_R . x) by A5, A13, VALUED_1:def_1 .= 1 + (exp_R . x) by A2, A13 ; (((2 / 3) (#) ((#R (3 / 2)) * (f + exp_R))) `| Z) . x = (2 / 3) * (diff (((#R (3 / 2)) * (f + exp_R)),x)) by A1, A12, A13, FDIFF_1:20 .= (2 / 3) * (((3 / 2) * (((f + exp_R) . x) #R ((3 / 2) - 1))) * (diff ((f + exp_R),x))) by A15, TAYLOR_1:22 .= (2 / 3) * (((3 / 2) * (((f + exp_R) . x) #R ((3 / 2) - 1))) * (((f + exp_R) `| Z) . x)) by A9, A13, FDIFF_1:def_7 .= (exp_R . x) * ((1 + (exp_R . x)) #R (1 / 2)) by A16, A14 ; hence (((2 / 3) (#) ((#R (3 / 2)) * (f + exp_R))) `| Z) . x = (exp_R . x) * ((1 + (exp_R . x)) #R (1 / 2)) ; ::_thesis: verum end; hence ( (2 / 3) (#) ((#R (3 / 2)) * (f + exp_R)) is_differentiable_on Z & ( for x being Real st x in Z holds (((2 / 3) (#) ((#R (3 / 2)) * (f + exp_R))) `| Z) . x = (exp_R . x) * ((1 + (exp_R . x)) #R (1 / 2)) ) ) by A1, A12, FDIFF_1:20; ::_thesis: verum end; theorem :: FDIFF_6:37 for a being Real for Z being open Subset of REAL for f, f1 being PartFunc of REAL,REAL st Z c= dom ((2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1)))) & ( for x being Real st x in Z holds ( f . x = 1 & f1 . x = x * (log (number_e,a)) ) ) & a > 0 & a <> 1 holds ( (2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1))) is_differentiable_on Z & ( for x being Real st x in Z holds (((2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1)))) `| Z) . x = (a #R x) * ((1 + (a #R x)) #R (1 / 2)) ) ) proof let a be Real; ::_thesis: for Z being open Subset of REAL for f, f1 being PartFunc of REAL,REAL st Z c= dom ((2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1)))) & ( for x being Real st x in Z holds ( f . x = 1 & f1 . x = x * (log (number_e,a)) ) ) & a > 0 & a <> 1 holds ( (2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1))) is_differentiable_on Z & ( for x being Real st x in Z holds (((2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1)))) `| Z) . x = (a #R x) * ((1 + (a #R x)) #R (1 / 2)) ) ) let Z be open Subset of REAL; ::_thesis: for f, f1 being PartFunc of REAL,REAL st Z c= dom ((2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1)))) & ( for x being Real st x in Z holds ( f . x = 1 & f1 . x = x * (log (number_e,a)) ) ) & a > 0 & a <> 1 holds ( (2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1))) is_differentiable_on Z & ( for x being Real st x in Z holds (((2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1)))) `| Z) . x = (a #R x) * ((1 + (a #R x)) #R (1 / 2)) ) ) let f, f1 be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom ((2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1)))) & ( for x being Real st x in Z holds ( f . x = 1 & f1 . x = x * (log (number_e,a)) ) ) & a > 0 & a <> 1 implies ( (2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1))) is_differentiable_on Z & ( for x being Real st x in Z holds (((2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1)))) `| Z) . x = (a #R x) * ((1 + (a #R x)) #R (1 / 2)) ) ) ) assume that A1: Z c= dom ((2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1)))) and A2: for x being Real st x in Z holds ( f . x = 1 & f1 . x = x * (log (number_e,a)) ) and A3: a > 0 and A4: a <> 1 ; ::_thesis: ( (2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1))) is_differentiable_on Z & ( for x being Real st x in Z holds (((2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1)))) `| Z) . x = (a #R x) * ((1 + (a #R x)) #R (1 / 2)) ) ) A5: for x being Real st x in Z holds f . x = (0 * x) + 1 by A2; A6: Z c= dom ((#R (3 / 2)) * (f + (exp_R * f1))) by A1, VALUED_1:def_5; then for y being set st y in Z holds y in dom (f + (exp_R * f1)) by FUNCT_1:11; then A7: Z c= dom (f + (exp_R * f1)) by TARSKI:def_3; then A8: Z c= (dom (exp_R * f1)) /\ (dom f) by VALUED_1:def_1; then A9: Z c= dom (exp_R * f1) by XBOOLE_1:18; A10: for x being Real st x in Z holds f1 . x = x * (log (number_e,a)) by A2; then A11: exp_R * f1 is_differentiable_on Z by A3, A9, Th11; A12: Z c= dom f by A8, XBOOLE_1:18; then A13: f is_differentiable_on Z by A5, FDIFF_1:23; then A14: f + (exp_R * f1) is_differentiable_on Z by A7, A11, FDIFF_1:18; A15: for x being Real st x in Z holds (f + (exp_R * f1)) . x > 0 proof let x be Real; ::_thesis: ( x in Z implies (f + (exp_R * f1)) . x > 0 ) assume A16: x in Z ; ::_thesis: (f + (exp_R * f1)) . x > 0 then (f + (exp_R * f1)) . x = (f . x) + ((exp_R * f1) . x) by A7, VALUED_1:def_1 .= (f . x) + (exp_R . (f1 . x)) by A9, A16, FUNCT_1:12 .= 1 + (exp_R . (f1 . x)) by A2, A16 .= 1 + (exp_R . (x * (log (number_e,a)))) by A2, A16 ; hence (f + (exp_R * f1)) . x > 0 by SIN_COS:54, XREAL_1:34; ::_thesis: verum end; now__::_thesis:_for_x_being_Real_st_x_in_Z_holds_ (#R_(3_/_2))_*_(f_+_(exp_R_*_f1))_is_differentiable_in_x let x be Real; ::_thesis: ( x in Z implies (#R (3 / 2)) * (f + (exp_R * f1)) is_differentiable_in x ) assume x in Z ; ::_thesis: (#R (3 / 2)) * (f + (exp_R * f1)) is_differentiable_in x then ( f + (exp_R * f1) is_differentiable_in x & (f + (exp_R * f1)) . x > 0 ) by A14, A15, FDIFF_1:9; hence (#R (3 / 2)) * (f + (exp_R * f1)) is_differentiable_in x by TAYLOR_1:22; ::_thesis: verum end; then A17: (#R (3 / 2)) * (f + (exp_R * f1)) is_differentiable_on Z by A6, FDIFF_1:9; A18: log (number_e,a) <> 0 proof A19: number_e <> 1 by TAYLOR_1:11; assume log (number_e,a) = 0 ; ::_thesis: contradiction then log (number_e,a) = log (number_e,1) by SIN_COS2:13, TAYLOR_1:13; then a = number_e to_power (log (number_e,1)) by A3, A19, POWER:def_3, TAYLOR_1:11 .= 1 by A19, POWER:def_3, TAYLOR_1:11 ; hence contradiction by A4; ::_thesis: verum end; for x being Real st x in Z holds (((2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1)))) `| Z) . x = (a #R x) * ((1 + (a #R x)) #R (1 / 2)) proof let x be Real; ::_thesis: ( x in Z implies (((2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1)))) `| Z) . x = (a #R x) * ((1 + (a #R x)) #R (1 / 2)) ) A20: 3 * (log (number_e,a)) <> 0 by A18; assume A21: x in Z ; ::_thesis: (((2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1)))) `| Z) . x = (a #R x) * ((1 + (a #R x)) #R (1 / 2)) then A22: ((f + (exp_R * f1)) `| Z) . x = (diff (f,x)) + (diff ((exp_R * f1),x)) by A7, A13, A11, FDIFF_1:18 .= (diff (f,x)) + (((exp_R * f1) `| Z) . x) by A11, A21, FDIFF_1:def_7 .= ((f `| Z) . x) + (((exp_R * f1) `| Z) . x) by A13, A21, FDIFF_1:def_7 .= 0 + (((exp_R * f1) `| Z) . x) by A12, A5, A21, FDIFF_1:23 .= (a #R x) * (log (number_e,a)) by A3, A10, A9, A21, Th11 ; A23: (f + (exp_R * f1)) . x = (f . x) + ((exp_R * f1) . x) by A7, A21, VALUED_1:def_1 .= (f . x) + (exp_R . (f1 . x)) by A9, A21, FUNCT_1:12 .= 1 + (exp_R . (f1 . x)) by A2, A21 .= 1 + (exp_R . (x * (log (number_e,a)))) by A2, A21 .= 1 + (a #R x) by A3, Th1 ; ( f + (exp_R * f1) is_differentiable_in x & (f + (exp_R * f1)) . x > 0 ) by A14, A15, A21, FDIFF_1:9; then diff (((#R (3 / 2)) * (f + (exp_R * f1))),x) = ((3 / 2) * (((f + (exp_R * f1)) . x) #R ((3 / 2) - 1))) * (diff ((f + (exp_R * f1)),x)) by TAYLOR_1:22 .= ((3 / 2) * ((1 + (a #R x)) #R (1 / 2))) * ((a #R x) * (log (number_e,a))) by A14, A21, A23, A22, FDIFF_1:def_7 .= (((3 * (log (number_e,a))) / 2) * (a #R x)) * ((1 + (a #R x)) #R (1 / 2)) ; then (((2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1)))) `| Z) . x = (2 / (3 * (log (number_e,a)))) * ((((3 * (log (number_e,a))) / 2) * (a #R x)) * ((1 + (a #R x)) #R (1 / 2))) by A1, A17, A21, FDIFF_1:20 .= (((2 / (3 * (log (number_e,a)))) * ((3 * (log (number_e,a))) / 2)) * (a #R x)) * ((1 + (a #R x)) #R (1 / 2)) .= (1 * (a #R x)) * ((1 + (a #R x)) #R (1 / 2)) by A20, XCMPLX_1:112 .= (a #R x) * ((1 + (a #R x)) #R (1 / 2)) ; hence (((2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1)))) `| Z) . x = (a #R x) * ((1 + (a #R x)) #R (1 / 2)) ; ::_thesis: verum end; hence ( (2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1))) is_differentiable_on Z & ( for x being Real st x in Z holds (((2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1)))) `| Z) . x = (a #R x) * ((1 + (a #R x)) #R (1 / 2)) ) ) by A1, A17, FDIFF_1:20; ::_thesis: verum end; theorem :: FDIFF_6:38 for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom ((- (1 / 2)) (#) (cos * f)) & ( for x being Real st x in Z holds f . x = 2 * x ) holds ( (- (1 / 2)) (#) (cos * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((- (1 / 2)) (#) (cos * f)) `| Z) . x = sin (2 * x) ) ) proof let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st Z c= dom ((- (1 / 2)) (#) (cos * f)) & ( for x being Real st x in Z holds f . x = 2 * x ) holds ( (- (1 / 2)) (#) (cos * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((- (1 / 2)) (#) (cos * f)) `| Z) . x = sin (2 * x) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom ((- (1 / 2)) (#) (cos * f)) & ( for x being Real st x in Z holds f . x = 2 * x ) implies ( (- (1 / 2)) (#) (cos * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((- (1 / 2)) (#) (cos * f)) `| Z) . x = sin (2 * x) ) ) ) assume that A1: Z c= dom ((- (1 / 2)) (#) (cos * f)) and A2: for x being Real st x in Z holds f . x = 2 * x ; ::_thesis: ( (- (1 / 2)) (#) (cos * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((- (1 / 2)) (#) (cos * f)) `| Z) . x = sin (2 * x) ) ) A3: ( Z c= dom (cos * f) & ( for x being Real st x in Z holds f . x = (2 * x) + 0 ) ) by A1, A2, VALUED_1:def_5; then A4: cos * f is_differentiable_on Z by FDIFF_4:38; for x being Real st x in Z holds (((- (1 / 2)) (#) (cos * f)) `| Z) . x = sin (2 * x) proof let x be Real; ::_thesis: ( x in Z implies (((- (1 / 2)) (#) (cos * f)) `| Z) . x = sin (2 * x) ) assume A5: x in Z ; ::_thesis: (((- (1 / 2)) (#) (cos * f)) `| Z) . x = sin (2 * x) then (((- (1 / 2)) (#) (cos * f)) `| Z) . x = (- (1 / 2)) * (diff ((cos * f),x)) by A1, A4, FDIFF_1:20 .= (- (1 / 2)) * (((cos * f) `| Z) . x) by A4, A5, FDIFF_1:def_7 .= (- (1 / 2)) * (- (2 * (sin . ((2 * x) + 0)))) by A3, A5, FDIFF_4:38 .= sin (2 * x) by SIN_COS:def_17 ; hence (((- (1 / 2)) (#) (cos * f)) `| Z) . x = sin (2 * x) ; ::_thesis: verum end; hence ( (- (1 / 2)) (#) (cos * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((- (1 / 2)) (#) (cos * f)) `| Z) . x = sin (2 * x) ) ) by A1, A4, FDIFF_1:20; ::_thesis: verum end; theorem :: FDIFF_6:39 for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom (2 (#) ((#R (1 / 2)) * (f - cos))) & ( for x being Real st x in Z holds ( f . x = 1 & sin . x > 0 & cos . x < 1 & cos . x > - 1 ) ) holds ( 2 (#) ((#R (1 / 2)) * (f - cos)) is_differentiable_on Z & ( for x being Real st x in Z holds ((2 (#) ((#R (1 / 2)) * (f - cos))) `| Z) . x = (1 + (cos . x)) #R (1 / 2) ) ) proof let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st Z c= dom (2 (#) ((#R (1 / 2)) * (f - cos))) & ( for x being Real st x in Z holds ( f . x = 1 & sin . x > 0 & cos . x < 1 & cos . x > - 1 ) ) holds ( 2 (#) ((#R (1 / 2)) * (f - cos)) is_differentiable_on Z & ( for x being Real st x in Z holds ((2 (#) ((#R (1 / 2)) * (f - cos))) `| Z) . x = (1 + (cos . x)) #R (1 / 2) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (2 (#) ((#R (1 / 2)) * (f - cos))) & ( for x being Real st x in Z holds ( f . x = 1 & sin . x > 0 & cos . x < 1 & cos . x > - 1 ) ) implies ( 2 (#) ((#R (1 / 2)) * (f - cos)) is_differentiable_on Z & ( for x being Real st x in Z holds ((2 (#) ((#R (1 / 2)) * (f - cos))) `| Z) . x = (1 + (cos . x)) #R (1 / 2) ) ) ) assume that A1: Z c= dom (2 (#) ((#R (1 / 2)) * (f - cos))) and A2: for x being Real st x in Z holds ( f . x = 1 & sin . x > 0 & cos . x < 1 & cos . x > - 1 ) ; ::_thesis: ( 2 (#) ((#R (1 / 2)) * (f - cos)) is_differentiable_on Z & ( for x being Real st x in Z holds ((2 (#) ((#R (1 / 2)) * (f - cos))) `| Z) . x = (1 + (cos . x)) #R (1 / 2) ) ) A3: for x being Real st x in Z holds f . x = (0 * x) + 1 by A2; A4: Z c= dom ((#R (1 / 2)) * (f - cos)) by A1, VALUED_1:def_5; then for y being set st y in Z holds y in dom (f - cos) by FUNCT_1:11; then A5: Z c= dom (f - cos) by TARSKI:def_3; then Z c= (dom cos) /\ (dom f) by VALUED_1:12; then A6: Z c= dom f by XBOOLE_1:18; then A7: f is_differentiable_on Z by A3, FDIFF_1:23; A8: cos is_differentiable_on Z by FDIFF_1:26, SIN_COS:67; then A9: f - cos is_differentiable_on Z by A5, A7, FDIFF_1:19; A10: for x being Real st x in Z holds (f - cos) . x > 0 proof let x be Real; ::_thesis: ( x in Z implies (f - cos) . x > 0 ) assume A11: x in Z ; ::_thesis: (f - cos) . x > 0 then cos . x < 1 by A2; then A12: 1 - (cos . x) > 1 - 1 by XREAL_1:15; (f - cos) . x = (f . x) - (cos . x) by A5, A11, VALUED_1:13 .= 1 - (cos . x) by A2, A11 ; hence (f - cos) . x > 0 by A12; ::_thesis: verum end; now__::_thesis:_for_x_being_Real_st_x_in_Z_holds_ (#R_(1_/_2))_*_(f_-_cos)_is_differentiable_in_x let x be Real; ::_thesis: ( x in Z implies (#R (1 / 2)) * (f - cos) is_differentiable_in x ) assume x in Z ; ::_thesis: (#R (1 / 2)) * (f - cos) is_differentiable_in x then ( f - cos is_differentiable_in x & (f - cos) . x > 0 ) by A9, A10, FDIFF_1:9; hence (#R (1 / 2)) * (f - cos) is_differentiable_in x by TAYLOR_1:22; ::_thesis: verum end; then A13: (#R (1 / 2)) * (f - cos) is_differentiable_on Z by A4, FDIFF_1:9; for x being Real st x in Z holds ((2 (#) ((#R (1 / 2)) * (f - cos))) `| Z) . x = (1 + (cos . x)) #R (1 / 2) proof let x be Real; ::_thesis: ( x in Z implies ((2 (#) ((#R (1 / 2)) * (f - cos))) `| Z) . x = (1 + (cos . x)) #R (1 / 2) ) assume A14: x in Z ; ::_thesis: ((2 (#) ((#R (1 / 2)) * (f - cos))) `| Z) . x = (1 + (cos . x)) #R (1 / 2) then A15: diff ((f - cos),x) = ((f - cos) `| Z) . x by A9, FDIFF_1:def_7 .= (diff (f,x)) - (diff (cos,x)) by A5, A7, A8, A14, FDIFF_1:19 .= ((f `| Z) . x) - (diff (cos,x)) by A7, A14, FDIFF_1:def_7 .= 0 - (diff (cos,x)) by A6, A3, A14, FDIFF_1:23 .= 0 - (- (sin . x)) by SIN_COS:63 .= sin . x ; A16: cos . x > - 1 by A2, A14; A17: (f - cos) . x = (f . x) - (cos . x) by A5, A14, VALUED_1:13 .= 1 - (cos . x) by A2, A14 ; A18: ( f - cos is_differentiable_in x & (f - cos) . x > 0 ) by A9, A10, A14, FDIFF_1:9; A19: ( sin . x > 0 & cos . x < 1 ) by A2, A14; ((2 (#) ((#R (1 / 2)) * (f - cos))) `| Z) . x = 2 * (diff (((#R (1 / 2)) * (f - cos)),x)) by A1, A13, A14, FDIFF_1:20 .= 2 * (((1 / 2) * (((f - cos) . x) #R ((1 / 2) - 1))) * (diff ((f - cos),x))) by A18, TAYLOR_1:22 .= (sin . x) * ((1 - (cos . x)) #R (- (1 / 2))) by A17, A15 .= (sin . x) * (1 / ((1 - (cos . x)) #R (1 / 2))) by A10, A14, A17, PREPOWER:76 .= (sin . x) / ((1 - (cos . x)) #R (1 / 2)) by XCMPLX_1:99 .= (1 + (cos . x)) #R (1 / 2) by A19, A16, Lm4 ; hence ((2 (#) ((#R (1 / 2)) * (f - cos))) `| Z) . x = (1 + (cos . x)) #R (1 / 2) ; ::_thesis: verum end; hence ( 2 (#) ((#R (1 / 2)) * (f - cos)) is_differentiable_on Z & ( for x being Real st x in Z holds ((2 (#) ((#R (1 / 2)) * (f - cos))) `| Z) . x = (1 + (cos . x)) #R (1 / 2) ) ) by A1, A13, FDIFF_1:20; ::_thesis: verum end; theorem :: FDIFF_6:40 for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom ((- 2) (#) ((#R (1 / 2)) * (f + cos))) & ( for x being Real st x in Z holds ( f . x = 1 & sin . x > 0 & cos . x < 1 & cos . x > - 1 ) ) holds ( (- 2) (#) ((#R (1 / 2)) * (f + cos)) is_differentiable_on Z & ( for x being Real st x in Z holds (((- 2) (#) ((#R (1 / 2)) * (f + cos))) `| Z) . x = (1 - (cos . x)) #R (1 / 2) ) ) proof let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st Z c= dom ((- 2) (#) ((#R (1 / 2)) * (f + cos))) & ( for x being Real st x in Z holds ( f . x = 1 & sin . x > 0 & cos . x < 1 & cos . x > - 1 ) ) holds ( (- 2) (#) ((#R (1 / 2)) * (f + cos)) is_differentiable_on Z & ( for x being Real st x in Z holds (((- 2) (#) ((#R (1 / 2)) * (f + cos))) `| Z) . x = (1 - (cos . x)) #R (1 / 2) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom ((- 2) (#) ((#R (1 / 2)) * (f + cos))) & ( for x being Real st x in Z holds ( f . x = 1 & sin . x > 0 & cos . x < 1 & cos . x > - 1 ) ) implies ( (- 2) (#) ((#R (1 / 2)) * (f + cos)) is_differentiable_on Z & ( for x being Real st x in Z holds (((- 2) (#) ((#R (1 / 2)) * (f + cos))) `| Z) . x = (1 - (cos . x)) #R (1 / 2) ) ) ) assume that A1: Z c= dom ((- 2) (#) ((#R (1 / 2)) * (f + cos))) and A2: for x being Real st x in Z holds ( f . x = 1 & sin . x > 0 & cos . x < 1 & cos . x > - 1 ) ; ::_thesis: ( (- 2) (#) ((#R (1 / 2)) * (f + cos)) is_differentiable_on Z & ( for x being Real st x in Z holds (((- 2) (#) ((#R (1 / 2)) * (f + cos))) `| Z) . x = (1 - (cos . x)) #R (1 / 2) ) ) A3: for x being Real st x in Z holds f . x = (0 * x) + 1 by A2; A4: Z c= dom ((#R (1 / 2)) * (f + cos)) by A1, VALUED_1:def_5; then for y being set st y in Z holds y in dom (f + cos) by FUNCT_1:11; then A5: Z c= dom (f + cos) by TARSKI:def_3; then Z c= (dom cos) /\ (dom f) by VALUED_1:def_1; then A6: Z c= dom f by XBOOLE_1:18; then A7: f is_differentiable_on Z by A3, FDIFF_1:23; A8: cos is_differentiable_on Z by FDIFF_1:26, SIN_COS:67; then A9: f + cos is_differentiable_on Z by A5, A7, FDIFF_1:18; A10: for x being Real st x in Z holds (f + cos) . x > 0 proof let x be Real; ::_thesis: ( x in Z implies (f + cos) . x > 0 ) assume A11: x in Z ; ::_thesis: (f + cos) . x > 0 then cos . x > - 1 by A2; then A12: 1 + (cos . x) > 1 + (- 1) by XREAL_1:8; (f + cos) . x = (f . x) + (cos . x) by A5, A11, VALUED_1:def_1 .= 1 + (cos . x) by A2, A11 ; hence (f + cos) . x > 0 by A12; ::_thesis: verum end; now__::_thesis:_for_x_being_Real_st_x_in_Z_holds_ (#R_(1_/_2))_*_(f_+_cos)_is_differentiable_in_x let x be Real; ::_thesis: ( x in Z implies (#R (1 / 2)) * (f + cos) is_differentiable_in x ) assume x in Z ; ::_thesis: (#R (1 / 2)) * (f + cos) is_differentiable_in x then ( f + cos is_differentiable_in x & (f + cos) . x > 0 ) by A9, A10, FDIFF_1:9; hence (#R (1 / 2)) * (f + cos) is_differentiable_in x by TAYLOR_1:22; ::_thesis: verum end; then A13: (#R (1 / 2)) * (f + cos) is_differentiable_on Z by A4, FDIFF_1:9; for x being Real st x in Z holds (((- 2) (#) ((#R (1 / 2)) * (f + cos))) `| Z) . x = (1 - (cos . x)) #R (1 / 2) proof let x be Real; ::_thesis: ( x in Z implies (((- 2) (#) ((#R (1 / 2)) * (f + cos))) `| Z) . x = (1 - (cos . x)) #R (1 / 2) ) assume A14: x in Z ; ::_thesis: (((- 2) (#) ((#R (1 / 2)) * (f + cos))) `| Z) . x = (1 - (cos . x)) #R (1 / 2) then A15: diff ((f + cos),x) = ((f + cos) `| Z) . x by A9, FDIFF_1:def_7 .= (diff (f,x)) + (diff (cos,x)) by A5, A7, A8, A14, FDIFF_1:18 .= ((f `| Z) . x) + (diff (cos,x)) by A7, A14, FDIFF_1:def_7 .= 0 + (diff (cos,x)) by A6, A3, A14, FDIFF_1:23 .= - (sin . x) by SIN_COS:63 ; A16: cos . x > - 1 by A2, A14; A17: (f + cos) . x = (f . x) + (cos . x) by A5, A14, VALUED_1:def_1 .= 1 + (cos . x) by A2, A14 ; A18: ( f + cos is_differentiable_in x & (f + cos) . x > 0 ) by A9, A10, A14, FDIFF_1:9; A19: ( sin . x > 0 & cos . x < 1 ) by A2, A14; (((- 2) (#) ((#R (1 / 2)) * (f + cos))) `| Z) . x = (- 2) * (diff (((#R (1 / 2)) * (f + cos)),x)) by A1, A13, A14, FDIFF_1:20 .= (- 2) * (((1 / 2) * (((f + cos) . x) #R ((1 / 2) - 1))) * (diff ((f + cos),x))) by A18, TAYLOR_1:22 .= - (- ((sin . x) * ((1 + (cos . x)) #R (- (1 / 2))))) by A17, A15 .= (sin . x) * (1 / ((1 + (cos . x)) #R (1 / 2))) by A10, A14, A17, PREPOWER:76 .= (sin . x) / ((1 + (cos . x)) #R (1 / 2)) by XCMPLX_1:99 .= (1 - (cos . x)) #R (1 / 2) by A19, A16, Lm5 ; hence (((- 2) (#) ((#R (1 / 2)) * (f + cos))) `| Z) . x = (1 - (cos . x)) #R (1 / 2) ; ::_thesis: verum end; hence ( (- 2) (#) ((#R (1 / 2)) * (f + cos)) is_differentiable_on Z & ( for x being Real st x in Z holds (((- 2) (#) ((#R (1 / 2)) * (f + cos))) `| Z) . x = (1 - (cos . x)) #R (1 / 2) ) ) by A1, A13, FDIFF_1:20; ::_thesis: verum end; Lm6: for Z being open Subset of REAL for f1 being PartFunc of REAL,REAL st Z c= dom (f1 + (2 (#) sin)) & ( for x being Real st x in Z holds f1 . x = 1 ) holds ( f1 + (2 (#) sin) is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 + (2 (#) sin)) `| Z) . x = 2 * (cos . x) ) ) proof let Z be open Subset of REAL; ::_thesis: for f1 being PartFunc of REAL,REAL st Z c= dom (f1 + (2 (#) sin)) & ( for x being Real st x in Z holds f1 . x = 1 ) holds ( f1 + (2 (#) sin) is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 + (2 (#) sin)) `| Z) . x = 2 * (cos . x) ) ) let f1 be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (f1 + (2 (#) sin)) & ( for x being Real st x in Z holds f1 . x = 1 ) implies ( f1 + (2 (#) sin) is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 + (2 (#) sin)) `| Z) . x = 2 * (cos . x) ) ) ) assume that A1: Z c= dom (f1 + (2 (#) sin)) and A2: for x being Real st x in Z holds f1 . x = 1 ; ::_thesis: ( f1 + (2 (#) sin) is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 + (2 (#) sin)) `| Z) . x = 2 * (cos . x) ) ) A3: Z c= (dom f1) /\ (dom (2 (#) sin)) by A1, VALUED_1:def_1; then A4: Z c= dom f1 by XBOOLE_1:18; A5: sin is_differentiable_on Z by FDIFF_1:26, SIN_COS:68; A6: for x being Real st x in Z holds f1 . x = (0 * x) + 1 by A2; then A7: f1 is_differentiable_on Z by A4, FDIFF_1:23; Z c= dom (2 (#) sin) by A3, XBOOLE_1:18; then A8: 2 (#) sin is_differentiable_on Z by A5, FDIFF_1:20; for x being Real st x in Z holds ((f1 + (2 (#) sin)) `| Z) . x = 2 * (cos . x) proof let x be Real; ::_thesis: ( x in Z implies ((f1 + (2 (#) sin)) `| Z) . x = 2 * (cos . x) ) A9: sin is_differentiable_in x by SIN_COS:64; assume A10: x in Z ; ::_thesis: ((f1 + (2 (#) sin)) `| Z) . x = 2 * (cos . x) then ((f1 + (2 (#) sin)) `| Z) . x = (diff (f1,x)) + (diff ((2 (#) sin),x)) by A1, A7, A8, FDIFF_1:18 .= ((f1 `| Z) . x) + (diff ((2 (#) sin),x)) by A7, A10, FDIFF_1:def_7 .= ((f1 `| Z) . x) + (2 * (diff (sin,x))) by A9, FDIFF_1:15 .= 0 + (2 * (diff (sin,x))) by A4, A6, A10, FDIFF_1:23 .= 2 * (cos . x) by SIN_COS:64 ; hence ((f1 + (2 (#) sin)) `| Z) . x = 2 * (cos . x) ; ::_thesis: verum end; hence ( f1 + (2 (#) sin) is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 + (2 (#) sin)) `| Z) . x = 2 * (cos . x) ) ) by A1, A7, A8, FDIFF_1:18; ::_thesis: verum end; theorem :: FDIFF_6:41 for Z being open Subset of REAL for f, f1 being PartFunc of REAL,REAL st Z c= dom ((1 / 2) (#) (ln * f)) & f = f1 + (2 (#) sin) & ( for x being Real st x in Z holds ( f1 . x = 1 & f . x > 0 ) ) holds ( (1 / 2) (#) (ln * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) (ln * f)) `| Z) . x = (cos . x) / (1 + (2 * (sin . x))) ) ) proof let Z be open Subset of REAL; ::_thesis: for f, f1 being PartFunc of REAL,REAL st Z c= dom ((1 / 2) (#) (ln * f)) & f = f1 + (2 (#) sin) & ( for x being Real st x in Z holds ( f1 . x = 1 & f . x > 0 ) ) holds ( (1 / 2) (#) (ln * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) (ln * f)) `| Z) . x = (cos . x) / (1 + (2 * (sin . x))) ) ) let f, f1 be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom ((1 / 2) (#) (ln * f)) & f = f1 + (2 (#) sin) & ( for x being Real st x in Z holds ( f1 . x = 1 & f . x > 0 ) ) implies ( (1 / 2) (#) (ln * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) (ln * f)) `| Z) . x = (cos . x) / (1 + (2 * (sin . x))) ) ) ) assume that A1: Z c= dom ((1 / 2) (#) (ln * f)) and A2: f = f1 + (2 (#) sin) and A3: for x being Real st x in Z holds ( f1 . x = 1 & f . x > 0 ) ; ::_thesis: ( (1 / 2) (#) (ln * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) (ln * f)) `| Z) . x = (cos . x) / (1 + (2 * (sin . x))) ) ) A4: Z c= dom (ln * f) by A1, VALUED_1:def_5; then for y being set st y in Z holds y in dom f by FUNCT_1:11; then A5: Z c= dom (f1 + (2 (#) sin)) by A2, TARSKI:def_3; A6: for x being Real st x in Z holds f1 . x = 1 by A3; then A7: f is_differentiable_on Z by A2, A5, Lm6; for x being Real st x in Z holds ln * f is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies ln * f is_differentiable_in x ) assume x in Z ; ::_thesis: ln * f is_differentiable_in x then ( f is_differentiable_in x & f . x > 0 ) by A3, A7, FDIFF_1:9; hence ln * f is_differentiable_in x by TAYLOR_1:20; ::_thesis: verum end; then A8: ln * f is_differentiable_on Z by A4, FDIFF_1:9; Z c= (dom f1) /\ (dom (2 (#) sin)) by A5, VALUED_1:def_1; then A9: Z c= dom (2 (#) sin) by XBOOLE_1:18; for x being Real st x in Z holds (((1 / 2) (#) (ln * f)) `| Z) . x = (cos . x) / (1 + (2 * (sin . x))) proof let x be Real; ::_thesis: ( x in Z implies (((1 / 2) (#) (ln * f)) `| Z) . x = (cos . x) / (1 + (2 * (sin . x))) ) assume A10: x in Z ; ::_thesis: (((1 / 2) (#) (ln * f)) `| Z) . x = (cos . x) / (1 + (2 * (sin . x))) then A11: f . x = (f1 . x) + ((2 (#) sin) . x) by A2, A5, VALUED_1:def_1 .= 1 + ((2 (#) sin) . x) by A3, A10 .= 1 + (2 * (sin . x)) by A9, A10, VALUED_1:def_5 ; A12: ( f is_differentiable_in x & f . x > 0 ) by A3, A7, A10, FDIFF_1:9; (((1 / 2) (#) (ln * f)) `| Z) . x = (1 / 2) * (diff ((ln * f),x)) by A1, A8, A10, FDIFF_1:20 .= (1 / 2) * ((diff (f,x)) / (f . x)) by A12, TAYLOR_1:20 .= (1 / 2) * (((f `| Z) . x) / (f . x)) by A7, A10, FDIFF_1:def_7 .= (1 / 2) * ((2 * (cos . x)) / (f . x)) by A2, A6, A5, A10, Lm6 .= ((1 / 2) * (2 * (cos . x))) / (f . x) by XCMPLX_1:74 .= (cos . x) / (1 + (2 * (sin . x))) by A11 ; hence (((1 / 2) (#) (ln * f)) `| Z) . x = (cos . x) / (1 + (2 * (sin . x))) ; ::_thesis: verum end; hence ( (1 / 2) (#) (ln * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) (ln * f)) `| Z) . x = (cos . x) / (1 + (2 * (sin . x))) ) ) by A1, A8, FDIFF_1:20; ::_thesis: verum end; Lm7: for Z being open Subset of REAL for f1 being PartFunc of REAL,REAL st Z c= dom (f1 + (2 (#) cos)) & ( for x being Real st x in Z holds f1 . x = 1 ) holds ( f1 + (2 (#) cos) is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 + (2 (#) cos)) `| Z) . x = - (2 * (sin . x)) ) ) proof let Z be open Subset of REAL; ::_thesis: for f1 being PartFunc of REAL,REAL st Z c= dom (f1 + (2 (#) cos)) & ( for x being Real st x in Z holds f1 . x = 1 ) holds ( f1 + (2 (#) cos) is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 + (2 (#) cos)) `| Z) . x = - (2 * (sin . x)) ) ) let f1 be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (f1 + (2 (#) cos)) & ( for x being Real st x in Z holds f1 . x = 1 ) implies ( f1 + (2 (#) cos) is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 + (2 (#) cos)) `| Z) . x = - (2 * (sin . x)) ) ) ) assume that A1: Z c= dom (f1 + (2 (#) cos)) and A2: for x being Real st x in Z holds f1 . x = 1 ; ::_thesis: ( f1 + (2 (#) cos) is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 + (2 (#) cos)) `| Z) . x = - (2 * (sin . x)) ) ) A3: Z c= (dom f1) /\ (dom (2 (#) cos)) by A1, VALUED_1:def_1; then A4: Z c= dom f1 by XBOOLE_1:18; A5: cos is_differentiable_on Z by FDIFF_1:26, SIN_COS:67; A6: for x being Real st x in Z holds f1 . x = (0 * x) + 1 by A2; then A7: f1 is_differentiable_on Z by A4, FDIFF_1:23; Z c= dom (2 (#) cos) by A3, XBOOLE_1:18; then A8: 2 (#) cos is_differentiable_on Z by A5, FDIFF_1:20; for x being Real st x in Z holds ((f1 + (2 (#) cos)) `| Z) . x = - (2 * (sin . x)) proof let x be Real; ::_thesis: ( x in Z implies ((f1 + (2 (#) cos)) `| Z) . x = - (2 * (sin . x)) ) A9: cos is_differentiable_in x by SIN_COS:63; assume A10: x in Z ; ::_thesis: ((f1 + (2 (#) cos)) `| Z) . x = - (2 * (sin . x)) then ((f1 + (2 (#) cos)) `| Z) . x = (diff (f1,x)) + (diff ((2 (#) cos),x)) by A1, A7, A8, FDIFF_1:18 .= ((f1 `| Z) . x) + (diff ((2 (#) cos),x)) by A7, A10, FDIFF_1:def_7 .= ((f1 `| Z) . x) + (2 * (diff (cos,x))) by A9, FDIFF_1:15 .= 0 + (2 * (diff (cos,x))) by A4, A6, A10, FDIFF_1:23 .= 0 + (2 * (- (sin . x))) by SIN_COS:63 .= - (2 * (sin . x)) ; hence ((f1 + (2 (#) cos)) `| Z) . x = - (2 * (sin . x)) ; ::_thesis: verum end; hence ( f1 + (2 (#) cos) is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 + (2 (#) cos)) `| Z) . x = - (2 * (sin . x)) ) ) by A1, A7, A8, FDIFF_1:18; ::_thesis: verum end; theorem :: FDIFF_6:42 for Z being open Subset of REAL for f, f1 being PartFunc of REAL,REAL st Z c= dom ((- (1 / 2)) (#) (ln * f)) & f = f1 + (2 (#) cos) & ( for x being Real st x in Z holds ( f1 . x = 1 & f . x > 0 ) ) holds ( (- (1 / 2)) (#) (ln * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((- (1 / 2)) (#) (ln * f)) `| Z) . x = (sin . x) / (1 + (2 * (cos . x))) ) ) proof let Z be open Subset of REAL; ::_thesis: for f, f1 being PartFunc of REAL,REAL st Z c= dom ((- (1 / 2)) (#) (ln * f)) & f = f1 + (2 (#) cos) & ( for x being Real st x in Z holds ( f1 . x = 1 & f . x > 0 ) ) holds ( (- (1 / 2)) (#) (ln * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((- (1 / 2)) (#) (ln * f)) `| Z) . x = (sin . x) / (1 + (2 * (cos . x))) ) ) let f, f1 be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom ((- (1 / 2)) (#) (ln * f)) & f = f1 + (2 (#) cos) & ( for x being Real st x in Z holds ( f1 . x = 1 & f . x > 0 ) ) implies ( (- (1 / 2)) (#) (ln * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((- (1 / 2)) (#) (ln * f)) `| Z) . x = (sin . x) / (1 + (2 * (cos . x))) ) ) ) assume that A1: Z c= dom ((- (1 / 2)) (#) (ln * f)) and A2: f = f1 + (2 (#) cos) and A3: for x being Real st x in Z holds ( f1 . x = 1 & f . x > 0 ) ; ::_thesis: ( (- (1 / 2)) (#) (ln * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((- (1 / 2)) (#) (ln * f)) `| Z) . x = (sin . x) / (1 + (2 * (cos . x))) ) ) A4: Z c= dom (ln * f) by A1, VALUED_1:def_5; then for y being set st y in Z holds y in dom f by FUNCT_1:11; then A5: Z c= dom (f1 + (2 (#) cos)) by A2, TARSKI:def_3; A6: for x being Real st x in Z holds f1 . x = 1 by A3; then A7: f is_differentiable_on Z by A2, A5, Lm7; for x being Real st x in Z holds ln * f is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies ln * f is_differentiable_in x ) assume x in Z ; ::_thesis: ln * f is_differentiable_in x then ( f is_differentiable_in x & f . x > 0 ) by A3, A7, FDIFF_1:9; hence ln * f is_differentiable_in x by TAYLOR_1:20; ::_thesis: verum end; then A8: ln * f is_differentiable_on Z by A4, FDIFF_1:9; Z c= (dom f1) /\ (dom (2 (#) cos)) by A5, VALUED_1:def_1; then A9: Z c= dom (2 (#) cos) by XBOOLE_1:18; for x being Real st x in Z holds (((- (1 / 2)) (#) (ln * f)) `| Z) . x = (sin . x) / (1 + (2 * (cos . x))) proof let x be Real; ::_thesis: ( x in Z implies (((- (1 / 2)) (#) (ln * f)) `| Z) . x = (sin . x) / (1 + (2 * (cos . x))) ) assume A10: x in Z ; ::_thesis: (((- (1 / 2)) (#) (ln * f)) `| Z) . x = (sin . x) / (1 + (2 * (cos . x))) then A11: f . x = (f1 . x) + ((2 (#) cos) . x) by A2, A5, VALUED_1:def_1 .= 1 + ((2 (#) cos) . x) by A3, A10 .= 1 + (2 * (cos . x)) by A9, A10, VALUED_1:def_5 ; A12: ( f is_differentiable_in x & f . x > 0 ) by A3, A7, A10, FDIFF_1:9; (((- (1 / 2)) (#) (ln * f)) `| Z) . x = (- (1 / 2)) * (diff ((ln * f),x)) by A1, A8, A10, FDIFF_1:20 .= (- (1 / 2)) * ((diff (f,x)) / (f . x)) by A12, TAYLOR_1:20 .= (- (1 / 2)) * (((f `| Z) . x) / (f . x)) by A7, A10, FDIFF_1:def_7 .= (- (1 / 2)) * ((- (2 * (sin . x))) / (f . x)) by A2, A6, A5, A10, Lm7 .= ((- (1 / 2)) * (- (2 * (sin . x)))) / (f . x) by XCMPLX_1:74 .= (sin . x) / (1 + (2 * (cos . x))) by A11 ; hence (((- (1 / 2)) (#) (ln * f)) `| Z) . x = (sin . x) / (1 + (2 * (cos . x))) ; ::_thesis: verum end; hence ( (- (1 / 2)) (#) (ln * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((- (1 / 2)) (#) (ln * f)) `| Z) . x = (sin . x) / (1 + (2 * (cos . x))) ) ) by A1, A8, FDIFF_1:20; ::_thesis: verum end; theorem Th43: :: FDIFF_6:43 for a being Real for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom ((1 / (4 * a)) (#) (sin * f)) & ( for x being Real st x in Z holds f . x = (2 * a) * x ) & a <> 0 holds ( (1 / (4 * a)) (#) (sin * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / (4 * a)) (#) (sin * f)) `| Z) . x = (1 / 2) * (cos ((2 * a) * x)) ) ) proof let a be Real; ::_thesis: for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom ((1 / (4 * a)) (#) (sin * f)) & ( for x being Real st x in Z holds f . x = (2 * a) * x ) & a <> 0 holds ( (1 / (4 * a)) (#) (sin * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / (4 * a)) (#) (sin * f)) `| Z) . x = (1 / 2) * (cos ((2 * a) * x)) ) ) let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st Z c= dom ((1 / (4 * a)) (#) (sin * f)) & ( for x being Real st x in Z holds f . x = (2 * a) * x ) & a <> 0 holds ( (1 / (4 * a)) (#) (sin * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / (4 * a)) (#) (sin * f)) `| Z) . x = (1 / 2) * (cos ((2 * a) * x)) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom ((1 / (4 * a)) (#) (sin * f)) & ( for x being Real st x in Z holds f . x = (2 * a) * x ) & a <> 0 implies ( (1 / (4 * a)) (#) (sin * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / (4 * a)) (#) (sin * f)) `| Z) . x = (1 / 2) * (cos ((2 * a) * x)) ) ) ) assume that A1: Z c= dom ((1 / (4 * a)) (#) (sin * f)) and A2: for x being Real st x in Z holds f . x = (2 * a) * x and A3: a <> 0 ; ::_thesis: ( (1 / (4 * a)) (#) (sin * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / (4 * a)) (#) (sin * f)) `| Z) . x = (1 / 2) * (cos ((2 * a) * x)) ) ) A4: ( Z c= dom (sin * f) & ( for x being Real st x in Z holds f . x = ((2 * a) * x) + 0 ) ) by A1, A2, VALUED_1:def_5; then A5: sin * f is_differentiable_on Z by FDIFF_4:37; for x being Real st x in Z holds (((1 / (4 * a)) (#) (sin * f)) `| Z) . x = (1 / 2) * (cos ((2 * a) * x)) proof let x be Real; ::_thesis: ( x in Z implies (((1 / (4 * a)) (#) (sin * f)) `| Z) . x = (1 / 2) * (cos ((2 * a) * x)) ) assume A6: x in Z ; ::_thesis: (((1 / (4 * a)) (#) (sin * f)) `| Z) . x = (1 / 2) * (cos ((2 * a) * x)) then (((1 / (4 * a)) (#) (sin * f)) `| Z) . x = (1 / (4 * a)) * (diff ((sin * f),x)) by A1, A5, FDIFF_1:20 .= (1 / (4 * a)) * (((sin * f) `| Z) . x) by A5, A6, FDIFF_1:def_7 .= (1 / (4 * a)) * ((2 * a) * (cos . (((2 * a) * x) + 0))) by A4, A6, FDIFF_4:37 .= ((1 / (4 * a)) * (2 * a)) * (cos . (((2 * a) * x) + 0)) .= (((1 / 4) * (1 / a)) * (2 * a)) * (cos . ((2 * a) * x)) by XCMPLX_1:102 .= (((1 / 4) * 2) * ((1 / a) * a)) * (cos . ((2 * a) * x)) .= ((1 / 2) * 1) * (cos . ((2 * a) * x)) by A3, XCMPLX_1:106 .= (1 / 2) * (cos ((2 * a) * x)) by SIN_COS:def_19 ; hence (((1 / (4 * a)) (#) (sin * f)) `| Z) . x = (1 / 2) * (cos ((2 * a) * x)) ; ::_thesis: verum end; hence ( (1 / (4 * a)) (#) (sin * f) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / (4 * a)) (#) (sin * f)) `| Z) . x = (1 / 2) * (cos ((2 * a) * x)) ) ) by A1, A5, FDIFF_1:20; ::_thesis: verum end; theorem :: FDIFF_6:44 for a being Real for Z being open Subset of REAL for f1, f being PartFunc of REAL,REAL st Z c= dom (f1 - ((1 / (4 * a)) (#) (sin * f))) & ( for x being Real st x in Z holds ( f1 . x = x / 2 & f . x = (2 * a) * x ) ) & a <> 0 holds ( f1 - ((1 / (4 * a)) (#) (sin * f)) is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 - ((1 / (4 * a)) (#) (sin * f))) `| Z) . x = (sin (a * x)) ^2 ) ) proof let a be Real; ::_thesis: for Z being open Subset of REAL for f1, f being PartFunc of REAL,REAL st Z c= dom (f1 - ((1 / (4 * a)) (#) (sin * f))) & ( for x being Real st x in Z holds ( f1 . x = x / 2 & f . x = (2 * a) * x ) ) & a <> 0 holds ( f1 - ((1 / (4 * a)) (#) (sin * f)) is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 - ((1 / (4 * a)) (#) (sin * f))) `| Z) . x = (sin (a * x)) ^2 ) ) let Z be open Subset of REAL; ::_thesis: for f1, f being PartFunc of REAL,REAL st Z c= dom (f1 - ((1 / (4 * a)) (#) (sin * f))) & ( for x being Real st x in Z holds ( f1 . x = x / 2 & f . x = (2 * a) * x ) ) & a <> 0 holds ( f1 - ((1 / (4 * a)) (#) (sin * f)) is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 - ((1 / (4 * a)) (#) (sin * f))) `| Z) . x = (sin (a * x)) ^2 ) ) let f1, f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (f1 - ((1 / (4 * a)) (#) (sin * f))) & ( for x being Real st x in Z holds ( f1 . x = x / 2 & f . x = (2 * a) * x ) ) & a <> 0 implies ( f1 - ((1 / (4 * a)) (#) (sin * f)) is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 - ((1 / (4 * a)) (#) (sin * f))) `| Z) . x = (sin (a * x)) ^2 ) ) ) assume that A1: Z c= dom (f1 - ((1 / (4 * a)) (#) (sin * f))) and A2: for x being Real st x in Z holds ( f1 . x = x / 2 & f . x = (2 * a) * x ) and A3: a <> 0 ; ::_thesis: ( f1 - ((1 / (4 * a)) (#) (sin * f)) is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 - ((1 / (4 * a)) (#) (sin * f))) `| Z) . x = (sin (a * x)) ^2 ) ) A4: Z c= (dom f1) /\ (dom ((1 / (4 * a)) (#) (sin * f))) by A1, VALUED_1:12; then A5: Z c= dom ((1 / (4 * a)) (#) (sin * f)) by XBOOLE_1:18; A6: for x being Real st x in Z holds f1 . x = ((1 / 2) * x) + 0 proof let x be Real; ::_thesis: ( x in Z implies f1 . x = ((1 / 2) * x) + 0 ) assume x in Z ; ::_thesis: f1 . x = ((1 / 2) * x) + 0 then f1 . x = x / 2 by A2 .= ((1 / 2) * x) + 0 ; hence f1 . x = ((1 / 2) * x) + 0 ; ::_thesis: verum end; A7: for x being Real st x in Z holds f . x = (2 * a) * x by A2; then A8: (1 / (4 * a)) (#) (sin * f) is_differentiable_on Z by A3, A5, Th43; A9: Z c= dom f1 by A4, XBOOLE_1:18; then A10: f1 is_differentiable_on Z by A6, FDIFF_1:23; for x being Real st x in Z holds ((f1 - ((1 / (4 * a)) (#) (sin * f))) `| Z) . x = (sin (a * x)) ^2 proof let x be Real; ::_thesis: ( x in Z implies ((f1 - ((1 / (4 * a)) (#) (sin * f))) `| Z) . x = (sin (a * x)) ^2 ) assume A11: x in Z ; ::_thesis: ((f1 - ((1 / (4 * a)) (#) (sin * f))) `| Z) . x = (sin (a * x)) ^2 then ((f1 - ((1 / (4 * a)) (#) (sin * f))) `| Z) . x = (diff (f1,x)) - (diff (((1 / (4 * a)) (#) (sin * f)),x)) by A1, A8, A10, FDIFF_1:19 .= ((f1 `| Z) . x) - (diff (((1 / (4 * a)) (#) (sin * f)),x)) by A10, A11, FDIFF_1:def_7 .= ((f1 `| Z) . x) - ((((1 / (4 * a)) (#) (sin * f)) `| Z) . x) by A8, A11, FDIFF_1:def_7 .= ((f1 `| Z) . x) - ((1 / 2) * (cos ((2 * a) * x))) by A3, A7, A5, A11, Th43 .= (1 / 2) - ((1 / 2) * (cos ((2 * a) * x))) by A9, A6, A11, FDIFF_1:23 .= (1 / 2) * (1 - (cos (2 * (a * x)))) .= (1 / 2) * (2 * ((sin (a * x)) ^2)) by Lm1 .= (sin (a * x)) ^2 ; hence ((f1 - ((1 / (4 * a)) (#) (sin * f))) `| Z) . x = (sin (a * x)) ^2 ; ::_thesis: verum end; hence ( f1 - ((1 / (4 * a)) (#) (sin * f)) is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 - ((1 / (4 * a)) (#) (sin * f))) `| Z) . x = (sin (a * x)) ^2 ) ) by A1, A8, A10, FDIFF_1:19; ::_thesis: verum end; theorem :: FDIFF_6:45 for a being Real for Z being open Subset of REAL for f1, f being PartFunc of REAL,REAL st Z c= dom (f1 + ((1 / (4 * a)) (#) (sin * f))) & ( for x being Real st x in Z holds ( f1 . x = x / 2 & f . x = (2 * a) * x ) ) & a <> 0 holds ( f1 + ((1 / (4 * a)) (#) (sin * f)) is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 + ((1 / (4 * a)) (#) (sin * f))) `| Z) . x = (cos (a * x)) ^2 ) ) proof let a be Real; ::_thesis: for Z being open Subset of REAL for f1, f being PartFunc of REAL,REAL st Z c= dom (f1 + ((1 / (4 * a)) (#) (sin * f))) & ( for x being Real st x in Z holds ( f1 . x = x / 2 & f . x = (2 * a) * x ) ) & a <> 0 holds ( f1 + ((1 / (4 * a)) (#) (sin * f)) is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 + ((1 / (4 * a)) (#) (sin * f))) `| Z) . x = (cos (a * x)) ^2 ) ) let Z be open Subset of REAL; ::_thesis: for f1, f being PartFunc of REAL,REAL st Z c= dom (f1 + ((1 / (4 * a)) (#) (sin * f))) & ( for x being Real st x in Z holds ( f1 . x = x / 2 & f . x = (2 * a) * x ) ) & a <> 0 holds ( f1 + ((1 / (4 * a)) (#) (sin * f)) is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 + ((1 / (4 * a)) (#) (sin * f))) `| Z) . x = (cos (a * x)) ^2 ) ) let f1, f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (f1 + ((1 / (4 * a)) (#) (sin * f))) & ( for x being Real st x in Z holds ( f1 . x = x / 2 & f . x = (2 * a) * x ) ) & a <> 0 implies ( f1 + ((1 / (4 * a)) (#) (sin * f)) is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 + ((1 / (4 * a)) (#) (sin * f))) `| Z) . x = (cos (a * x)) ^2 ) ) ) assume that A1: Z c= dom (f1 + ((1 / (4 * a)) (#) (sin * f))) and A2: for x being Real st x in Z holds ( f1 . x = x / 2 & f . x = (2 * a) * x ) and A3: a <> 0 ; ::_thesis: ( f1 + ((1 / (4 * a)) (#) (sin * f)) is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 + ((1 / (4 * a)) (#) (sin * f))) `| Z) . x = (cos (a * x)) ^2 ) ) A4: Z c= (dom f1) /\ (dom ((1 / (4 * a)) (#) (sin * f))) by A1, VALUED_1:def_1; then A5: Z c= dom ((1 / (4 * a)) (#) (sin * f)) by XBOOLE_1:18; A6: for x being Real st x in Z holds f1 . x = ((1 / 2) * x) + 0 proof let x be Real; ::_thesis: ( x in Z implies f1 . x = ((1 / 2) * x) + 0 ) assume x in Z ; ::_thesis: f1 . x = ((1 / 2) * x) + 0 then f1 . x = x / 2 by A2 .= ((1 / 2) * x) + 0 ; hence f1 . x = ((1 / 2) * x) + 0 ; ::_thesis: verum end; A7: for x being Real st x in Z holds f . x = (2 * a) * x by A2; then A8: (1 / (4 * a)) (#) (sin * f) is_differentiable_on Z by A3, A5, Th43; A9: Z c= dom f1 by A4, XBOOLE_1:18; then A10: f1 is_differentiable_on Z by A6, FDIFF_1:23; for x being Real st x in Z holds ((f1 + ((1 / (4 * a)) (#) (sin * f))) `| Z) . x = (cos (a * x)) ^2 proof let x be Real; ::_thesis: ( x in Z implies ((f1 + ((1 / (4 * a)) (#) (sin * f))) `| Z) . x = (cos (a * x)) ^2 ) assume A11: x in Z ; ::_thesis: ((f1 + ((1 / (4 * a)) (#) (sin * f))) `| Z) . x = (cos (a * x)) ^2 then ((f1 + ((1 / (4 * a)) (#) (sin * f))) `| Z) . x = (diff (f1,x)) + (diff (((1 / (4 * a)) (#) (sin * f)),x)) by A1, A8, A10, FDIFF_1:18 .= ((f1 `| Z) . x) + (diff (((1 / (4 * a)) (#) (sin * f)),x)) by A10, A11, FDIFF_1:def_7 .= ((f1 `| Z) . x) + ((((1 / (4 * a)) (#) (sin * f)) `| Z) . x) by A8, A11, FDIFF_1:def_7 .= ((f1 `| Z) . x) + ((1 / 2) * (cos ((2 * a) * x))) by A3, A7, A5, A11, Th43 .= (1 / 2) + ((1 / 2) * (cos ((2 * a) * x))) by A9, A6, A11, FDIFF_1:23 .= (1 / 2) * (1 + (cos (2 * (a * x)))) .= (1 / 2) * (2 * ((cos (a * x)) ^2)) by Lm2 .= (cos (a * x)) ^2 ; hence ((f1 + ((1 / (4 * a)) (#) (sin * f))) `| Z) . x = (cos (a * x)) ^2 ; ::_thesis: verum end; hence ( f1 + ((1 / (4 * a)) (#) (sin * f)) is_differentiable_on Z & ( for x being Real st x in Z holds ((f1 + ((1 / (4 * a)) (#) (sin * f))) `| Z) . x = (cos (a * x)) ^2 ) ) by A1, A8, A10, FDIFF_1:18; ::_thesis: verum end; theorem Th46: :: FDIFF_6:46 for n being Element of NAT for Z being open Subset of REAL st Z c= dom ((1 / n) (#) ((#Z n) * cos)) & n > 0 holds ( (1 / n) (#) ((#Z n) * cos) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / n) (#) ((#Z n) * cos)) `| Z) . x = - (((cos . x) #Z (n - 1)) * (sin . x)) ) ) proof let n be Element of NAT ; ::_thesis: for Z being open Subset of REAL st Z c= dom ((1 / n) (#) ((#Z n) * cos)) & n > 0 holds ( (1 / n) (#) ((#Z n) * cos) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / n) (#) ((#Z n) * cos)) `| Z) . x = - (((cos . x) #Z (n - 1)) * (sin . x)) ) ) let Z be open Subset of REAL; ::_thesis: ( Z c= dom ((1 / n) (#) ((#Z n) * cos)) & n > 0 implies ( (1 / n) (#) ((#Z n) * cos) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / n) (#) ((#Z n) * cos)) `| Z) . x = - (((cos . x) #Z (n - 1)) * (sin . x)) ) ) ) assume that A1: Z c= dom ((1 / n) (#) ((#Z n) * cos)) and A2: n > 0 ; ::_thesis: ( (1 / n) (#) ((#Z n) * cos) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / n) (#) ((#Z n) * cos)) `| Z) . x = - (((cos . x) #Z (n - 1)) * (sin . x)) ) ) A3: now__::_thesis:_for_x_being_Real_st_x_in_Z_holds_ (#Z_n)_*_cos_is_differentiable_in_x let x be Real; ::_thesis: ( x in Z implies (#Z n) * cos is_differentiable_in x ) assume x in Z ; ::_thesis: (#Z n) * cos is_differentiable_in x cos is_differentiable_in x by SIN_COS:63; hence (#Z n) * cos is_differentiable_in x by TAYLOR_1:3; ::_thesis: verum end; Z c= dom ((#Z n) * cos) by A1, VALUED_1:def_5; then A4: (#Z n) * cos is_differentiable_on Z by A3, FDIFF_1:9; for x being Real st x in Z holds (((1 / n) (#) ((#Z n) * cos)) `| Z) . x = - (((cos . x) #Z (n - 1)) * (sin . x)) proof let x be Real; ::_thesis: ( x in Z implies (((1 / n) (#) ((#Z n) * cos)) `| Z) . x = - (((cos . x) #Z (n - 1)) * (sin . x)) ) A5: cos is_differentiable_in x by SIN_COS:63; assume x in Z ; ::_thesis: (((1 / n) (#) ((#Z n) * cos)) `| Z) . x = - (((cos . x) #Z (n - 1)) * (sin . x)) then (((1 / n) (#) ((#Z n) * cos)) `| Z) . x = (1 / n) * (diff (((#Z n) * cos),x)) by A1, A4, FDIFF_1:20 .= (1 / n) * ((n * ((cos . x) #Z (n - 1))) * (diff (cos,x))) by A5, TAYLOR_1:3 .= (1 / n) * ((n * ((cos . x) #Z (n - 1))) * (- (sin . x))) by SIN_COS:63 .= (((1 / n) * n) * ((cos . x) #Z (n - 1))) * (- (sin . x)) .= (((n ") * n) * ((cos . x) #Z (n - 1))) * (- (sin . x)) by XCMPLX_1:215 .= (1 * ((cos . x) #Z (n - 1))) * (- (sin . x)) by A2, XCMPLX_0:def_7 .= - (((cos . x) #Z (n - 1)) * (sin . x)) ; hence (((1 / n) (#) ((#Z n) * cos)) `| Z) . x = - (((cos . x) #Z (n - 1)) * (sin . x)) ; ::_thesis: verum end; hence ( (1 / n) (#) ((#Z n) * cos) is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / n) (#) ((#Z n) * cos)) `| Z) . x = - (((cos . x) #Z (n - 1)) * (sin . x)) ) ) by A1, A4, FDIFF_1:20; ::_thesis: verum end; theorem :: FDIFF_6:47 for Z being open Subset of REAL st Z c= dom (((1 / 3) (#) ((#Z 3) * cos)) - cos) holds ( ((1 / 3) (#) ((#Z 3) * cos)) - cos is_differentiable_on Z & ( for x being Real st x in Z holds ((((1 / 3) (#) ((#Z 3) * cos)) - cos) `| Z) . x = (sin . x) |^ 3 ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (((1 / 3) (#) ((#Z 3) * cos)) - cos) implies ( ((1 / 3) (#) ((#Z 3) * cos)) - cos is_differentiable_on Z & ( for x being Real st x in Z holds ((((1 / 3) (#) ((#Z 3) * cos)) - cos) `| Z) . x = (sin . x) |^ 3 ) ) ) assume A1: Z c= dom (((1 / 3) (#) ((#Z 3) * cos)) - cos) ; ::_thesis: ( ((1 / 3) (#) ((#Z 3) * cos)) - cos is_differentiable_on Z & ( for x being Real st x in Z holds ((((1 / 3) (#) ((#Z 3) * cos)) - cos) `| Z) . x = (sin . x) |^ 3 ) ) then Z c= (dom ((1 / 3) (#) ((#Z 3) * cos))) /\ (dom cos) by VALUED_1:12; then A2: Z c= dom ((1 / 3) (#) ((#Z 3) * cos)) by XBOOLE_1:18; then A3: (1 / 3) (#) ((#Z 3) * cos) is_differentiable_on Z by Th46; A4: cos is_differentiable_on Z by FDIFF_1:26, SIN_COS:67; now__::_thesis:_for_x_being_Real_st_x_in_Z_holds_ ((((1_/_3)_(#)_((#Z_3)_*_cos))_-_cos)_`|_Z)_._x_=_(sin_._x)_|^_3 let x be Real; ::_thesis: ( x in Z implies ((((1 / 3) (#) ((#Z 3) * cos)) - cos) `| Z) . x = (sin . x) |^ 3 ) assume A5: x in Z ; ::_thesis: ((((1 / 3) (#) ((#Z 3) * cos)) - cos) `| Z) . x = (sin . x) |^ 3 then ((((1 / 3) (#) ((#Z 3) * cos)) - cos) `| Z) . x = (diff (((1 / 3) (#) ((#Z 3) * cos)),x)) - (diff (cos,x)) by A1, A3, A4, FDIFF_1:19 .= (diff (((1 / 3) (#) ((#Z 3) * cos)),x)) - (- (sin . x)) by SIN_COS:63 .= ((((1 / 3) (#) ((#Z 3) * cos)) `| Z) . x) - (- (sin . x)) by A3, A5, FDIFF_1:def_7 .= (- (((cos . x) #Z (3 - 1)) * (sin . x))) - (- (sin . x)) by A2, A5, Th46 .= (sin . x) * (1 - ((cos . x) #Z 2)) .= (sin . x) * (1 - ((cos . x) |^ (abs 2))) by PREPOWER:def_3 .= (sin . x) * (1 - ((cos . x) |^ 2)) by ABSVALUE:def_1 .= (sin . x) * (1 - ((cos . x) * (cos . x))) by WSIERP_1:1 .= (sin . x) * ((((cos . x) * (cos . x)) + ((sin . x) * (sin . x))) - ((cos . x) * (cos . x))) by SIN_COS:28 .= (sin . x) * ((sin . x) |^ 2) by WSIERP_1:1 .= (sin . x) |^ (2 + 1) by NEWTON:6 .= (sin . x) |^ 3 ; hence ((((1 / 3) (#) ((#Z 3) * cos)) - cos) `| Z) . x = (sin . x) |^ 3 ; ::_thesis: verum end; hence ( ((1 / 3) (#) ((#Z 3) * cos)) - cos is_differentiable_on Z & ( for x being Real st x in Z holds ((((1 / 3) (#) ((#Z 3) * cos)) - cos) `| Z) . x = (sin . x) |^ 3 ) ) by A1, A3, A4, FDIFF_1:19; ::_thesis: verum end; theorem :: FDIFF_6:48 for Z being open Subset of REAL st Z c= dom (sin - ((1 / 3) (#) ((#Z 3) * sin))) holds ( sin - ((1 / 3) (#) ((#Z 3) * sin)) is_differentiable_on Z & ( for x being Real st x in Z holds ((sin - ((1 / 3) (#) ((#Z 3) * sin))) `| Z) . x = (cos . x) |^ 3 ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (sin - ((1 / 3) (#) ((#Z 3) * sin))) implies ( sin - ((1 / 3) (#) ((#Z 3) * sin)) is_differentiable_on Z & ( for x being Real st x in Z holds ((sin - ((1 / 3) (#) ((#Z 3) * sin))) `| Z) . x = (cos . x) |^ 3 ) ) ) assume A1: Z c= dom (sin - ((1 / 3) (#) ((#Z 3) * sin))) ; ::_thesis: ( sin - ((1 / 3) (#) ((#Z 3) * sin)) is_differentiable_on Z & ( for x being Real st x in Z holds ((sin - ((1 / 3) (#) ((#Z 3) * sin))) `| Z) . x = (cos . x) |^ 3 ) ) then Z c= (dom ((1 / 3) (#) ((#Z 3) * sin))) /\ (dom sin) by VALUED_1:12; then A2: Z c= dom ((1 / 3) (#) ((#Z 3) * sin)) by XBOOLE_1:18; then A3: (1 / 3) (#) ((#Z 3) * sin) is_differentiable_on Z by FDIFF_4:54; A4: sin is_differentiable_on Z by FDIFF_1:26, SIN_COS:68; now__::_thesis:_for_x_being_Real_st_x_in_Z_holds_ ((sin_-_((1_/_3)_(#)_((#Z_3)_*_sin)))_`|_Z)_._x_=_(cos_._x)_|^_3 let x be Real; ::_thesis: ( x in Z implies ((sin - ((1 / 3) (#) ((#Z 3) * sin))) `| Z) . x = (cos . x) |^ 3 ) assume A5: x in Z ; ::_thesis: ((sin - ((1 / 3) (#) ((#Z 3) * sin))) `| Z) . x = (cos . x) |^ 3 then ((sin - ((1 / 3) (#) ((#Z 3) * sin))) `| Z) . x = (diff (sin,x)) - (diff (((1 / 3) (#) ((#Z 3) * sin)),x)) by A1, A3, A4, FDIFF_1:19 .= (cos . x) - (diff (((1 / 3) (#) ((#Z 3) * sin)),x)) by SIN_COS:64 .= (cos . x) - ((((1 / 3) (#) ((#Z 3) * sin)) `| Z) . x) by A3, A5, FDIFF_1:def_7 .= (cos . x) - (((sin . x) #Z (3 - 1)) * (cos . x)) by A2, A5, FDIFF_4:54 .= (cos . x) * (1 - ((sin . x) #Z 2)) .= (cos . x) * (1 - ((sin . x) |^ (abs 2))) by PREPOWER:def_3 .= (cos . x) * (1 - ((sin . x) |^ 2)) by ABSVALUE:def_1 .= (cos . x) * (1 - ((sin . x) * (sin . x))) by WSIERP_1:1 .= (cos . x) * ((((cos . x) * (cos . x)) + ((sin . x) * (sin . x))) - ((sin . x) * (sin . x))) by SIN_COS:28 .= (cos . x) * ((cos . x) |^ 2) by WSIERP_1:1 .= (cos . x) |^ (2 + 1) by NEWTON:6 .= (cos . x) |^ 3 ; hence ((sin - ((1 / 3) (#) ((#Z 3) * sin))) `| Z) . x = (cos . x) |^ 3 ; ::_thesis: verum end; hence ( sin - ((1 / 3) (#) ((#Z 3) * sin)) is_differentiable_on Z & ( for x being Real st x in Z holds ((sin - ((1 / 3) (#) ((#Z 3) * sin))) `| Z) . x = (cos . x) |^ 3 ) ) by A1, A3, A4, FDIFF_1:19; ::_thesis: verum end; theorem :: FDIFF_6:49 for Z being open Subset of REAL st Z c= dom (sin * ln) holds ( sin * ln is_differentiable_on Z & ( for x being Real st x in Z holds ((sin * ln) `| Z) . x = (cos . (log (number_e,x))) / x ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (sin * ln) implies ( sin * ln is_differentiable_on Z & ( for x being Real st x in Z holds ((sin * ln) `| Z) . x = (cos . (log (number_e,x))) / x ) ) ) assume A1: Z c= dom (sin * ln) ; ::_thesis: ( sin * ln is_differentiable_on Z & ( for x being Real st x in Z holds ((sin * ln) `| Z) . x = (cos . (log (number_e,x))) / x ) ) then for y being set st y in Z holds y in dom ln by FUNCT_1:11; then A2: Z c= dom ln by TARSKI:def_3; then A3: ln is_differentiable_on Z by FDIFF_1:26, TAYLOR_1:18; A4: for x being Real st x in Z holds sin * ln is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies sin * ln is_differentiable_in x ) assume x in Z ; ::_thesis: sin * ln is_differentiable_in x then A5: ln is_differentiable_in x by A3, FDIFF_1:9; sin is_differentiable_in ln . x by SIN_COS:64; hence sin * ln is_differentiable_in x by A5, FDIFF_2:13; ::_thesis: verum end; then A6: sin * ln is_differentiable_on Z by A1, FDIFF_1:9; for x being Real st x in Z holds ((sin * ln) `| Z) . x = (cos . (log (number_e,x))) / x proof let x be Real; ::_thesis: ( x in Z implies ((sin * ln) `| Z) . x = (cos . (log (number_e,x))) / x ) A7: sin is_differentiable_in ln . x by SIN_COS:64; assume A8: x in Z ; ::_thesis: ((sin * ln) `| Z) . x = (cos . (log (number_e,x))) / x then A9: x in right_open_halfline 0 by A1, FUNCT_1:11, TAYLOR_1:18; ln is_differentiable_in x by A3, A8, FDIFF_1:9; then diff ((sin * ln),x) = (diff (sin,(ln . x))) * (diff (ln,x)) by A7, FDIFF_2:13 .= (cos . (ln . x)) * (diff (ln,x)) by SIN_COS:64 .= (cos . (log (number_e,x))) * (diff (ln,x)) by A9, TAYLOR_1:def_2 .= (cos . (log (number_e,x))) * (1 / x) by A2, A8, TAYLOR_1:18 .= (cos . (log (number_e,x))) / x by XCMPLX_1:99 ; hence ((sin * ln) `| Z) . x = (cos . (log (number_e,x))) / x by A6, A8, FDIFF_1:def_7; ::_thesis: verum end; hence ( sin * ln is_differentiable_on Z & ( for x being Real st x in Z holds ((sin * ln) `| Z) . x = (cos . (log (number_e,x))) / x ) ) by A1, A4, FDIFF_1:9; ::_thesis: verum end; theorem :: FDIFF_6:50 for Z being open Subset of REAL st Z c= dom (- (cos * ln)) holds ( - (cos * ln) is_differentiable_on Z & ( for x being Real st x in Z holds ((- (cos * ln)) `| Z) . x = (sin . (log (number_e,x))) / x ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (- (cos * ln)) implies ( - (cos * ln) is_differentiable_on Z & ( for x being Real st x in Z holds ((- (cos * ln)) `| Z) . x = (sin . (log (number_e,x))) / x ) ) ) assume A1: Z c= dom (- (cos * ln)) ; ::_thesis: ( - (cos * ln) is_differentiable_on Z & ( for x being Real st x in Z holds ((- (cos * ln)) `| Z) . x = (sin . (log (number_e,x))) / x ) ) then A2: Z c= dom (cos * ln) by VALUED_1:8; then for y being set st y in Z holds y in dom ln by FUNCT_1:11; then A3: Z c= dom ln by TARSKI:def_3; then A4: ln is_differentiable_on Z by FDIFF_1:26, TAYLOR_1:18; for x being Real st x in Z holds cos * ln is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies cos * ln is_differentiable_in x ) assume x in Z ; ::_thesis: cos * ln is_differentiable_in x then A5: ln is_differentiable_in x by A4, FDIFF_1:9; cos is_differentiable_in ln . x by SIN_COS:63; hence cos * ln is_differentiable_in x by A5, FDIFF_2:13; ::_thesis: verum end; then A6: cos * ln is_differentiable_on Z by A2, FDIFF_1:9; A7: for x being Real st x in Z holds ((- (cos * ln)) `| Z) . x = (sin . (log (number_e,x))) / x proof let x be Real; ::_thesis: ( x in Z implies ((- (cos * ln)) `| Z) . x = (sin . (log (number_e,x))) / x ) A8: cos is_differentiable_in ln . x by SIN_COS:63; assume A9: x in Z ; ::_thesis: ((- (cos * ln)) `| Z) . x = (sin . (log (number_e,x))) / x then A10: x in right_open_halfline 0 by A2, FUNCT_1:11, TAYLOR_1:18; A11: ln is_differentiable_in x by A4, A9, FDIFF_1:9; ((- (cos * ln)) `| Z) . x = (- 1) * (diff ((cos * ln),x)) by A1, A6, A9, FDIFF_1:20 .= (- 1) * ((diff (cos,(ln . x))) * (diff (ln,x))) by A11, A8, FDIFF_2:13 .= (- 1) * ((- (sin . (ln . x))) * (diff (ln,x))) by SIN_COS:63 .= ((- 1) * (- (sin . (ln . x)))) * (diff (ln,x)) .= ((- 1) * (- (sin . (log (number_e,x))))) * (diff (ln,x)) by A10, TAYLOR_1:def_2 .= ((- 1) * (- (sin . (log (number_e,x))))) * (1 / x) by A3, A9, TAYLOR_1:18 .= (sin . (log (number_e,x))) / x by XCMPLX_1:99 ; hence ((- (cos * ln)) `| Z) . x = (sin . (log (number_e,x))) / x ; ::_thesis: verum end; Z c= dom ((- 1) (#) (cos * ln)) by A1; hence ( - (cos * ln) is_differentiable_on Z & ( for x being Real st x in Z holds ((- (cos * ln)) `| Z) . x = (sin . (log (number_e,x))) / x ) ) by A6, A7, FDIFF_1:20; ::_thesis: verum end;