:: INTEGR11 semantic presentation begin Lm1: [#] REAL = dom (AffineMap ((1 / 2),0)) by FUNCT_2:def_1; Lm2: [#] REAL = dom (sin * (AffineMap (2,0))) by FUNCT_2:def_1; Lm3: dom ((1 / 4) (#) (sin * (AffineMap (2,0)))) = [#] REAL by FUNCT_2:def_1; theorem Th1: :: INTEGR11:1 ( (AffineMap ((1 / 2),0)) - ((1 / 4) (#) (sin * (AffineMap (2,0)))) is_differentiable_on REAL & ( for x being Real holds (((AffineMap ((1 / 2),0)) - ((1 / 4) (#) (sin * (AffineMap (2,0))))) `| REAL) . x = (sin . x) ^2 ) ) proof A1: for x being Real st x in REAL holds (AffineMap (2,0)) . x = (2 * x) + 0 by FCONT_1:def_4; then A2: sin * (AffineMap (2,0)) is_differentiable_on REAL by Lm2, FDIFF_4:37; then A3: (1 / 4) (#) (sin * (AffineMap (2,0))) is_differentiable_on REAL by Lm3, FDIFF_1:20; A4: dom ((AffineMap ((1 / 2),0)) - ((1 / 4) (#) (sin * (AffineMap (2,0))))) = [#] REAL by FUNCT_2:def_1; A5: for x being Real st x in REAL holds (AffineMap ((1 / 2),0)) . x = ((1 / 2) * x) + 0 by FCONT_1:def_4; then A6: AffineMap ((1 / 2),0) is_differentiable_on REAL by Lm1, FDIFF_1:23; A7: for x being Real st x in REAL holds (((1 / 4) (#) (sin * (AffineMap (2,0)))) `| REAL) . x = (1 / 2) * (cos (2 * x)) proof let x be Real; ::_thesis: ( x in REAL implies (((1 / 4) (#) (sin * (AffineMap (2,0)))) `| REAL) . x = (1 / 2) * (cos (2 * x)) ) assume x in REAL ; ::_thesis: (((1 / 4) (#) (sin * (AffineMap (2,0)))) `| REAL) . x = (1 / 2) * (cos (2 * x)) (((1 / 4) (#) (sin * (AffineMap (2,0)))) `| REAL) . x = (1 / 4) * (diff ((sin * (AffineMap (2,0))),x)) by A2, Lm3, FDIFF_1:20 .= (1 / 4) * (((sin * (AffineMap (2,0))) `| REAL) . x) by A2, FDIFF_1:def_7 .= (1 / 4) * (2 * (cos . ((2 * x) + 0))) by A1, Lm2, FDIFF_4:37 .= (1 / 2) * (cos (2 * x)) ; hence (((1 / 4) (#) (sin * (AffineMap (2,0)))) `| REAL) . x = (1 / 2) * (cos (2 * x)) ; ::_thesis: verum end; for x being Real st x in REAL holds (((AffineMap ((1 / 2),0)) - ((1 / 4) (#) (sin * (AffineMap (2,0))))) `| REAL) . x = (sin . x) ^2 proof let x be Real; ::_thesis: ( x in REAL implies (((AffineMap ((1 / 2),0)) - ((1 / 4) (#) (sin * (AffineMap (2,0))))) `| REAL) . x = (sin . x) ^2 ) assume x in REAL ; ::_thesis: (((AffineMap ((1 / 2),0)) - ((1 / 4) (#) (sin * (AffineMap (2,0))))) `| REAL) . x = (sin . x) ^2 (((AffineMap ((1 / 2),0)) - ((1 / 4) (#) (sin * (AffineMap (2,0))))) `| REAL) . x = (diff ((AffineMap ((1 / 2),0)),x)) - (diff (((1 / 4) (#) (sin * (AffineMap (2,0)))),x)) by A4, A6, A3, FDIFF_1:19 .= (((AffineMap ((1 / 2),0)) `| REAL) . x) - (diff (((1 / 4) (#) (sin * (AffineMap (2,0)))),x)) by A6, FDIFF_1:def_7 .= (1 / 2) - (diff (((1 / 4) (#) (sin * (AffineMap (2,0)))),x)) by A5, Lm1, FDIFF_1:23 .= (1 / 2) - ((((1 / 4) (#) (sin * (AffineMap (2,0)))) `| REAL) . x) by A3, FDIFF_1:def_7 .= (1 / 2) - ((1 / 2) * (cos (2 * x))) by A7 .= (1 - (cos (2 * x))) / 2 .= (sin x) ^2 by SIN_COS5:20 ; hence (((AffineMap ((1 / 2),0)) - ((1 / 4) (#) (sin * (AffineMap (2,0))))) `| REAL) . x = (sin . x) ^2 ; ::_thesis: verum end; hence ( (AffineMap ((1 / 2),0)) - ((1 / 4) (#) (sin * (AffineMap (2,0)))) is_differentiable_on REAL & ( for x being Real holds (((AffineMap ((1 / 2),0)) - ((1 / 4) (#) (sin * (AffineMap (2,0))))) `| REAL) . x = (sin . x) ^2 ) ) by A4, A6, A3, FDIFF_1:19; ::_thesis: verum end; theorem Th2: :: INTEGR11:2 ( (AffineMap ((1 / 2),0)) + ((1 / 4) (#) (sin * (AffineMap (2,0)))) is_differentiable_on REAL & ( for x being Real holds (((AffineMap ((1 / 2),0)) + ((1 / 4) (#) (sin * (AffineMap (2,0))))) `| REAL) . x = (cos . x) ^2 ) ) proof A1: for x being Real st x in REAL holds (AffineMap (2,0)) . x = (2 * x) + 0 by FCONT_1:def_4; then A2: sin * (AffineMap (2,0)) is_differentiable_on REAL by Lm2, FDIFF_4:37; then A3: (1 / 4) (#) (sin * (AffineMap (2,0))) is_differentiable_on REAL by Lm3, FDIFF_1:20; A4: dom ((AffineMap ((1 / 2),0)) + ((1 / 4) (#) (sin * (AffineMap (2,0))))) = [#] REAL by FUNCT_2:def_1; A5: for x being Real st x in REAL holds (AffineMap ((1 / 2),0)) . x = ((1 / 2) * x) + 0 by FCONT_1:def_4; then A6: AffineMap ((1 / 2),0) is_differentiable_on REAL by Lm1, FDIFF_1:23; A7: for x being Real st x in REAL holds (((1 / 4) (#) (sin * (AffineMap (2,0)))) `| REAL) . x = (1 / 2) * (cos (2 * x)) proof let x be Real; ::_thesis: ( x in REAL implies (((1 / 4) (#) (sin * (AffineMap (2,0)))) `| REAL) . x = (1 / 2) * (cos (2 * x)) ) assume x in REAL ; ::_thesis: (((1 / 4) (#) (sin * (AffineMap (2,0)))) `| REAL) . x = (1 / 2) * (cos (2 * x)) (((1 / 4) (#) (sin * (AffineMap (2,0)))) `| REAL) . x = (1 / 4) * (diff ((sin * (AffineMap (2,0))),x)) by A2, Lm3, FDIFF_1:20 .= (1 / 4) * (((sin * (AffineMap (2,0))) `| REAL) . x) by A2, FDIFF_1:def_7 .= (1 / 4) * (2 * (cos . ((2 * x) + 0))) by A1, Lm2, FDIFF_4:37 .= (1 / 2) * (cos (2 * x)) ; hence (((1 / 4) (#) (sin * (AffineMap (2,0)))) `| REAL) . x = (1 / 2) * (cos (2 * x)) ; ::_thesis: verum end; for x being Real st x in REAL holds (((AffineMap ((1 / 2),0)) + ((1 / 4) (#) (sin * (AffineMap (2,0))))) `| REAL) . x = (cos . x) ^2 proof let x be Real; ::_thesis: ( x in REAL implies (((AffineMap ((1 / 2),0)) + ((1 / 4) (#) (sin * (AffineMap (2,0))))) `| REAL) . x = (cos . x) ^2 ) assume x in REAL ; ::_thesis: (((AffineMap ((1 / 2),0)) + ((1 / 4) (#) (sin * (AffineMap (2,0))))) `| REAL) . x = (cos . x) ^2 (((AffineMap ((1 / 2),0)) + ((1 / 4) (#) (sin * (AffineMap (2,0))))) `| REAL) . x = (diff ((AffineMap ((1 / 2),0)),x)) + (diff (((1 / 4) (#) (sin * (AffineMap (2,0)))),x)) by A4, A6, A3, FDIFF_1:18 .= (((AffineMap ((1 / 2),0)) `| REAL) . x) + (diff (((1 / 4) (#) (sin * (AffineMap (2,0)))),x)) by A6, FDIFF_1:def_7 .= (1 / 2) + (diff (((1 / 4) (#) (sin * (AffineMap (2,0)))),x)) by A5, Lm1, FDIFF_1:23 .= (1 / 2) + ((((1 / 4) (#) (sin * (AffineMap (2,0)))) `| REAL) . x) by A3, FDIFF_1:def_7 .= (1 / 2) + ((1 / 2) * (cos (2 * x))) by A7 .= (1 + (cos (2 * x))) / 2 .= (cos x) ^2 by SIN_COS5:21 ; hence (((AffineMap ((1 / 2),0)) + ((1 / 4) (#) (sin * (AffineMap (2,0))))) `| REAL) . x = (cos . x) ^2 ; ::_thesis: verum end; hence ( (AffineMap ((1 / 2),0)) + ((1 / 4) (#) (sin * (AffineMap (2,0)))) is_differentiable_on REAL & ( for x being Real holds (((AffineMap ((1 / 2),0)) + ((1 / 4) (#) (sin * (AffineMap (2,0))))) `| REAL) . x = (cos . x) ^2 ) ) by A4, A6, A3, FDIFF_1:18; ::_thesis: verum end; theorem Th3: :: INTEGR11:3 for n being Element of NAT holds ( (1 / (n + 1)) (#) ((#Z (n + 1)) * sin) is_differentiable_on REAL & ( for x being Real holds (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) `| REAL) . x = ((sin . x) #Z n) * (cos . x) ) ) proof let n be Element of NAT ; ::_thesis: ( (1 / (n + 1)) (#) ((#Z (n + 1)) * sin) is_differentiable_on REAL & ( for x being Real holds (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) `| REAL) . x = ((sin . x) #Z n) * (cos . x) ) ) A1: [#] REAL = dom ((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) by FUNCT_2:def_1; for x0 being Real holds (#Z (n + 1)) * sin is_differentiable_in x0 proof let x0 be Real; ::_thesis: (#Z (n + 1)) * sin is_differentiable_in x0 sin is_differentiable_in x0 by SIN_COS:64; hence (#Z (n + 1)) * sin is_differentiable_in x0 by TAYLOR_1:3; ::_thesis: verum end; then ( [#] REAL = dom ((#Z (n + 1)) * sin) & ( for x0 being Real st x0 in REAL holds (#Z (n + 1)) * sin is_differentiable_in x0 ) ) by FUNCT_2:def_1; then A2: (#Z (n + 1)) * sin is_differentiable_on REAL by FDIFF_1:9; A3: for x being Real st x in REAL holds (((#Z (n + 1)) * sin) `| REAL) . x = ((n + 1) * ((sin . x) #Z n)) * (cos . x) proof set m = n + 1; let x be Real; ::_thesis: ( x in REAL implies (((#Z (n + 1)) * sin) `| REAL) . x = ((n + 1) * ((sin . x) #Z n)) * (cos . x) ) assume x in REAL ; ::_thesis: (((#Z (n + 1)) * sin) `| REAL) . x = ((n + 1) * ((sin . x) #Z n)) * (cos . x) sin is_differentiable_in x by SIN_COS:64; then diff (((#Z (n + 1)) * sin),x) = ((n + 1) * ((sin . x) #Z ((n + 1) - 1))) * (diff (sin,x)) by TAYLOR_1:3 .= ((n + 1) * ((sin . x) #Z ((n + 1) - 1))) * (cos . x) by SIN_COS:64 ; hence (((#Z (n + 1)) * sin) `| REAL) . x = ((n + 1) * ((sin . x) #Z n)) * (cos . x) by A2, FDIFF_1:def_7; ::_thesis: verum end; for x being Real st x in REAL holds (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) `| REAL) . x = ((sin . x) #Z n) * (cos . x) proof let x be Real; ::_thesis: ( x in REAL implies (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) `| REAL) . x = ((sin . x) #Z n) * (cos . x) ) assume x in REAL ; ::_thesis: (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) `| REAL) . x = ((sin . x) #Z n) * (cos . x) (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) `| REAL) . x = (1 / (n + 1)) * (diff (((#Z (n + 1)) * sin),x)) by A1, A2, FDIFF_1:20 .= (1 / (n + 1)) * ((((#Z (n + 1)) * sin) `| REAL) . x) by A2, FDIFF_1:def_7 .= (1 / (n + 1)) * (((n + 1) * ((sin . x) #Z n)) * (cos . x)) by A3 .= (((1 / (n + 1)) * (n + 1)) * ((sin . x) #Z n)) * (cos . x) .= (((n + 1) / (n + 1)) * ((sin . x) #Z n)) * (cos . x) by XCMPLX_1:99 .= (1 * ((sin . x) #Z n)) * (cos . x) by XCMPLX_1:60 ; hence (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) `| REAL) . x = ((sin . x) #Z n) * (cos . x) ; ::_thesis: verum end; hence ( (1 / (n + 1)) (#) ((#Z (n + 1)) * sin) is_differentiable_on REAL & ( for x being Real holds (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) `| REAL) . x = ((sin . x) #Z n) * (cos . x) ) ) by A1, A2, FDIFF_1:20; ::_thesis: verum end; theorem Th4: :: INTEGR11:4 for n being Element of NAT holds ( (- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos) is_differentiable_on REAL & ( for x being Real holds (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) `| REAL) . x = ((cos . x) #Z n) * (sin . x) ) ) proof let n be Element of NAT ; ::_thesis: ( (- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos) is_differentiable_on REAL & ( for x being Real holds (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) `| REAL) . x = ((cos . x) #Z n) * (sin . x) ) ) A1: [#] REAL = dom ((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) by FUNCT_2:def_1; for x0 being Real holds (#Z (n + 1)) * cos is_differentiable_in x0 proof let x0 be Real; ::_thesis: (#Z (n + 1)) * cos is_differentiable_in x0 cos is_differentiable_in x0 by SIN_COS:63; hence (#Z (n + 1)) * cos is_differentiable_in x0 by TAYLOR_1:3; ::_thesis: verum end; then A2: for x0 being Real st x0 in REAL holds (#Z (n + 1)) * cos is_differentiable_in x0 ; ( [#] REAL = dom (#Z (n + 1)) & REAL = dom ((#Z (n + 1)) * cos) ) by FUNCT_2:def_1; then A3: (#Z (n + 1)) * cos is_differentiable_on REAL by A2, FDIFF_1:9; A4: for x being Real st x in REAL holds (((#Z (n + 1)) * cos) `| REAL) . x = ((- (n + 1)) * ((cos . x) #Z n)) * (sin . x) proof set m = n + 1; let x be Real; ::_thesis: ( x in REAL implies (((#Z (n + 1)) * cos) `| REAL) . x = ((- (n + 1)) * ((cos . x) #Z n)) * (sin . x) ) assume x in REAL ; ::_thesis: (((#Z (n + 1)) * cos) `| REAL) . x = ((- (n + 1)) * ((cos . x) #Z n)) * (sin . x) cos is_differentiable_in x by SIN_COS:63; then diff (((#Z (n + 1)) * cos),x) = ((n + 1) * ((cos . x) #Z ((n + 1) - 1))) * (diff (cos,x)) by TAYLOR_1:3 .= ((n + 1) * ((cos . x) #Z ((n + 1) - 1))) * (- (sin . x)) by SIN_COS:63 .= ((- (n + 1)) * ((cos . x) #Z ((n + 1) - 1))) * (sin . x) ; hence (((#Z (n + 1)) * cos) `| REAL) . x = ((- (n + 1)) * ((cos . x) #Z n)) * (sin . x) by A3, FDIFF_1:def_7; ::_thesis: verum end; for x being Real st x in REAL holds (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) `| REAL) . x = ((cos . x) #Z n) * (sin . x) proof let x be Real; ::_thesis: ( x in REAL implies (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) `| REAL) . x = ((cos . x) #Z n) * (sin . x) ) assume x in REAL ; ::_thesis: (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) `| REAL) . x = ((cos . x) #Z n) * (sin . x) (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) `| REAL) . x = (- (1 / (n + 1))) * (diff (((#Z (n + 1)) * cos),x)) by A1, A3, FDIFF_1:20 .= (- (1 / (n + 1))) * ((((#Z (n + 1)) * cos) `| REAL) . x) by A3, FDIFF_1:def_7 .= (- (1 / (n + 1))) * (((- (n + 1)) * ((cos . x) #Z n)) * (sin . x)) by A4 .= (((1 / (n + 1)) * (n + 1)) * ((cos . x) #Z n)) * (sin . x) .= (((n + 1) / (n + 1)) * ((cos . x) #Z n)) * (sin . x) by XCMPLX_1:99 .= (1 * ((cos . x) #Z n)) * (sin . x) by XCMPLX_1:60 ; hence (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) `| REAL) . x = ((cos . x) #Z n) * (sin . x) ; ::_thesis: verum end; hence ( (- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos) is_differentiable_on REAL & ( for x being Real holds (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) `| REAL) . x = ((cos . x) #Z n) * (sin . x) ) ) by A1, A3, FDIFF_1:20; ::_thesis: verum end; theorem Th5: :: INTEGR11:5 for m, n being Element of NAT st m + n <> 0 & m - n <> 0 holds ( ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) + ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) is_differentiable_on REAL & ( for x being Real holds ((((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) + ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0))))) `| REAL) . x = (cos . (m * x)) * (cos . (n * x)) ) ) proof let m, n be Element of NAT ; ::_thesis: ( m + n <> 0 & m - n <> 0 implies ( ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) + ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) is_differentiable_on REAL & ( for x being Real holds ((((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) + ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0))))) `| REAL) . x = (cos . (m * x)) * (cos . (n * x)) ) ) ) assume that A1: m + n <> 0 and A2: m - n <> 0 ; ::_thesis: ( ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) + ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) is_differentiable_on REAL & ( for x being Real holds ((((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) + ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0))))) `| REAL) . x = (cos . (m * x)) * (cos . (n * x)) ) ) A3: ( dom (sin * (AffineMap ((m - n),0))) = [#] REAL & ( for x being Real st x in REAL holds (AffineMap ((m - n),0)) . x = ((m - n) * x) + 0 ) ) by FCONT_1:def_4, FUNCT_2:def_1; then A4: sin * (AffineMap ((m - n),0)) is_differentiable_on REAL by FDIFF_4:37; A5: dom ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) = [#] REAL by FUNCT_2:def_1; then A6: (1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0))) is_differentiable_on REAL by A4, FDIFF_1:20; A7: for x being Real st x in REAL holds (((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) `| REAL) . x = (1 / 2) * (cos ((m - n) * x)) proof let x be Real; ::_thesis: ( x in REAL implies (((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) `| REAL) . x = (1 / 2) * (cos ((m - n) * x)) ) assume x in REAL ; ::_thesis: (((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) `| REAL) . x = (1 / 2) * (cos ((m - n) * x)) (((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) `| REAL) . x = (1 / (2 * (m - n))) * (diff ((sin * (AffineMap ((m - n),0))),x)) by A5, A4, FDIFF_1:20 .= (1 / (2 * (m - n))) * (((sin * (AffineMap ((m - n),0))) `| REAL) . x) by A4, FDIFF_1:def_7 .= (1 / (2 * (m - n))) * ((m - n) * (cos . (((m - n) * x) + 0))) by A3, FDIFF_4:37 .= ((m - n) * (1 / (2 * (m - n)))) * (cos . (((m - n) * x) + 0)) .= ((1 * (m - n)) / (2 * (m - n))) * (cos . (((m - n) * x) + 0)) by XCMPLX_1:74 .= (1 / 2) * (cos ((m - n) * x)) by A2, XCMPLX_1:91 ; hence (((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) `| REAL) . x = (1 / 2) * (cos ((m - n) * x)) ; ::_thesis: verum end; A8: dom (((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) + ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0))))) = [#] REAL by FUNCT_2:def_1; A9: ( dom (sin * (AffineMap ((m + n),0))) = [#] REAL & ( for x being Real st x in REAL holds (AffineMap ((m + n),0)) . x = ((m + n) * x) + 0 ) ) by FCONT_1:def_4, FUNCT_2:def_1; then A10: sin * (AffineMap ((m + n),0)) is_differentiable_on REAL by FDIFF_4:37; A11: [#] REAL = dom ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) by FUNCT_2:def_1; then A12: (1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0))) is_differentiable_on REAL by A10, FDIFF_1:20; A13: for x being Real st x in REAL holds (((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) `| REAL) . x = (1 / 2) * (cos ((m + n) * x)) proof let x be Real; ::_thesis: ( x in REAL implies (((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) `| REAL) . x = (1 / 2) * (cos ((m + n) * x)) ) assume x in REAL ; ::_thesis: (((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) `| REAL) . x = (1 / 2) * (cos ((m + n) * x)) (((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) `| REAL) . x = (1 / (2 * (m + n))) * (diff ((sin * (AffineMap ((m + n),0))),x)) by A11, A10, FDIFF_1:20 .= (1 / (2 * (m + n))) * (((sin * (AffineMap ((m + n),0))) `| REAL) . x) by A10, FDIFF_1:def_7 .= (1 / (2 * (m + n))) * ((m + n) * (cos . (((m + n) * x) + 0))) by A9, FDIFF_4:37 .= ((m + n) * (1 / (2 * (m + n)))) * (cos . (((m + n) * x) + 0)) .= ((1 * (m + n)) / (2 * (m + n))) * (cos . (((m + n) * x) + 0)) by XCMPLX_1:74 .= (1 / 2) * (cos ((m + n) * x)) by A1, XCMPLX_1:91 ; hence (((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) `| REAL) . x = (1 / 2) * (cos ((m + n) * x)) ; ::_thesis: verum end; for x being Real st x in REAL holds ((((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) + ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0))))) `| REAL) . x = (cos . (m * x)) * (cos . (n * x)) proof let x be Real; ::_thesis: ( x in REAL implies ((((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) + ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0))))) `| REAL) . x = (cos . (m * x)) * (cos . (n * x)) ) assume x in REAL ; ::_thesis: ((((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) + ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0))))) `| REAL) . x = (cos . (m * x)) * (cos . (n * x)) ((((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) + ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0))))) `| REAL) . x = (diff (((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))),x)) + (diff (((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))),x)) by A8, A12, A6, FDIFF_1:18 .= ((((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) `| REAL) . x) + (diff (((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))),x)) by A12, FDIFF_1:def_7 .= ((((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) `| REAL) . x) + ((((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) `| REAL) . x) by A6, FDIFF_1:def_7 .= ((1 / 2) * (cos ((m + n) * x))) + ((((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) `| REAL) . x) by A13 .= ((1 / 2) * (cos ((m + n) * x))) + ((1 / 2) * (cos ((m - n) * x))) by A7 .= (1 / 2) * ((cos ((m + n) * x)) + (cos ((m - n) * x))) .= (1 / 2) * (2 * ((cos ((((m + n) * x) + ((m - n) * x)) / 2)) * (cos ((((m + n) * x) - ((m - n) * x)) / 2)))) by SIN_COS4:17 .= (cos . (m * x)) * (cos . (n * x)) ; hence ((((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) + ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0))))) `| REAL) . x = (cos . (m * x)) * (cos . (n * x)) ; ::_thesis: verum end; hence ( ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) + ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) is_differentiable_on REAL & ( for x being Real holds ((((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) + ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0))))) `| REAL) . x = (cos . (m * x)) * (cos . (n * x)) ) ) by A8, A12, A6, FDIFF_1:18; ::_thesis: verum end; theorem Th6: :: INTEGR11:6 for m, n being Element of NAT st m + n <> 0 & m - n <> 0 holds ( ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) - ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) is_differentiable_on REAL & ( for x being Real holds ((((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) - ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0))))) `| REAL) . x = (sin . (m * x)) * (sin . (n * x)) ) ) proof let m, n be Element of NAT ; ::_thesis: ( m + n <> 0 & m - n <> 0 implies ( ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) - ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) is_differentiable_on REAL & ( for x being Real holds ((((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) - ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0))))) `| REAL) . x = (sin . (m * x)) * (sin . (n * x)) ) ) ) assume that A1: m + n <> 0 and A2: m - n <> 0 ; ::_thesis: ( ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) - ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) is_differentiable_on REAL & ( for x being Real holds ((((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) - ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0))))) `| REAL) . x = (sin . (m * x)) * (sin . (n * x)) ) ) A3: dom ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) = [#] REAL by FUNCT_2:def_1; A4: ( dom (sin * (AffineMap ((m - n),0))) = [#] REAL & ( for x being Real st x in REAL holds (AffineMap ((m - n),0)) . x = ((m - n) * x) + 0 ) ) by FCONT_1:def_4, FUNCT_2:def_1; then A5: sin * (AffineMap ((m - n),0)) is_differentiable_on REAL by FDIFF_4:37; then A6: (1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0))) is_differentiable_on REAL by A3, FDIFF_1:20; A7: for x being Real st x in REAL holds (((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) `| REAL) . x = (1 / 2) * (cos ((m - n) * x)) proof let x be Real; ::_thesis: ( x in REAL implies (((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) `| REAL) . x = (1 / 2) * (cos ((m - n) * x)) ) assume x in REAL ; ::_thesis: (((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) `| REAL) . x = (1 / 2) * (cos ((m - n) * x)) (((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) `| REAL) . x = (1 / (2 * (m - n))) * (diff ((sin * (AffineMap ((m - n),0))),x)) by A3, A5, FDIFF_1:20 .= (1 / (2 * (m - n))) * (((sin * (AffineMap ((m - n),0))) `| REAL) . x) by A5, FDIFF_1:def_7 .= (1 / (2 * (m - n))) * ((m - n) * (cos . (((m - n) * x) + 0))) by A4, FDIFF_4:37 .= ((m - n) * (1 / (2 * (m - n)))) * (cos . (((m - n) * x) + 0)) .= ((1 * (m - n)) / (2 * (m - n))) * (cos . (((m - n) * x) + 0)) by XCMPLX_1:74 .= (1 / 2) * (cos ((m - n) * x)) by A2, XCMPLX_1:91 ; hence (((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) `| REAL) . x = (1 / 2) * (cos ((m - n) * x)) ; ::_thesis: verum end; A8: dom (((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) - ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0))))) = [#] REAL by FUNCT_2:def_1; A9: ( dom (sin * (AffineMap ((m + n),0))) = [#] REAL & ( for x being Real st x in REAL holds (AffineMap ((m + n),0)) . x = ((m + n) * x) + 0 ) ) by FCONT_1:def_4, FUNCT_2:def_1; then A10: sin * (AffineMap ((m + n),0)) is_differentiable_on REAL by FDIFF_4:37; A11: REAL = dom ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) by FUNCT_2:def_1; then A12: (1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0))) is_differentiable_on REAL by A3, A10, FDIFF_1:20; A13: for x being Real st x in REAL holds (((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) `| REAL) . x = (1 / 2) * (cos ((m + n) * x)) proof let x be Real; ::_thesis: ( x in REAL implies (((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) `| REAL) . x = (1 / 2) * (cos ((m + n) * x)) ) assume x in REAL ; ::_thesis: (((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) `| REAL) . x = (1 / 2) * (cos ((m + n) * x)) (((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) `| REAL) . x = (1 / (2 * (m + n))) * (diff ((sin * (AffineMap ((m + n),0))),x)) by A11, A3, A10, FDIFF_1:20 .= (1 / (2 * (m + n))) * (((sin * (AffineMap ((m + n),0))) `| REAL) . x) by A10, FDIFF_1:def_7 .= (1 / (2 * (m + n))) * ((m + n) * (cos . (((m + n) * x) + 0))) by A9, FDIFF_4:37 .= ((m + n) * (1 / (2 * (m + n)))) * (cos . (((m + n) * x) + 0)) .= ((1 * (m + n)) / (2 * (m + n))) * (cos . (((m + n) * x) + 0)) by XCMPLX_1:74 .= (1 / 2) * (cos ((m + n) * x)) by A1, XCMPLX_1:91 ; hence (((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) `| REAL) . x = (1 / 2) * (cos ((m + n) * x)) ; ::_thesis: verum end; for x being Real st x in REAL holds ((((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) - ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0))))) `| REAL) . x = (sin . (m * x)) * (sin . (n * x)) proof let x be Real; ::_thesis: ( x in REAL implies ((((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) - ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0))))) `| REAL) . x = (sin . (m * x)) * (sin . (n * x)) ) assume x in REAL ; ::_thesis: ((((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) - ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0))))) `| REAL) . x = (sin . (m * x)) * (sin . (n * x)) ((((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) - ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0))))) `| REAL) . x = (diff (((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))),x)) - (diff (((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))),x)) by A8, A12, A6, FDIFF_1:19 .= ((((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) `| REAL) . x) - (diff (((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))),x)) by A6, FDIFF_1:def_7 .= ((((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) `| REAL) . x) - ((((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) `| REAL) . x) by A12, FDIFF_1:def_7 .= ((1 / 2) * (cos ((m - n) * x))) - ((((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) `| REAL) . x) by A7 .= ((1 / 2) * (cos ((m - n) * x))) - ((1 / 2) * (cos ((m + n) * x))) by A13 .= (1 / 2) * ((cos ((m - n) * x)) - (cos ((m + n) * x))) .= (1 / 2) * (- (2 * ((sin ((((m - n) * x) + ((m + n) * x)) / 2)) * (sin ((((m - n) * x) - ((m + n) * x)) / 2))))) by SIN_COS4:18 .= (1 / 2) * (- (2 * ((sin (m * x)) * (sin (- (n * x)))))) .= (1 / 2) * (- (2 * ((sin (m * x)) * (- (sin (n * x)))))) by SIN_COS:31 .= (sin . (m * x)) * (sin . (n * x)) ; hence ((((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) - ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0))))) `| REAL) . x = (sin . (m * x)) * (sin . (n * x)) ; ::_thesis: verum end; hence ( ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) - ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) is_differentiable_on REAL & ( for x being Real holds ((((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) - ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0))))) `| REAL) . x = (sin . (m * x)) * (sin . (n * x)) ) ) by A8, A12, A6, FDIFF_1:19; ::_thesis: verum end; theorem Th7: :: INTEGR11:7 for m, n being Element of NAT st m + n <> 0 & m - n <> 0 holds ( (- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0)))) is_differentiable_on REAL & ( for x being Real holds (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) `| REAL) . x = (sin . (m * x)) * (cos . (n * x)) ) ) proof let m, n be Element of NAT ; ::_thesis: ( m + n <> 0 & m - n <> 0 implies ( (- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0)))) is_differentiable_on REAL & ( for x being Real holds (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) `| REAL) . x = (sin . (m * x)) * (cos . (n * x)) ) ) ) assume that A1: m + n <> 0 and A2: m - n <> 0 ; ::_thesis: ( (- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0)))) is_differentiable_on REAL & ( for x being Real holds (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) `| REAL) . x = (sin . (m * x)) * (cos . (n * x)) ) ) A3: ( dom (cos * (AffineMap ((m + n),0))) = [#] REAL & ( for x being Real st x in REAL holds (AffineMap ((m + n),0)) . x = ((m + n) * x) + 0 ) ) by FCONT_1:def_4, FUNCT_2:def_1; then A4: cos * (AffineMap ((m + n),0)) is_differentiable_on REAL by FDIFF_4:38; A5: for x being Real st x in REAL holds ((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) `| REAL) . x = (1 / 2) * (sin ((m + n) * x)) proof let x be Real; ::_thesis: ( x in REAL implies ((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) `| REAL) . x = (1 / 2) * (sin ((m + n) * x)) ) assume x in REAL ; ::_thesis: ((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) `| REAL) . x = (1 / 2) * (sin ((m + n) * x)) A6: dom (((- 1) / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0)))) = [#] REAL by FUNCT_2:def_1; ((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) `| REAL) . x = ((((- 1) * (1 / (2 * (m + n)))) (#) (cos * (AffineMap ((m + n),0)))) `| REAL) . x by RFUNCT_1:17 .= (((- (1 / (2 * (m + n)))) (#) (cos * (AffineMap ((m + n),0)))) `| REAL) . x .= ((((- 1) / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0)))) `| REAL) . x by XCMPLX_1:187 .= ((- 1) / (2 * (m + n))) * (diff ((cos * (AffineMap ((m + n),0))),x)) by A4, A6, FDIFF_1:20 .= ((- 1) / (2 * (m + n))) * (((cos * (AffineMap ((m + n),0))) `| REAL) . x) by A4, FDIFF_1:def_7 .= ((- 1) / (2 * (m + n))) * (- ((m + n) * (sin . (((m + n) * x) + 0)))) by A3, FDIFF_4:38 .= ((- ((- 1) / (2 * (m + n)))) * (m + n)) * (sin . (((m + n) * x) + 0)) .= ((1 / (2 * (m + n))) * (m + n)) * (sin . (((m + n) * x) + 0)) by XCMPLX_1:190 .= ((1 * (m + n)) / (2 * (m + n))) * (sin . (((m + n) * x) + 0)) by XCMPLX_1:74 .= (1 / 2) * (sin ((m + n) * x)) by A1, XCMPLX_1:91 ; hence ((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) `| REAL) . x = (1 / 2) * (sin ((m + n) * x)) ; ::_thesis: verum end; A7: dom (- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) = [#] REAL by FUNCT_2:def_1; dom ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0)))) = [#] REAL by FUNCT_2:def_1; then ( - ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0)))) = (- 1) (#) ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0)))) & (1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))) is_differentiable_on REAL ) by A4, FDIFF_1:20; then A8: - ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0)))) is_differentiable_on REAL by A7, FDIFF_1:20; A9: REAL = dom ((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) by FUNCT_2:def_1; A10: ( dom (cos * (AffineMap ((m - n),0))) = [#] REAL & ( for x being Real st x in REAL holds (AffineMap ((m - n),0)) . x = ((m - n) * x) + 0 ) ) by FCONT_1:def_4, FUNCT_2:def_1; then A11: cos * (AffineMap ((m - n),0)) is_differentiable_on REAL by FDIFF_4:38; A12: dom ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0)))) = [#] REAL by FUNCT_2:def_1; then A13: (1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))) is_differentiable_on REAL by A11, FDIFF_1:20; A14: for x being Real st x in REAL holds (((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0)))) `| REAL) . x = - ((1 / 2) * (sin ((m - n) * x))) proof let x be Real; ::_thesis: ( x in REAL implies (((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0)))) `| REAL) . x = - ((1 / 2) * (sin ((m - n) * x))) ) assume x in REAL ; ::_thesis: (((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0)))) `| REAL) . x = - ((1 / 2) * (sin ((m - n) * x))) (((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0)))) `| REAL) . x = (1 / (2 * (m - n))) * (diff ((cos * (AffineMap ((m - n),0))),x)) by A12, A11, FDIFF_1:20 .= (1 / (2 * (m - n))) * (((cos * (AffineMap ((m - n),0))) `| REAL) . x) by A11, FDIFF_1:def_7 .= (1 / (2 * (m - n))) * (- ((m - n) * (sin . (((m - n) * x) + 0)))) by A10, FDIFF_4:38 .= ((- (1 / (2 * (m - n)))) * (m - n)) * (sin . (((m - n) * x) + 0)) .= (((- 1) / (2 * (m - n))) * (m - n)) * (sin . (((m - n) * x) + 0)) by XCMPLX_1:187 .= (((- 1) * (m - n)) / (2 * (m - n))) * (sin . (((m - n) * x) + 0)) by XCMPLX_1:74 .= ((- 1) / 2) * (sin ((m - n) * x)) by A2, XCMPLX_1:91 ; hence (((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0)))) `| REAL) . x = - ((1 / 2) * (sin ((m - n) * x))) ; ::_thesis: verum end; for x being Real st x in REAL holds (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) `| REAL) . x = (sin . (m * x)) * (cos . (n * x)) proof let x be Real; ::_thesis: ( x in REAL implies (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) `| REAL) . x = (sin . (m * x)) * (cos . (n * x)) ) assume x in REAL ; ::_thesis: (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) `| REAL) . x = (sin . (m * x)) * (cos . (n * x)) (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) `| REAL) . x = (diff ((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))),x)) - (diff (((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0)))),x)) by A7, A9, A8, A13, FDIFF_1:19 .= (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) `| REAL) . x) - (diff (((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0)))),x)) by A8, FDIFF_1:def_7 .= (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) `| REAL) . x) - ((((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0)))) `| REAL) . x) by A13, FDIFF_1:def_7 .= ((1 / 2) * (sin ((m + n) * x))) - ((((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0)))) `| REAL) . x) by A5 .= ((1 / 2) * (sin ((m + n) * x))) - (- ((1 / 2) * (sin ((m - n) * x)))) by A14 .= (1 / 2) * ((sin ((m + n) * x)) + (sin ((m - n) * x))) .= (1 / 2) * (2 * ((cos ((((m + n) * x) - ((m - n) * x)) / 2)) * (sin ((((m + n) * x) + ((m - n) * x)) / 2)))) by SIN_COS4:15 .= (sin . (m * x)) * (cos . (n * x)) ; hence (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) `| REAL) . x = (sin . (m * x)) * (cos . (n * x)) ; ::_thesis: verum end; hence ( (- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0)))) is_differentiable_on REAL & ( for x being Real holds (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) `| REAL) . x = (sin . (m * x)) * (cos . (n * x)) ) ) by A7, A9, A8, A13, FDIFF_1:19; ::_thesis: verum end; theorem Th8: :: INTEGR11:8 for n being Element of NAT st n <> 0 holds ( ((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) - ((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0)))) is_differentiable_on REAL & ( for x being Real holds ((((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) - ((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0))))) `| REAL) . x = x * (sin . (n * x)) ) ) proof let n be Element of NAT ; ::_thesis: ( n <> 0 implies ( ((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) - ((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0)))) is_differentiable_on REAL & ( for x being Real holds ((((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) - ((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0))))) `| REAL) . x = x * (sin . (n * x)) ) ) ) A1: dom (((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) - ((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0))))) = [#] REAL by FUNCT_2:def_1; A2: ( dom (AffineMap ((1 / n),0)) = REAL & ( for x being Real st x in REAL holds (AffineMap ((1 / n),0)) . x = ((1 / n) * x) + 0 ) ) by FCONT_1:def_4, FUNCT_2:def_1; then A3: AffineMap ((1 / n),0) is_differentiable_on REAL by A1, FDIFF_1:23; A4: for x being Real st x in REAL holds (AffineMap (n,0)) . x = (n * x) + 0 by FCONT_1:def_4; A5: dom (sin * (AffineMap (n,0))) = [#] REAL by FUNCT_2:def_1; then A6: sin * (AffineMap (n,0)) is_differentiable_on REAL by A4, FDIFF_4:37; A7: dom ((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) = REAL by FUNCT_2:def_1; then A8: (1 / (n ^2)) (#) (sin * (AffineMap (n,0))) is_differentiable_on REAL by A1, A6, FDIFF_1:20; assume A9: n <> 0 ; ::_thesis: ( ((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) - ((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0)))) is_differentiable_on REAL & ( for x being Real holds ((((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) - ((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0))))) `| REAL) . x = x * (sin . (n * x)) ) ) A10: for x being Real st x in REAL holds (((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) `| REAL) . x = (1 / n) * (cos (n * x)) proof let x be Real; ::_thesis: ( x in REAL implies (((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) `| REAL) . x = (1 / n) * (cos (n * x)) ) assume x in REAL ; ::_thesis: (((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) `| REAL) . x = (1 / n) * (cos (n * x)) (((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) `| REAL) . x = (1 / (n ^2)) * (diff ((sin * (AffineMap (n,0))),x)) by A7, A1, A6, FDIFF_1:20 .= (1 / (n ^2)) * (((sin * (AffineMap (n,0))) `| REAL) . x) by A6, FDIFF_1:def_7 .= (1 / (n ^2)) * (n * (cos . ((n * x) + 0))) by A5, A4, FDIFF_4:37 .= (n * (1 / (n * n))) * (cos . ((n * x) + 0)) .= ((n * 1) / (n * n)) * (cos . ((n * x) + 0)) by XCMPLX_1:74 .= (1 / n) * (cos . ((n * x) + 0)) by A9, XCMPLX_1:91 ; hence (((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) `| REAL) . x = (1 / n) * (cos (n * x)) ; ::_thesis: verum end; A11: dom (cos * (AffineMap (n,0))) = [#] REAL by FUNCT_2:def_1; then A12: cos * (AffineMap (n,0)) is_differentiable_on REAL by A4, FDIFF_4:38; A13: dom ((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0)))) = REAL by FUNCT_2:def_1; then A14: (AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0))) is_differentiable_on REAL by A1, A3, A12, FDIFF_1:21; A15: for x being Real st x in REAL holds (((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0)))) `| REAL) . x = ((1 / n) * (cos . (n * x))) - (x * (sin . (n * x))) proof let x be Real; ::_thesis: ( x in REAL implies (((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0)))) `| REAL) . x = ((1 / n) * (cos . (n * x))) - (x * (sin . (n * x))) ) assume x in REAL ; ::_thesis: (((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0)))) `| REAL) . x = ((1 / n) * (cos . (n * x))) - (x * (sin . (n * x))) (((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0)))) `| REAL) . x = (((cos * (AffineMap (n,0))) . x) * (diff ((AffineMap ((1 / n),0)),x))) + (((AffineMap ((1 / n),0)) . x) * (diff ((cos * (AffineMap (n,0))),x))) by A13, A1, A3, A12, FDIFF_1:21 .= (((cos * (AffineMap (n,0))) . x) * (((AffineMap ((1 / n),0)) `| REAL) . x)) + (((AffineMap ((1 / n),0)) . x) * (diff ((cos * (AffineMap (n,0))),x))) by A3, FDIFF_1:def_7 .= (((cos * (AffineMap (n,0))) . x) * (1 / n)) + (((AffineMap ((1 / n),0)) . x) * (diff ((cos * (AffineMap (n,0))),x))) by A1, A2, FDIFF_1:23 .= (((cos * (AffineMap (n,0))) . x) * (1 / n)) + (((AffineMap ((1 / n),0)) . x) * (((cos * (AffineMap (n,0))) `| REAL) . x)) by A12, FDIFF_1:def_7 .= (((cos * (AffineMap (n,0))) . x) * (1 / n)) + (((AffineMap ((1 / n),0)) . x) * (- (n * (sin . ((n * x) + 0))))) by A11, A4, FDIFF_4:38 .= (((cos * (AffineMap (n,0))) . x) * (1 / n)) + ((((1 / n) * x) + 0) * (- (n * (sin . ((n * x) + 0))))) by FCONT_1:def_4 .= ((cos . ((AffineMap (n,0)) . x)) * (1 / n)) + (((1 / n) * x) * (- (n * (sin . ((n * x) + 0))))) by A11, FUNCT_1:12 .= ((1 / n) * (cos . (n * x))) + (- ((((1 / n) * n) * x) * (sin . (n * x)))) by FCONT_1:def_4 .= ((1 / n) * (cos . (n * x))) + (- ((1 * x) * (sin . (n * x)))) by A9, XCMPLX_1:87 .= ((1 / n) * (cos . (n * x))) - (x * (sin . (n * x))) ; hence (((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0)))) `| REAL) . x = ((1 / n) * (cos . (n * x))) - (x * (sin . (n * x))) ; ::_thesis: verum end; for x being Real st x in REAL holds ((((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) - ((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0))))) `| REAL) . x = x * (sin . (n * x)) proof let x be Real; ::_thesis: ( x in REAL implies ((((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) - ((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0))))) `| REAL) . x = x * (sin . (n * x)) ) assume x in REAL ; ::_thesis: ((((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) - ((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0))))) `| REAL) . x = x * (sin . (n * x)) ((((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) - ((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0))))) `| REAL) . x = (diff (((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))),x)) - (diff (((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0)))),x)) by A1, A8, A14, FDIFF_1:19 .= ((((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) `| REAL) . x) - (diff (((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0)))),x)) by A8, FDIFF_1:def_7 .= ((1 / n) * (cos (n * x))) - (diff (((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0)))),x)) by A10 .= ((1 / n) * (cos (n * x))) - ((((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0)))) `| REAL) . x) by A14, FDIFF_1:def_7 .= ((1 / n) * (cos (n * x))) - (((1 / n) * (cos . (n * x))) - (x * (sin . (n * x)))) by A15 .= x * (sin . (n * x)) ; hence ((((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) - ((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0))))) `| REAL) . x = x * (sin . (n * x)) ; ::_thesis: verum end; hence ( ((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) - ((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0)))) is_differentiable_on REAL & ( for x being Real holds ((((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) - ((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0))))) `| REAL) . x = x * (sin . (n * x)) ) ) by A1, A8, A14, FDIFF_1:19; ::_thesis: verum end; theorem Th9: :: INTEGR11:9 for n being Element of NAT st n <> 0 holds ( ((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0)))) is_differentiable_on REAL & ( for x being Real holds ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) `| REAL) . x = x * (cos . (n * x)) ) ) proof let n be Element of NAT ; ::_thesis: ( n <> 0 implies ( ((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0)))) is_differentiable_on REAL & ( for x being Real holds ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) `| REAL) . x = x * (cos . (n * x)) ) ) ) A1: dom (((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) = [#] REAL by FUNCT_2:def_1; A2: ( dom (AffineMap ((1 / n),0)) = REAL & ( for x being Real st x in REAL holds (AffineMap ((1 / n),0)) . x = ((1 / n) * x) + 0 ) ) by FCONT_1:def_4, FUNCT_2:def_1; then A3: AffineMap ((1 / n),0) is_differentiable_on REAL by A1, FDIFF_1:23; A4: for x being Real st x in REAL holds (AffineMap (n,0)) . x = (n * x) + 0 by FCONT_1:def_4; A5: dom (cos * (AffineMap (n,0))) = [#] REAL by FUNCT_2:def_1; then A6: cos * (AffineMap (n,0)) is_differentiable_on REAL by A4, FDIFF_4:38; A7: dom ((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) = REAL by FUNCT_2:def_1; then A8: (1 / (n ^2)) (#) (cos * (AffineMap (n,0))) is_differentiable_on REAL by A1, A6, FDIFF_1:20; assume A9: n <> 0 ; ::_thesis: ( ((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0)))) is_differentiable_on REAL & ( for x being Real holds ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) `| REAL) . x = x * (cos . (n * x)) ) ) A10: for x being Real st x in REAL holds (((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) `| REAL) . x = - ((1 / n) * (sin (n * x))) proof let x be Real; ::_thesis: ( x in REAL implies (((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) `| REAL) . x = - ((1 / n) * (sin (n * x))) ) assume x in REAL ; ::_thesis: (((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) `| REAL) . x = - ((1 / n) * (sin (n * x))) (((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) `| REAL) . x = (1 / (n ^2)) * (diff ((cos * (AffineMap (n,0))),x)) by A7, A1, A6, FDIFF_1:20 .= (1 / (n ^2)) * (((cos * (AffineMap (n,0))) `| REAL) . x) by A6, FDIFF_1:def_7 .= (1 / (n ^2)) * (- (n * (sin . ((n * x) + 0)))) by A5, A4, FDIFF_4:38 .= ((- 1) * (n * (1 / (n * n)))) * (sin . ((n * x) + 0)) .= ((- 1) * (n / ((n * n) / 1))) * (sin . ((n * x) + 0)) by XCMPLX_1:79 .= ((- 1) * ((n * 1) / (n * n))) * (sin . ((n * x) + 0)) .= ((- 1) * (1 / n)) * (sin . ((n * x) + 0)) by A9, XCMPLX_1:91 ; hence (((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) `| REAL) . x = - ((1 / n) * (sin (n * x))) ; ::_thesis: verum end; A11: dom (sin * (AffineMap (n,0))) = [#] REAL by FUNCT_2:def_1; then A12: sin * (AffineMap (n,0)) is_differentiable_on REAL by A4, FDIFF_4:37; A13: dom ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0)))) = REAL by FUNCT_2:def_1; then A14: (AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))) is_differentiable_on REAL by A1, A3, A12, FDIFF_1:21; A15: for x being Real st x in REAL holds (((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0)))) `| REAL) . x = ((1 / n) * (sin . (n * x))) + (x * (cos . (n * x))) proof let x be Real; ::_thesis: ( x in REAL implies (((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0)))) `| REAL) . x = ((1 / n) * (sin . (n * x))) + (x * (cos . (n * x))) ) assume x in REAL ; ::_thesis: (((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0)))) `| REAL) . x = ((1 / n) * (sin . (n * x))) + (x * (cos . (n * x))) (((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0)))) `| REAL) . x = (((sin * (AffineMap (n,0))) . x) * (diff ((AffineMap ((1 / n),0)),x))) + (((AffineMap ((1 / n),0)) . x) * (diff ((sin * (AffineMap (n,0))),x))) by A13, A1, A3, A12, FDIFF_1:21 .= (((sin * (AffineMap (n,0))) . x) * (((AffineMap ((1 / n),0)) `| REAL) . x)) + (((AffineMap ((1 / n),0)) . x) * (diff ((sin * (AffineMap (n,0))),x))) by A3, FDIFF_1:def_7 .= (((sin * (AffineMap (n,0))) . x) * (1 / n)) + (((AffineMap ((1 / n),0)) . x) * (diff ((sin * (AffineMap (n,0))),x))) by A1, A2, FDIFF_1:23 .= (((sin * (AffineMap (n,0))) . x) * (1 / n)) + (((AffineMap ((1 / n),0)) . x) * (((sin * (AffineMap (n,0))) `| REAL) . x)) by A12, FDIFF_1:def_7 .= (((sin * (AffineMap (n,0))) . x) * (1 / n)) + (((AffineMap ((1 / n),0)) . x) * (n * (cos . ((n * x) + 0)))) by A11, A4, FDIFF_4:37 .= (((sin * (AffineMap (n,0))) . x) * (1 / n)) + ((((1 / n) * x) + 0) * (n * (cos . ((n * x) + 0)))) by FCONT_1:def_4 .= ((sin . ((AffineMap (n,0)) . x)) * (1 / n)) + (((1 / n) * x) * (n * (cos . ((n * x) + 0)))) by A11, FUNCT_1:12 .= ((1 / n) * (sin . (n * x))) + ((((1 / n) * n) * x) * (cos . (n * x))) by FCONT_1:def_4 .= ((1 / n) * (sin . (n * x))) + ((1 * x) * (cos . (n * x))) by A9, XCMPLX_1:87 .= ((1 / n) * (sin . (n * x))) + (x * (cos . (n * x))) ; hence (((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0)))) `| REAL) . x = ((1 / n) * (sin . (n * x))) + (x * (cos . (n * x))) ; ::_thesis: verum end; for x being Real st x in REAL holds ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) `| REAL) . x = x * (cos . (n * x)) proof let x be Real; ::_thesis: ( x in REAL implies ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) `| REAL) . x = x * (cos . (n * x)) ) assume x in REAL ; ::_thesis: ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) `| REAL) . x = x * (cos . (n * x)) ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) `| REAL) . x = (diff (((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))),x)) + (diff (((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0)))),x)) by A1, A8, A14, FDIFF_1:18 .= ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) `| REAL) . x) + (diff (((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0)))),x)) by A8, FDIFF_1:def_7 .= (- ((1 / n) * (sin (n * x)))) + (diff (((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0)))),x)) by A10 .= (- ((1 / n) * (sin (n * x)))) + ((((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0)))) `| REAL) . x) by A14, FDIFF_1:def_7 .= (- ((1 / n) * (sin (n * x)))) + (((1 / n) * (sin . (n * x))) + (x * (cos . (n * x)))) by A15 .= x * (cos . (n * x)) ; hence ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) `| REAL) . x = x * (cos . (n * x)) ; ::_thesis: verum end; hence ( ((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0)))) is_differentiable_on REAL & ( for x being Real holds ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) `| REAL) . x = x * (cos . (n * x)) ) ) by A1, A8, A14, FDIFF_1:18; ::_thesis: verum end; theorem Th10: :: INTEGR11:10 ( ((AffineMap (1,0)) (#) cosh) - sinh is_differentiable_on REAL & ( for x being Real holds ((((AffineMap (1,0)) (#) cosh) - sinh) `| REAL) . x = x * (sinh . x) ) ) proof A1: dom (((AffineMap (1,0)) (#) cosh) - sinh) = [#] REAL by FUNCT_2:def_1; A2: ( dom (AffineMap (1,0)) = [#] REAL & ( for x being Real st x in REAL holds (AffineMap (1,0)) . x = (1 * x) + 0 ) ) by FCONT_1:def_4, FUNCT_2:def_1; then A3: AffineMap (1,0) is_differentiable_on REAL by FDIFF_1:23; A4: dom ((AffineMap (1,0)) (#) cosh) = [#] REAL by FUNCT_2:def_1; then A5: (AffineMap (1,0)) (#) cosh is_differentiable_on REAL by A3, FDIFF_1:21, SIN_COS2:35; A6: for x being Real st x in REAL holds (((AffineMap (1,0)) (#) cosh) `| REAL) . x = (cosh . x) + (x * (sinh . x)) proof let x be Real; ::_thesis: ( x in REAL implies (((AffineMap (1,0)) (#) cosh) `| REAL) . x = (cosh . x) + (x * (sinh . x)) ) assume x in REAL ; ::_thesis: (((AffineMap (1,0)) (#) cosh) `| REAL) . x = (cosh . x) + (x * (sinh . x)) (((AffineMap (1,0)) (#) cosh) `| REAL) . x = ((cosh . x) * (diff ((AffineMap (1,0)),x))) + (((AffineMap (1,0)) . x) * (diff (cosh,x))) by A4, A3, FDIFF_1:21, SIN_COS2:35 .= ((cosh . x) * (((AffineMap (1,0)) `| REAL) . x)) + (((AffineMap (1,0)) . x) * (diff (cosh,x))) by A3, FDIFF_1:def_7 .= ((cosh . x) * 1) + (((AffineMap (1,0)) . x) * (diff (cosh,x))) by A2, FDIFF_1:23 .= (cosh . x) + (((AffineMap (1,0)) . x) * (sinh . x)) by SIN_COS2:35 .= (cosh . x) + (((1 * x) + 0) * (sinh . x)) by FCONT_1:def_4 .= (cosh . x) + (x * (sinh . x)) ; hence (((AffineMap (1,0)) (#) cosh) `| REAL) . x = (cosh . x) + (x * (sinh . x)) ; ::_thesis: verum end; for x being Real st x in REAL holds ((((AffineMap (1,0)) (#) cosh) - sinh) `| REAL) . x = x * (sinh . x) proof let x be Real; ::_thesis: ( x in REAL implies ((((AffineMap (1,0)) (#) cosh) - sinh) `| REAL) . x = x * (sinh . x) ) assume x in REAL ; ::_thesis: ((((AffineMap (1,0)) (#) cosh) - sinh) `| REAL) . x = x * (sinh . x) ((((AffineMap (1,0)) (#) cosh) - sinh) `| REAL) . x = (diff (((AffineMap (1,0)) (#) cosh),x)) - (diff (sinh,x)) by A1, A5, FDIFF_1:19, SIN_COS2:34 .= ((((AffineMap (1,0)) (#) cosh) `| REAL) . x) - (diff (sinh,x)) by A5, FDIFF_1:def_7 .= ((cosh . x) + (x * (sinh . x))) - (diff (sinh,x)) by A6 .= ((cosh . x) + (x * (sinh . x))) - (cosh . x) by SIN_COS2:34 .= x * (sinh . x) ; hence ((((AffineMap (1,0)) (#) cosh) - sinh) `| REAL) . x = x * (sinh . x) ; ::_thesis: verum end; hence ( ((AffineMap (1,0)) (#) cosh) - sinh is_differentiable_on REAL & ( for x being Real holds ((((AffineMap (1,0)) (#) cosh) - sinh) `| REAL) . x = x * (sinh . x) ) ) by A1, A5, FDIFF_1:19, SIN_COS2:34; ::_thesis: verum end; theorem Th11: :: INTEGR11:11 ( ((AffineMap (1,0)) (#) sinh) - cosh is_differentiable_on REAL & ( for x being Real holds ((((AffineMap (1,0)) (#) sinh) - cosh) `| REAL) . x = x * (cosh . x) ) ) proof A1: dom (((AffineMap (1,0)) (#) sinh) - cosh) = [#] REAL by FUNCT_2:def_1; A2: ( dom (AffineMap (1,0)) = [#] REAL & ( for x being Real st x in REAL holds (AffineMap (1,0)) . x = (1 * x) + 0 ) ) by FCONT_1:def_4, FUNCT_2:def_1; then A3: AffineMap (1,0) is_differentiable_on REAL by FDIFF_1:23; A4: dom ((AffineMap (1,0)) (#) sinh) = [#] REAL by FUNCT_2:def_1; then A5: (AffineMap (1,0)) (#) sinh is_differentiable_on REAL by A3, FDIFF_1:21, SIN_COS2:34; A6: for x being Real st x in REAL holds (((AffineMap (1,0)) (#) sinh) `| REAL) . x = (sinh . x) + (x * (cosh . x)) proof let x be Real; ::_thesis: ( x in REAL implies (((AffineMap (1,0)) (#) sinh) `| REAL) . x = (sinh . x) + (x * (cosh . x)) ) assume x in REAL ; ::_thesis: (((AffineMap (1,0)) (#) sinh) `| REAL) . x = (sinh . x) + (x * (cosh . x)) (((AffineMap (1,0)) (#) sinh) `| REAL) . x = ((sinh . x) * (diff ((AffineMap (1,0)),x))) + (((AffineMap (1,0)) . x) * (diff (sinh,x))) by A4, A3, FDIFF_1:21, SIN_COS2:34 .= ((sinh . x) * (((AffineMap (1,0)) `| REAL) . x)) + (((AffineMap (1,0)) . x) * (diff (sinh,x))) by A3, FDIFF_1:def_7 .= ((sinh . x) * 1) + (((AffineMap (1,0)) . x) * (diff (sinh,x))) by A2, FDIFF_1:23 .= (sinh . x) + (((AffineMap (1,0)) . x) * (cosh . x)) by SIN_COS2:34 .= (sinh . x) + (((1 * x) + 0) * (cosh . x)) by FCONT_1:def_4 .= (sinh . x) + (x * (cosh . x)) ; hence (((AffineMap (1,0)) (#) sinh) `| REAL) . x = (sinh . x) + (x * (cosh . x)) ; ::_thesis: verum end; for x being Real st x in REAL holds ((((AffineMap (1,0)) (#) sinh) - cosh) `| REAL) . x = x * (cosh . x) proof let x be Real; ::_thesis: ( x in REAL implies ((((AffineMap (1,0)) (#) sinh) - cosh) `| REAL) . x = x * (cosh . x) ) assume x in REAL ; ::_thesis: ((((AffineMap (1,0)) (#) sinh) - cosh) `| REAL) . x = x * (cosh . x) ((((AffineMap (1,0)) (#) sinh) - cosh) `| REAL) . x = (diff (((AffineMap (1,0)) (#) sinh),x)) - (diff (cosh,x)) by A1, A5, FDIFF_1:19, SIN_COS2:35 .= ((((AffineMap (1,0)) (#) sinh) `| REAL) . x) - (diff (cosh,x)) by A5, FDIFF_1:def_7 .= ((sinh . x) + (x * (cosh . x))) - (diff (cosh,x)) by A6 .= ((sinh . x) + (x * (cosh . x))) - (sinh . x) by SIN_COS2:35 .= x * (cosh . x) ; hence ((((AffineMap (1,0)) (#) sinh) - cosh) `| REAL) . x = x * (cosh . x) ; ::_thesis: verum end; hence ( ((AffineMap (1,0)) (#) sinh) - cosh is_differentiable_on REAL & ( for x being Real holds ((((AffineMap (1,0)) (#) sinh) - cosh) `| REAL) . x = x * (cosh . x) ) ) by A1, A5, FDIFF_1:19, SIN_COS2:35; ::_thesis: verum end; theorem Th12: :: INTEGR11:12 for a, b being Real for n being Element of NAT st a * (n + 1) <> 0 holds ( (1 / (a * (n + 1))) (#) ((#Z (n + 1)) * (AffineMap (a,b))) is_differentiable_on REAL & ( for x being Real holds (((1 / (a * (n + 1))) (#) ((#Z (n + 1)) * (AffineMap (a,b)))) `| REAL) . x = ((a * x) + b) #Z n ) ) proof let a, b be Real; ::_thesis: for n being Element of NAT st a * (n + 1) <> 0 holds ( (1 / (a * (n + 1))) (#) ((#Z (n + 1)) * (AffineMap (a,b))) is_differentiable_on REAL & ( for x being Real holds (((1 / (a * (n + 1))) (#) ((#Z (n + 1)) * (AffineMap (a,b)))) `| REAL) . x = ((a * x) + b) #Z n ) ) let n be Element of NAT ; ::_thesis: ( a * (n + 1) <> 0 implies ( (1 / (a * (n + 1))) (#) ((#Z (n + 1)) * (AffineMap (a,b))) is_differentiable_on REAL & ( for x being Real holds (((1 / (a * (n + 1))) (#) ((#Z (n + 1)) * (AffineMap (a,b)))) `| REAL) . x = ((a * x) + b) #Z n ) ) ) A1: [#] REAL = dom ((1 / (a * (n + 1))) (#) ((#Z (n + 1)) * (AffineMap (a,b)))) by FUNCT_2:def_1; A2: [#] REAL = dom (AffineMap (a,b)) by FUNCT_2:def_1; A3: for x being Real st x in REAL holds (AffineMap (a,b)) . x = (a * x) + b by FCONT_1:def_4; then A4: AffineMap (a,b) is_differentiable_on REAL by A2, FDIFF_1:23; for x being Real holds (#Z (n + 1)) * (AffineMap (a,b)) is_differentiable_in x proof let x be Real; ::_thesis: (#Z (n + 1)) * (AffineMap (a,b)) is_differentiable_in x AffineMap (a,b) is_differentiable_in x by A2, A4, FDIFF_1:9; hence (#Z (n + 1)) * (AffineMap (a,b)) is_differentiable_in x by TAYLOR_1:3; ::_thesis: verum end; then ( [#] REAL = dom ((#Z (n + 1)) * (AffineMap (a,b))) & ( for x being Real st x in REAL holds (#Z (n + 1)) * (AffineMap (a,b)) is_differentiable_in x ) ) by FUNCT_2:def_1; then A5: (#Z (n + 1)) * (AffineMap (a,b)) is_differentiable_on REAL by FDIFF_1:9; A6: for x being Real st x in REAL holds (((#Z (n + 1)) * (AffineMap (a,b))) `| REAL) . x = (a * (n + 1)) * (((AffineMap (a,b)) . x) #Z n) proof set m = n + 1; let x be Real; ::_thesis: ( x in REAL implies (((#Z (n + 1)) * (AffineMap (a,b))) `| REAL) . x = (a * (n + 1)) * (((AffineMap (a,b)) . x) #Z n) ) assume x in REAL ; ::_thesis: (((#Z (n + 1)) * (AffineMap (a,b))) `| REAL) . x = (a * (n + 1)) * (((AffineMap (a,b)) . x) #Z n) AffineMap (a,b) is_differentiable_in x by A2, A4, FDIFF_1:9; then diff (((#Z (n + 1)) * (AffineMap (a,b))),x) = ((n + 1) * (((AffineMap (a,b)) . x) #Z ((n + 1) - 1))) * (diff ((AffineMap (a,b)),x)) by TAYLOR_1:3 .= ((n + 1) * (((AffineMap (a,b)) . x) #Z ((n + 1) - 1))) * (((AffineMap (a,b)) `| REAL) . x) by A4, FDIFF_1:def_7 .= ((n + 1) * (((AffineMap (a,b)) . x) #Z ((n + 1) - 1))) * a by A2, A3, FDIFF_1:23 ; hence (((#Z (n + 1)) * (AffineMap (a,b))) `| REAL) . x = (a * (n + 1)) * (((AffineMap (a,b)) . x) #Z n) by A5, FDIFF_1:def_7; ::_thesis: verum end; assume A7: a * (n + 1) <> 0 ; ::_thesis: ( (1 / (a * (n + 1))) (#) ((#Z (n + 1)) * (AffineMap (a,b))) is_differentiable_on REAL & ( for x being Real holds (((1 / (a * (n + 1))) (#) ((#Z (n + 1)) * (AffineMap (a,b)))) `| REAL) . x = ((a * x) + b) #Z n ) ) for x being Real st x in REAL holds (((1 / (a * (n + 1))) (#) ((#Z (n + 1)) * (AffineMap (a,b)))) `| REAL) . x = ((a * x) + b) #Z n proof let x be Real; ::_thesis: ( x in REAL implies (((1 / (a * (n + 1))) (#) ((#Z (n + 1)) * (AffineMap (a,b)))) `| REAL) . x = ((a * x) + b) #Z n ) assume x in REAL ; ::_thesis: (((1 / (a * (n + 1))) (#) ((#Z (n + 1)) * (AffineMap (a,b)))) `| REAL) . x = ((a * x) + b) #Z n (((1 / (a * (n + 1))) (#) ((#Z (n + 1)) * (AffineMap (a,b)))) `| REAL) . x = (1 / (a * (n + 1))) * (diff (((#Z (n + 1)) * (AffineMap (a,b))),x)) by A1, A5, FDIFF_1:20 .= (1 / (a * (n + 1))) * ((((#Z (n + 1)) * (AffineMap (a,b))) `| REAL) . x) by A5, FDIFF_1:def_7 .= (1 / (a * (n + 1))) * ((a * (n + 1)) * (((AffineMap (a,b)) . x) #Z n)) by A6 .= ((1 / (a * (n + 1))) * (a * (n + 1))) * (((AffineMap (a,b)) . x) #Z n) .= ((a * (n + 1)) / (a * (n + 1))) * (((AffineMap (a,b)) . x) #Z n) by XCMPLX_1:99 .= 1 * (((AffineMap (a,b)) . x) #Z n) by A7, XCMPLX_1:60 .= ((a * x) + b) #Z n by FCONT_1:def_4 ; hence (((1 / (a * (n + 1))) (#) ((#Z (n + 1)) * (AffineMap (a,b)))) `| REAL) . x = ((a * x) + b) #Z n ; ::_thesis: verum end; hence ( (1 / (a * (n + 1))) (#) ((#Z (n + 1)) * (AffineMap (a,b))) is_differentiable_on REAL & ( for x being Real holds (((1 / (a * (n + 1))) (#) ((#Z (n + 1)) * (AffineMap (a,b)))) `| REAL) . x = ((a * x) + b) #Z n ) ) by A1, A5, FDIFF_1:20; ::_thesis: verum end; begin theorem Th13: :: INTEGR11:13 for A being non empty closed_interval Subset of REAL holds integral ((sin ^2),A) = (((AffineMap ((1 / 2),0)) - ((1 / 4) (#) (sin * (AffineMap (2,0))))) . (upper_bound A)) - (((AffineMap ((1 / 2),0)) - ((1 / 4) (#) (sin * (AffineMap (2,0))))) . (lower_bound A)) proof let A be non empty closed_interval Subset of REAL; ::_thesis: integral ((sin ^2),A) = (((AffineMap ((1 / 2),0)) - ((1 / 4) (#) (sin * (AffineMap (2,0))))) . (upper_bound A)) - (((AffineMap ((1 / 2),0)) - ((1 / 4) (#) (sin * (AffineMap (2,0))))) . (lower_bound A)) A1: for x being Real st x in dom (((AffineMap ((1 / 2),0)) - ((1 / 4) (#) (sin * (AffineMap (2,0))))) `| REAL) holds (((AffineMap ((1 / 2),0)) - ((1 / 4) (#) (sin * (AffineMap (2,0))))) `| REAL) . x = (sin ^2) . x proof let x be Real; ::_thesis: ( x in dom (((AffineMap ((1 / 2),0)) - ((1 / 4) (#) (sin * (AffineMap (2,0))))) `| REAL) implies (((AffineMap ((1 / 2),0)) - ((1 / 4) (#) (sin * (AffineMap (2,0))))) `| REAL) . x = (sin ^2) . x ) assume x in dom (((AffineMap ((1 / 2),0)) - ((1 / 4) (#) (sin * (AffineMap (2,0))))) `| REAL) ; ::_thesis: (((AffineMap ((1 / 2),0)) - ((1 / 4) (#) (sin * (AffineMap (2,0))))) `| REAL) . x = (sin ^2) . x (((AffineMap ((1 / 2),0)) - ((1 / 4) (#) (sin * (AffineMap (2,0))))) `| REAL) . x = (sin . x) ^2 by Th1 .= (sin ^2) . x by VALUED_1:11 ; hence (((AffineMap ((1 / 2),0)) - ((1 / 4) (#) (sin * (AffineMap (2,0))))) `| REAL) . x = (sin ^2) . x ; ::_thesis: verum end; A2: dom (sin ^2) = REAL by SIN_COS:24, VALUED_1:11; then dom (((AffineMap ((1 / 2),0)) - ((1 / 4) (#) (sin * (AffineMap (2,0))))) `| REAL) = dom (sin ^2) by Th1, FDIFF_1:def_7; then A3: ((AffineMap ((1 / 2),0)) - ((1 / 4) (#) (sin * (AffineMap (2,0))))) `| REAL = sin ^2 by A1, PARTFUN1:5; (sin ^2) | A is bounded by A2, INTEGRA5:10; hence integral ((sin ^2),A) = (((AffineMap ((1 / 2),0)) - ((1 / 4) (#) (sin * (AffineMap (2,0))))) . (upper_bound A)) - (((AffineMap ((1 / 2),0)) - ((1 / 4) (#) (sin * (AffineMap (2,0))))) . (lower_bound A)) by A2, A3, Th1, INTEGRA5:11, INTEGRA5:13; ::_thesis: verum end; Lm4: dom (AffineMap (2,0)) = [#] REAL by FUNCT_2:def_1; Lm5: dom ((AffineMap ((1 / 2),0)) - ((1 / 4) (#) (sin * (AffineMap (2,0))))) = REAL by FUNCT_2:def_1; theorem :: INTEGR11:14 for A being non empty closed_interval Subset of REAL st A = [.0,PI.] holds integral ((sin ^2),A) = PI / 2 proof let A be non empty closed_interval Subset of REAL; ::_thesis: ( A = [.0,PI.] implies integral ((sin ^2),A) = PI / 2 ) assume A = [.0,PI.] ; ::_thesis: integral ((sin ^2),A) = PI / 2 then ( upper_bound A = PI & lower_bound A = 0 ) by INTEGRA8:37; then integral ((sin ^2),A) = (((AffineMap ((1 / 2),0)) - ((1 / 4) (#) (sin * (AffineMap (2,0))))) . PI) - (((AffineMap ((1 / 2),0)) - ((1 / 4) (#) (sin * (AffineMap (2,0))))) . 0) by Th13 .= (((AffineMap ((1 / 2),0)) . PI) - (((1 / 4) (#) (sin * (AffineMap (2,0)))) . PI)) - (((AffineMap ((1 / 2),0)) - ((1 / 4) (#) (sin * (AffineMap (2,0))))) . 0) by Lm5, VALUED_1:13 .= (((AffineMap ((1 / 2),0)) . PI) - (((1 / 4) (#) (sin * (AffineMap (2,0)))) . PI)) - (((AffineMap ((1 / 2),0)) . 0) - (((1 / 4) (#) (sin * (AffineMap (2,0)))) . 0)) by Lm5, VALUED_1:13 .= ((((1 / 2) * PI) + 0) - (((1 / 4) (#) (sin * (AffineMap (2,0)))) . PI)) - (((AffineMap ((1 / 2),0)) . 0) - (((1 / 4) (#) (sin * (AffineMap (2,0)))) . 0)) by FCONT_1:def_4 .= (((1 / 2) * PI) - ((1 / 4) * ((sin * (AffineMap (2,0))) . PI))) - (((AffineMap ((1 / 2),0)) . 0) - (((1 / 4) (#) (sin * (AffineMap (2,0)))) . 0)) by VALUED_1:6 .= (((1 / 2) * PI) - ((1 / 4) * (sin . ((AffineMap (2,0)) . PI)))) - (((AffineMap ((1 / 2),0)) . 0) - (((1 / 4) (#) (sin * (AffineMap (2,0)))) . 0)) by Lm4, FUNCT_1:13 .= (((1 / 2) * PI) - ((1 / 4) * (sin . ((2 * PI) + 0)))) - (((AffineMap ((1 / 2),0)) . 0) - (((1 / 4) (#) (sin * (AffineMap (2,0)))) . 0)) by FCONT_1:def_4 .= (((1 / 2) * PI) - ((1 / 4) * (sin . ((2 * PI) + 0)))) - (0 - (((1 / 4) (#) (sin * (AffineMap (2,0)))) . 0)) by FCONT_1:48 .= ((((1 / 2) * PI) - ((1 / 4) * (sin . ((2 * PI) + 0)))) - 0) + (((1 / 4) (#) (sin * (AffineMap (2,0)))) . 0) .= (((1 / 2) * PI) - ((1 / 4) * (sin . ((2 * PI) + 0)))) + ((1 / 4) * ((sin * (AffineMap (2,0))) . 0)) by VALUED_1:6 .= (((1 / 2) * PI) - ((1 / 4) * (sin . ((2 * PI) + 0)))) + ((1 / 4) * (sin . ((AffineMap (2,0)) . 0))) by Lm4, FUNCT_1:13 .= (((1 / 2) * PI) - ((1 / 4) * (sin . (0 + ((2 * PI) * 1))))) + ((1 / 4) * (sin . 0)) by FCONT_1:48 .= (((1 / 2) * PI) - ((1 / 4) * (sin . 0))) + ((1 / 4) * (sin . 0)) by SIN_COS6:8 .= PI / 2 ; hence integral ((sin ^2),A) = PI / 2 ; ::_thesis: verum end; theorem :: INTEGR11:15 for A being non empty closed_interval Subset of REAL st A = [.0,(2 * PI).] holds integral ((sin ^2),A) = PI proof let A be non empty closed_interval Subset of REAL; ::_thesis: ( A = [.0,(2 * PI).] implies integral ((sin ^2),A) = PI ) assume A = [.0,(2 * PI).] ; ::_thesis: integral ((sin ^2),A) = PI then ( upper_bound A = 2 * PI & lower_bound A = 0 ) by INTEGRA8:37; then integral ((sin ^2),A) = (((AffineMap ((1 / 2),0)) - ((1 / 4) (#) (sin * (AffineMap (2,0))))) . (2 * PI)) - (((AffineMap ((1 / 2),0)) - ((1 / 4) (#) (sin * (AffineMap (2,0))))) . 0) by Th13 .= (((AffineMap ((1 / 2),0)) . (2 * PI)) - (((1 / 4) (#) (sin * (AffineMap (2,0)))) . (2 * PI))) - (((AffineMap ((1 / 2),0)) - ((1 / 4) (#) (sin * (AffineMap (2,0))))) . 0) by Lm5, VALUED_1:13 .= (((AffineMap ((1 / 2),0)) . (2 * PI)) - (((1 / 4) (#) (sin * (AffineMap (2,0)))) . (2 * PI))) - (((AffineMap ((1 / 2),0)) . 0) - (((1 / 4) (#) (sin * (AffineMap (2,0)))) . 0)) by Lm5, VALUED_1:13 .= ((((1 / 2) * (2 * PI)) + 0) - (((1 / 4) (#) (sin * (AffineMap (2,0)))) . (2 * PI))) - (((AffineMap ((1 / 2),0)) . 0) - (((1 / 4) (#) (sin * (AffineMap (2,0)))) . 0)) by FCONT_1:def_4 .= (((1 / 2) * (2 * PI)) - ((1 / 4) * ((sin * (AffineMap (2,0))) . (2 * PI)))) - (((AffineMap ((1 / 2),0)) . 0) - (((1 / 4) (#) (sin * (AffineMap (2,0)))) . 0)) by VALUED_1:6 .= (((1 / 2) * (2 * PI)) - ((1 / 4) * (sin . ((AffineMap (2,0)) . (2 * PI))))) - (((AffineMap ((1 / 2),0)) . 0) - (((1 / 4) (#) (sin * (AffineMap (2,0)))) . 0)) by Lm4, FUNCT_1:13 .= (((1 / 2) * (2 * PI)) - ((1 / 4) * (sin . ((2 * (2 * PI)) + 0)))) - (((AffineMap ((1 / 2),0)) . 0) - (((1 / 4) (#) (sin * (AffineMap (2,0)))) . 0)) by FCONT_1:def_4 .= (((1 / 2) * (2 * PI)) - ((1 / 4) * (sin . (((2 * 2) * PI) + 0)))) - (0 - (((1 / 4) (#) (sin * (AffineMap (2,0)))) . 0)) by FCONT_1:48 .= ((((1 / 2) * (2 * PI)) - ((1 / 4) * (sin . (((2 * 2) * PI) + 0)))) - 0) + (((1 / 4) (#) (sin * (AffineMap (2,0)))) . 0) .= (((1 / 2) * (2 * PI)) - ((1 / 4) * (sin . (((2 * 2) * PI) + 0)))) + ((1 / 4) * ((sin * (AffineMap (2,0))) . 0)) by VALUED_1:6 .= (((1 / 2) * (2 * PI)) - ((1 / 4) * (sin . (((2 * 2) * PI) + 0)))) + ((1 / 4) * (sin . ((AffineMap (2,0)) . 0))) by Lm4, FUNCT_1:13 .= (((1 / 2) * (2 * PI)) - ((1 / 4) * (sin . (0 + ((2 * PI) * 2))))) + ((1 / 4) * (sin . 0)) by FCONT_1:48 .= (((1 / 2) * (2 * PI)) - ((1 / 4) * (sin . 0))) + ((1 / 4) * (sin . 0)) by SIN_COS6:8 .= PI ; hence integral ((sin ^2),A) = PI ; ::_thesis: verum end; theorem Th16: :: INTEGR11:16 for A being non empty closed_interval Subset of REAL holds integral ((cos ^2),A) = (((AffineMap ((1 / 2),0)) + ((1 / 4) (#) (sin * (AffineMap (2,0))))) . (upper_bound A)) - (((AffineMap ((1 / 2),0)) + ((1 / 4) (#) (sin * (AffineMap (2,0))))) . (lower_bound A)) proof let A be non empty closed_interval Subset of REAL; ::_thesis: integral ((cos ^2),A) = (((AffineMap ((1 / 2),0)) + ((1 / 4) (#) (sin * (AffineMap (2,0))))) . (upper_bound A)) - (((AffineMap ((1 / 2),0)) + ((1 / 4) (#) (sin * (AffineMap (2,0))))) . (lower_bound A)) A1: for x being Real st x in dom (((AffineMap ((1 / 2),0)) + ((1 / 4) (#) (sin * (AffineMap (2,0))))) `| REAL) holds (((AffineMap ((1 / 2),0)) + ((1 / 4) (#) (sin * (AffineMap (2,0))))) `| REAL) . x = (cos ^2) . x proof let x be Real; ::_thesis: ( x in dom (((AffineMap ((1 / 2),0)) + ((1 / 4) (#) (sin * (AffineMap (2,0))))) `| REAL) implies (((AffineMap ((1 / 2),0)) + ((1 / 4) (#) (sin * (AffineMap (2,0))))) `| REAL) . x = (cos ^2) . x ) assume x in dom (((AffineMap ((1 / 2),0)) + ((1 / 4) (#) (sin * (AffineMap (2,0))))) `| REAL) ; ::_thesis: (((AffineMap ((1 / 2),0)) + ((1 / 4) (#) (sin * (AffineMap (2,0))))) `| REAL) . x = (cos ^2) . x (((AffineMap ((1 / 2),0)) + ((1 / 4) (#) (sin * (AffineMap (2,0))))) `| REAL) . x = (cos . x) ^2 by Th2 .= (cos ^2) . x by VALUED_1:11 ; hence (((AffineMap ((1 / 2),0)) + ((1 / 4) (#) (sin * (AffineMap (2,0))))) `| REAL) . x = (cos ^2) . x ; ::_thesis: verum end; A2: dom (cos ^2) = REAL by SIN_COS:24, VALUED_1:11; then dom (((AffineMap ((1 / 2),0)) + ((1 / 4) (#) (sin * (AffineMap (2,0))))) `| REAL) = dom (cos ^2) by Th2, FDIFF_1:def_7; then A3: ((AffineMap ((1 / 2),0)) + ((1 / 4) (#) (sin * (AffineMap (2,0))))) `| REAL = cos ^2 by A1, PARTFUN1:5; (cos ^2) | A is bounded by A2, INTEGRA5:10; hence integral ((cos ^2),A) = (((AffineMap ((1 / 2),0)) + ((1 / 4) (#) (sin * (AffineMap (2,0))))) . (upper_bound A)) - (((AffineMap ((1 / 2),0)) + ((1 / 4) (#) (sin * (AffineMap (2,0))))) . (lower_bound A)) by A2, A3, Th2, INTEGRA5:11, INTEGRA5:13; ::_thesis: verum end; theorem :: INTEGR11:17 for A being non empty closed_interval Subset of REAL st A = [.0,PI.] holds integral ((cos ^2),A) = PI / 2 proof let A be non empty closed_interval Subset of REAL; ::_thesis: ( A = [.0,PI.] implies integral ((cos ^2),A) = PI / 2 ) assume A = [.0,PI.] ; ::_thesis: integral ((cos ^2),A) = PI / 2 then ( upper_bound A = PI & lower_bound A = 0 ) by INTEGRA8:37; then integral ((cos ^2),A) = (((AffineMap ((1 / 2),0)) + ((1 / 4) (#) (sin * (AffineMap (2,0))))) . PI) - (((AffineMap ((1 / 2),0)) + ((1 / 4) (#) (sin * (AffineMap (2,0))))) . 0) by Th16 .= (((AffineMap ((1 / 2),0)) . PI) + (((1 / 4) (#) (sin * (AffineMap (2,0)))) . PI)) - (((AffineMap ((1 / 2),0)) + ((1 / 4) (#) (sin * (AffineMap (2,0))))) . 0) by VALUED_1:1 .= (((AffineMap ((1 / 2),0)) . PI) + (((1 / 4) (#) (sin * (AffineMap (2,0)))) . PI)) - (((AffineMap ((1 / 2),0)) . 0) + (((1 / 4) (#) (sin * (AffineMap (2,0)))) . 0)) by VALUED_1:1 .= ((((1 / 2) * PI) + 0) + (((1 / 4) (#) (sin * (AffineMap (2,0)))) . PI)) - (((AffineMap ((1 / 2),0)) . 0) + (((1 / 4) (#) (sin * (AffineMap (2,0)))) . 0)) by FCONT_1:def_4 .= (((1 / 2) * PI) + ((1 / 4) * ((sin * (AffineMap (2,0))) . PI))) - (((AffineMap ((1 / 2),0)) . 0) + (((1 / 4) (#) (sin * (AffineMap (2,0)))) . 0)) by VALUED_1:6 .= (((1 / 2) * PI) + ((1 / 4) * (sin . ((AffineMap (2,0)) . PI)))) - (((AffineMap ((1 / 2),0)) . 0) + (((1 / 4) (#) (sin * (AffineMap (2,0)))) . 0)) by Lm4, FUNCT_1:13 .= (((1 / 2) * PI) + ((1 / 4) * (sin . ((2 * PI) + 0)))) - (((AffineMap ((1 / 2),0)) . 0) + (((1 / 4) (#) (sin * (AffineMap (2,0)))) . 0)) by FCONT_1:def_4 .= (((1 / 2) * PI) + ((1 / 4) * (sin . ((2 * PI) + 0)))) - (0 + (((1 / 4) (#) (sin * (AffineMap (2,0)))) . 0)) by FCONT_1:48 .= (((1 / 2) * PI) + ((1 / 4) * (sin . ((2 * PI) + 0)))) - ((1 / 4) * ((sin * (AffineMap (2,0))) . 0)) by VALUED_1:6 .= (((1 / 2) * PI) + ((1 / 4) * (sin . ((2 * PI) + 0)))) - ((1 / 4) * (sin . ((AffineMap (2,0)) . 0))) by Lm4, FUNCT_1:13 .= (((1 / 2) * PI) + ((1 / 4) * (sin . (0 + ((2 * PI) * 1))))) - ((1 / 4) * (sin . 0)) by FCONT_1:48 .= (((1 / 2) * PI) + ((1 / 4) * (sin . 0))) - ((1 / 4) * (sin . 0)) by SIN_COS6:8 .= PI / 2 ; hence integral ((cos ^2),A) = PI / 2 ; ::_thesis: verum end; theorem :: INTEGR11:18 for A being non empty closed_interval Subset of REAL st A = [.0,(2 * PI).] holds integral ((cos ^2),A) = PI proof let A be non empty closed_interval Subset of REAL; ::_thesis: ( A = [.0,(2 * PI).] implies integral ((cos ^2),A) = PI ) assume A = [.0,(2 * PI).] ; ::_thesis: integral ((cos ^2),A) = PI then ( upper_bound A = 2 * PI & lower_bound A = 0 ) by INTEGRA8:37; then integral ((cos ^2),A) = (((AffineMap ((1 / 2),0)) + ((1 / 4) (#) (sin * (AffineMap (2,0))))) . (2 * PI)) - (((AffineMap ((1 / 2),0)) + ((1 / 4) (#) (sin * (AffineMap (2,0))))) . 0) by Th16 .= (((AffineMap ((1 / 2),0)) . (2 * PI)) + (((1 / 4) (#) (sin * (AffineMap (2,0)))) . (2 * PI))) - (((AffineMap ((1 / 2),0)) + ((1 / 4) (#) (sin * (AffineMap (2,0))))) . 0) by VALUED_1:1 .= (((AffineMap ((1 / 2),0)) . (2 * PI)) + (((1 / 4) (#) (sin * (AffineMap (2,0)))) . (2 * PI))) - (((AffineMap ((1 / 2),0)) . 0) + (((1 / 4) (#) (sin * (AffineMap (2,0)))) . 0)) by VALUED_1:1 .= ((((1 / 2) * (2 * PI)) + 0) + (((1 / 4) (#) (sin * (AffineMap (2,0)))) . (2 * PI))) - (((AffineMap ((1 / 2),0)) . 0) + (((1 / 4) (#) (sin * (AffineMap (2,0)))) . 0)) by FCONT_1:def_4 .= (((1 / 2) * (2 * PI)) + ((1 / 4) * ((sin * (AffineMap (2,0))) . (2 * PI)))) - (((AffineMap ((1 / 2),0)) . 0) + (((1 / 4) (#) (sin * (AffineMap (2,0)))) . 0)) by VALUED_1:6 .= (((1 / 2) * (2 * PI)) + ((1 / 4) * (sin . ((AffineMap (2,0)) . (2 * PI))))) - (((AffineMap ((1 / 2),0)) . 0) + (((1 / 4) (#) (sin * (AffineMap (2,0)))) . 0)) by Lm4, FUNCT_1:13 .= (((1 / 2) * (2 * PI)) + ((1 / 4) * (sin . ((2 * (2 * PI)) + 0)))) - (((AffineMap ((1 / 2),0)) . 0) + (((1 / 4) (#) (sin * (AffineMap (2,0)))) . 0)) by FCONT_1:def_4 .= (((1 / 2) * (2 * PI)) + ((1 / 4) * (sin . (((2 * 2) * PI) + 0)))) - (0 + (((1 / 4) (#) (sin * (AffineMap (2,0)))) . 0)) by FCONT_1:48 .= (((1 / 2) * (2 * PI)) + ((1 / 4) * (sin . (((2 * 2) * PI) + 0)))) - ((1 / 4) * ((sin * (AffineMap (2,0))) . 0)) by VALUED_1:6 .= (((1 / 2) * (2 * PI)) + ((1 / 4) * (sin . (((2 * 2) * PI) + 0)))) - ((1 / 4) * (sin . ((AffineMap (2,0)) . 0))) by Lm4, FUNCT_1:13 .= (((1 / 2) * (2 * PI)) + ((1 / 4) * (sin . (0 + ((2 * PI) * 2))))) - ((1 / 4) * (sin . 0)) by FCONT_1:48 .= (((1 / 2) * (2 * PI)) + ((1 / 4) * (sin . 0))) - ((1 / 4) * (sin . 0)) by SIN_COS6:8 .= PI ; hence integral ((cos ^2),A) = PI ; ::_thesis: verum end; theorem Th19: :: INTEGR11:19 for n being Element of NAT for A being non empty closed_interval Subset of REAL holds integral ((((#Z n) * sin) (#) cos),A) = (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) . (upper_bound A)) - (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) . (lower_bound A)) proof let n be Element of NAT ; ::_thesis: for A being non empty closed_interval Subset of REAL holds integral ((((#Z n) * sin) (#) cos),A) = (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) . (upper_bound A)) - (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) . (lower_bound A)) let A be non empty closed_interval Subset of REAL; ::_thesis: integral ((((#Z n) * sin) (#) cos),A) = (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) . (upper_bound A)) - (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) . (lower_bound A)) A1: [#] REAL = dom (((#Z n) * sin) (#) cos) by FUNCT_2:def_1; A2: for x being Real st x in dom (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) `| REAL) holds (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) `| REAL) . x = (((#Z n) * sin) (#) cos) . x proof let x be Real; ::_thesis: ( x in dom (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) `| REAL) implies (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) `| REAL) . x = (((#Z n) * sin) (#) cos) . x ) assume x in dom (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) `| REAL) ; ::_thesis: (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) `| REAL) . x = (((#Z n) * sin) (#) cos) . x (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) `| REAL) . x = ((sin . x) #Z n) * (cos . x) by Th3 .= ((#Z n) . (sin . x)) * (cos . x) by TAYLOR_1:def_1 .= (((#Z n) * sin) . x) * (cos . x) by FUNCT_1:13, SIN_COS:24 .= (((#Z n) * sin) (#) cos) . x by A1, VALUED_1:def_4 ; hence (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) `| REAL) . x = (((#Z n) * sin) (#) cos) . x ; ::_thesis: verum end; for x0 being Real holds (#Z n) * sin is_differentiable_in x0 proof let x0 be Real; ::_thesis: (#Z n) * sin is_differentiable_in x0 sin is_differentiable_in x0 by SIN_COS:64; hence (#Z n) * sin is_differentiable_in x0 by TAYLOR_1:3; ::_thesis: verum end; then ( dom ((#Z n) * sin) = REAL & ( for x0 being Real st x0 in REAL holds (#Z n) * sin is_differentiable_in x0 ) ) by FUNCT_2:def_1; then (#Z n) * sin is_differentiable_on REAL by A1, FDIFF_1:9; then A3: (((#Z n) * sin) (#) cos) | REAL is continuous by A1, FDIFF_1:21, FDIFF_1:25, SIN_COS:67; then (((#Z n) * sin) (#) cos) | A is continuous by FCONT_1:16; then A4: ((#Z n) * sin) (#) cos is_integrable_on A by A1, INTEGRA5:11; (1 / (n + 1)) (#) ((#Z (n + 1)) * sin) is_differentiable_on REAL by Th3; then dom (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) `| REAL) = dom (((#Z n) * sin) (#) cos) by A1, FDIFF_1:def_7; then A5: ((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) `| REAL = ((#Z n) * sin) (#) cos by A2, PARTFUN1:5; (((#Z n) * sin) (#) cos) | A is bounded by A1, A3, FCONT_1:16, INTEGRA5:10; hence integral ((((#Z n) * sin) (#) cos),A) = (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) . (upper_bound A)) - (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) . (lower_bound A)) by A4, A5, Th3, INTEGRA5:13; ::_thesis: verum end; theorem :: INTEGR11:20 for n being Element of NAT for A being non empty closed_interval Subset of REAL st A = [.0,PI.] holds integral ((((#Z n) * sin) (#) cos),A) = 0 proof let n be Element of NAT ; ::_thesis: for A being non empty closed_interval Subset of REAL st A = [.0,PI.] holds integral ((((#Z n) * sin) (#) cos),A) = 0 let A be non empty closed_interval Subset of REAL; ::_thesis: ( A = [.0,PI.] implies integral ((((#Z n) * sin) (#) cos),A) = 0 ) assume A = [.0,PI.] ; ::_thesis: integral ((((#Z n) * sin) (#) cos),A) = 0 then ( upper_bound A = PI & lower_bound A = 0 ) by INTEGRA8:37; then integral ((((#Z n) * sin) (#) cos),A) = (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) . PI) - (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) . 0) by Th19 .= ((1 / (n + 1)) * (((#Z (n + 1)) * sin) . PI)) - (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) . 0) by VALUED_1:6 .= ((1 / (n + 1)) * (((#Z (n + 1)) * sin) . PI)) - ((1 / (n + 1)) * (((#Z (n + 1)) * sin) . 0)) by VALUED_1:6 .= ((1 / (n + 1)) * ((#Z (n + 1)) . (sin . PI))) - ((1 / (n + 1)) * (((#Z (n + 1)) * sin) . 0)) by FUNCT_1:13, SIN_COS:24 .= ((1 / (n + 1)) * ((#Z (n + 1)) . (sin . PI))) - ((1 / (n + 1)) * ((#Z (n + 1)) . (sin . 0))) by FUNCT_1:13, SIN_COS:24 .= 0 by SIN_COS:30, SIN_COS:76 ; hence integral ((((#Z n) * sin) (#) cos),A) = 0 ; ::_thesis: verum end; theorem :: INTEGR11:21 for n being Element of NAT for A being non empty closed_interval Subset of REAL st A = [.0,(2 * PI).] holds integral ((((#Z n) * sin) (#) cos),A) = 0 proof let n be Element of NAT ; ::_thesis: for A being non empty closed_interval Subset of REAL st A = [.0,(2 * PI).] holds integral ((((#Z n) * sin) (#) cos),A) = 0 let A be non empty closed_interval Subset of REAL; ::_thesis: ( A = [.0,(2 * PI).] implies integral ((((#Z n) * sin) (#) cos),A) = 0 ) assume A = [.0,(2 * PI).] ; ::_thesis: integral ((((#Z n) * sin) (#) cos),A) = 0 then ( upper_bound A = 2 * PI & lower_bound A = 0 ) by INTEGRA8:37; then integral ((((#Z n) * sin) (#) cos),A) = (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) . (2 * PI)) - (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) . 0) by Th19 .= ((1 / (n + 1)) * (((#Z (n + 1)) * sin) . (2 * PI))) - (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) . 0) by VALUED_1:6 .= ((1 / (n + 1)) * (((#Z (n + 1)) * sin) . (2 * PI))) - ((1 / (n + 1)) * (((#Z (n + 1)) * sin) . 0)) by VALUED_1:6 .= ((1 / (n + 1)) * ((#Z (n + 1)) . (sin . (2 * PI)))) - ((1 / (n + 1)) * (((#Z (n + 1)) * sin) . 0)) by FUNCT_1:13, SIN_COS:24 .= ((1 / (n + 1)) * ((#Z (n + 1)) . (sin . (2 * PI)))) - ((1 / (n + 1)) * ((#Z (n + 1)) . (sin . 0))) by FUNCT_1:13, SIN_COS:24 .= 0 by SIN_COS:30, SIN_COS:76 ; hence integral ((((#Z n) * sin) (#) cos),A) = 0 ; ::_thesis: verum end; theorem Th22: :: INTEGR11:22 for n being Element of NAT for A being non empty closed_interval Subset of REAL holds integral ((((#Z n) * cos) (#) sin),A) = (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) . (upper_bound A)) - (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) . (lower_bound A)) proof let n be Element of NAT ; ::_thesis: for A being non empty closed_interval Subset of REAL holds integral ((((#Z n) * cos) (#) sin),A) = (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) . (upper_bound A)) - (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) . (lower_bound A)) let A be non empty closed_interval Subset of REAL; ::_thesis: integral ((((#Z n) * cos) (#) sin),A) = (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) . (upper_bound A)) - (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) . (lower_bound A)) A1: [#] REAL = dom (((#Z n) * cos) (#) sin) by FUNCT_2:def_1; A2: for x being Real st x in dom (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) `| REAL) holds (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) `| REAL) . x = (((#Z n) * cos) (#) sin) . x proof let x be Real; ::_thesis: ( x in dom (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) `| REAL) implies (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) `| REAL) . x = (((#Z n) * cos) (#) sin) . x ) assume x in dom (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) `| REAL) ; ::_thesis: (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) `| REAL) . x = (((#Z n) * cos) (#) sin) . x (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) `| REAL) . x = ((cos . x) #Z n) * (sin . x) by Th4 .= ((#Z n) . (cos . x)) * (sin . x) by TAYLOR_1:def_1 .= (((#Z n) * cos) . x) * (sin . x) by FUNCT_1:13, SIN_COS:24 .= (((#Z n) * cos) (#) sin) . x by A1, VALUED_1:def_4 ; hence (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) `| REAL) . x = (((#Z n) * cos) (#) sin) . x ; ::_thesis: verum end; for x0 being Real holds (#Z n) * cos is_differentiable_in x0 proof let x0 be Real; ::_thesis: (#Z n) * cos is_differentiable_in x0 cos is_differentiable_in x0 by SIN_COS:63; hence (#Z n) * cos is_differentiable_in x0 by TAYLOR_1:3; ::_thesis: verum end; then ( dom ((#Z n) * cos) = REAL & ( for x0 being Real st x0 in REAL holds (#Z n) * cos is_differentiable_in x0 ) ) by FUNCT_2:def_1; then (#Z n) * cos is_differentiable_on REAL by A1, FDIFF_1:9; then A3: (((#Z n) * cos) (#) sin) | REAL is continuous by A1, FDIFF_1:21, FDIFF_1:25, SIN_COS:68; then (((#Z n) * cos) (#) sin) | A is continuous by FCONT_1:16; then A4: ((#Z n) * cos) (#) sin is_integrable_on A by A1, INTEGRA5:11; (- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos) is_differentiable_on REAL by Th4; then dom (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) `| REAL) = dom (((#Z n) * cos) (#) sin) by A1, FDIFF_1:def_7; then A5: ((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) `| REAL = ((#Z n) * cos) (#) sin by A2, PARTFUN1:5; (((#Z n) * cos) (#) sin) | A is bounded by A1, A3, FCONT_1:16, INTEGRA5:10; hence integral ((((#Z n) * cos) (#) sin),A) = (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) . (upper_bound A)) - (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) . (lower_bound A)) by A4, A5, Th4, INTEGRA5:13; ::_thesis: verum end; theorem :: INTEGR11:23 for n being Element of NAT for A being non empty closed_interval Subset of REAL st A = [.0,(2 * PI).] holds integral ((((#Z n) * cos) (#) sin),A) = 0 proof let n be Element of NAT ; ::_thesis: for A being non empty closed_interval Subset of REAL st A = [.0,(2 * PI).] holds integral ((((#Z n) * cos) (#) sin),A) = 0 let A be non empty closed_interval Subset of REAL; ::_thesis: ( A = [.0,(2 * PI).] implies integral ((((#Z n) * cos) (#) sin),A) = 0 ) assume A = [.0,(2 * PI).] ; ::_thesis: integral ((((#Z n) * cos) (#) sin),A) = 0 then ( upper_bound A = 2 * PI & lower_bound A = 0 ) by INTEGRA8:37; then integral ((((#Z n) * cos) (#) sin),A) = (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) . (2 * PI)) - (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) . 0) by Th22 .= ((- (1 / (n + 1))) * (((#Z (n + 1)) * cos) . (2 * PI))) - (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) . 0) by VALUED_1:6 .= ((- (1 / (n + 1))) * (((#Z (n + 1)) * cos) . (2 * PI))) - ((- (1 / (n + 1))) * (((#Z (n + 1)) * cos) . 0)) by VALUED_1:6 .= ((- (1 / (n + 1))) * ((#Z (n + 1)) . (cos . (2 * PI)))) - ((- (1 / (n + 1))) * (((#Z (n + 1)) * cos) . 0)) by FUNCT_1:13, SIN_COS:24 .= ((- (1 / (n + 1))) * ((#Z (n + 1)) . (cos . (2 * PI)))) - ((- (1 / (n + 1))) * ((#Z (n + 1)) . (cos . 0))) by FUNCT_1:13, SIN_COS:24 .= 0 by SIN_COS:30, SIN_COS:76 ; hence integral ((((#Z n) * cos) (#) sin),A) = 0 ; ::_thesis: verum end; theorem :: INTEGR11:24 for n being Element of NAT for A being non empty closed_interval Subset of REAL st A = [.(- (PI / 2)),(PI / 2).] holds integral ((((#Z n) * cos) (#) sin),A) = 0 proof let n be Element of NAT ; ::_thesis: for A being non empty closed_interval Subset of REAL st A = [.(- (PI / 2)),(PI / 2).] holds integral ((((#Z n) * cos) (#) sin),A) = 0 let A be non empty closed_interval Subset of REAL; ::_thesis: ( A = [.(- (PI / 2)),(PI / 2).] implies integral ((((#Z n) * cos) (#) sin),A) = 0 ) assume A = [.(- (PI / 2)),(PI / 2).] ; ::_thesis: integral ((((#Z n) * cos) (#) sin),A) = 0 then ( upper_bound A = PI / 2 & lower_bound A = - (PI / 2) ) by INTEGRA8:37; then integral ((((#Z n) * cos) (#) sin),A) = (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) . (PI / 2)) - (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) . (- (PI / 2))) by Th22 .= ((- (1 / (n + 1))) * (((#Z (n + 1)) * cos) . (PI / 2))) - (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) . (- (PI / 2))) by VALUED_1:6 .= ((- (1 / (n + 1))) * (((#Z (n + 1)) * cos) . (PI / 2))) - ((- (1 / (n + 1))) * (((#Z (n + 1)) * cos) . (- (PI / 2)))) by VALUED_1:6 .= ((- (1 / (n + 1))) * ((#Z (n + 1)) . (cos . (PI / 2)))) - ((- (1 / (n + 1))) * (((#Z (n + 1)) * cos) . (- (PI / 2)))) by FUNCT_1:13, SIN_COS:24 .= ((- (1 / (n + 1))) * ((#Z (n + 1)) . (cos . (PI / 2)))) - ((- (1 / (n + 1))) * ((#Z (n + 1)) . (cos . (- (PI / 2))))) by FUNCT_1:13, SIN_COS:24 .= ((- (1 / (n + 1))) * ((#Z (n + 1)) . (cos . (PI / 2)))) - ((- (1 / (n + 1))) * ((#Z (n + 1)) . (cos . (PI / 2)))) by SIN_COS:30 .= 0 ; hence integral ((((#Z n) * cos) (#) sin),A) = 0 ; ::_thesis: verum end; theorem :: INTEGR11:25 for m, n being Element of NAT for A being non empty closed_interval Subset of REAL st m + n <> 0 & m - n <> 0 holds integral (((cos * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))),A) = ((((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) + ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0))))) . (upper_bound A)) - ((((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) + ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0))))) . (lower_bound A)) proof let m, n be Element of NAT ; ::_thesis: for A being non empty closed_interval Subset of REAL st m + n <> 0 & m - n <> 0 holds integral (((cos * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))),A) = ((((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) + ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0))))) . (upper_bound A)) - ((((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) + ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0))))) . (lower_bound A)) let A be non empty closed_interval Subset of REAL; ::_thesis: ( m + n <> 0 & m - n <> 0 implies integral (((cos * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))),A) = ((((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) + ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0))))) . (upper_bound A)) - ((((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) + ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0))))) . (lower_bound A)) ) assume A1: ( m + n <> 0 & m - n <> 0 ) ; ::_thesis: integral (((cos * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))),A) = ((((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) + ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0))))) . (upper_bound A)) - ((((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) + ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0))))) . (lower_bound A)) A2: for x being Real st x in REAL holds (AffineMap (n,0)) . x = n * x proof let x be Real; ::_thesis: ( x in REAL implies (AffineMap (n,0)) . x = n * x ) assume x in REAL ; ::_thesis: (AffineMap (n,0)) . x = n * x (AffineMap (n,0)) . x = (n * x) + 0 by FCONT_1:def_4 .= n * x ; hence (AffineMap (n,0)) . x = n * x ; ::_thesis: verum end; A3: dom (cos * (AffineMap (n,0))) = [#] REAL by FUNCT_2:def_1; A4: dom (cos * (AffineMap (m,0))) = [#] REAL by FUNCT_2:def_1; A5: for x being Real st x in REAL holds (AffineMap (m,0)) . x = m * x proof let x be Real; ::_thesis: ( x in REAL implies (AffineMap (m,0)) . x = m * x ) assume x in REAL ; ::_thesis: (AffineMap (m,0)) . x = m * x (AffineMap (m,0)) . x = (m * x) + 0 by FCONT_1:def_4 .= m * x ; hence (AffineMap (m,0)) . x = m * x ; ::_thesis: verum end; A6: for x being Real st x in dom ((((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) + ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0))))) `| REAL) holds ((((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) + ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0))))) `| REAL) . x = ((cos * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))) . x proof let x be Real; ::_thesis: ( x in dom ((((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) + ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0))))) `| REAL) implies ((((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) + ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0))))) `| REAL) . x = ((cos * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))) . x ) assume x in dom ((((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) + ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0))))) `| REAL) ; ::_thesis: ((((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) + ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0))))) `| REAL) . x = ((cos * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))) . x ((((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) + ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0))))) `| REAL) . x = (cos . (m * x)) * (cos . (n * x)) by A1, Th5 .= (cos . ((AffineMap (m,0)) . x)) * (cos . (n * x)) by A5 .= (cos . ((AffineMap (m,0)) . x)) * (cos . ((AffineMap (n,0)) . x)) by A2 .= ((cos * (AffineMap (m,0))) . x) * (cos . ((AffineMap (n,0)) . x)) by A4, FUNCT_1:12 .= ((cos * (AffineMap (m,0))) . x) * ((cos * (AffineMap (n,0))) . x) by A3, FUNCT_1:12 .= ((cos * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))) . x by VALUED_1:5 ; hence ((((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) + ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0))))) `| REAL) . x = ((cos * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))) . x ; ::_thesis: verum end; A7: [#] REAL = dom ((cos * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))) by FUNCT_2:def_1; ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) + ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) is_differentiable_on REAL by A1, Th5; then dom ((((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) + ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0))))) `| REAL) = dom ((cos * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))) by A7, FDIFF_1:def_7; then A8: (((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) + ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0))))) `| REAL = (cos * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0))) by A6, PARTFUN1:5; ((cos * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))) | A is continuous ; then A9: (cos * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0))) is_integrable_on A by A7, INTEGRA5:11; ((cos * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))) | A is bounded by A7, INTEGRA5:10; hence integral (((cos * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))),A) = ((((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) + ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0))))) . (upper_bound A)) - ((((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) + ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0))))) . (lower_bound A)) by A1, A9, A8, Th5, INTEGRA5:13; ::_thesis: verum end; theorem :: INTEGR11:26 for m, n being Element of NAT for A being non empty closed_interval Subset of REAL st m + n <> 0 & m - n <> 0 holds integral (((sin * (AffineMap (m,0))) (#) (sin * (AffineMap (n,0)))),A) = ((((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) - ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0))))) . (upper_bound A)) - ((((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) - ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0))))) . (lower_bound A)) proof let m, n be Element of NAT ; ::_thesis: for A being non empty closed_interval Subset of REAL st m + n <> 0 & m - n <> 0 holds integral (((sin * (AffineMap (m,0))) (#) (sin * (AffineMap (n,0)))),A) = ((((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) - ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0))))) . (upper_bound A)) - ((((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) - ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0))))) . (lower_bound A)) let A be non empty closed_interval Subset of REAL; ::_thesis: ( m + n <> 0 & m - n <> 0 implies integral (((sin * (AffineMap (m,0))) (#) (sin * (AffineMap (n,0)))),A) = ((((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) - ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0))))) . (upper_bound A)) - ((((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) - ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0))))) . (lower_bound A)) ) assume A1: ( m + n <> 0 & m - n <> 0 ) ; ::_thesis: integral (((sin * (AffineMap (m,0))) (#) (sin * (AffineMap (n,0)))),A) = ((((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) - ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0))))) . (upper_bound A)) - ((((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) - ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0))))) . (lower_bound A)) A2: for x being Real st x in REAL holds (AffineMap (n,0)) . x = n * x proof let x be Real; ::_thesis: ( x in REAL implies (AffineMap (n,0)) . x = n * x ) assume x in REAL ; ::_thesis: (AffineMap (n,0)) . x = n * x (AffineMap (n,0)) . x = (n * x) + 0 by FCONT_1:def_4 .= n * x ; hence (AffineMap (n,0)) . x = n * x ; ::_thesis: verum end; A3: dom (sin * (AffineMap (n,0))) = [#] REAL by FUNCT_2:def_1; A4: dom (sin * (AffineMap (m,0))) = [#] REAL by FUNCT_2:def_1; A5: for x being Real st x in REAL holds (AffineMap (m,0)) . x = m * x proof let x be Real; ::_thesis: ( x in REAL implies (AffineMap (m,0)) . x = m * x ) assume x in REAL ; ::_thesis: (AffineMap (m,0)) . x = m * x (AffineMap (m,0)) . x = (m * x) + 0 by FCONT_1:def_4 .= m * x ; hence (AffineMap (m,0)) . x = m * x ; ::_thesis: verum end; A6: for x being Real st x in dom ((((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) - ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0))))) `| REAL) holds ((((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) - ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0))))) `| REAL) . x = ((sin * (AffineMap (m,0))) (#) (sin * (AffineMap (n,0)))) . x proof let x be Real; ::_thesis: ( x in dom ((((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) - ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0))))) `| REAL) implies ((((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) - ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0))))) `| REAL) . x = ((sin * (AffineMap (m,0))) (#) (sin * (AffineMap (n,0)))) . x ) assume x in dom ((((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) - ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0))))) `| REAL) ; ::_thesis: ((((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) - ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0))))) `| REAL) . x = ((sin * (AffineMap (m,0))) (#) (sin * (AffineMap (n,0)))) . x ((((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) - ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0))))) `| REAL) . x = (sin . (m * x)) * (sin . (n * x)) by A1, Th6 .= (sin . ((AffineMap (m,0)) . x)) * (sin . (n * x)) by A5 .= (sin . ((AffineMap (m,0)) . x)) * (sin . ((AffineMap (n,0)) . x)) by A2 .= ((sin * (AffineMap (m,0))) . x) * (sin . ((AffineMap (n,0)) . x)) by A4, FUNCT_1:12 .= ((sin * (AffineMap (m,0))) . x) * ((sin * (AffineMap (n,0))) . x) by A3, FUNCT_1:12 .= ((sin * (AffineMap (m,0))) (#) (sin * (AffineMap (n,0)))) . x by VALUED_1:5 ; hence ((((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) - ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0))))) `| REAL) . x = ((sin * (AffineMap (m,0))) (#) (sin * (AffineMap (n,0)))) . x ; ::_thesis: verum end; A7: REAL = dom ((sin * (AffineMap (m,0))) (#) (sin * (AffineMap (n,0)))) by FUNCT_2:def_1; ((sin * (AffineMap (m,0))) (#) (sin * (AffineMap (n,0)))) | A is continuous ; then A8: (sin * (AffineMap (m,0))) (#) (sin * (AffineMap (n,0))) is_integrable_on A by A7, INTEGRA5:11; ((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) - ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0)))) is_differentiable_on REAL by A1, Th6; then dom ((((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) - ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0))))) `| REAL) = dom ((sin * (AffineMap (m,0))) (#) (sin * (AffineMap (n,0)))) by A7, FDIFF_1:def_7; then A9: (((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) - ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0))))) `| REAL = (sin * (AffineMap (m,0))) (#) (sin * (AffineMap (n,0))) by A6, PARTFUN1:5; ((sin * (AffineMap (m,0))) (#) (sin * (AffineMap (n,0)))) | A is bounded by A7, INTEGRA5:10; hence integral (((sin * (AffineMap (m,0))) (#) (sin * (AffineMap (n,0)))),A) = ((((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) - ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0))))) . (upper_bound A)) - ((((1 / (2 * (m - n))) (#) (sin * (AffineMap ((m - n),0)))) - ((1 / (2 * (m + n))) (#) (sin * (AffineMap ((m + n),0))))) . (lower_bound A)) by A1, A8, A9, Th6, INTEGRA5:13; ::_thesis: verum end; theorem :: INTEGR11:27 for m, n being Element of NAT for A being non empty closed_interval Subset of REAL st m + n <> 0 & m - n <> 0 holds integral (((sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))),A) = (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) . (upper_bound A)) - (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) . (lower_bound A)) proof let m, n be Element of NAT ; ::_thesis: for A being non empty closed_interval Subset of REAL st m + n <> 0 & m - n <> 0 holds integral (((sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))),A) = (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) . (upper_bound A)) - (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) . (lower_bound A)) let A be non empty closed_interval Subset of REAL; ::_thesis: ( m + n <> 0 & m - n <> 0 implies integral (((sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))),A) = (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) . (upper_bound A)) - (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) . (lower_bound A)) ) assume A1: ( m + n <> 0 & m - n <> 0 ) ; ::_thesis: integral (((sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))),A) = (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) . (upper_bound A)) - (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) . (lower_bound A)) A2: for x being Real st x in REAL holds (AffineMap (n,0)) . x = n * x proof let x be Real; ::_thesis: ( x in REAL implies (AffineMap (n,0)) . x = n * x ) assume x in REAL ; ::_thesis: (AffineMap (n,0)) . x = n * x (AffineMap (n,0)) . x = (n * x) + 0 by FCONT_1:def_4 .= n * x ; hence (AffineMap (n,0)) . x = n * x ; ::_thesis: verum end; A3: dom (cos * (AffineMap (n,0))) = [#] REAL by FUNCT_2:def_1; A4: dom (sin * (AffineMap (m,0))) = [#] REAL by FUNCT_2:def_1; A5: for x being Real st x in REAL holds (AffineMap (m,0)) . x = m * x proof let x be Real; ::_thesis: ( x in REAL implies (AffineMap (m,0)) . x = m * x ) assume x in REAL ; ::_thesis: (AffineMap (m,0)) . x = m * x (AffineMap (m,0)) . x = (m * x) + 0 by FCONT_1:def_4 .= m * x ; hence (AffineMap (m,0)) . x = m * x ; ::_thesis: verum end; A6: for x being Real st x in dom (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) `| REAL) holds (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) `| REAL) . x = ((sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))) . x proof let x be Real; ::_thesis: ( x in dom (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) `| REAL) implies (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) `| REAL) . x = ((sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))) . x ) assume x in dom (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) `| REAL) ; ::_thesis: (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) `| REAL) . x = ((sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))) . x (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) `| REAL) . x = (sin . (m * x)) * (cos . (n * x)) by A1, Th7 .= (sin . ((AffineMap (m,0)) . x)) * (cos . (n * x)) by A5 .= (sin . ((AffineMap (m,0)) . x)) * (cos . ((AffineMap (n,0)) . x)) by A2 .= ((sin * (AffineMap (m,0))) . x) * (cos . ((AffineMap (n,0)) . x)) by A4, FUNCT_1:12 .= ((sin * (AffineMap (m,0))) . x) * ((cos * (AffineMap (n,0))) . x) by A3, FUNCT_1:12 .= ((sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))) . x by VALUED_1:5 ; hence (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) `| REAL) . x = ((sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))) . x ; ::_thesis: verum end; A7: [#] REAL = dom ((sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))) by FUNCT_2:def_1; ((sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))) | A is continuous ; then A8: (sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0))) is_integrable_on A by A7, INTEGRA5:11; (- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0)))) is_differentiable_on REAL by A1, Th7; then dom (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) `| REAL) = dom ((sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))) by A7, FDIFF_1:def_7; then A9: ((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) `| REAL = (sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0))) by A6, PARTFUN1:5; ((sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))) | A is bounded by A7, INTEGRA5:10; hence integral (((sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))),A) = (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) . (upper_bound A)) - (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) . (lower_bound A)) by A1, A8, A9, Th7, INTEGRA5:13; ::_thesis: verum end; theorem :: INTEGR11:28 for n being Element of NAT for A being non empty closed_interval Subset of REAL st n <> 0 holds integral (((AffineMap (1,0)) (#) (sin * (AffineMap (n,0)))),A) = ((((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) - ((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0))))) . (upper_bound A)) - ((((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) - ((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0))))) . (lower_bound A)) proof let n be Element of NAT ; ::_thesis: for A being non empty closed_interval Subset of REAL st n <> 0 holds integral (((AffineMap (1,0)) (#) (sin * (AffineMap (n,0)))),A) = ((((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) - ((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0))))) . (upper_bound A)) - ((((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) - ((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0))))) . (lower_bound A)) let A be non empty closed_interval Subset of REAL; ::_thesis: ( n <> 0 implies integral (((AffineMap (1,0)) (#) (sin * (AffineMap (n,0)))),A) = ((((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) - ((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0))))) . (upper_bound A)) - ((((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) - ((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0))))) . (lower_bound A)) ) assume A1: n <> 0 ; ::_thesis: integral (((AffineMap (1,0)) (#) (sin * (AffineMap (n,0)))),A) = ((((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) - ((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0))))) . (upper_bound A)) - ((((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) - ((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0))))) . (lower_bound A)) A2: for x being Real st x in REAL holds (AffineMap (n,0)) . x = n * x proof let x be Real; ::_thesis: ( x in REAL implies (AffineMap (n,0)) . x = n * x ) assume x in REAL ; ::_thesis: (AffineMap (n,0)) . x = n * x (AffineMap (n,0)) . x = (n * x) + 0 by FCONT_1:def_4 .= n * x ; hence (AffineMap (n,0)) . x = n * x ; ::_thesis: verum end; A3: dom (sin * (AffineMap (n,0))) = [#] REAL by FUNCT_2:def_1; A4: for x being Real st x in REAL holds (AffineMap (1,0)) . x = x proof let x be Real; ::_thesis: ( x in REAL implies (AffineMap (1,0)) . x = x ) assume x in REAL ; ::_thesis: (AffineMap (1,0)) . x = x (AffineMap (1,0)) . x = (1 * x) + 0 by FCONT_1:def_4 .= x ; hence (AffineMap (1,0)) . x = x ; ::_thesis: verum end; A5: for x being Real st x in dom ((((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) - ((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0))))) `| REAL) holds ((((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) - ((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0))))) `| REAL) . x = ((AffineMap (1,0)) (#) (sin * (AffineMap (n,0)))) . x proof let x be Real; ::_thesis: ( x in dom ((((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) - ((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0))))) `| REAL) implies ((((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) - ((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0))))) `| REAL) . x = ((AffineMap (1,0)) (#) (sin * (AffineMap (n,0)))) . x ) assume x in dom ((((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) - ((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0))))) `| REAL) ; ::_thesis: ((((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) - ((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0))))) `| REAL) . x = ((AffineMap (1,0)) (#) (sin * (AffineMap (n,0)))) . x ((((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) - ((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0))))) `| REAL) . x = x * (sin . (n * x)) by A1, Th8 .= x * (sin . ((AffineMap (n,0)) . x)) by A2 .= x * ((sin * (AffineMap (n,0))) . x) by A3, FUNCT_1:12 .= ((AffineMap (1,0)) . x) * ((sin * (AffineMap (n,0))) . x) by A4 .= ((AffineMap (1,0)) (#) (sin * (AffineMap (n,0)))) . x by VALUED_1:5 ; hence ((((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) - ((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0))))) `| REAL) . x = ((AffineMap (1,0)) (#) (sin * (AffineMap (n,0)))) . x ; ::_thesis: verum end; A6: dom ((AffineMap (1,0)) (#) (sin * (AffineMap (n,0)))) = [#] REAL by FUNCT_2:def_1; ((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) - ((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0)))) is_differentiable_on REAL by A1, Th8; then dom ((((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) - ((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0))))) `| REAL) = dom ((AffineMap (1,0)) (#) (sin * (AffineMap (n,0)))) by A6, FDIFF_1:def_7; then A7: (((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) - ((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0))))) `| REAL = (AffineMap (1,0)) (#) (sin * (AffineMap (n,0))) by A5, PARTFUN1:5; ((AffineMap (1,0)) (#) (sin * (AffineMap (n,0)))) | A is continuous ; then A8: (AffineMap (1,0)) (#) (sin * (AffineMap (n,0))) is_integrable_on A by A6, INTEGRA5:11; ((AffineMap (1,0)) (#) (sin * (AffineMap (n,0)))) | A is bounded by A6, INTEGRA5:10; hence integral (((AffineMap (1,0)) (#) (sin * (AffineMap (n,0)))),A) = ((((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) - ((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0))))) . (upper_bound A)) - ((((1 / (n ^2)) (#) (sin * (AffineMap (n,0)))) - ((AffineMap ((1 / n),0)) (#) (cos * (AffineMap (n,0))))) . (lower_bound A)) by A1, A8, A7, Th8, INTEGRA5:13; ::_thesis: verum end; theorem :: INTEGR11:29 for n being Element of NAT for A being non empty closed_interval Subset of REAL st n <> 0 holds integral (((AffineMap (1,0)) (#) (cos * (AffineMap (n,0)))),A) = ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) . (upper_bound A)) - ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) . (lower_bound A)) proof let n be Element of NAT ; ::_thesis: for A being non empty closed_interval Subset of REAL st n <> 0 holds integral (((AffineMap (1,0)) (#) (cos * (AffineMap (n,0)))),A) = ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) . (upper_bound A)) - ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) . (lower_bound A)) let A be non empty closed_interval Subset of REAL; ::_thesis: ( n <> 0 implies integral (((AffineMap (1,0)) (#) (cos * (AffineMap (n,0)))),A) = ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) . (upper_bound A)) - ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) . (lower_bound A)) ) assume A1: n <> 0 ; ::_thesis: integral (((AffineMap (1,0)) (#) (cos * (AffineMap (n,0)))),A) = ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) . (upper_bound A)) - ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) . (lower_bound A)) A2: for x being Real st x in REAL holds (AffineMap (n,0)) . x = n * x proof let x be Real; ::_thesis: ( x in REAL implies (AffineMap (n,0)) . x = n * x ) assume x in REAL ; ::_thesis: (AffineMap (n,0)) . x = n * x (AffineMap (n,0)) . x = (n * x) + 0 by FCONT_1:def_4 .= n * x ; hence (AffineMap (n,0)) . x = n * x ; ::_thesis: verum end; A3: dom (cos * (AffineMap (n,0))) = [#] REAL by FUNCT_2:def_1; A4: for x being Real st x in REAL holds (AffineMap (1,0)) . x = x proof let x be Real; ::_thesis: ( x in REAL implies (AffineMap (1,0)) . x = x ) assume x in REAL ; ::_thesis: (AffineMap (1,0)) . x = x (AffineMap (1,0)) . x = (1 * x) + 0 by FCONT_1:def_4 .= x ; hence (AffineMap (1,0)) . x = x ; ::_thesis: verum end; A5: for x being Real st x in dom ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) `| REAL) holds ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) `| REAL) . x = ((AffineMap (1,0)) (#) (cos * (AffineMap (n,0)))) . x proof let x be Real; ::_thesis: ( x in dom ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) `| REAL) implies ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) `| REAL) . x = ((AffineMap (1,0)) (#) (cos * (AffineMap (n,0)))) . x ) assume x in dom ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) `| REAL) ; ::_thesis: ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) `| REAL) . x = ((AffineMap (1,0)) (#) (cos * (AffineMap (n,0)))) . x ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) `| REAL) . x = x * (cos . (n * x)) by A1, Th9 .= x * (cos . ((AffineMap (n,0)) . x)) by A2 .= x * ((cos * (AffineMap (n,0))) . x) by A3, FUNCT_1:12 .= ((AffineMap (1,0)) . x) * ((cos * (AffineMap (n,0))) . x) by A4 .= ((AffineMap (1,0)) (#) (cos * (AffineMap (n,0)))) . x by VALUED_1:5 ; hence ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) `| REAL) . x = ((AffineMap (1,0)) (#) (cos * (AffineMap (n,0)))) . x ; ::_thesis: verum end; A6: dom ((AffineMap (1,0)) (#) (cos * (AffineMap (n,0)))) = [#] REAL by FUNCT_2:def_1; ((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0)))) is_differentiable_on REAL by A1, Th9; then dom ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) `| REAL) = dom ((AffineMap (1,0)) (#) (cos * (AffineMap (n,0)))) by A6, FDIFF_1:def_7; then A7: (((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) `| REAL = (AffineMap (1,0)) (#) (cos * (AffineMap (n,0))) by A5, PARTFUN1:5; ((AffineMap (1,0)) (#) (cos * (AffineMap (n,0)))) | A is continuous ; then A8: (AffineMap (1,0)) (#) (cos * (AffineMap (n,0))) is_integrable_on A by A6, INTEGRA5:11; ((AffineMap (1,0)) (#) (cos * (AffineMap (n,0)))) | A is bounded by A6, INTEGRA5:10; hence integral (((AffineMap (1,0)) (#) (cos * (AffineMap (n,0)))),A) = ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) . (upper_bound A)) - ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) . (lower_bound A)) by A1, A8, A7, Th9, INTEGRA5:13; ::_thesis: verum end; theorem :: INTEGR11:30 for A being non empty closed_interval Subset of REAL holds integral (((AffineMap (1,0)) (#) sinh),A) = ((((AffineMap (1,0)) (#) cosh) - sinh) . (upper_bound A)) - ((((AffineMap (1,0)) (#) cosh) - sinh) . (lower_bound A)) proof let A be non empty closed_interval Subset of REAL; ::_thesis: integral (((AffineMap (1,0)) (#) sinh),A) = ((((AffineMap (1,0)) (#) cosh) - sinh) . (upper_bound A)) - ((((AffineMap (1,0)) (#) cosh) - sinh) . (lower_bound A)) A1: for x being Real st x in dom ((((AffineMap (1,0)) (#) cosh) - sinh) `| REAL) holds ((((AffineMap (1,0)) (#) cosh) - sinh) `| REAL) . x = ((AffineMap (1,0)) (#) sinh) . x proof let x be Real; ::_thesis: ( x in dom ((((AffineMap (1,0)) (#) cosh) - sinh) `| REAL) implies ((((AffineMap (1,0)) (#) cosh) - sinh) `| REAL) . x = ((AffineMap (1,0)) (#) sinh) . x ) assume x in dom ((((AffineMap (1,0)) (#) cosh) - sinh) `| REAL) ; ::_thesis: ((((AffineMap (1,0)) (#) cosh) - sinh) `| REAL) . x = ((AffineMap (1,0)) (#) sinh) . x ((((AffineMap (1,0)) (#) cosh) - sinh) `| REAL) . x = ((1 * x) + 0) * (sinh . x) by Th10 .= ((AffineMap (1,0)) . x) * (sinh . x) by FCONT_1:def_4 .= ((AffineMap (1,0)) (#) sinh) . x by VALUED_1:5 ; hence ((((AffineMap (1,0)) (#) cosh) - sinh) `| REAL) . x = ((AffineMap (1,0)) (#) sinh) . x ; ::_thesis: verum end; A2: dom ((AffineMap (1,0)) (#) sinh) = [#] REAL by FUNCT_2:def_1; then dom ((((AffineMap (1,0)) (#) cosh) - sinh) `| REAL) = dom ((AffineMap (1,0)) (#) sinh) by Th10, FDIFF_1:def_7; then A3: (((AffineMap (1,0)) (#) cosh) - sinh) `| REAL = (AffineMap (1,0)) (#) sinh by A1, PARTFUN1:5; ( dom (AffineMap (1,0)) = [#] REAL & ( for x being Real st x in REAL holds (AffineMap (1,0)) . x = (1 * x) + 0 ) ) by FCONT_1:def_4, FUNCT_2:def_1; then AffineMap (1,0) is_differentiable_on REAL by FDIFF_1:23; then A4: ((AffineMap (1,0)) (#) sinh) | REAL is continuous by A2, FDIFF_1:21, FDIFF_1:25, SIN_COS2:34; then A5: ((AffineMap (1,0)) (#) sinh) | A is continuous by FCONT_1:16; ((AffineMap (1,0)) (#) sinh) | A is bounded by A2, A4, FCONT_1:16, INTEGRA5:10; hence integral (((AffineMap (1,0)) (#) sinh),A) = ((((AffineMap (1,0)) (#) cosh) - sinh) . (upper_bound A)) - ((((AffineMap (1,0)) (#) cosh) - sinh) . (lower_bound A)) by A2, A5, A3, Th10, INTEGRA5:11, INTEGRA5:13; ::_thesis: verum end; theorem :: INTEGR11:31 for A being non empty closed_interval Subset of REAL holds integral (((AffineMap (1,0)) (#) cosh),A) = ((((AffineMap (1,0)) (#) sinh) - cosh) . (upper_bound A)) - ((((AffineMap (1,0)) (#) sinh) - cosh) . (lower_bound A)) proof let A be non empty closed_interval Subset of REAL; ::_thesis: integral (((AffineMap (1,0)) (#) cosh),A) = ((((AffineMap (1,0)) (#) sinh) - cosh) . (upper_bound A)) - ((((AffineMap (1,0)) (#) sinh) - cosh) . (lower_bound A)) A1: for x being Real st x in dom ((((AffineMap (1,0)) (#) sinh) - cosh) `| REAL) holds ((((AffineMap (1,0)) (#) sinh) - cosh) `| REAL) . x = ((AffineMap (1,0)) (#) cosh) . x proof let x be Real; ::_thesis: ( x in dom ((((AffineMap (1,0)) (#) sinh) - cosh) `| REAL) implies ((((AffineMap (1,0)) (#) sinh) - cosh) `| REAL) . x = ((AffineMap (1,0)) (#) cosh) . x ) assume x in dom ((((AffineMap (1,0)) (#) sinh) - cosh) `| REAL) ; ::_thesis: ((((AffineMap (1,0)) (#) sinh) - cosh) `| REAL) . x = ((AffineMap (1,0)) (#) cosh) . x ((((AffineMap (1,0)) (#) sinh) - cosh) `| REAL) . x = ((1 * x) + 0) * (cosh . x) by Th11 .= ((AffineMap (1,0)) . x) * (cosh . x) by FCONT_1:def_4 .= ((AffineMap (1,0)) (#) cosh) . x by VALUED_1:5 ; hence ((((AffineMap (1,0)) (#) sinh) - cosh) `| REAL) . x = ((AffineMap (1,0)) (#) cosh) . x ; ::_thesis: verum end; A2: dom ((AffineMap (1,0)) (#) cosh) = [#] REAL by FUNCT_2:def_1; then dom ((((AffineMap (1,0)) (#) sinh) - cosh) `| REAL) = dom ((AffineMap (1,0)) (#) cosh) by Th11, FDIFF_1:def_7; then A3: (((AffineMap (1,0)) (#) sinh) - cosh) `| REAL = (AffineMap (1,0)) (#) cosh by A1, PARTFUN1:5; ( dom (AffineMap (1,0)) = [#] REAL & ( for x being Real st x in REAL holds (AffineMap (1,0)) . x = (1 * x) + 0 ) ) by FCONT_1:def_4, FUNCT_2:def_1; then AffineMap (1,0) is_differentiable_on REAL by FDIFF_1:23; then A4: ((AffineMap (1,0)) (#) cosh) | REAL is continuous by A2, FDIFF_1:21, FDIFF_1:25, SIN_COS2:35; then A5: ((AffineMap (1,0)) (#) cosh) | A is continuous by FCONT_1:16; ((AffineMap (1,0)) (#) cosh) | A is bounded by A2, A4, FCONT_1:16, INTEGRA5:10; hence integral (((AffineMap (1,0)) (#) cosh),A) = ((((AffineMap (1,0)) (#) sinh) - cosh) . (upper_bound A)) - ((((AffineMap (1,0)) (#) sinh) - cosh) . (lower_bound A)) by A2, A5, A3, Th11, INTEGRA5:11, INTEGRA5:13; ::_thesis: verum end; theorem :: INTEGR11:32 for a, b being Real for n being Element of NAT for A being non empty closed_interval Subset of REAL st a * (n + 1) <> 0 holds integral (((#Z n) * (AffineMap (a,b))),A) = (((1 / (a * (n + 1))) (#) ((#Z (n + 1)) * (AffineMap (a,b)))) . (upper_bound A)) - (((1 / (a * (n + 1))) (#) ((#Z (n + 1)) * (AffineMap (a,b)))) . (lower_bound A)) proof let a, b be Real; ::_thesis: for n being Element of NAT for A being non empty closed_interval Subset of REAL st a * (n + 1) <> 0 holds integral (((#Z n) * (AffineMap (a,b))),A) = (((1 / (a * (n + 1))) (#) ((#Z (n + 1)) * (AffineMap (a,b)))) . (upper_bound A)) - (((1 / (a * (n + 1))) (#) ((#Z (n + 1)) * (AffineMap (a,b)))) . (lower_bound A)) let n be Element of NAT ; ::_thesis: for A being non empty closed_interval Subset of REAL st a * (n + 1) <> 0 holds integral (((#Z n) * (AffineMap (a,b))),A) = (((1 / (a * (n + 1))) (#) ((#Z (n + 1)) * (AffineMap (a,b)))) . (upper_bound A)) - (((1 / (a * (n + 1))) (#) ((#Z (n + 1)) * (AffineMap (a,b)))) . (lower_bound A)) let A be non empty closed_interval Subset of REAL; ::_thesis: ( a * (n + 1) <> 0 implies integral (((#Z n) * (AffineMap (a,b))),A) = (((1 / (a * (n + 1))) (#) ((#Z (n + 1)) * (AffineMap (a,b)))) . (upper_bound A)) - (((1 / (a * (n + 1))) (#) ((#Z (n + 1)) * (AffineMap (a,b)))) . (lower_bound A)) ) assume A1: a * (n + 1) <> 0 ; ::_thesis: integral (((#Z n) * (AffineMap (a,b))),A) = (((1 / (a * (n + 1))) (#) ((#Z (n + 1)) * (AffineMap (a,b)))) . (upper_bound A)) - (((1 / (a * (n + 1))) (#) ((#Z (n + 1)) * (AffineMap (a,b)))) . (lower_bound A)) A2: [#] REAL = dom (AffineMap (a,b)) by FUNCT_2:def_1; A3: for x being Real st x in dom (((1 / (a * (n + 1))) (#) ((#Z (n + 1)) * (AffineMap (a,b)))) `| REAL) holds (((1 / (a * (n + 1))) (#) ((#Z (n + 1)) * (AffineMap (a,b)))) `| REAL) . x = ((#Z n) * (AffineMap (a,b))) . x proof let x be Real; ::_thesis: ( x in dom (((1 / (a * (n + 1))) (#) ((#Z (n + 1)) * (AffineMap (a,b)))) `| REAL) implies (((1 / (a * (n + 1))) (#) ((#Z (n + 1)) * (AffineMap (a,b)))) `| REAL) . x = ((#Z n) * (AffineMap (a,b))) . x ) assume x in dom (((1 / (a * (n + 1))) (#) ((#Z (n + 1)) * (AffineMap (a,b)))) `| REAL) ; ::_thesis: (((1 / (a * (n + 1))) (#) ((#Z (n + 1)) * (AffineMap (a,b)))) `| REAL) . x = ((#Z n) * (AffineMap (a,b))) . x (((1 / (a * (n + 1))) (#) ((#Z (n + 1)) * (AffineMap (a,b)))) `| REAL) . x = ((a * x) + b) #Z n by A1, Th12 .= ((AffineMap (a,b)) . x) #Z n by FCONT_1:def_4 .= (#Z n) . ((AffineMap (a,b)) . x) by TAYLOR_1:def_1 .= ((#Z n) * (AffineMap (a,b))) . x by A2, FUNCT_1:13 ; hence (((1 / (a * (n + 1))) (#) ((#Z (n + 1)) * (AffineMap (a,b)))) `| REAL) . x = ((#Z n) * (AffineMap (a,b))) . x ; ::_thesis: verum end; A4: [#] REAL = dom ((#Z n) * (AffineMap (a,b))) by FUNCT_2:def_1; for x being Real st x in REAL holds (AffineMap (a,b)) . x = (a * x) + b by FCONT_1:def_4; then A5: AffineMap (a,b) is_differentiable_on REAL by A2, FDIFF_1:23; for x being Real holds (#Z n) * (AffineMap (a,b)) is_differentiable_in x proof let x be Real; ::_thesis: (#Z n) * (AffineMap (a,b)) is_differentiable_in x AffineMap (a,b) is_differentiable_in x by A2, A5, FDIFF_1:9; hence (#Z n) * (AffineMap (a,b)) is_differentiable_in x by TAYLOR_1:3; ::_thesis: verum end; then for x being Real st x in REAL holds (#Z n) * (AffineMap (a,b)) is_differentiable_in x ; then (#Z n) * (AffineMap (a,b)) is_differentiable_on REAL by A4, FDIFF_1:9; then A6: ((#Z n) * (AffineMap (a,b))) | REAL is continuous by FDIFF_1:25; then ((#Z n) * (AffineMap (a,b))) | A is continuous by FCONT_1:16; then A7: (#Z n) * (AffineMap (a,b)) is_integrable_on A by A4, INTEGRA5:11; (1 / (a * (n + 1))) (#) ((#Z (n + 1)) * (AffineMap (a,b))) is_differentiable_on REAL by A1, Th12; then dom (((1 / (a * (n + 1))) (#) ((#Z (n + 1)) * (AffineMap (a,b)))) `| REAL) = dom ((#Z n) * (AffineMap (a,b))) by A4, FDIFF_1:def_7; then A8: ((1 / (a * (n + 1))) (#) ((#Z (n + 1)) * (AffineMap (a,b)))) `| REAL = (#Z n) * (AffineMap (a,b)) by A3, PARTFUN1:5; ((#Z n) * (AffineMap (a,b))) | A is bounded by A4, A6, FCONT_1:16, INTEGRA5:10; hence integral (((#Z n) * (AffineMap (a,b))),A) = (((1 / (a * (n + 1))) (#) ((#Z (n + 1)) * (AffineMap (a,b)))) . (upper_bound A)) - (((1 / (a * (n + 1))) (#) ((#Z (n + 1)) * (AffineMap (a,b)))) . (lower_bound A)) by A1, A7, A8, Th12, INTEGRA5:13; ::_thesis: verum end; begin theorem Th33: :: INTEGR11:33 for Z being open Subset of REAL for f being PartFunc of REAL,REAL st Z c= dom ((1 / 2) (#) f) & f = #Z 2 holds ( (1 / 2) (#) f is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) f) `| Z) . x = x ) ) proof let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st Z c= dom ((1 / 2) (#) f) & f = #Z 2 holds ( (1 / 2) (#) f is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) f) `| Z) . x = x ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom ((1 / 2) (#) f) & f = #Z 2 implies ( (1 / 2) (#) f is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) f) `| Z) . x = x ) ) ) assume that A1: Z c= dom ((1 / 2) (#) f) and A2: f = #Z 2 ; ::_thesis: ( (1 / 2) (#) f is_differentiable_on Z & ( for x being Real st x in Z holds (((1 / 2) (#) f) `| Z) . x = x ) ) ( Z c= dom f & ( for x being Real st x in Z holds f is_differentiable_in x ) ) by A1, A2, TAYLOR_1:2, VALUED_1:def_5; then A3: f is_differentiable_on Z by FDIFF_1:9; A4: for x being Real st x in Z holds (f `| Z) . x = 2 * x proof let x be Real; ::_thesis: ( x in Z implies (f `| Z) . x = 2 * x ) 2 * (x #Z (2 - 1)) = 2 * x by PREPOWER:35; then A5: diff (f,x) = 2 * x by A2, TAYLOR_1:2; assume x in Z ; ::_thesis: (f `| Z) . x = 2 * x hence (f `| Z) . x = 2 * x by A3, A5, FDIFF_1:def_7; ::_thesis: verum end; for x being Real st x in Z holds (((1 / 2) (#) f) `| Z) . x = x proof let x be Real; ::_thesis: ( x in Z implies (((1 / 2) (#) f) `| Z) . x = x ) assume A6: x in Z ; ::_thesis: (((1 / 2) (#) f) `| Z) . x = x then (((1 / 2) (#) f) `| Z) . x = (1 / 2) * (diff (f,x)) by A1, A3, FDIFF_1:20 .= (1 / 2) * ((f `| Z) . x) by A3, A6, FDIFF_1:def_7 .= (1 / 2) * (2 * x) by A4, A6 .= x ; hence (((1 / 2) (#) f) `| Z) . x = x ; ::_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 = x ) ) by A1, A3, FDIFF_1:20; ::_thesis: verum end; theorem :: INTEGR11:34 for A being non empty closed_interval Subset of REAL for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & f = #Z 2 & Z = dom ((1 / 2) (#) f) holds integral ((id Z),A) = (((1 / 2) (#) f) . (upper_bound A)) - (((1 / 2) (#) f) . (lower_bound A)) proof let A be non empty closed_interval Subset of REAL; ::_thesis: for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & f = #Z 2 & Z = dom ((1 / 2) (#) f) holds integral ((id Z),A) = (((1 / 2) (#) f) . (upper_bound A)) - (((1 / 2) (#) f) . (lower_bound A)) let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st A c= Z & f = #Z 2 & Z = dom ((1 / 2) (#) f) holds integral ((id Z),A) = (((1 / 2) (#) f) . (upper_bound A)) - (((1 / 2) (#) f) . (lower_bound A)) let f be PartFunc of REAL,REAL; ::_thesis: ( A c= Z & f = #Z 2 & Z = dom ((1 / 2) (#) f) implies integral ((id Z),A) = (((1 / 2) (#) f) . (upper_bound A)) - (((1 / 2) (#) f) . (lower_bound A)) ) assume that A1: A c= Z and A2: ( f = #Z 2 & Z = dom ((1 / 2) (#) f) ) ; ::_thesis: integral ((id Z),A) = (((1 / 2) (#) f) . (upper_bound A)) - (((1 / 2) (#) f) . (lower_bound A)) A3: A c= dom (id Z) by A1; then A4: (id Z) | A is bounded by INTEGRA5:10; A5: (1 / 2) (#) f is_differentiable_on Z by A2, Th33; A6: for x being Real st x in dom (((1 / 2) (#) f) `| Z) holds (((1 / 2) (#) f) `| Z) . x = (id Z) . x proof let x be Real; ::_thesis: ( x in dom (((1 / 2) (#) f) `| Z) implies (((1 / 2) (#) f) `| Z) . x = (id Z) . x ) assume x in dom (((1 / 2) (#) f) `| Z) ; ::_thesis: (((1 / 2) (#) f) `| Z) . x = (id Z) . x then A7: x in Z by A5, FDIFF_1:def_7; then (((1 / 2) (#) f) `| Z) . x = x by A2, Th33 .= (id Z) . x by A7, FUNCT_1:18 ; hence (((1 / 2) (#) f) `| Z) . x = (id Z) . x ; ::_thesis: verum end; dom (((1 / 2) (#) f) `| Z) = dom (id Z) by A5, FDIFF_1:def_7; then A8: ((1 / 2) (#) f) `| Z = id Z by A6, PARTFUN1:5; (id Z) | A is continuous ; then id Z is_integrable_on A by A3, INTEGRA5:11; hence integral ((id Z),A) = (((1 / 2) (#) f) . (upper_bound A)) - (((1 / 2) (#) f) . (lower_bound A)) by A1, A2, A4, A8, Th33, INTEGRA5:13; ::_thesis: verum end; theorem :: INTEGR11:35 for A being non empty closed_interval Subset of REAL for Z being open Subset of REAL for f being PartFunc of REAL,REAL st not 0 in Z & A c= Z & ( for x being Real st x in Z holds ( x <> 0 & f . x = - (1 / (x ^2)) ) ) & dom f = Z & f | A is continuous holds integral (f,A) = (((id Z) ^) . (upper_bound A)) - (((id Z) ^) . (lower_bound A)) proof let A be non empty closed_interval Subset of REAL; ::_thesis: for Z being open Subset of REAL for f being PartFunc of REAL,REAL st not 0 in Z & A c= Z & ( for x being Real st x in Z holds ( x <> 0 & f . x = - (1 / (x ^2)) ) ) & dom f = Z & f | A is continuous holds integral (f,A) = (((id Z) ^) . (upper_bound A)) - (((id Z) ^) . (lower_bound A)) let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st not 0 in Z & A c= Z & ( for x being Real st x in Z holds ( x <> 0 & f . x = - (1 / (x ^2)) ) ) & dom f = Z & f | A is continuous holds integral (f,A) = (((id Z) ^) . (upper_bound A)) - (((id Z) ^) . (lower_bound A)) let f be PartFunc of REAL,REAL; ::_thesis: ( not 0 in Z & A c= Z & ( for x being Real st x in Z holds ( x <> 0 & f . x = - (1 / (x ^2)) ) ) & dom f = Z & f | A is continuous implies integral (f,A) = (((id Z) ^) . (upper_bound A)) - (((id Z) ^) . (lower_bound A)) ) set g = id Z; assume that A1: not 0 in Z and A2: A c= Z and A3: for x being Real st x in Z holds ( x <> 0 & f . x = - (1 / (x ^2)) ) and A4: dom f = Z and A5: f | A is continuous ; ::_thesis: integral (f,A) = (((id Z) ^) . (upper_bound A)) - (((id Z) ^) . (lower_bound A)) A6: f is_integrable_on A by A2, A4, A5, INTEGRA5:11; A7: (id Z) ^ is_differentiable_on Z by A1, FDIFF_5:4; A8: for x being Real st x in dom (((id Z) ^) `| Z) holds (((id Z) ^) `| Z) . x = f . x proof let x be Real; ::_thesis: ( x in dom (((id Z) ^) `| Z) implies (((id Z) ^) `| Z) . x = f . x ) assume x in dom (((id Z) ^) `| Z) ; ::_thesis: (((id Z) ^) `| Z) . x = f . x then A9: x in Z by A7, FDIFF_1:def_7; then (((id Z) ^) `| Z) . x = - (1 / (x ^2)) by A1, FDIFF_5:4 .= f . x by A3, A9 ; hence (((id Z) ^) `| Z) . x = f . x ; ::_thesis: verum end; dom (((id Z) ^) `| Z) = dom f by A4, A7, FDIFF_1:def_7; then ((id Z) ^) `| Z = f by A8, PARTFUN1:5; hence integral (f,A) = (((id Z) ^) . (upper_bound A)) - (((id Z) ^) . (lower_bound A)) by A2, A4, A5, A6, A7, INTEGRA5:10, INTEGRA5:13; ::_thesis: verum end; theorem :: INTEGR11:36 for A being non empty closed_interval Subset of REAL for Z being open Subset of REAL for f1, f2, f being PartFunc of REAL,REAL st A c= Z & f1 = #Z 2 & ( for x being Real st x in Z holds ( f2 . x = 1 & x <> 0 & f . x = (2 * x) / ((1 + (x ^2)) ^2) ) ) & dom (f1 / (f2 + f1)) = Z & Z = dom f & f | A is continuous holds integral (f,A) = ((f1 / (f2 + f1)) . (upper_bound A)) - ((f1 / (f2 + f1)) . (lower_bound A)) proof let A be non empty closed_interval Subset of REAL; ::_thesis: for Z being open Subset of REAL for f1, f2, f being PartFunc of REAL,REAL st A c= Z & f1 = #Z 2 & ( for x being Real st x in Z holds ( f2 . x = 1 & x <> 0 & f . x = (2 * x) / ((1 + (x ^2)) ^2) ) ) & dom (f1 / (f2 + f1)) = Z & Z = dom f & f | A is continuous holds integral (f,A) = ((f1 / (f2 + f1)) . (upper_bound A)) - ((f1 / (f2 + f1)) . (lower_bound A)) let Z be open Subset of REAL; ::_thesis: for f1, f2, f being PartFunc of REAL,REAL st A c= Z & f1 = #Z 2 & ( for x being Real st x in Z holds ( f2 . x = 1 & x <> 0 & f . x = (2 * x) / ((1 + (x ^2)) ^2) ) ) & dom (f1 / (f2 + f1)) = Z & Z = dom f & f | A is continuous holds integral (f,A) = ((f1 / (f2 + f1)) . (upper_bound A)) - ((f1 / (f2 + f1)) . (lower_bound A)) let f1, f2, f be PartFunc of REAL,REAL; ::_thesis: ( A c= Z & f1 = #Z 2 & ( for x being Real st x in Z holds ( f2 . x = 1 & x <> 0 & f . x = (2 * x) / ((1 + (x ^2)) ^2) ) ) & dom (f1 / (f2 + f1)) = Z & Z = dom f & f | A is continuous implies integral (f,A) = ((f1 / (f2 + f1)) . (upper_bound A)) - ((f1 / (f2 + f1)) . (lower_bound A)) ) assume that A1: A c= Z and A2: f1 = #Z 2 and A3: for x being Real st x in Z holds ( f2 . x = 1 & x <> 0 & f . x = (2 * x) / ((1 + (x ^2)) ^2) ) and A4: dom (f1 / (f2 + f1)) = Z and A5: Z = dom f and A6: f | A is continuous ; ::_thesis: integral (f,A) = ((f1 / (f2 + f1)) . (upper_bound A)) - ((f1 / (f2 + f1)) . (lower_bound A)) A7: f is_integrable_on A by A1, A5, A6, INTEGRA5:11; A8: for x being Real st x in Z holds ( f2 . x = 1 & x <> 0 ) by A3; then A9: f1 / (f2 + f1) is_differentiable_on Z by A2, A4, FDIFF_6:7; A10: for x being Real st x in dom ((f1 / (f2 + f1)) `| Z) holds ((f1 / (f2 + f1)) `| Z) . x = f . x proof let x be Real; ::_thesis: ( x in dom ((f1 / (f2 + f1)) `| Z) implies ((f1 / (f2 + f1)) `| Z) . x = f . x ) assume x in dom ((f1 / (f2 + f1)) `| Z) ; ::_thesis: ((f1 / (f2 + f1)) `| Z) . x = f . x then A11: x in Z by A9, FDIFF_1:def_7; then ((f1 / (f2 + f1)) `| Z) . x = (2 * x) / ((1 + (x ^2)) ^2) by A2, A4, A8, FDIFF_6:7 .= f . x by A3, A11 ; hence ((f1 / (f2 + f1)) `| Z) . x = f . x ; ::_thesis: verum end; dom ((f1 / (f2 + f1)) `| Z) = dom f by A5, A9, FDIFF_1:def_7; then (f1 / (f2 + f1)) `| Z = f by A10, PARTFUN1:5; hence integral (f,A) = ((f1 / (f2 + f1)) . (upper_bound A)) - ((f1 / (f2 + f1)) . (lower_bound A)) by A1, A5, A6, A7, A9, INTEGRA5:10, INTEGRA5:13; ::_thesis: verum end; theorem Th37: :: INTEGR11:37 for Z being open Subset of REAL st Z c= dom (tan + sec) & ( for x being Real st x in Z holds ( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 ) ) holds ( tan + sec is_differentiable_on Z & ( for x being Real st x in Z holds ((tan + sec) `| Z) . x = 1 / (1 - (sin . x)) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (tan + sec) & ( for x being Real st x in Z holds ( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 ) ) implies ( tan + sec is_differentiable_on Z & ( for x being Real st x in Z holds ((tan + sec) `| Z) . x = 1 / (1 - (sin . x)) ) ) ) assume that A1: Z c= dom (tan + sec) and A2: for x being Real st x in Z holds ( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 ) ; ::_thesis: ( tan + sec is_differentiable_on Z & ( for x being Real st x in Z holds ((tan + sec) `| Z) . x = 1 / (1 - (sin . x)) ) ) Z c= (dom tan) /\ (dom (cos ^)) by A1, VALUED_1:def_1; then A3: Z c= dom tan by XBOOLE_1:18; then A4: for x being Real st x in Z holds cos . x <> 0 by FDIFF_8:1; then A5: cos ^ is_differentiable_on Z by FDIFF_4:39; for x being Real st x in Z holds tan is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies tan is_differentiable_in x ) assume x in Z ; ::_thesis: tan is_differentiable_in x then cos . x <> 0 by A3, FDIFF_8:1; hence tan is_differentiable_in x by FDIFF_7:46; ::_thesis: verum end; then A6: tan is_differentiable_on Z by A3, FDIFF_1:9; for x being Real st x in Z holds ((tan + sec) `| Z) . x = 1 / (1 - (sin . x)) proof let x be Real; ::_thesis: ( x in Z implies ((tan + sec) `| Z) . x = 1 / (1 - (sin . x)) ) assume A7: x in Z ; ::_thesis: ((tan + sec) `| Z) . x = 1 / (1 - (sin . x)) then A8: cos . x <> 0 by A3, FDIFF_8:1; A9: 1 + (sin . x) <> 0 by A2, A7; ((tan + sec) `| Z) . x = (diff (tan,x)) + (diff ((cos ^),x)) by A1, A5, A6, A7, FDIFF_1:18 .= (1 / ((cos . x) ^2)) + (diff ((cos ^),x)) by A8, FDIFF_7:46 .= (1 / ((cos . x) ^2)) + (((cos ^) `| Z) . x) by A5, A7, FDIFF_1:def_7 .= (1 / ((cos . x) ^2)) + ((sin . x) / ((cos . x) ^2)) by A4, A7, FDIFF_4:39 .= (1 + (sin . x)) / ((((cos . x) ^2) + ((sin . x) ^2)) - ((sin . x) ^2)) by XCMPLX_1:62 .= (1 + (sin . x)) / (1 - ((sin . x) ^2)) by SIN_COS:28 .= (1 + (sin . x)) / ((1 + (sin . x)) * (1 - (sin . x))) .= ((1 + (sin . x)) / (1 + (sin . x))) / (1 - (sin . x)) by XCMPLX_1:78 .= 1 / (1 - (sin . x)) by A9, XCMPLX_1:60 ; hence ((tan + sec) `| Z) . x = 1 / (1 - (sin . x)) ; ::_thesis: verum end; hence ( tan + sec is_differentiable_on Z & ( for x being Real st x in Z holds ((tan + sec) `| Z) . x = 1 / (1 - (sin . x)) ) ) by A1, A5, A6, FDIFF_1:18; ::_thesis: verum end; theorem :: INTEGR11:38 for A being non empty closed_interval Subset of REAL for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds ( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 & f . x = 1 / (1 - (sin . x)) ) ) & dom (tan + sec) = Z & Z = dom f & f | A is continuous holds integral (f,A) = ((tan + sec) . (upper_bound A)) - ((tan + sec) . (lower_bound A)) proof let A be non empty closed_interval Subset of REAL; ::_thesis: for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds ( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 & f . x = 1 / (1 - (sin . x)) ) ) & dom (tan + sec) = Z & Z = dom f & f | A is continuous holds integral (f,A) = ((tan + sec) . (upper_bound A)) - ((tan + sec) . (lower_bound A)) let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds ( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 & f . x = 1 / (1 - (sin . x)) ) ) & dom (tan + sec) = Z & Z = dom f & f | A is continuous holds integral (f,A) = ((tan + sec) . (upper_bound A)) - ((tan + sec) . (lower_bound A)) let f be PartFunc of REAL,REAL; ::_thesis: ( A c= Z & ( for x being Real st x in Z holds ( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 & f . x = 1 / (1 - (sin . x)) ) ) & dom (tan + sec) = Z & Z = dom f & f | A is continuous implies integral (f,A) = ((tan + sec) . (upper_bound A)) - ((tan + sec) . (lower_bound A)) ) assume that A1: A c= Z and A2: for x being Real st x in Z holds ( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 & f . x = 1 / (1 - (sin . x)) ) and A3: dom (tan + sec) = Z and A4: Z = dom f and A5: f | A is continuous ; ::_thesis: integral (f,A) = ((tan + sec) . (upper_bound A)) - ((tan + sec) . (lower_bound A)) A6: f is_integrable_on A by A1, A4, A5, INTEGRA5:11; A7: for x being Real st x in Z holds ( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 ) by A2; then A8: tan + sec is_differentiable_on Z by A3, Th37; A9: for x being Real st x in dom ((tan + sec) `| Z) holds ((tan + sec) `| Z) . x = f . x proof let x be Real; ::_thesis: ( x in dom ((tan + sec) `| Z) implies ((tan + sec) `| Z) . x = f . x ) assume x in dom ((tan + sec) `| Z) ; ::_thesis: ((tan + sec) `| Z) . x = f . x then A10: x in Z by A8, FDIFF_1:def_7; then ((tan + sec) `| Z) . x = 1 / (1 - (sin . x)) by A3, A7, Th37 .= f . x by A2, A10 ; hence ((tan + sec) `| Z) . x = f . x ; ::_thesis: verum end; dom ((tan + sec) `| Z) = dom f by A4, A8, FDIFF_1:def_7; then (tan + sec) `| Z = f by A9, PARTFUN1:5; hence integral (f,A) = ((tan + sec) . (upper_bound A)) - ((tan + sec) . (lower_bound A)) by A1, A4, A5, A6, A8, INTEGRA5:10, INTEGRA5:13; ::_thesis: verum end; theorem Th39: :: INTEGR11:39 for Z being open Subset of REAL st Z c= dom (tan - sec) & ( for x being Real st x in Z holds ( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 ) ) holds ( tan - sec is_differentiable_on Z & ( for x being Real st x in Z holds ((tan - sec) `| Z) . x = 1 / (1 + (sin . x)) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (tan - sec) & ( for x being Real st x in Z holds ( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 ) ) implies ( tan - sec is_differentiable_on Z & ( for x being Real st x in Z holds ((tan - sec) `| Z) . x = 1 / (1 + (sin . x)) ) ) ) assume that A1: Z c= dom (tan - sec) and A2: for x being Real st x in Z holds ( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 ) ; ::_thesis: ( tan - sec is_differentiable_on Z & ( for x being Real st x in Z holds ((tan - sec) `| Z) . x = 1 / (1 + (sin . x)) ) ) Z c= (dom tan) /\ (dom (cos ^)) by A1, VALUED_1:12; then A3: Z c= dom tan by XBOOLE_1:18; then A4: for x being Real st x in Z holds cos . x <> 0 by FDIFF_8:1; then A5: cos ^ is_differentiable_on Z by FDIFF_4:39; for x being Real st x in Z holds tan is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies tan is_differentiable_in x ) assume x in Z ; ::_thesis: tan is_differentiable_in x then cos . x <> 0 by A3, FDIFF_8:1; hence tan is_differentiable_in x by FDIFF_7:46; ::_thesis: verum end; then A6: tan is_differentiable_on Z by A3, FDIFF_1:9; for x being Real st x in Z holds ((tan - sec) `| Z) . x = 1 / (1 + (sin . x)) proof let x be Real; ::_thesis: ( x in Z implies ((tan - sec) `| Z) . x = 1 / (1 + (sin . x)) ) assume A7: x in Z ; ::_thesis: ((tan - sec) `| Z) . x = 1 / (1 + (sin . x)) then A8: cos . x <> 0 by A3, FDIFF_8:1; A9: 1 - (sin . x) <> 0 by A2, A7; ((tan - sec) `| Z) . x = (diff (tan,x)) - (diff ((cos ^),x)) by A1, A5, A6, A7, FDIFF_1:19 .= (1 / ((cos . x) ^2)) - (diff ((cos ^),x)) by A8, FDIFF_7:46 .= (1 / ((cos . x) ^2)) - (((cos ^) `| Z) . x) by A5, A7, FDIFF_1:def_7 .= (1 / ((cos . x) ^2)) - ((sin . x) / ((cos . x) ^2)) by A4, A7, FDIFF_4:39 .= (1 - (sin . x)) / ((((cos . x) ^2) + ((sin . x) ^2)) - ((sin . x) ^2)) by XCMPLX_1:120 .= (1 - (sin . x)) / (1 - ((sin . x) ^2)) by SIN_COS:28 .= (1 - (sin . x)) / ((1 + (sin . x)) * (1 - (sin . x))) .= ((1 - (sin . x)) / (1 - (sin . x))) / (1 + (sin . x)) by XCMPLX_1:78 .= 1 / (1 + (sin . x)) by A9, XCMPLX_1:60 ; hence ((tan - sec) `| Z) . x = 1 / (1 + (sin . x)) ; ::_thesis: verum end; hence ( tan - sec is_differentiable_on Z & ( for x being Real st x in Z holds ((tan - sec) `| Z) . x = 1 / (1 + (sin . x)) ) ) by A1, A5, A6, FDIFF_1:19; ::_thesis: verum end; theorem :: INTEGR11:40 for A being non empty closed_interval Subset of REAL for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds ( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 & f . x = 1 / (1 + (sin . x)) ) ) & dom (tan - sec) = Z & Z = dom f & f | A is continuous holds integral (f,A) = ((tan - sec) . (upper_bound A)) - ((tan - sec) . (lower_bound A)) proof let A be non empty closed_interval Subset of REAL; ::_thesis: for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds ( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 & f . x = 1 / (1 + (sin . x)) ) ) & dom (tan - sec) = Z & Z = dom f & f | A is continuous holds integral (f,A) = ((tan - sec) . (upper_bound A)) - ((tan - sec) . (lower_bound A)) let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds ( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 & f . x = 1 / (1 + (sin . x)) ) ) & dom (tan - sec) = Z & Z = dom f & f | A is continuous holds integral (f,A) = ((tan - sec) . (upper_bound A)) - ((tan - sec) . (lower_bound A)) let f be PartFunc of REAL,REAL; ::_thesis: ( A c= Z & ( for x being Real st x in Z holds ( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 & f . x = 1 / (1 + (sin . x)) ) ) & dom (tan - sec) = Z & Z = dom f & f | A is continuous implies integral (f,A) = ((tan - sec) . (upper_bound A)) - ((tan - sec) . (lower_bound A)) ) assume that A1: A c= Z and A2: for x being Real st x in Z holds ( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 & f . x = 1 / (1 + (sin . x)) ) and A3: dom (tan - sec) = Z and A4: Z = dom f and A5: f | A is continuous ; ::_thesis: integral (f,A) = ((tan - sec) . (upper_bound A)) - ((tan - sec) . (lower_bound A)) A6: f is_integrable_on A by A1, A4, A5, INTEGRA5:11; A7: for x being Real st x in Z holds ( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 ) by A2; then A8: tan - sec is_differentiable_on Z by A3, Th39; A9: for x being Real st x in dom ((tan - sec) `| Z) holds ((tan - sec) `| Z) . x = f . x proof let x be Real; ::_thesis: ( x in dom ((tan - sec) `| Z) implies ((tan - sec) `| Z) . x = f . x ) assume x in dom ((tan - sec) `| Z) ; ::_thesis: ((tan - sec) `| Z) . x = f . x then A10: x in Z by A8, FDIFF_1:def_7; then ((tan - sec) `| Z) . x = 1 / (1 + (sin . x)) by A3, A7, Th39 .= f . x by A2, A10 ; hence ((tan - sec) `| Z) . x = f . x ; ::_thesis: verum end; dom ((tan - sec) `| Z) = dom f by A4, A8, FDIFF_1:def_7; then (tan - sec) `| Z = f by A9, PARTFUN1:5; hence integral (f,A) = ((tan - sec) . (upper_bound A)) - ((tan - sec) . (lower_bound A)) by A1, A4, A5, A6, A8, INTEGRA5:10, INTEGRA5:13; ::_thesis: verum end; theorem Th41: :: INTEGR11:41 for Z being open Subset of REAL st Z c= dom ((- cot) + cosec) & ( for x being Real st x in Z holds ( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 ) ) holds ( (- cot) + cosec is_differentiable_on Z & ( for x being Real st x in Z holds (((- cot) + cosec) `| Z) . x = 1 / (1 + (cos . x)) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom ((- cot) + cosec) & ( for x being Real st x in Z holds ( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 ) ) implies ( (- cot) + cosec is_differentiable_on Z & ( for x being Real st x in Z holds (((- cot) + cosec) `| Z) . x = 1 / (1 + (cos . x)) ) ) ) assume that A1: Z c= dom ((- cot) + cosec) and A2: for x being Real st x in Z holds ( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 ) ; ::_thesis: ( (- cot) + cosec is_differentiable_on Z & ( for x being Real st x in Z holds (((- cot) + cosec) `| Z) . x = 1 / (1 + (cos . x)) ) ) Z c= (dom (- cot)) /\ (dom (sin ^)) by A1, VALUED_1:def_1; then A3: Z c= dom (- cot) by XBOOLE_1:18; then A4: Z c= dom cot by VALUED_1:8; for x being Real st x in Z holds cot is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies cot is_differentiable_in x ) assume x in Z ; ::_thesis: cot is_differentiable_in x then sin . x <> 0 by A4, FDIFF_8:2; hence cot is_differentiable_in x by FDIFF_7:47; ::_thesis: verum end; then A5: cot is_differentiable_on Z by A4, FDIFF_1:9; then A6: (- 1) (#) cot is_differentiable_on Z by A3, FDIFF_1:20; A7: for x being Real st x in Z holds sin . x <> 0 by A4, FDIFF_8:2; then A8: sin ^ is_differentiable_on Z by FDIFF_4:40; for x being Real st x in Z holds (((- cot) + cosec) `| Z) . x = 1 / (1 + (cos . x)) proof let x be Real; ::_thesis: ( x in Z implies (((- cot) + cosec) `| Z) . x = 1 / (1 + (cos . x)) ) assume A9: x in Z ; ::_thesis: (((- cot) + cosec) `| Z) . x = 1 / (1 + (cos . x)) then A10: sin . x <> 0 by A4, FDIFF_8:2; A11: 1 - (cos . x) <> 0 by A2, A9; (((- cot) + cosec) `| Z) . x = (diff ((- cot),x)) + (diff ((sin ^),x)) by A1, A8, A6, A9, FDIFF_1:18 .= ((((- 1) (#) cot) `| Z) . x) + (diff ((sin ^),x)) by A6, A9, FDIFF_1:def_7 .= ((- 1) * (diff (cot,x))) + (diff ((sin ^),x)) by A3, A5, A9, FDIFF_1:20 .= ((- 1) * (- (1 / ((sin . x) ^2)))) + (diff ((sin ^),x)) by A10, FDIFF_7:47 .= (1 / ((sin . x) ^2)) + (((sin ^) `| Z) . x) by A8, A9, FDIFF_1:def_7 .= (1 / ((sin . x) ^2)) + (- ((cos . x) / ((sin . x) ^2))) by A7, A9, FDIFF_4:40 .= (1 / ((sin . x) ^2)) - ((cos . x) / ((sin . x) ^2)) .= (1 - (cos . x)) / ((((sin . x) ^2) + ((cos . x) ^2)) - ((cos . x) ^2)) by XCMPLX_1:120 .= (1 - (cos . x)) / (1 - ((cos . x) ^2)) by SIN_COS:28 .= (1 - (cos . x)) / ((1 - (cos . x)) * (1 + (cos . x))) .= ((1 - (cos . x)) / (1 - (cos . x))) / (1 + (cos . x)) by XCMPLX_1:78 .= 1 / (1 + (cos . x)) by A11, XCMPLX_1:60 ; hence (((- cot) + cosec) `| Z) . x = 1 / (1 + (cos . x)) ; ::_thesis: verum end; hence ( (- cot) + cosec is_differentiable_on Z & ( for x being Real st x in Z holds (((- cot) + cosec) `| Z) . x = 1 / (1 + (cos . x)) ) ) by A1, A8, A6, FDIFF_1:18; ::_thesis: verum end; theorem :: INTEGR11:42 for A being non empty closed_interval Subset of REAL for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds ( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 & f . x = 1 / (1 + (cos . x)) ) ) & dom ((- cot) + cosec) = Z & Z = dom f & f | A is continuous holds integral (f,A) = (((- cot) + cosec) . (upper_bound A)) - (((- cot) + cosec) . (lower_bound A)) proof let A be non empty closed_interval Subset of REAL; ::_thesis: for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds ( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 & f . x = 1 / (1 + (cos . x)) ) ) & dom ((- cot) + cosec) = Z & Z = dom f & f | A is continuous holds integral (f,A) = (((- cot) + cosec) . (upper_bound A)) - (((- cot) + cosec) . (lower_bound A)) let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds ( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 & f . x = 1 / (1 + (cos . x)) ) ) & dom ((- cot) + cosec) = Z & Z = dom f & f | A is continuous holds integral (f,A) = (((- cot) + cosec) . (upper_bound A)) - (((- cot) + cosec) . (lower_bound A)) let f be PartFunc of REAL,REAL; ::_thesis: ( A c= Z & ( for x being Real st x in Z holds ( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 & f . x = 1 / (1 + (cos . x)) ) ) & dom ((- cot) + cosec) = Z & Z = dom f & f | A is continuous implies integral (f,A) = (((- cot) + cosec) . (upper_bound A)) - (((- cot) + cosec) . (lower_bound A)) ) assume that A1: A c= Z and A2: for x being Real st x in Z holds ( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 & f . x = 1 / (1 + (cos . x)) ) and A3: dom ((- cot) + cosec) = Z and A4: Z = dom f and A5: f | A is continuous ; ::_thesis: integral (f,A) = (((- cot) + cosec) . (upper_bound A)) - (((- cot) + cosec) . (lower_bound A)) A6: f is_integrable_on A by A1, A4, A5, INTEGRA5:11; A7: for x being Real st x in Z holds ( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 ) by A2; then A8: (- cot) + cosec is_differentiable_on Z by A3, Th41; A9: for x being Real st x in dom (((- cot) + cosec) `| Z) holds (((- cot) + cosec) `| Z) . x = f . x proof let x be Real; ::_thesis: ( x in dom (((- cot) + cosec) `| Z) implies (((- cot) + cosec) `| Z) . x = f . x ) assume x in dom (((- cot) + cosec) `| Z) ; ::_thesis: (((- cot) + cosec) `| Z) . x = f . x then A10: x in Z by A8, FDIFF_1:def_7; then (((- cot) + cosec) `| Z) . x = 1 / (1 + (cos . x)) by A3, A7, Th41 .= f . x by A2, A10 ; hence (((- cot) + cosec) `| Z) . x = f . x ; ::_thesis: verum end; dom (((- cot) + cosec) `| Z) = dom f by A4, A8, FDIFF_1:def_7; then ((- cot) + cosec) `| Z = f by A9, PARTFUN1:5; hence integral (f,A) = (((- cot) + cosec) . (upper_bound A)) - (((- cot) + cosec) . (lower_bound A)) by A1, A4, A5, A6, A8, INTEGRA5:10, INTEGRA5:13; ::_thesis: verum end; theorem Th43: :: INTEGR11:43 for Z being open Subset of REAL st Z c= dom ((- cot) - cosec) & ( for x being Real st x in Z holds ( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 ) ) holds ( (- cot) - cosec is_differentiable_on Z & ( for x being Real st x in Z holds (((- cot) - cosec) `| Z) . x = 1 / (1 - (cos . x)) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom ((- cot) - cosec) & ( for x being Real st x in Z holds ( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 ) ) implies ( (- cot) - cosec is_differentiable_on Z & ( for x being Real st x in Z holds (((- cot) - cosec) `| Z) . x = 1 / (1 - (cos . x)) ) ) ) assume that A1: Z c= dom ((- cot) - cosec) and A2: for x being Real st x in Z holds ( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 ) ; ::_thesis: ( (- cot) - cosec is_differentiable_on Z & ( for x being Real st x in Z holds (((- cot) - cosec) `| Z) . x = 1 / (1 - (cos . x)) ) ) Z c= (dom (- cot)) /\ (dom (sin ^)) by A1, VALUED_1:12; then A3: Z c= dom (- cot) by XBOOLE_1:18; then A4: Z c= dom cot by VALUED_1:8; for x being Real st x in Z holds cot is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies cot is_differentiable_in x ) assume x in Z ; ::_thesis: cot is_differentiable_in x then sin . x <> 0 by A4, FDIFF_8:2; hence cot is_differentiable_in x by FDIFF_7:47; ::_thesis: verum end; then A5: cot is_differentiable_on Z by A4, FDIFF_1:9; then A6: (- 1) (#) cot is_differentiable_on Z by A3, FDIFF_1:20; A7: for x being Real st x in Z holds sin . x <> 0 by A4, FDIFF_8:2; then A8: sin ^ is_differentiable_on Z by FDIFF_4:40; for x being Real st x in Z holds (((- cot) - cosec) `| Z) . x = 1 / (1 - (cos . x)) proof let x be Real; ::_thesis: ( x in Z implies (((- cot) - cosec) `| Z) . x = 1 / (1 - (cos . x)) ) assume A9: x in Z ; ::_thesis: (((- cot) - cosec) `| Z) . x = 1 / (1 - (cos . x)) then A10: sin . x <> 0 by A4, FDIFF_8:2; A11: 1 + (cos . x) <> 0 by A2, A9; (((- cot) - cosec) `| Z) . x = (diff ((- cot),x)) - (diff ((sin ^),x)) by A1, A8, A6, A9, FDIFF_1:19 .= ((((- 1) (#) cot) `| Z) . x) - (diff ((sin ^),x)) by A6, A9, FDIFF_1:def_7 .= ((- 1) * (diff (cot,x))) - (diff ((sin ^),x)) by A3, A5, A9, FDIFF_1:20 .= ((- 1) * (- (1 / ((sin . x) ^2)))) - (diff ((sin ^),x)) by A10, FDIFF_7:47 .= (1 / ((sin . x) ^2)) - (((sin ^) `| Z) . x) by A8, A9, FDIFF_1:def_7 .= (1 / ((sin . x) ^2)) - (- ((cos . x) / ((sin . x) ^2))) by A7, A9, FDIFF_4:40 .= (1 / ((sin . x) ^2)) + ((cos . x) / ((sin . x) ^2)) .= (1 + (cos . x)) / ((((sin . x) ^2) + ((cos . x) ^2)) - ((cos . x) ^2)) by XCMPLX_1:62 .= (1 + (cos . x)) / (1 - ((cos . x) ^2)) by SIN_COS:28 .= (1 + (cos . x)) / ((1 + (cos . x)) * (1 - (cos . x))) .= ((1 + (cos . x)) / (1 + (cos . x))) / (1 - (cos . x)) by XCMPLX_1:78 .= 1 / (1 - (cos . x)) by A11, XCMPLX_1:60 ; hence (((- cot) - cosec) `| Z) . x = 1 / (1 - (cos . x)) ; ::_thesis: verum end; hence ( (- cot) - cosec is_differentiable_on Z & ( for x being Real st x in Z holds (((- cot) - cosec) `| Z) . x = 1 / (1 - (cos . x)) ) ) by A1, A8, A6, FDIFF_1:19; ::_thesis: verum end; theorem :: INTEGR11:44 for A being non empty closed_interval Subset of REAL for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds ( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 & f . x = 1 / (1 - (cos . x)) ) ) & dom ((- cot) - cosec) = Z & Z = dom f & f | A is continuous holds integral (f,A) = (((- cot) - cosec) . (upper_bound A)) - (((- cot) - cosec) . (lower_bound A)) proof let A be non empty closed_interval Subset of REAL; ::_thesis: for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds ( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 & f . x = 1 / (1 - (cos . x)) ) ) & dom ((- cot) - cosec) = Z & Z = dom f & f | A is continuous holds integral (f,A) = (((- cot) - cosec) . (upper_bound A)) - (((- cot) - cosec) . (lower_bound A)) let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds ( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 & f . x = 1 / (1 - (cos . x)) ) ) & dom ((- cot) - cosec) = Z & Z = dom f & f | A is continuous holds integral (f,A) = (((- cot) - cosec) . (upper_bound A)) - (((- cot) - cosec) . (lower_bound A)) let f be PartFunc of REAL,REAL; ::_thesis: ( A c= Z & ( for x being Real st x in Z holds ( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 & f . x = 1 / (1 - (cos . x)) ) ) & dom ((- cot) - cosec) = Z & Z = dom f & f | A is continuous implies integral (f,A) = (((- cot) - cosec) . (upper_bound A)) - (((- cot) - cosec) . (lower_bound A)) ) assume that A1: A c= Z and A2: for x being Real st x in Z holds ( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 & f . x = 1 / (1 - (cos . x)) ) and A3: dom ((- cot) - cosec) = Z and A4: Z = dom f and A5: f | A is continuous ; ::_thesis: integral (f,A) = (((- cot) - cosec) . (upper_bound A)) - (((- cot) - cosec) . (lower_bound A)) A6: f is_integrable_on A by A1, A4, A5, INTEGRA5:11; A7: for x being Real st x in Z holds ( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 ) by A2; then A8: (- cot) - cosec is_differentiable_on Z by A3, Th43; A9: for x being Real st x in dom (((- cot) - cosec) `| Z) holds (((- cot) - cosec) `| Z) . x = f . x proof let x be Real; ::_thesis: ( x in dom (((- cot) - cosec) `| Z) implies (((- cot) - cosec) `| Z) . x = f . x ) assume x in dom (((- cot) - cosec) `| Z) ; ::_thesis: (((- cot) - cosec) `| Z) . x = f . x then A10: x in Z by A8, FDIFF_1:def_7; then (((- cot) - cosec) `| Z) . x = 1 / (1 - (cos . x)) by A3, A7, Th43 .= f . x by A2, A10 ; hence (((- cot) - cosec) `| Z) . x = f . x ; ::_thesis: verum end; dom (((- cot) - cosec) `| Z) = dom f by A4, A8, FDIFF_1:def_7; then ((- cot) - cosec) `| Z = f by A9, PARTFUN1:5; hence integral (f,A) = (((- cot) - cosec) . (upper_bound A)) - (((- cot) - cosec) . (lower_bound A)) by A1, A4, A5, A6, A8, INTEGRA5:10, INTEGRA5:13; ::_thesis: verum end; theorem :: INTEGR11:45 for A being non empty closed_interval Subset of REAL for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds f . x = 1 / (1 + (x ^2)) ) & Z = dom f & f | A is continuous holds integral (f,A) = (arctan . (upper_bound A)) - (arctan . (lower_bound A)) proof let A be non empty closed_interval Subset of REAL; ::_thesis: for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds f . x = 1 / (1 + (x ^2)) ) & Z = dom f & f | A is continuous holds integral (f,A) = (arctan . (upper_bound A)) - (arctan . (lower_bound A)) let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds f . x = 1 / (1 + (x ^2)) ) & Z = dom f & f | A is continuous holds integral (f,A) = (arctan . (upper_bound A)) - (arctan . (lower_bound A)) let f be PartFunc of REAL,REAL; ::_thesis: ( A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds f . x = 1 / (1 + (x ^2)) ) & Z = dom f & f | A is continuous implies integral (f,A) = (arctan . (upper_bound A)) - (arctan . (lower_bound A)) ) assume that A1: A c= Z and A2: Z c= ].(- 1),1.[ and A3: for x being Real st x in Z holds f . x = 1 / (1 + (x ^2)) and A4: Z = dom f and A5: f | A is continuous ; ::_thesis: integral (f,A) = (arctan . (upper_bound A)) - (arctan . (lower_bound A)) A6: arctan is_differentiable_on Z by A2, SIN_COS9:81; A7: for x being Real st x in dom (arctan `| Z) holds (arctan `| Z) . x = f . x proof let x be Real; ::_thesis: ( x in dom (arctan `| Z) implies (arctan `| Z) . x = f . x ) assume x in dom (arctan `| Z) ; ::_thesis: (arctan `| Z) . x = f . x then A8: x in Z by A6, FDIFF_1:def_7; then (arctan `| Z) . x = 1 / (1 + (x ^2)) by A2, SIN_COS9:81 .= f . x by A3, A8 ; hence (arctan `| Z) . x = f . x ; ::_thesis: verum end; dom (arctan `| Z) = dom f by A4, A6, FDIFF_1:def_7; then A9: arctan `| Z = f by A7, PARTFUN1:5; ( f is_integrable_on A & f | A is bounded ) by A1, A4, A5, INTEGRA5:10, INTEGRA5:11; hence integral (f,A) = (arctan . (upper_bound A)) - (arctan . (lower_bound A)) by A1, A2, A9, INTEGRA5:13, SIN_COS9:81; ::_thesis: verum end; theorem :: INTEGR11:46 for r being Real for A being non empty closed_interval Subset of REAL for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds f . x = r / (1 + (x ^2)) ) & Z = dom f & f | A is continuous holds integral (f,A) = ((r (#) arctan) . (upper_bound A)) - ((r (#) arctan) . (lower_bound A)) proof let r be Real; ::_thesis: for A being non empty closed_interval Subset of REAL for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds f . x = r / (1 + (x ^2)) ) & Z = dom f & f | A is continuous holds integral (f,A) = ((r (#) arctan) . (upper_bound A)) - ((r (#) arctan) . (lower_bound A)) let A be non empty closed_interval Subset of REAL; ::_thesis: for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds f . x = r / (1 + (x ^2)) ) & Z = dom f & f | A is continuous holds integral (f,A) = ((r (#) arctan) . (upper_bound A)) - ((r (#) arctan) . (lower_bound A)) let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds f . x = r / (1 + (x ^2)) ) & Z = dom f & f | A is continuous holds integral (f,A) = ((r (#) arctan) . (upper_bound A)) - ((r (#) arctan) . (lower_bound A)) let f be PartFunc of REAL,REAL; ::_thesis: ( A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds f . x = r / (1 + (x ^2)) ) & Z = dom f & f | A is continuous implies integral (f,A) = ((r (#) arctan) . (upper_bound A)) - ((r (#) arctan) . (lower_bound A)) ) assume that A1: A c= Z and A2: Z c= ].(- 1),1.[ and A3: for x being Real st x in Z holds f . x = r / (1 + (x ^2)) and A4: Z = dom f and A5: f | A is continuous ; ::_thesis: integral (f,A) = ((r (#) arctan) . (upper_bound A)) - ((r (#) arctan) . (lower_bound A)) A6: r (#) arctan is_differentiable_on Z by A2, SIN_COS9:83; A7: for x being Real st x in dom ((r (#) arctan) `| Z) holds ((r (#) arctan) `| Z) . x = f . x proof let x be Real; ::_thesis: ( x in dom ((r (#) arctan) `| Z) implies ((r (#) arctan) `| Z) . x = f . x ) assume x in dom ((r (#) arctan) `| Z) ; ::_thesis: ((r (#) arctan) `| Z) . x = f . x then A8: x in Z by A6, FDIFF_1:def_7; then ((r (#) arctan) `| Z) . x = r / (1 + (x ^2)) by A2, SIN_COS9:83 .= f . x by A3, A8 ; hence ((r (#) arctan) `| Z) . x = f . x ; ::_thesis: verum end; dom ((r (#) arctan) `| Z) = dom f by A4, A6, FDIFF_1:def_7; then A9: (r (#) arctan) `| Z = f by A7, PARTFUN1:5; ( f is_integrable_on A & f | A is bounded ) by A1, A4, A5, INTEGRA5:10, INTEGRA5:11; hence integral (f,A) = ((r (#) arctan) . (upper_bound A)) - ((r (#) arctan) . (lower_bound A)) by A1, A2, A9, INTEGRA5:13, SIN_COS9:83; ::_thesis: verum end; theorem :: INTEGR11:47 for A being non empty closed_interval Subset of REAL for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds f . x = - (1 / (1 + (x ^2))) ) & Z = dom f & f | A is continuous holds integral (f,A) = (arccot . (upper_bound A)) - (arccot . (lower_bound A)) proof let A be non empty closed_interval Subset of REAL; ::_thesis: for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds f . x = - (1 / (1 + (x ^2))) ) & Z = dom f & f | A is continuous holds integral (f,A) = (arccot . (upper_bound A)) - (arccot . (lower_bound A)) let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds f . x = - (1 / (1 + (x ^2))) ) & Z = dom f & f | A is continuous holds integral (f,A) = (arccot . (upper_bound A)) - (arccot . (lower_bound A)) let f be PartFunc of REAL,REAL; ::_thesis: ( A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds f . x = - (1 / (1 + (x ^2))) ) & Z = dom f & f | A is continuous implies integral (f,A) = (arccot . (upper_bound A)) - (arccot . (lower_bound A)) ) assume that A1: A c= Z and A2: Z c= ].(- 1),1.[ and A3: for x being Real st x in Z holds f . x = - (1 / (1 + (x ^2))) and A4: Z = dom f and A5: f | A is continuous ; ::_thesis: integral (f,A) = (arccot . (upper_bound A)) - (arccot . (lower_bound A)) A6: arccot is_differentiable_on Z by A2, SIN_COS9:82; A7: for x being Real st x in dom (arccot `| Z) holds (arccot `| Z) . x = f . x proof let x be Real; ::_thesis: ( x in dom (arccot `| Z) implies (arccot `| Z) . x = f . x ) assume x in dom (arccot `| Z) ; ::_thesis: (arccot `| Z) . x = f . x then A8: x in Z by A6, FDIFF_1:def_7; then (arccot `| Z) . x = - (1 / (1 + (x ^2))) by A2, SIN_COS9:82 .= f . x by A3, A8 ; hence (arccot `| Z) . x = f . x ; ::_thesis: verum end; dom (arccot `| Z) = dom f by A4, A6, FDIFF_1:def_7; then A9: arccot `| Z = f by A7, PARTFUN1:5; ( f is_integrable_on A & f | A is bounded ) by A1, A4, A5, INTEGRA5:10, INTEGRA5:11; hence integral (f,A) = (arccot . (upper_bound A)) - (arccot . (lower_bound A)) by A1, A2, A9, INTEGRA5:13, SIN_COS9:82; ::_thesis: verum end; theorem :: INTEGR11:48 for r being Real for A being non empty closed_interval Subset of REAL for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds f . x = - (r / (1 + (x ^2))) ) & Z = dom f & f | A is continuous holds integral (f,A) = ((r (#) arccot) . (upper_bound A)) - ((r (#) arccot) . (lower_bound A)) proof let r be Real; ::_thesis: for A being non empty closed_interval Subset of REAL for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds f . x = - (r / (1 + (x ^2))) ) & Z = dom f & f | A is continuous holds integral (f,A) = ((r (#) arccot) . (upper_bound A)) - ((r (#) arccot) . (lower_bound A)) let A be non empty closed_interval Subset of REAL; ::_thesis: for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds f . x = - (r / (1 + (x ^2))) ) & Z = dom f & f | A is continuous holds integral (f,A) = ((r (#) arccot) . (upper_bound A)) - ((r (#) arccot) . (lower_bound A)) let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds f . x = - (r / (1 + (x ^2))) ) & Z = dom f & f | A is continuous holds integral (f,A) = ((r (#) arccot) . (upper_bound A)) - ((r (#) arccot) . (lower_bound A)) let f be PartFunc of REAL,REAL; ::_thesis: ( A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds f . x = - (r / (1 + (x ^2))) ) & Z = dom f & f | A is continuous implies integral (f,A) = ((r (#) arccot) . (upper_bound A)) - ((r (#) arccot) . (lower_bound A)) ) assume that A1: A c= Z and A2: Z c= ].(- 1),1.[ and A3: for x being Real st x in Z holds f . x = - (r / (1 + (x ^2))) and A4: Z = dom f and A5: f | A is continuous ; ::_thesis: integral (f,A) = ((r (#) arccot) . (upper_bound A)) - ((r (#) arccot) . (lower_bound A)) A6: r (#) arccot is_differentiable_on Z by A2, SIN_COS9:84; A7: for x being Real st x in dom ((r (#) arccot) `| Z) holds ((r (#) arccot) `| Z) . x = f . x proof let x be Real; ::_thesis: ( x in dom ((r (#) arccot) `| Z) implies ((r (#) arccot) `| Z) . x = f . x ) assume x in dom ((r (#) arccot) `| Z) ; ::_thesis: ((r (#) arccot) `| Z) . x = f . x then A8: x in Z by A6, FDIFF_1:def_7; then ((r (#) arccot) `| Z) . x = - (r / (1 + (x ^2))) by A2, SIN_COS9:84 .= f . x by A3, A8 ; hence ((r (#) arccot) `| Z) . x = f . x ; ::_thesis: verum end; dom ((r (#) arccot) `| Z) = dom f by A4, A6, FDIFF_1:def_7; then A9: (r (#) arccot) `| Z = f by A7, PARTFUN1:5; ( f is_integrable_on A & f | A is bounded ) by A1, A4, A5, INTEGRA5:10, INTEGRA5:11; hence integral (f,A) = ((r (#) arccot) . (upper_bound A)) - ((r (#) arccot) . (lower_bound A)) by A1, A2, A9, INTEGRA5:13, SIN_COS9:84; ::_thesis: verum end; theorem Th49: :: INTEGR11:49 for Z being open Subset of REAL st Z c= dom (((id Z) + cot) - cosec) & ( for x being Real st x in Z holds ( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 ) ) holds ( ((id Z) + cot) - cosec is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) + cot) - cosec) `| Z) . x = (cos . x) / (1 + (cos . x)) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (((id Z) + cot) - cosec) & ( for x being Real st x in Z holds ( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 ) ) implies ( ((id Z) + cot) - cosec is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) + cot) - cosec) `| Z) . x = (cos . x) / (1 + (cos . x)) ) ) ) assume that A1: Z c= dom (((id Z) + cot) - cosec) and A2: for x being Real st x in Z holds ( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 ) ; ::_thesis: ( ((id Z) + cot) - cosec is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) + cot) - cosec) `| Z) . x = (cos . x) / (1 + (cos . x)) ) ) A3: Z c= (dom ((id Z) + cot)) /\ (dom cosec) by A1, VALUED_1:12; then A4: Z c= dom ((id Z) + cot) by XBOOLE_1:18; then Z c= (dom (id Z)) /\ (dom cot) by VALUED_1:def_1; then A5: Z c= dom cot by XBOOLE_1:18; A6: for x being Real st x in Z holds (id Z) . x = (1 * x) + 0 by FUNCT_1:18; A7: Z c= dom (id Z) ; then A8: id Z is_differentiable_on Z by A6, FDIFF_1:23; for x being Real st x in Z holds cot is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies cot is_differentiable_in x ) assume x in Z ; ::_thesis: cot is_differentiable_in x then sin . x <> 0 by A5, FDIFF_8:2; hence cot is_differentiable_in x by FDIFF_7:47; ::_thesis: verum end; then A9: cot is_differentiable_on Z by A5, FDIFF_1:9; then A10: (id Z) + cot is_differentiable_on Z by A4, A8, FDIFF_1:18; A11: Z c= dom cosec by A3, XBOOLE_1:18; then A12: cosec is_differentiable_on Z by FDIFF_9:5; A13: for x being Real st x in Z holds (((id Z) + cot) `| Z) . x = - (((cos . x) ^2) / ((sin . x) ^2)) proof let x be Real; ::_thesis: ( x in Z implies (((id Z) + cot) `| Z) . x = - (((cos . x) ^2) / ((sin . x) ^2)) ) assume A14: x in Z ; ::_thesis: (((id Z) + cot) `| Z) . x = - (((cos . x) ^2) / ((sin . x) ^2)) then A15: sin . x <> 0 by A5, FDIFF_8:2; then A16: (sin . x) ^2 > 0 by SQUARE_1:12; (((id Z) + cot) `| Z) . x = (diff ((id Z),x)) + (diff (cot,x)) by A4, A9, A8, A14, FDIFF_1:18 .= (((id Z) `| Z) . x) + (diff (cot,x)) by A8, A14, FDIFF_1:def_7 .= 1 + (diff (cot,x)) by A7, A6, A14, FDIFF_1:23 .= 1 + (- (1 / ((sin . x) ^2))) by A15, FDIFF_7:47 .= 1 - (1 / ((sin . x) ^2)) .= (((sin . x) ^2) / ((sin . x) ^2)) - (1 / ((sin . x) ^2)) by A16, XCMPLX_1:60 .= (((sin . x) ^2) - 1) / ((sin . x) ^2) by XCMPLX_1:120 .= (((sin . x) ^2) - (((sin . x) ^2) + ((cos . x) ^2))) / ((sin . x) ^2) by SIN_COS:28 .= (- ((cos . x) ^2)) / ((sin . x) ^2) .= - (((cos . x) ^2) / ((sin . x) ^2)) by XCMPLX_1:187 ; hence (((id Z) + cot) `| Z) . x = - (((cos . x) ^2) / ((sin . x) ^2)) ; ::_thesis: verum end; for x being Real st x in Z holds ((((id Z) + cot) - cosec) `| Z) . x = (cos . x) / (1 + (cos . x)) proof let x be Real; ::_thesis: ( x in Z implies ((((id Z) + cot) - cosec) `| Z) . x = (cos . x) / (1 + (cos . x)) ) assume A17: x in Z ; ::_thesis: ((((id Z) + cot) - cosec) `| Z) . x = (cos . x) / (1 + (cos . x)) then A18: 1 - (cos . x) <> 0 by A2; ((((id Z) + cot) - cosec) `| Z) . x = (diff (((id Z) + cot),x)) - (diff (cosec,x)) by A1, A12, A10, A17, FDIFF_1:19 .= ((((id Z) + cot) `| Z) . x) - (diff (cosec,x)) by A10, A17, FDIFF_1:def_7 .= (- (((cos . x) ^2) / ((sin . x) ^2))) - (diff (cosec,x)) by A13, A17 .= (- (((cos . x) ^2) / ((sin . x) ^2))) - ((cosec `| Z) . x) by A12, A17, FDIFF_1:def_7 .= (- (((cos . x) ^2) / ((sin . x) ^2))) - (- ((cos . x) / ((sin . x) ^2))) by A11, A17, FDIFF_9:5 .= ((cos . x) / ((sin . x) ^2)) - (((cos . x) ^2) / ((sin . x) ^2)) .= ((cos . x) - ((cos . x) * (cos . x))) / ((sin . x) ^2) by XCMPLX_1:120 .= ((cos . x) * (1 - (cos . x))) / ((((sin . x) ^2) + ((cos . x) ^2)) - ((cos . x) ^2)) .= ((cos . x) * (1 - (cos . x))) / (1 - ((cos . x) ^2)) by SIN_COS:28 .= ((cos . x) * (1 - (cos . x))) / ((1 - (cos . x)) * (1 + (cos . x))) .= (((cos . x) * (1 - (cos . x))) / (1 - (cos . x))) / (1 + (cos . x)) by XCMPLX_1:78 .= ((cos . x) * ((1 - (cos . x)) / (1 - (cos . x)))) / (1 + (cos . x)) by XCMPLX_1:74 .= ((cos . x) * 1) / (1 + (cos . x)) by A18, XCMPLX_1:60 .= (cos . x) / (1 + (cos . x)) ; hence ((((id Z) + cot) - cosec) `| Z) . x = (cos . x) / (1 + (cos . x)) ; ::_thesis: verum end; hence ( ((id Z) + cot) - cosec is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) + cot) - cosec) `| Z) . x = (cos . x) / (1 + (cos . x)) ) ) by A1, A12, A10, FDIFF_1:19; ::_thesis: verum end; theorem :: INTEGR11:50 for A being non empty closed_interval Subset of REAL for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds ( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 & f . x = (cos . x) / (1 + (cos . x)) ) ) & dom (((id Z) + cot) - cosec) = Z & Z = dom f & f | A is continuous holds integral (f,A) = ((((id Z) + cot) - cosec) . (upper_bound A)) - ((((id Z) + cot) - cosec) . (lower_bound A)) proof let A be non empty closed_interval Subset of REAL; ::_thesis: for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds ( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 & f . x = (cos . x) / (1 + (cos . x)) ) ) & dom (((id Z) + cot) - cosec) = Z & Z = dom f & f | A is continuous holds integral (f,A) = ((((id Z) + cot) - cosec) . (upper_bound A)) - ((((id Z) + cot) - cosec) . (lower_bound A)) let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds ( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 & f . x = (cos . x) / (1 + (cos . x)) ) ) & dom (((id Z) + cot) - cosec) = Z & Z = dom f & f | A is continuous holds integral (f,A) = ((((id Z) + cot) - cosec) . (upper_bound A)) - ((((id Z) + cot) - cosec) . (lower_bound A)) let f be PartFunc of REAL,REAL; ::_thesis: ( A c= Z & ( for x being Real st x in Z holds ( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 & f . x = (cos . x) / (1 + (cos . x)) ) ) & dom (((id Z) + cot) - cosec) = Z & Z = dom f & f | A is continuous implies integral (f,A) = ((((id Z) + cot) - cosec) . (upper_bound A)) - ((((id Z) + cot) - cosec) . (lower_bound A)) ) assume that A1: A c= Z and A2: for x being Real st x in Z holds ( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 & f . x = (cos . x) / (1 + (cos . x)) ) and A3: dom (((id Z) + cot) - cosec) = Z and A4: Z = dom f and A5: f | A is continuous ; ::_thesis: integral (f,A) = ((((id Z) + cot) - cosec) . (upper_bound A)) - ((((id Z) + cot) - cosec) . (lower_bound A)) A6: f is_integrable_on A by A1, A4, A5, INTEGRA5:11; A7: for x being Real st x in Z holds ( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 ) by A2; then A8: ((id Z) + cot) - cosec is_differentiable_on Z by A3, Th49; A9: for x being Real st x in dom ((((id Z) + cot) - cosec) `| Z) holds ((((id Z) + cot) - cosec) `| Z) . x = f . x proof let x be Real; ::_thesis: ( x in dom ((((id Z) + cot) - cosec) `| Z) implies ((((id Z) + cot) - cosec) `| Z) . x = f . x ) assume x in dom ((((id Z) + cot) - cosec) `| Z) ; ::_thesis: ((((id Z) + cot) - cosec) `| Z) . x = f . x then A10: x in Z by A8, FDIFF_1:def_7; then ((((id Z) + cot) - cosec) `| Z) . x = (cos . x) / (1 + (cos . x)) by A3, A7, Th49 .= f . x by A2, A10 ; hence ((((id Z) + cot) - cosec) `| Z) . x = f . x ; ::_thesis: verum end; dom ((((id Z) + cot) - cosec) `| Z) = dom f by A4, A8, FDIFF_1:def_7; then (((id Z) + cot) - cosec) `| Z = f by A9, PARTFUN1:5; hence integral (f,A) = ((((id Z) + cot) - cosec) . (upper_bound A)) - ((((id Z) + cot) - cosec) . (lower_bound A)) by A1, A4, A5, A6, A8, INTEGRA5:10, INTEGRA5:13; ::_thesis: verum end; theorem Th51: :: INTEGR11:51 for Z being open Subset of REAL st Z c= dom (((id Z) + cot) + cosec) & ( for x being Real st x in Z holds ( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 ) ) holds ( ((id Z) + cot) + cosec is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) + cot) + cosec) `| Z) . x = (cos . x) / ((cos . x) - 1) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (((id Z) + cot) + cosec) & ( for x being Real st x in Z holds ( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 ) ) implies ( ((id Z) + cot) + cosec is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) + cot) + cosec) `| Z) . x = (cos . x) / ((cos . x) - 1) ) ) ) assume that A1: Z c= dom (((id Z) + cot) + cosec) and A2: for x being Real st x in Z holds ( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 ) ; ::_thesis: ( ((id Z) + cot) + cosec is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) + cot) + cosec) `| Z) . x = (cos . x) / ((cos . x) - 1) ) ) A3: Z c= (dom ((id Z) + cot)) /\ (dom cosec) by A1, VALUED_1:def_1; then A4: Z c= dom ((id Z) + cot) by XBOOLE_1:18; then Z c= (dom (id Z)) /\ (dom cot) by VALUED_1:def_1; then A5: Z c= dom cot by XBOOLE_1:18; A6: for x being Real st x in Z holds (id Z) . x = (1 * x) + 0 by FUNCT_1:18; A7: Z c= dom (id Z) ; then A8: id Z is_differentiable_on Z by A6, FDIFF_1:23; for x being Real st x in Z holds cot is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies cot is_differentiable_in x ) assume x in Z ; ::_thesis: cot is_differentiable_in x then sin . x <> 0 by A5, FDIFF_8:2; hence cot is_differentiable_in x by FDIFF_7:47; ::_thesis: verum end; then A9: cot is_differentiable_on Z by A5, FDIFF_1:9; then A10: (id Z) + cot is_differentiable_on Z by A4, A8, FDIFF_1:18; A11: Z c= dom cosec by A3, XBOOLE_1:18; then A12: cosec is_differentiable_on Z by FDIFF_9:5; A13: for x being Real st x in Z holds (((id Z) + cot) `| Z) . x = - (((cos . x) ^2) / ((sin . x) ^2)) proof let x be Real; ::_thesis: ( x in Z implies (((id Z) + cot) `| Z) . x = - (((cos . x) ^2) / ((sin . x) ^2)) ) assume A14: x in Z ; ::_thesis: (((id Z) + cot) `| Z) . x = - (((cos . x) ^2) / ((sin . x) ^2)) then A15: sin . x <> 0 by A5, FDIFF_8:2; then A16: (sin . x) ^2 > 0 by SQUARE_1:12; (((id Z) + cot) `| Z) . x = (diff ((id Z),x)) + (diff (cot,x)) by A4, A9, A8, A14, FDIFF_1:18 .= (((id Z) `| Z) . x) + (diff (cot,x)) by A8, A14, FDIFF_1:def_7 .= 1 + (diff (cot,x)) by A7, A6, A14, FDIFF_1:23 .= 1 + (- (1 / ((sin . x) ^2))) by A15, FDIFF_7:47 .= 1 - (1 / ((sin . x) ^2)) .= (((sin . x) ^2) / ((sin . x) ^2)) - (1 / ((sin . x) ^2)) by A16, XCMPLX_1:60 .= (((sin . x) ^2) - 1) / ((sin . x) ^2) by XCMPLX_1:120 .= (((sin . x) ^2) - (((sin . x) ^2) + ((cos . x) ^2))) / ((sin . x) ^2) by SIN_COS:28 .= (- ((cos . x) ^2)) / ((sin . x) ^2) .= - (((cos . x) ^2) / ((sin . x) ^2)) by XCMPLX_1:187 ; hence (((id Z) + cot) `| Z) . x = - (((cos . x) ^2) / ((sin . x) ^2)) ; ::_thesis: verum end; for x being Real st x in Z holds ((((id Z) + cot) + cosec) `| Z) . x = (cos . x) / ((cos . x) - 1) proof let x be Real; ::_thesis: ( x in Z implies ((((id Z) + cot) + cosec) `| Z) . x = (cos . x) / ((cos . x) - 1) ) assume A17: x in Z ; ::_thesis: ((((id Z) + cot) + cosec) `| Z) . x = (cos . x) / ((cos . x) - 1) then A18: 1 + (cos . x) <> 0 by A2; ((((id Z) + cot) + cosec) `| Z) . x = (diff (((id Z) + cot),x)) + (diff (cosec,x)) by A1, A12, A10, A17, FDIFF_1:18 .= ((((id Z) + cot) `| Z) . x) + (diff (cosec,x)) by A10, A17, FDIFF_1:def_7 .= (- (((cos . x) ^2) / ((sin . x) ^2))) + (diff (cosec,x)) by A13, A17 .= (- (((cos . x) ^2) / ((sin . x) ^2))) + ((cosec `| Z) . x) by A12, A17, FDIFF_1:def_7 .= (- (((cos . x) ^2) / ((sin . x) ^2))) + (- ((cos . x) / ((sin . x) ^2))) by A11, A17, FDIFF_9:5 .= - ((((cos . x) ^2) / ((sin . x) ^2)) + ((cos . x) / ((sin . x) ^2))) .= - ((((cos . x) * (cos . x)) + (cos . x)) / ((sin . x) ^2)) by XCMPLX_1:62 .= - (((cos . x) * ((cos . x) + 1)) / ((((sin . x) ^2) + ((cos . x) ^2)) - ((cos . x) ^2))) .= - (((cos . x) * ((cos . x) + 1)) / (1 - ((cos . x) ^2))) by SIN_COS:28 .= - (((cos . x) * ((cos . x) + 1)) / ((1 + (cos . x)) * (1 - (cos . x)))) .= - ((((cos . x) * ((cos . x) + 1)) / (1 + (cos . x))) / (1 - (cos . x))) by XCMPLX_1:78 .= - (((cos . x) * ((1 + (cos . x)) / (1 + (cos . x)))) / (1 - (cos . x))) by XCMPLX_1:74 .= - (((cos . x) * 1) / (1 - (cos . x))) by A18, XCMPLX_1:60 .= (cos . x) / (- (1 - (cos . x))) by XCMPLX_1:188 .= (cos . x) / ((cos . x) - 1) ; hence ((((id Z) + cot) + cosec) `| Z) . x = (cos . x) / ((cos . x) - 1) ; ::_thesis: verum end; hence ( ((id Z) + cot) + cosec is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) + cot) + cosec) `| Z) . x = (cos . x) / ((cos . x) - 1) ) ) by A1, A12, A10, FDIFF_1:18; ::_thesis: verum end; theorem :: INTEGR11:52 for A being non empty closed_interval Subset of REAL for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds ( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 & f . x = (cos . x) / ((cos . x) - 1) ) ) & dom (((id Z) + cot) + cosec) = Z & Z = dom f & f | A is continuous holds integral (f,A) = ((((id Z) + cot) + cosec) . (upper_bound A)) - ((((id Z) + cot) + cosec) . (lower_bound A)) proof let A be non empty closed_interval Subset of REAL; ::_thesis: for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds ( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 & f . x = (cos . x) / ((cos . x) - 1) ) ) & dom (((id Z) + cot) + cosec) = Z & Z = dom f & f | A is continuous holds integral (f,A) = ((((id Z) + cot) + cosec) . (upper_bound A)) - ((((id Z) + cot) + cosec) . (lower_bound A)) let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds ( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 & f . x = (cos . x) / ((cos . x) - 1) ) ) & dom (((id Z) + cot) + cosec) = Z & Z = dom f & f | A is continuous holds integral (f,A) = ((((id Z) + cot) + cosec) . (upper_bound A)) - ((((id Z) + cot) + cosec) . (lower_bound A)) let f be PartFunc of REAL,REAL; ::_thesis: ( A c= Z & ( for x being Real st x in Z holds ( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 & f . x = (cos . x) / ((cos . x) - 1) ) ) & dom (((id Z) + cot) + cosec) = Z & Z = dom f & f | A is continuous implies integral (f,A) = ((((id Z) + cot) + cosec) . (upper_bound A)) - ((((id Z) + cot) + cosec) . (lower_bound A)) ) assume that A1: A c= Z and A2: for x being Real st x in Z holds ( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 & f . x = (cos . x) / ((cos . x) - 1) ) and A3: dom (((id Z) + cot) + cosec) = Z and A4: Z = dom f and A5: f | A is continuous ; ::_thesis: integral (f,A) = ((((id Z) + cot) + cosec) . (upper_bound A)) - ((((id Z) + cot) + cosec) . (lower_bound A)) A6: f is_integrable_on A by A1, A4, A5, INTEGRA5:11; A7: for x being Real st x in Z holds ( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 ) by A2; then A8: ((id Z) + cot) + cosec is_differentiable_on Z by A3, Th51; A9: for x being Real st x in dom ((((id Z) + cot) + cosec) `| Z) holds ((((id Z) + cot) + cosec) `| Z) . x = f . x proof let x be Real; ::_thesis: ( x in dom ((((id Z) + cot) + cosec) `| Z) implies ((((id Z) + cot) + cosec) `| Z) . x = f . x ) assume x in dom ((((id Z) + cot) + cosec) `| Z) ; ::_thesis: ((((id Z) + cot) + cosec) `| Z) . x = f . x then A10: x in Z by A8, FDIFF_1:def_7; then ((((id Z) + cot) + cosec) `| Z) . x = (cos . x) / ((cos . x) - 1) by A3, A7, Th51 .= f . x by A2, A10 ; hence ((((id Z) + cot) + cosec) `| Z) . x = f . x ; ::_thesis: verum end; dom ((((id Z) + cot) + cosec) `| Z) = dom f by A4, A8, FDIFF_1:def_7; then (((id Z) + cot) + cosec) `| Z = f by A9, PARTFUN1:5; hence integral (f,A) = ((((id Z) + cot) + cosec) . (upper_bound A)) - ((((id Z) + cot) + cosec) . (lower_bound A)) by A1, A4, A5, A6, A8, INTEGRA5:10, INTEGRA5:13; ::_thesis: verum end; theorem Th53: :: INTEGR11:53 for Z being open Subset of REAL st Z c= dom (((id Z) - tan) + sec) & ( for x being Real st x in Z holds ( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 ) ) holds ( ((id Z) - tan) + sec is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) - tan) + sec) `| Z) . x = (sin . x) / ((sin . x) + 1) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (((id Z) - tan) + sec) & ( for x being Real st x in Z holds ( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 ) ) implies ( ((id Z) - tan) + sec is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) - tan) + sec) `| Z) . x = (sin . x) / ((sin . x) + 1) ) ) ) assume that A1: Z c= dom (((id Z) - tan) + sec) and A2: for x being Real st x in Z holds ( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 ) ; ::_thesis: ( ((id Z) - tan) + sec is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) - tan) + sec) `| Z) . x = (sin . x) / ((sin . x) + 1) ) ) A3: Z c= (dom ((id Z) - tan)) /\ (dom sec) by A1, VALUED_1:def_1; then A4: Z c= dom ((id Z) - tan) by XBOOLE_1:18; then A5: Z c= (dom (id Z)) /\ (dom tan) by VALUED_1:12; A6: Z c= dom (id Z) ; A7: for x being Real st x in Z holds (id Z) . x = (1 * x) + 0 by FUNCT_1:18; then A8: id Z is_differentiable_on Z by A6, FDIFF_1:23; A9: Z c= dom tan by A5, XBOOLE_1:18; for x being Real st x in Z holds tan is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies tan is_differentiable_in x ) assume x in Z ; ::_thesis: tan is_differentiable_in x then cos . x <> 0 by A9, FDIFF_8:1; hence tan is_differentiable_in x by FDIFF_7:46; ::_thesis: verum end; then A10: tan is_differentiable_on Z by A9, FDIFF_1:9; then A11: (id Z) - tan is_differentiable_on Z by A4, A8, FDIFF_1:19; A12: Z c= dom sec by A3, XBOOLE_1:18; then A13: sec is_differentiable_on Z by FDIFF_9:4; A14: for x being Real st x in Z holds (((id Z) - tan) `| Z) . x = - (((sin . x) ^2) / ((cos . x) ^2)) proof let x be Real; ::_thesis: ( x in Z implies (((id Z) - tan) `| Z) . x = - (((sin . x) ^2) / ((cos . x) ^2)) ) assume A15: x in Z ; ::_thesis: (((id Z) - tan) `| Z) . x = - (((sin . x) ^2) / ((cos . x) ^2)) then A16: cos . x <> 0 by A9, FDIFF_8:1; then A17: (cos . x) ^2 > 0 by SQUARE_1:12; (((id Z) - tan) `| Z) . x = (diff ((id Z),x)) - (diff (tan,x)) by A4, A10, A8, A15, FDIFF_1:19 .= (((id Z) `| Z) . x) - (diff (tan,x)) by A8, A15, FDIFF_1:def_7 .= 1 - (diff (tan,x)) by A6, A7, A15, FDIFF_1:23 .= 1 - (1 / ((cos . x) ^2)) by A16, FDIFF_7:46 .= 1 - ((((cos . x) ^2) + ((sin . x) ^2)) / ((cos . x) ^2)) by SIN_COS:28 .= 1 - ((((cos . x) ^2) / ((cos . x) ^2)) + (((sin . x) ^2) / ((cos . x) ^2))) by XCMPLX_1:62 .= 1 - (1 + (((sin . x) ^2) / ((cos . x) ^2))) by A17, XCMPLX_1:60 .= - (((sin . x) ^2) / ((cos . x) ^2)) ; hence (((id Z) - tan) `| Z) . x = - (((sin . x) ^2) / ((cos . x) ^2)) ; ::_thesis: verum end; for x being Real st x in Z holds ((((id Z) - tan) + sec) `| Z) . x = (sin . x) / ((sin . x) + 1) proof let x be Real; ::_thesis: ( x in Z implies ((((id Z) - tan) + sec) `| Z) . x = (sin . x) / ((sin . x) + 1) ) assume A18: x in Z ; ::_thesis: ((((id Z) - tan) + sec) `| Z) . x = (sin . x) / ((sin . x) + 1) then A19: 1 - (sin . x) <> 0 by A2; ((((id Z) - tan) + sec) `| Z) . x = (diff (((id Z) - tan),x)) + (diff (sec,x)) by A1, A13, A11, A18, FDIFF_1:18 .= ((((id Z) - tan) `| Z) . x) + (diff (sec,x)) by A11, A18, FDIFF_1:def_7 .= (- (((sin . x) ^2) / ((cos . x) ^2))) + (diff (sec,x)) by A14, A18 .= (- (((sin . x) ^2) / ((cos . x) ^2))) + ((sec `| Z) . x) by A13, A18, FDIFF_1:def_7 .= (- (((sin . x) ^2) / ((cos . x) ^2))) + ((sin . x) / ((cos . x) ^2)) by A12, A18, FDIFF_9:4 .= ((sin . x) / ((cos . x) ^2)) - (((sin . x) ^2) / ((cos . x) ^2)) .= ((sin . x) - ((sin . x) * (sin . x))) / ((cos . x) ^2) by XCMPLX_1:120 .= ((sin . x) * (1 - (sin . x))) / ((((cos . x) ^2) + ((sin . x) ^2)) - ((sin . x) ^2)) .= ((sin . x) * (1 - (sin . x))) / (1 - ((sin . x) ^2)) by SIN_COS:28 .= ((sin . x) * (1 - (sin . x))) / ((1 - (sin . x)) * (1 + (sin . x))) .= (((sin . x) * (1 - (sin . x))) / (1 - (sin . x))) / (1 + (sin . x)) by XCMPLX_1:78 .= ((sin . x) * ((1 - (sin . x)) / (1 - (sin . x)))) / (1 + (sin . x)) by XCMPLX_1:74 .= ((sin . x) * 1) / (1 + (sin . x)) by A19, XCMPLX_1:60 .= (sin . x) / (1 + (sin . x)) ; hence ((((id Z) - tan) + sec) `| Z) . x = (sin . x) / ((sin . x) + 1) ; ::_thesis: verum end; hence ( ((id Z) - tan) + sec is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) - tan) + sec) `| Z) . x = (sin . x) / ((sin . x) + 1) ) ) by A1, A13, A11, FDIFF_1:18; ::_thesis: verum end; theorem :: INTEGR11:54 for A being non empty closed_interval Subset of REAL for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds ( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 & f . x = (sin . x) / (1 + (sin . x)) ) ) & Z c= dom (((id Z) - tan) + sec) & Z = dom f & f | A is continuous holds integral (f,A) = ((((id Z) - tan) + sec) . (upper_bound A)) - ((((id Z) - tan) + sec) . (lower_bound A)) proof let A be non empty closed_interval Subset of REAL; ::_thesis: for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds ( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 & f . x = (sin . x) / (1 + (sin . x)) ) ) & Z c= dom (((id Z) - tan) + sec) & Z = dom f & f | A is continuous holds integral (f,A) = ((((id Z) - tan) + sec) . (upper_bound A)) - ((((id Z) - tan) + sec) . (lower_bound A)) let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds ( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 & f . x = (sin . x) / (1 + (sin . x)) ) ) & Z c= dom (((id Z) - tan) + sec) & Z = dom f & f | A is continuous holds integral (f,A) = ((((id Z) - tan) + sec) . (upper_bound A)) - ((((id Z) - tan) + sec) . (lower_bound A)) let f be PartFunc of REAL,REAL; ::_thesis: ( A c= Z & ( for x being Real st x in Z holds ( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 & f . x = (sin . x) / (1 + (sin . x)) ) ) & Z c= dom (((id Z) - tan) + sec) & Z = dom f & f | A is continuous implies integral (f,A) = ((((id Z) - tan) + sec) . (upper_bound A)) - ((((id Z) - tan) + sec) . (lower_bound A)) ) assume that A1: A c= Z and A2: for x being Real st x in Z holds ( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 & f . x = (sin . x) / (1 + (sin . x)) ) and A3: Z c= dom (((id Z) - tan) + sec) and A4: Z = dom f and A5: f | A is continuous ; ::_thesis: integral (f,A) = ((((id Z) - tan) + sec) . (upper_bound A)) - ((((id Z) - tan) + sec) . (lower_bound A)) A6: f is_integrable_on A by A1, A4, A5, INTEGRA5:11; A7: for x being Real st x in Z holds ( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 ) by A2; then A8: ((id Z) - tan) + sec is_differentiable_on Z by A3, Th53; A9: for x being Real st x in dom ((((id Z) - tan) + sec) `| Z) holds ((((id Z) - tan) + sec) `| Z) . x = f . x proof let x be Real; ::_thesis: ( x in dom ((((id Z) - tan) + sec) `| Z) implies ((((id Z) - tan) + sec) `| Z) . x = f . x ) assume x in dom ((((id Z) - tan) + sec) `| Z) ; ::_thesis: ((((id Z) - tan) + sec) `| Z) . x = f . x then A10: x in Z by A8, FDIFF_1:def_7; then ((((id Z) - tan) + sec) `| Z) . x = (sin . x) / (1 + (sin . x)) by A3, A7, Th53 .= f . x by A2, A10 ; hence ((((id Z) - tan) + sec) `| Z) . x = f . x ; ::_thesis: verum end; dom ((((id Z) - tan) + sec) `| Z) = dom f by A4, A8, FDIFF_1:def_7; then (((id Z) - tan) + sec) `| Z = f by A9, PARTFUN1:5; hence integral (f,A) = ((((id Z) - tan) + sec) . (upper_bound A)) - ((((id Z) - tan) + sec) . (lower_bound A)) by A1, A4, A5, A6, A8, INTEGRA5:10, INTEGRA5:13; ::_thesis: verum end; theorem Th55: :: INTEGR11:55 for Z being open Subset of REAL st Z c= dom (((id Z) - tan) - sec) & ( for x being Real st x in Z holds ( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 ) ) holds ( ((id Z) - tan) - sec is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) - tan) - sec) `| Z) . x = (sin . x) / ((sin . x) - 1) ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (((id Z) - tan) - sec) & ( for x being Real st x in Z holds ( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 ) ) implies ( ((id Z) - tan) - sec is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) - tan) - sec) `| Z) . x = (sin . x) / ((sin . x) - 1) ) ) ) assume that A1: Z c= dom (((id Z) - tan) - sec) and A2: for x being Real st x in Z holds ( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 ) ; ::_thesis: ( ((id Z) - tan) - sec is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) - tan) - sec) `| Z) . x = (sin . x) / ((sin . x) - 1) ) ) A3: Z c= (dom ((id Z) - tan)) /\ (dom sec) by A1, VALUED_1:12; then A4: Z c= dom ((id Z) - tan) by XBOOLE_1:18; then Z c= (dom (id Z)) /\ (dom tan) by VALUED_1:12; then A5: Z c= dom tan by XBOOLE_1:18; A6: for x being Real st x in Z holds (id Z) . x = (1 * x) + 0 by FUNCT_1:18; A7: Z c= dom (id Z) ; then A8: id Z is_differentiable_on Z by A6, FDIFF_1:23; for x being Real st x in Z holds tan is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies tan is_differentiable_in x ) assume x in Z ; ::_thesis: tan is_differentiable_in x then cos . x <> 0 by A5, FDIFF_8:1; hence tan is_differentiable_in x by FDIFF_7:46; ::_thesis: verum end; then A9: tan is_differentiable_on Z by A5, FDIFF_1:9; then A10: (id Z) - tan is_differentiable_on Z by A4, A8, FDIFF_1:19; A11: Z c= dom sec by A3, XBOOLE_1:18; then A12: sec is_differentiable_on Z by FDIFF_9:4; A13: for x being Real st x in Z holds (((id Z) - tan) `| Z) . x = - (((sin . x) ^2) / ((cos . x) ^2)) proof let x be Real; ::_thesis: ( x in Z implies (((id Z) - tan) `| Z) . x = - (((sin . x) ^2) / ((cos . x) ^2)) ) assume A14: x in Z ; ::_thesis: (((id Z) - tan) `| Z) . x = - (((sin . x) ^2) / ((cos . x) ^2)) then A15: cos . x <> 0 by A5, FDIFF_8:1; then A16: (cos . x) ^2 > 0 by SQUARE_1:12; (((id Z) - tan) `| Z) . x = (diff ((id Z),x)) - (diff (tan,x)) by A4, A9, A8, A14, FDIFF_1:19 .= (((id Z) `| Z) . x) - (diff (tan,x)) by A8, A14, FDIFF_1:def_7 .= 1 - (diff (tan,x)) by A7, A6, A14, FDIFF_1:23 .= 1 - (1 / ((cos . x) ^2)) by A15, FDIFF_7:46 .= 1 - ((((cos . x) ^2) + ((sin . x) ^2)) / ((cos . x) ^2)) by SIN_COS:28 .= 1 - ((((cos . x) ^2) / ((cos . x) ^2)) + (((sin . x) ^2) / ((cos . x) ^2))) by XCMPLX_1:62 .= 1 - (1 + (((sin . x) ^2) / ((cos . x) ^2))) by A16, XCMPLX_1:60 .= - (((sin . x) ^2) / ((cos . x) ^2)) ; hence (((id Z) - tan) `| Z) . x = - (((sin . x) ^2) / ((cos . x) ^2)) ; ::_thesis: verum end; for x being Real st x in Z holds ((((id Z) - tan) - sec) `| Z) . x = (sin . x) / ((sin . x) - 1) proof let x be Real; ::_thesis: ( x in Z implies ((((id Z) - tan) - sec) `| Z) . x = (sin . x) / ((sin . x) - 1) ) assume A17: x in Z ; ::_thesis: ((((id Z) - tan) - sec) `| Z) . x = (sin . x) / ((sin . x) - 1) then A18: 1 + (sin . x) <> 0 by A2; ((((id Z) - tan) - sec) `| Z) . x = (diff (((id Z) - tan),x)) - (diff (sec,x)) by A1, A12, A10, A17, FDIFF_1:19 .= ((((id Z) - tan) `| Z) . x) - (diff (sec,x)) by A10, A17, FDIFF_1:def_7 .= (- (((sin . x) ^2) / ((cos . x) ^2))) - (diff (sec,x)) by A13, A17 .= (- (((sin . x) ^2) / ((cos . x) ^2))) - ((sec `| Z) . x) by A12, A17, FDIFF_1:def_7 .= (- (((sin . x) ^2) / ((cos . x) ^2))) - ((sin . x) / ((cos . x) ^2)) by A11, A17, FDIFF_9:4 .= - (((sin . x) / ((cos . x) ^2)) + (((sin . x) ^2) / ((cos . x) ^2))) .= - (((sin . x) + ((sin . x) ^2)) / ((cos . x) ^2)) by XCMPLX_1:62 .= - (((sin . x) * (1 + (sin . x))) / ((((cos . x) ^2) + ((sin . x) ^2)) - ((sin . x) ^2))) .= - (((sin . x) * (1 + (sin . x))) / (1 - ((sin . x) ^2))) by SIN_COS:28 .= - (((sin . x) * (1 + (sin . x))) / ((1 + (sin . x)) * (1 - (sin . x)))) .= - ((((sin . x) * (1 + (sin . x))) / (1 + (sin . x))) / (1 - (sin . x))) by XCMPLX_1:78 .= - (((sin . x) * ((1 + (sin . x)) / (1 + (sin . x)))) / (1 - (sin . x))) by XCMPLX_1:74 .= - (((sin . x) * 1) / (1 - (sin . x))) by A18, XCMPLX_1:60 .= (sin . x) / (- (1 - (sin . x))) by XCMPLX_1:188 .= (sin . x) / ((sin . x) - 1) ; hence ((((id Z) - tan) - sec) `| Z) . x = (sin . x) / ((sin . x) - 1) ; ::_thesis: verum end; hence ( ((id Z) - tan) - sec is_differentiable_on Z & ( for x being Real st x in Z holds ((((id Z) - tan) - sec) `| Z) . x = (sin . x) / ((sin . x) - 1) ) ) by A1, A12, A10, FDIFF_1:19; ::_thesis: verum end; theorem :: INTEGR11:56 for A being non empty closed_interval Subset of REAL for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds ( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 & f . x = (sin . x) / ((sin . x) - 1) ) ) & Z c= dom (((id Z) - tan) - sec) & Z = dom f & f | A is continuous holds integral (f,A) = ((((id Z) - tan) - sec) . (upper_bound A)) - ((((id Z) - tan) - sec) . (lower_bound A)) proof let A be non empty closed_interval Subset of REAL; ::_thesis: for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds ( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 & f . x = (sin . x) / ((sin . x) - 1) ) ) & Z c= dom (((id Z) - tan) - sec) & Z = dom f & f | A is continuous holds integral (f,A) = ((((id Z) - tan) - sec) . (upper_bound A)) - ((((id Z) - tan) - sec) . (lower_bound A)) let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds ( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 & f . x = (sin . x) / ((sin . x) - 1) ) ) & Z c= dom (((id Z) - tan) - sec) & Z = dom f & f | A is continuous holds integral (f,A) = ((((id Z) - tan) - sec) . (upper_bound A)) - ((((id Z) - tan) - sec) . (lower_bound A)) let f be PartFunc of REAL,REAL; ::_thesis: ( A c= Z & ( for x being Real st x in Z holds ( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 & f . x = (sin . x) / ((sin . x) - 1) ) ) & Z c= dom (((id Z) - tan) - sec) & Z = dom f & f | A is continuous implies integral (f,A) = ((((id Z) - tan) - sec) . (upper_bound A)) - ((((id Z) - tan) - sec) . (lower_bound A)) ) assume that A1: A c= Z and A2: for x being Real st x in Z holds ( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 & f . x = (sin . x) / ((sin . x) - 1) ) and A3: Z c= dom (((id Z) - tan) - sec) and A4: Z = dom f and A5: f | A is continuous ; ::_thesis: integral (f,A) = ((((id Z) - tan) - sec) . (upper_bound A)) - ((((id Z) - tan) - sec) . (lower_bound A)) A6: f is_integrable_on A by A1, A4, A5, INTEGRA5:11; A7: for x being Real st x in Z holds ( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 ) by A2; then A8: ((id Z) - tan) - sec is_differentiable_on Z by A3, Th55; A9: for x being Real st x in dom ((((id Z) - tan) - sec) `| Z) holds ((((id Z) - tan) - sec) `| Z) . x = f . x proof let x be Real; ::_thesis: ( x in dom ((((id Z) - tan) - sec) `| Z) implies ((((id Z) - tan) - sec) `| Z) . x = f . x ) assume x in dom ((((id Z) - tan) - sec) `| Z) ; ::_thesis: ((((id Z) - tan) - sec) `| Z) . x = f . x then A10: x in Z by A8, FDIFF_1:def_7; then ((((id Z) - tan) - sec) `| Z) . x = (sin . x) / ((sin . x) - 1) by A3, A7, Th55 .= f . x by A2, A10 ; hence ((((id Z) - tan) - sec) `| Z) . x = f . x ; ::_thesis: verum end; dom ((((id Z) - tan) - sec) `| Z) = dom f by A4, A8, FDIFF_1:def_7; then (((id Z) - tan) - sec) `| Z = f by A9, PARTFUN1:5; hence integral (f,A) = ((((id Z) - tan) - sec) . (upper_bound A)) - ((((id Z) - tan) - sec) . (lower_bound A)) by A1, A4, A5, A6, A8, INTEGRA5:10, INTEGRA5:13; ::_thesis: verum end; theorem Th57: :: INTEGR11:57 for Z being open Subset of REAL st Z c= dom (tan - (id Z)) holds ( tan - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds ((tan - (id Z)) `| Z) . x = (tan . x) ^2 ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom (tan - (id Z)) implies ( tan - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds ((tan - (id Z)) `| Z) . x = (tan . x) ^2 ) ) ) A1: for x being Real st x in Z holds (id Z) . x = (1 * x) + 0 by FUNCT_1:18; assume A2: Z c= dom (tan - (id Z)) ; ::_thesis: ( tan - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds ((tan - (id Z)) `| Z) . x = (tan . x) ^2 ) ) then Z c= (dom tan) /\ (dom (id Z)) by VALUED_1:12; then A3: Z c= dom tan by XBOOLE_1:18; A4: Z c= dom (id Z) ; then A5: id Z is_differentiable_on Z by A1, FDIFF_1:23; for x being Real st x in Z holds tan is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies tan is_differentiable_in x ) assume x in Z ; ::_thesis: tan is_differentiable_in x then cos . x <> 0 by A3, FDIFF_8:1; hence tan is_differentiable_in x by FDIFF_7:46; ::_thesis: verum end; then A6: tan is_differentiable_on Z by A3, FDIFF_1:9; for x being Real st x in Z holds ((tan - (id Z)) `| Z) . x = (tan . x) ^2 proof let x be Real; ::_thesis: ( x in Z implies ((tan - (id Z)) `| Z) . x = (tan . x) ^2 ) assume A7: x in Z ; ::_thesis: ((tan - (id Z)) `| Z) . x = (tan . x) ^2 then A8: cos . x <> 0 by A3, FDIFF_8:1; then A9: (cos . x) ^2 > 0 by SQUARE_1:12; ((tan - (id Z)) `| Z) . x = (diff (tan,x)) - (diff ((id Z),x)) by A2, A5, A6, A7, FDIFF_1:19 .= (1 / ((cos . x) ^2)) - (diff ((id Z),x)) by A8, FDIFF_7:46 .= (1 / ((cos . x) ^2)) - (((id Z) `| Z) . x) by A5, A7, FDIFF_1:def_7 .= (1 / ((cos . x) ^2)) - 1 by A4, A1, A7, FDIFF_1:23 .= (1 / ((cos . x) ^2)) - (((cos . x) ^2) / ((cos . x) ^2)) by A9, XCMPLX_1:60 .= (1 - ((cos . x) ^2)) / ((cos . x) ^2) by XCMPLX_1:120 .= ((((sin . x) ^2) + ((cos . x) ^2)) - ((cos . x) ^2)) / ((cos . x) ^2) by SIN_COS:28 .= ((sin x) / (cos x)) * ((sin . x) / (cos . x)) by XCMPLX_1:76 .= (tan . x) * (tan x) by A3, A7, FDIFF_8:1, SIN_COS9:15 .= (tan . x) ^2 by A3, A7, FDIFF_8:1, SIN_COS9:15 ; hence ((tan - (id Z)) `| Z) . x = (tan . x) ^2 ; ::_thesis: verum end; hence ( tan - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds ((tan - (id Z)) `| Z) . x = (tan . x) ^2 ) ) by A2, A5, A6, FDIFF_1:19; ::_thesis: verum end; theorem :: INTEGR11:58 for A being non empty closed_interval Subset of REAL for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds ( cos . x > 0 & f . x = (tan . x) ^2 ) ) & Z c= dom (tan - (id Z)) & Z = dom f & f | A is continuous holds integral (f,A) = ((tan - (id Z)) . (upper_bound A)) - ((tan - (id Z)) . (lower_bound A)) proof let A be non empty closed_interval Subset of REAL; ::_thesis: for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds ( cos . x > 0 & f . x = (tan . x) ^2 ) ) & Z c= dom (tan - (id Z)) & Z = dom f & f | A is continuous holds integral (f,A) = ((tan - (id Z)) . (upper_bound A)) - ((tan - (id Z)) . (lower_bound A)) let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds ( cos . x > 0 & f . x = (tan . x) ^2 ) ) & Z c= dom (tan - (id Z)) & Z = dom f & f | A is continuous holds integral (f,A) = ((tan - (id Z)) . (upper_bound A)) - ((tan - (id Z)) . (lower_bound A)) let f be PartFunc of REAL,REAL; ::_thesis: ( A c= Z & ( for x being Real st x in Z holds ( cos . x > 0 & f . x = (tan . x) ^2 ) ) & Z c= dom (tan - (id Z)) & Z = dom f & f | A is continuous implies integral (f,A) = ((tan - (id Z)) . (upper_bound A)) - ((tan - (id Z)) . (lower_bound A)) ) assume that A1: A c= Z and A2: for x being Real st x in Z holds ( cos . x > 0 & f . x = (tan . x) ^2 ) and A3: Z c= dom (tan - (id Z)) and A4: Z = dom f and A5: f | A is continuous ; ::_thesis: integral (f,A) = ((tan - (id Z)) . (upper_bound A)) - ((tan - (id Z)) . (lower_bound A)) A6: f is_integrable_on A by A1, A4, A5, INTEGRA5:11; A7: tan - (id Z) is_differentiable_on Z by A3, Th57; A8: for x being Real st x in dom ((tan - (id Z)) `| Z) holds ((tan - (id Z)) `| Z) . x = f . x proof let x be Real; ::_thesis: ( x in dom ((tan - (id Z)) `| Z) implies ((tan - (id Z)) `| Z) . x = f . x ) assume x in dom ((tan - (id Z)) `| Z) ; ::_thesis: ((tan - (id Z)) `| Z) . x = f . x then A9: x in Z by A7, FDIFF_1:def_7; then ((tan - (id Z)) `| Z) . x = (tan . x) ^2 by A3, Th57 .= f . x by A2, A9 ; hence ((tan - (id Z)) `| Z) . x = f . x ; ::_thesis: verum end; dom ((tan - (id Z)) `| Z) = dom f by A4, A7, FDIFF_1:def_7; then (tan - (id Z)) `| Z = f by A8, PARTFUN1:5; hence integral (f,A) = ((tan - (id Z)) . (upper_bound A)) - ((tan - (id Z)) . (lower_bound A)) by A1, A4, A5, A6, A7, INTEGRA5:10, INTEGRA5:13; ::_thesis: verum end; theorem Th59: :: INTEGR11:59 for Z being open Subset of REAL st Z c= dom ((- cot) - (id Z)) holds ( (- cot) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds (((- cot) - (id Z)) `| Z) . x = (cot . x) ^2 ) ) proof let Z be open Subset of REAL; ::_thesis: ( Z c= dom ((- cot) - (id Z)) implies ( (- cot) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds (((- cot) - (id Z)) `| Z) . x = (cot . x) ^2 ) ) ) set f = - cot; A1: for x being Real st x in Z holds (id Z) . x = (1 * x) + 0 by FUNCT_1:18; assume A2: Z c= dom ((- cot) - (id Z)) ; ::_thesis: ( (- cot) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds (((- cot) - (id Z)) `| Z) . x = (cot . x) ^2 ) ) then Z c= (dom (- cot)) /\ (dom (id Z)) by VALUED_1:12; then A3: Z c= dom (- cot) by XBOOLE_1:18; then A4: Z c= dom cot by VALUED_1:8; for x being Real st x in Z holds cot is_differentiable_in x proof let x be Real; ::_thesis: ( x in Z implies cot is_differentiable_in x ) assume x in Z ; ::_thesis: cot is_differentiable_in x then sin . x <> 0 by A4, FDIFF_8:2; hence cot is_differentiable_in x by FDIFF_7:47; ::_thesis: verum end; then A5: cot is_differentiable_on Z by A4, FDIFF_1:9; then A6: (- 1) (#) cot is_differentiable_on Z by A3, FDIFF_1:20; A7: Z c= dom (id Z) ; then A8: id Z is_differentiable_on Z by A1, FDIFF_1:23; for x being Real st x in Z holds (((- cot) - (id Z)) `| Z) . x = (cot . x) ^2 proof let x be Real; ::_thesis: ( x in Z implies (((- cot) - (id Z)) `| Z) . x = (cot . x) ^2 ) assume A9: x in Z ; ::_thesis: (((- cot) - (id Z)) `| Z) . x = (cot . x) ^2 then A10: sin . x <> 0 by A4, FDIFF_8:2; then A11: (sin . x) ^2 > 0 by SQUARE_1:12; (((- cot) - (id Z)) `| Z) . x = (diff ((- cot),x)) - (diff ((id Z),x)) by A2, A8, A6, A9, FDIFF_1:19 .= ((((- 1) (#) cot) `| Z) . x) - (diff ((id Z),x)) by A6, A9, FDIFF_1:def_7 .= ((- 1) * (diff (cot,x))) - (diff ((id Z),x)) by A3, A5, A9, FDIFF_1:20 .= ((- 1) * (- (1 / ((sin . x) ^2)))) - (diff ((id Z),x)) by A10, FDIFF_7:47 .= (1 / ((sin . x) ^2)) - (((id Z) `| Z) . x) by A8, A9, FDIFF_1:def_7 .= (1 / ((sin . x) ^2)) - 1 by A7, A1, A9, FDIFF_1:23 .= (1 / ((sin . x) ^2)) - (((sin . x) ^2) / ((sin . x) ^2)) by A11, XCMPLX_1:60 .= (1 - ((sin . x) ^2)) / ((sin . x) ^2) by XCMPLX_1:120 .= ((((cos . x) ^2) + ((sin . x) ^2)) - ((sin . x) ^2)) / ((sin . x) ^2) by SIN_COS:28 .= ((cos x) / (sin x)) * ((cos . x) / (sin . x)) by XCMPLX_1:76 .= (cot . x) * (cot x) by A4, A9, FDIFF_8:2, SIN_COS9:16 .= (cot . x) ^2 by A4, A9, FDIFF_8:2, SIN_COS9:16 ; hence (((- cot) - (id Z)) `| Z) . x = (cot . x) ^2 ; ::_thesis: verum end; hence ( (- cot) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds (((- cot) - (id Z)) `| Z) . x = (cot . x) ^2 ) ) by A2, A8, A6, FDIFF_1:19; ::_thesis: verum end; theorem :: INTEGR11:60 for A being non empty closed_interval Subset of REAL for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds ( sin . x > 0 & f . x = (cot . x) ^2 ) ) & Z c= dom ((- cot) - (id Z)) & Z = dom f & f | A is continuous holds integral (f,A) = (((- cot) - (id Z)) . (upper_bound A)) - (((- cot) - (id Z)) . (lower_bound A)) proof let A be non empty closed_interval Subset of REAL; ::_thesis: for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds ( sin . x > 0 & f . x = (cot . x) ^2 ) ) & Z c= dom ((- cot) - (id Z)) & Z = dom f & f | A is continuous holds integral (f,A) = (((- cot) - (id Z)) . (upper_bound A)) - (((- cot) - (id Z)) . (lower_bound A)) let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds ( sin . x > 0 & f . x = (cot . x) ^2 ) ) & Z c= dom ((- cot) - (id Z)) & Z = dom f & f | A is continuous holds integral (f,A) = (((- cot) - (id Z)) . (upper_bound A)) - (((- cot) - (id Z)) . (lower_bound A)) let f be PartFunc of REAL,REAL; ::_thesis: ( A c= Z & ( for x being Real st x in Z holds ( sin . x > 0 & f . x = (cot . x) ^2 ) ) & Z c= dom ((- cot) - (id Z)) & Z = dom f & f | A is continuous implies integral (f,A) = (((- cot) - (id Z)) . (upper_bound A)) - (((- cot) - (id Z)) . (lower_bound A)) ) assume that A1: A c= Z and A2: for x being Real st x in Z holds ( sin . x > 0 & f . x = (cot . x) ^2 ) and A3: Z c= dom ((- cot) - (id Z)) and A4: Z = dom f and A5: f | A is continuous ; ::_thesis: integral (f,A) = (((- cot) - (id Z)) . (upper_bound A)) - (((- cot) - (id Z)) . (lower_bound A)) A6: f is_integrable_on A by A1, A4, A5, INTEGRA5:11; A7: (- cot) - (id Z) is_differentiable_on Z by A3, Th59; A8: for x being Real st x in dom (((- cot) - (id Z)) `| Z) holds (((- cot) - (id Z)) `| Z) . x = f . x proof let x be Real; ::_thesis: ( x in dom (((- cot) - (id Z)) `| Z) implies (((- cot) - (id Z)) `| Z) . x = f . x ) assume x in dom (((- cot) - (id Z)) `| Z) ; ::_thesis: (((- cot) - (id Z)) `| Z) . x = f . x then A9: x in Z by A7, FDIFF_1:def_7; then (((- cot) - (id Z)) `| Z) . x = (cot . x) ^2 by A3, Th59 .= f . x by A2, A9 ; hence (((- cot) - (id Z)) `| Z) . x = f . x ; ::_thesis: verum end; dom (((- cot) - (id Z)) `| Z) = dom f by A4, A7, FDIFF_1:def_7; then ((- cot) - (id Z)) `| Z = f by A8, PARTFUN1:5; hence integral (f,A) = (((- cot) - (id Z)) . (upper_bound A)) - (((- cot) - (id Z)) . (lower_bound A)) by A1, A4, A5, A6, A7, INTEGRA5:10, INTEGRA5:13; ::_thesis: verum end; theorem :: INTEGR11:61 for A being non empty closed_interval Subset of REAL for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds ( f . x = 1 / ((cos . x) ^2) & cos . x <> 0 ) ) & dom tan = Z & Z = dom f & f | A is continuous holds integral (f,A) = (tan . (upper_bound A)) - (tan . (lower_bound A)) proof let A be non empty closed_interval Subset of REAL; ::_thesis: for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds ( f . x = 1 / ((cos . x) ^2) & cos . x <> 0 ) ) & dom tan = Z & Z = dom f & f | A is continuous holds integral (f,A) = (tan . (upper_bound A)) - (tan . (lower_bound A)) let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds ( f . x = 1 / ((cos . x) ^2) & cos . x <> 0 ) ) & dom tan = Z & Z = dom f & f | A is continuous holds integral (f,A) = (tan . (upper_bound A)) - (tan . (lower_bound A)) let f be PartFunc of REAL,REAL; ::_thesis: ( A c= Z & ( for x being Real st x in Z holds ( f . x = 1 / ((cos . x) ^2) & cos . x <> 0 ) ) & dom tan = Z & Z = dom f & f | A is continuous implies integral (f,A) = (tan . (upper_bound A)) - (tan . (lower_bound A)) ) assume that A1: A c= Z and A2: for x being Real st x in Z holds ( f . x = 1 / ((cos . x) ^2) & cos . x <> 0 ) and A3: dom tan = Z and A4: Z = dom f and A5: f | A is continuous ; ::_thesis: integral (f,A) = (tan . (upper_bound A)) - (tan . (lower_bound A)) A6: f is_integrable_on A by A1, A4, A5, INTEGRA5:11; A7: tan is_differentiable_on Z by A3, INTEGRA8:33; A8: for x being Real st x in dom (tan `| Z) holds (tan `| Z) . x = f . x proof let x be Real; ::_thesis: ( x in dom (tan `| Z) implies (tan `| Z) . x = f . x ) assume x in dom (tan `| Z) ; ::_thesis: (tan `| Z) . x = f . x then A9: x in Z by A7, FDIFF_1:def_7; then (tan `| Z) . x = 1 / ((cos . x) ^2) by A3, INTEGRA8:33 .= f . x by A2, A9 ; hence (tan `| Z) . x = f . x ; ::_thesis: verum end; dom (tan `| Z) = dom f by A4, A7, FDIFF_1:def_7; then tan `| Z = f by A8, PARTFUN1:5; hence integral (f,A) = (tan . (upper_bound A)) - (tan . (lower_bound A)) by A1, A4, A5, A6, A7, INTEGRA5:10, INTEGRA5:13; ::_thesis: verum end; theorem :: INTEGR11:62 for A being non empty closed_interval Subset of REAL for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds ( f . x = - (1 / ((sin . x) ^2)) & sin . x <> 0 ) ) & dom cot = Z & Z = dom f & f | A is continuous holds integral (f,A) = (cot . (upper_bound A)) - (cot . (lower_bound A)) proof let A be non empty closed_interval Subset of REAL; ::_thesis: for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds ( f . x = - (1 / ((sin . x) ^2)) & sin . x <> 0 ) ) & dom cot = Z & Z = dom f & f | A is continuous holds integral (f,A) = (cot . (upper_bound A)) - (cot . (lower_bound A)) let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds ( f . x = - (1 / ((sin . x) ^2)) & sin . x <> 0 ) ) & dom cot = Z & Z = dom f & f | A is continuous holds integral (f,A) = (cot . (upper_bound A)) - (cot . (lower_bound A)) let f be PartFunc of REAL,REAL; ::_thesis: ( A c= Z & ( for x being Real st x in Z holds ( f . x = - (1 / ((sin . x) ^2)) & sin . x <> 0 ) ) & dom cot = Z & Z = dom f & f | A is continuous implies integral (f,A) = (cot . (upper_bound A)) - (cot . (lower_bound A)) ) assume that A1: A c= Z and A2: for x being Real st x in Z holds ( f . x = - (1 / ((sin . x) ^2)) & sin . x <> 0 ) and A3: dom cot = Z and A4: Z = dom f and A5: f | A is continuous ; ::_thesis: integral (f,A) = (cot . (upper_bound A)) - (cot . (lower_bound A)) A6: f is_integrable_on A by A1, A4, A5, INTEGRA5:11; A7: cot is_differentiable_on Z by A3, INTEGRA8:34; A8: for x being Real st x in dom (cot `| Z) holds (cot `| Z) . x = f . x proof let x be Real; ::_thesis: ( x in dom (cot `| Z) implies (cot `| Z) . x = f . x ) assume x in dom (cot `| Z) ; ::_thesis: (cot `| Z) . x = f . x then A9: x in Z by A7, FDIFF_1:def_7; then (cot `| Z) . x = - (1 / ((sin . x) ^2)) by A3, INTEGRA8:34 .= f . x by A2, A9 ; hence (cot `| Z) . x = f . x ; ::_thesis: verum end; dom (cot `| Z) = dom f by A4, A7, FDIFF_1:def_7; then cot `| Z = f by A8, PARTFUN1:5; hence integral (f,A) = (cot . (upper_bound A)) - (cot . (lower_bound A)) by A1, A4, A5, A6, A7, INTEGRA5:10, INTEGRA5:13; ::_thesis: verum end; theorem :: INTEGR11:63 for A being non empty closed_interval Subset of REAL for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds f . x = ((sin . x) - ((cos . x) ^2)) / ((cos . x) ^2) ) & Z c= dom (sec - (id Z)) & Z = dom f & f | A is continuous holds integral (f,A) = ((sec - (id Z)) . (upper_bound A)) - ((sec - (id Z)) . (lower_bound A)) proof let A be non empty closed_interval Subset of REAL; ::_thesis: for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds f . x = ((sin . x) - ((cos . x) ^2)) / ((cos . x) ^2) ) & Z c= dom (sec - (id Z)) & Z = dom f & f | A is continuous holds integral (f,A) = ((sec - (id Z)) . (upper_bound A)) - ((sec - (id Z)) . (lower_bound A)) let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds f . x = ((sin . x) - ((cos . x) ^2)) / ((cos . x) ^2) ) & Z c= dom (sec - (id Z)) & Z = dom f & f | A is continuous holds integral (f,A) = ((sec - (id Z)) . (upper_bound A)) - ((sec - (id Z)) . (lower_bound A)) let f be PartFunc of REAL,REAL; ::_thesis: ( A c= Z & ( for x being Real st x in Z holds f . x = ((sin . x) - ((cos . x) ^2)) / ((cos . x) ^2) ) & Z c= dom (sec - (id Z)) & Z = dom f & f | A is continuous implies integral (f,A) = ((sec - (id Z)) . (upper_bound A)) - ((sec - (id Z)) . (lower_bound A)) ) assume that A1: A c= Z and A2: for x being Real st x in Z holds f . x = ((sin . x) - ((cos . x) ^2)) / ((cos . x) ^2) and A3: Z c= dom (sec - (id Z)) and A4: Z = dom f and A5: f | A is continuous ; ::_thesis: integral (f,A) = ((sec - (id Z)) . (upper_bound A)) - ((sec - (id Z)) . (lower_bound A)) A6: sec - (id Z) is_differentiable_on Z by A3, FDIFF_9:22; A7: for x being Real st x in dom ((sec - (id Z)) `| Z) holds ((sec - (id Z)) `| Z) . x = f . x proof let x be Real; ::_thesis: ( x in dom ((sec - (id Z)) `| Z) implies ((sec - (id Z)) `| Z) . x = f . x ) assume x in dom ((sec - (id Z)) `| Z) ; ::_thesis: ((sec - (id Z)) `| Z) . x = f . x then A8: x in Z by A6, FDIFF_1:def_7; then ((sec - (id Z)) `| Z) . x = ((sin . x) - ((cos . x) ^2)) / ((cos . x) ^2) by A3, FDIFF_9:22 .= f . x by A2, A8 ; hence ((sec - (id Z)) `| Z) . x = f . x ; ::_thesis: verum end; dom ((sec - (id Z)) `| Z) = dom f by A4, A6, FDIFF_1:def_7; then A9: (sec - (id Z)) `| Z = f by A7, PARTFUN1:5; ( f is_integrable_on A & f | A is bounded ) by A1, A4, A5, INTEGRA5:10, INTEGRA5:11; hence integral (f,A) = ((sec - (id Z)) . (upper_bound A)) - ((sec - (id Z)) . (lower_bound A)) by A1, A3, A9, FDIFF_9:22, INTEGRA5:13; ::_thesis: verum end; theorem :: INTEGR11:64 for A being non empty closed_interval Subset of REAL for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds f . x = ((cos . x) - ((sin . x) ^2)) / ((sin . x) ^2) ) & Z c= dom ((- cosec) - (id Z)) & Z = dom f & f | A is continuous holds integral (f,A) = (((- cosec) - (id Z)) . (upper_bound A)) - (((- cosec) - (id Z)) . (lower_bound A)) proof let A be non empty closed_interval Subset of REAL; ::_thesis: for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds f . x = ((cos . x) - ((sin . x) ^2)) / ((sin . x) ^2) ) & Z c= dom ((- cosec) - (id Z)) & Z = dom f & f | A is continuous holds integral (f,A) = (((- cosec) - (id Z)) . (upper_bound A)) - (((- cosec) - (id Z)) . (lower_bound A)) let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds f . x = ((cos . x) - ((sin . x) ^2)) / ((sin . x) ^2) ) & Z c= dom ((- cosec) - (id Z)) & Z = dom f & f | A is continuous holds integral (f,A) = (((- cosec) - (id Z)) . (upper_bound A)) - (((- cosec) - (id Z)) . (lower_bound A)) let f be PartFunc of REAL,REAL; ::_thesis: ( A c= Z & ( for x being Real st x in Z holds f . x = ((cos . x) - ((sin . x) ^2)) / ((sin . x) ^2) ) & Z c= dom ((- cosec) - (id Z)) & Z = dom f & f | A is continuous implies integral (f,A) = (((- cosec) - (id Z)) . (upper_bound A)) - (((- cosec) - (id Z)) . (lower_bound A)) ) assume that A1: A c= Z and A2: for x being Real st x in Z holds f . x = ((cos . x) - ((sin . x) ^2)) / ((sin . x) ^2) and A3: Z c= dom ((- cosec) - (id Z)) and A4: Z = dom f and A5: f | A is continuous ; ::_thesis: integral (f,A) = (((- cosec) - (id Z)) . (upper_bound A)) - (((- cosec) - (id Z)) . (lower_bound A)) A6: (- cosec) - (id Z) is_differentiable_on Z by A3, FDIFF_9:23; A7: for x being Real st x in dom (((- cosec) - (id Z)) `| Z) holds (((- cosec) - (id Z)) `| Z) . x = f . x proof let x be Real; ::_thesis: ( x in dom (((- cosec) - (id Z)) `| Z) implies (((- cosec) - (id Z)) `| Z) . x = f . x ) assume x in dom (((- cosec) - (id Z)) `| Z) ; ::_thesis: (((- cosec) - (id Z)) `| Z) . x = f . x then A8: x in Z by A6, FDIFF_1:def_7; then (((- cosec) - (id Z)) `| Z) . x = ((cos . x) - ((sin . x) ^2)) / ((sin . x) ^2) by A3, FDIFF_9:23 .= f . x by A2, A8 ; hence (((- cosec) - (id Z)) `| Z) . x = f . x ; ::_thesis: verum end; dom (((- cosec) - (id Z)) `| Z) = dom f by A4, A6, FDIFF_1:def_7; then A9: ((- cosec) - (id Z)) `| Z = f by A7, PARTFUN1:5; ( f is_integrable_on A & f | A is bounded ) by A1, A4, A5, INTEGRA5:10, INTEGRA5:11; hence integral (f,A) = (((- cosec) - (id Z)) . (upper_bound A)) - (((- cosec) - (id Z)) . (lower_bound A)) by A1, A3, A9, FDIFF_9:23, INTEGRA5:13; ::_thesis: verum end; theorem :: INTEGR11:65 for A being non empty closed_interval Subset of REAL for Z being open Subset of REAL st A c= Z & ( for x being Real st x in Z holds sin . x > 0 ) & Z c= dom (ln * sin) & Z = dom cot & cot | A is continuous holds integral (cot,A) = ((ln * sin) . (upper_bound A)) - ((ln * sin) . (lower_bound A)) proof let A be non empty closed_interval Subset of REAL; ::_thesis: for Z being open Subset of REAL st A c= Z & ( for x being Real st x in Z holds sin . x > 0 ) & Z c= dom (ln * sin) & Z = dom cot & cot | A is continuous holds integral (cot,A) = ((ln * sin) . (upper_bound A)) - ((ln * sin) . (lower_bound A)) let Z be open Subset of REAL; ::_thesis: ( A c= Z & ( for x being Real st x in Z holds sin . x > 0 ) & Z c= dom (ln * sin) & Z = dom cot & cot | A is continuous implies integral (cot,A) = ((ln * sin) . (upper_bound A)) - ((ln * sin) . (lower_bound A)) ) set f = cot ; assume that A1: A c= Z and A2: for x being Real st x in Z holds sin . x > 0 and A3: Z c= dom (ln * sin) and A4: Z = dom cot and A5: cot | A is continuous ; ::_thesis: integral (cot,A) = ((ln * sin) . (upper_bound A)) - ((ln * sin) . (lower_bound A)) A6: ln * sin is_differentiable_on Z by A2, A3, FDIFF_4:43; A7: for x being Real st x in dom ((ln * sin) `| Z) holds ((ln * sin) `| Z) . x = cot . x proof let x be Real; ::_thesis: ( x in dom ((ln * sin) `| Z) implies ((ln * sin) `| Z) . x = cot . x ) assume x in dom ((ln * sin) `| Z) ; ::_thesis: ((ln * sin) `| Z) . x = cot . x then A8: x in Z by A6, FDIFF_1:def_7; then A9: sin . x <> 0 by A2; ((ln * sin) `| Z) . x = cot x by A2, A3, A8, FDIFF_4:43 .= cot . x by A9, SIN_COS9:16 ; hence ((ln * sin) `| Z) . x = cot . x ; ::_thesis: verum end; dom ((ln * sin) `| Z) = dom cot by A4, A6, FDIFF_1:def_7; then A10: (ln * sin) `| Z = cot by A7, PARTFUN1:5; ( cot is_integrable_on A & cot | A is bounded ) by A1, A4, A5, INTEGRA5:10, INTEGRA5:11; hence integral (cot,A) = ((ln * sin) . (upper_bound A)) - ((ln * sin) . (lower_bound A)) by A1, A2, A3, A10, FDIFF_4:43, INTEGRA5:13; ::_thesis: verum end; theorem :: INTEGR11:66 for A being non empty closed_interval Subset of REAL for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds f . x = (arcsin . x) / (sqrt (1 - (x ^2))) ) & Z c= dom ((1 / 2) (#) ((#Z 2) * arcsin)) & Z = dom f & f | A is continuous holds integral (f,A) = (((1 / 2) (#) ((#Z 2) * arcsin)) . (upper_bound A)) - (((1 / 2) (#) ((#Z 2) * arcsin)) . (lower_bound A)) proof let A be non empty closed_interval Subset of REAL; ::_thesis: for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds f . x = (arcsin . x) / (sqrt (1 - (x ^2))) ) & Z c= dom ((1 / 2) (#) ((#Z 2) * arcsin)) & Z = dom f & f | A is continuous holds integral (f,A) = (((1 / 2) (#) ((#Z 2) * arcsin)) . (upper_bound A)) - (((1 / 2) (#) ((#Z 2) * arcsin)) . (lower_bound A)) let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds f . x = (arcsin . x) / (sqrt (1 - (x ^2))) ) & Z c= dom ((1 / 2) (#) ((#Z 2) * arcsin)) & Z = dom f & f | A is continuous holds integral (f,A) = (((1 / 2) (#) ((#Z 2) * arcsin)) . (upper_bound A)) - (((1 / 2) (#) ((#Z 2) * arcsin)) . (lower_bound A)) let f be PartFunc of REAL,REAL; ::_thesis: ( A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds f . x = (arcsin . x) / (sqrt (1 - (x ^2))) ) & Z c= dom ((1 / 2) (#) ((#Z 2) * arcsin)) & Z = dom f & f | A is continuous implies integral (f,A) = (((1 / 2) (#) ((#Z 2) * arcsin)) . (upper_bound A)) - (((1 / 2) (#) ((#Z 2) * arcsin)) . (lower_bound A)) ) assume that A1: A c= Z and A2: Z c= ].(- 1),1.[ and A3: for x being Real st x in Z holds f . x = (arcsin . x) / (sqrt (1 - (x ^2))) and A4: Z c= dom ((1 / 2) (#) ((#Z 2) * arcsin)) and A5: Z = dom f and A6: f | A is continuous ; ::_thesis: integral (f,A) = (((1 / 2) (#) ((#Z 2) * arcsin)) . (upper_bound A)) - (((1 / 2) (#) ((#Z 2) * arcsin)) . (lower_bound A)) A7: (1 / 2) (#) ((#Z 2) * arcsin) is_differentiable_on Z by A2, A4, FDIFF_7:12; A8: for x being Real st x in dom (((1 / 2) (#) ((#Z 2) * arcsin)) `| Z) holds (((1 / 2) (#) ((#Z 2) * arcsin)) `| Z) . x = f . x proof let x be Real; ::_thesis: ( x in dom (((1 / 2) (#) ((#Z 2) * arcsin)) `| Z) implies (((1 / 2) (#) ((#Z 2) * arcsin)) `| Z) . x = f . x ) assume x in dom (((1 / 2) (#) ((#Z 2) * arcsin)) `| Z) ; ::_thesis: (((1 / 2) (#) ((#Z 2) * arcsin)) `| Z) . x = f . x then A9: x in Z by A7, FDIFF_1:def_7; then (((1 / 2) (#) ((#Z 2) * arcsin)) `| Z) . x = (arcsin . x) / (sqrt (1 - (x ^2))) by A2, A4, FDIFF_7:12 .= f . x by A3, A9 ; hence (((1 / 2) (#) ((#Z 2) * arcsin)) `| Z) . x = f . x ; ::_thesis: verum end; dom (((1 / 2) (#) ((#Z 2) * arcsin)) `| Z) = dom f by A5, A7, FDIFF_1:def_7; then A10: ((1 / 2) (#) ((#Z 2) * arcsin)) `| Z = f by A8, PARTFUN1:5; ( f is_integrable_on A & f | A is bounded ) by A1, A5, A6, INTEGRA5:10, INTEGRA5:11; hence integral (f,A) = (((1 / 2) (#) ((#Z 2) * arcsin)) . (upper_bound A)) - (((1 / 2) (#) ((#Z 2) * arcsin)) . (lower_bound A)) by A1, A2, A4, A10, FDIFF_7:12, INTEGRA5:13; ::_thesis: verum end; theorem :: INTEGR11:67 for A being non empty closed_interval Subset of REAL for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds f . x = - ((arccos . x) / (sqrt (1 - (x ^2)))) ) & Z c= dom ((1 / 2) (#) ((#Z 2) * arccos)) & Z = dom f & f | A is continuous holds integral (f,A) = (((1 / 2) (#) ((#Z 2) * arccos)) . (upper_bound A)) - (((1 / 2) (#) ((#Z 2) * arccos)) . (lower_bound A)) proof let A be non empty closed_interval Subset of REAL; ::_thesis: for Z being open Subset of REAL for f being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds f . x = - ((arccos . x) / (sqrt (1 - (x ^2)))) ) & Z c= dom ((1 / 2) (#) ((#Z 2) * arccos)) & Z = dom f & f | A is continuous holds integral (f,A) = (((1 / 2) (#) ((#Z 2) * arccos)) . (upper_bound A)) - (((1 / 2) (#) ((#Z 2) * arccos)) . (lower_bound A)) let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds f . x = - ((arccos . x) / (sqrt (1 - (x ^2)))) ) & Z c= dom ((1 / 2) (#) ((#Z 2) * arccos)) & Z = dom f & f | A is continuous holds integral (f,A) = (((1 / 2) (#) ((#Z 2) * arccos)) . (upper_bound A)) - (((1 / 2) (#) ((#Z 2) * arccos)) . (lower_bound A)) let f be PartFunc of REAL,REAL; ::_thesis: ( A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds f . x = - ((arccos . x) / (sqrt (1 - (x ^2)))) ) & Z c= dom ((1 / 2) (#) ((#Z 2) * arccos)) & Z = dom f & f | A is continuous implies integral (f,A) = (((1 / 2) (#) ((#Z 2) * arccos)) . (upper_bound A)) - (((1 / 2) (#) ((#Z 2) * arccos)) . (lower_bound A)) ) assume that A1: A c= Z and A2: Z c= ].(- 1),1.[ and A3: for x being Real st x in Z holds f . x = - ((arccos . x) / (sqrt (1 - (x ^2)))) and A4: Z c= dom ((1 / 2) (#) ((#Z 2) * arccos)) and A5: Z = dom f and A6: f | A is continuous ; ::_thesis: integral (f,A) = (((1 / 2) (#) ((#Z 2) * arccos)) . (upper_bound A)) - (((1 / 2) (#) ((#Z 2) * arccos)) . (lower_bound A)) A7: (1 / 2) (#) ((#Z 2) * arccos) is_differentiable_on Z by A2, A4, FDIFF_7:13; A8: for x being Real st x in dom (((1 / 2) (#) ((#Z 2) * arccos)) `| Z) holds (((1 / 2) (#) ((#Z 2) * arccos)) `| Z) . x = f . x proof let x be Real; ::_thesis: ( x in dom (((1 / 2) (#) ((#Z 2) * arccos)) `| Z) implies (((1 / 2) (#) ((#Z 2) * arccos)) `| Z) . x = f . x ) assume x in dom (((1 / 2) (#) ((#Z 2) * arccos)) `| Z) ; ::_thesis: (((1 / 2) (#) ((#Z 2) * arccos)) `| Z) . x = f . x then A9: x in Z by A7, FDIFF_1:def_7; then (((1 / 2) (#) ((#Z 2) * arccos)) `| Z) . x = - ((arccos . x) / (sqrt (1 - (x ^2)))) by A2, A4, FDIFF_7:13 .= f . x by A3, A9 ; hence (((1 / 2) (#) ((#Z 2) * arccos)) `| Z) . x = f . x ; ::_thesis: verum end; dom (((1 / 2) (#) ((#Z 2) * arccos)) `| Z) = dom f by A5, A7, FDIFF_1:def_7; then A10: ((1 / 2) (#) ((#Z 2) * arccos)) `| Z = f by A8, PARTFUN1:5; ( f is_integrable_on A & f | A is bounded ) by A1, A5, A6, INTEGRA5:10, INTEGRA5:11; hence integral (f,A) = (((1 / 2) (#) ((#Z 2) * arccos)) . (upper_bound A)) - (((1 / 2) (#) ((#Z 2) * arccos)) . (lower_bound A)) by A1, A2, A4, A10, FDIFF_7:13, INTEGRA5:13; ::_thesis: verum end; theorem :: INTEGR11:68 for A being non empty closed_interval Subset of REAL for Z being open Subset of REAL for f, f1, f2 being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & f = f1 - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds ( f1 . x = 1 & f . x > 0 & x <> 0 ) ) & dom arcsin = Z & Z c= dom (((id Z) (#) arcsin) + ((#R (1 / 2)) * f)) holds integral (arcsin,A) = ((((id Z) (#) arcsin) + ((#R (1 / 2)) * f)) . (upper_bound A)) - ((((id Z) (#) arcsin) + ((#R (1 / 2)) * f)) . (lower_bound A)) proof let A be non empty closed_interval Subset of REAL; ::_thesis: for Z being open Subset of REAL for f, f1, f2 being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & f = f1 - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds ( f1 . x = 1 & f . x > 0 & x <> 0 ) ) & dom arcsin = Z & Z c= dom (((id Z) (#) arcsin) + ((#R (1 / 2)) * f)) holds integral (arcsin,A) = ((((id Z) (#) arcsin) + ((#R (1 / 2)) * f)) . (upper_bound A)) - ((((id Z) (#) arcsin) + ((#R (1 / 2)) * f)) . (lower_bound A)) let Z be open Subset of REAL; ::_thesis: for f, f1, f2 being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & f = f1 - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds ( f1 . x = 1 & f . x > 0 & x <> 0 ) ) & dom arcsin = Z & Z c= dom (((id Z) (#) arcsin) + ((#R (1 / 2)) * f)) holds integral (arcsin,A) = ((((id Z) (#) arcsin) + ((#R (1 / 2)) * f)) . (upper_bound A)) - ((((id Z) (#) arcsin) + ((#R (1 / 2)) * f)) . (lower_bound A)) let f, f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( A c= Z & Z c= ].(- 1),1.[ & f = f1 - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds ( f1 . x = 1 & f . x > 0 & x <> 0 ) ) & dom arcsin = Z & Z c= dom (((id Z) (#) arcsin) + ((#R (1 / 2)) * f)) implies integral (arcsin,A) = ((((id Z) (#) arcsin) + ((#R (1 / 2)) * f)) . (upper_bound A)) - ((((id Z) (#) arcsin) + ((#R (1 / 2)) * f)) . (lower_bound A)) ) assume that A1: A c= Z and A2: ( Z c= ].(- 1),1.[ & f = f1 - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds ( f1 . x = 1 & f . x > 0 & x <> 0 ) ) ) and A3: dom arcsin = Z and A4: Z c= dom (((id Z) (#) arcsin) + ((#R (1 / 2)) * f)) ; ::_thesis: integral (arcsin,A) = ((((id Z) (#) arcsin) + ((#R (1 / 2)) * f)) . (upper_bound A)) - ((((id Z) (#) arcsin) + ((#R (1 / 2)) * f)) . (lower_bound A)) A5: arcsin | A is bounded by A1, A3, INTEGRA5:10; A6: ((id Z) (#) arcsin) + ((#R (1 / 2)) * f) is_differentiable_on Z by A2, A4, FDIFF_7:23; A7: for x being Real st x in dom ((((id Z) (#) arcsin) + ((#R (1 / 2)) * f)) `| Z) holds ((((id Z) (#) arcsin) + ((#R (1 / 2)) * f)) `| Z) . x = arcsin . x proof let x be Real; ::_thesis: ( x in dom ((((id Z) (#) arcsin) + ((#R (1 / 2)) * f)) `| Z) implies ((((id Z) (#) arcsin) + ((#R (1 / 2)) * f)) `| Z) . x = arcsin . x ) assume x in dom ((((id Z) (#) arcsin) + ((#R (1 / 2)) * f)) `| Z) ; ::_thesis: ((((id Z) (#) arcsin) + ((#R (1 / 2)) * f)) `| Z) . x = arcsin . x then x in Z by A6, FDIFF_1:def_7; hence ((((id Z) (#) arcsin) + ((#R (1 / 2)) * f)) `| Z) . x = arcsin . x by A2, A4, FDIFF_7:23; ::_thesis: verum end; dom ((((id Z) (#) arcsin) + ((#R (1 / 2)) * f)) `| Z) = dom arcsin by A3, A6, FDIFF_1:def_7; then (((id Z) (#) arcsin) + ((#R (1 / 2)) * f)) `| Z = arcsin by A7, PARTFUN1:5; hence integral (arcsin,A) = ((((id Z) (#) arcsin) + ((#R (1 / 2)) * f)) . (upper_bound A)) - ((((id Z) (#) arcsin) + ((#R (1 / 2)) * f)) . (lower_bound A)) by A1, A3, A5, A6, INTEGRA5:11, INTEGRA5:13; ::_thesis: verum end; theorem :: INTEGR11:69 for a being Real for A being non empty closed_interval Subset of REAL for Z being open Subset of REAL for f, f1, f2, f3 being PartFunc of REAL,REAL st A c= Z & f = f1 - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds ( f1 . x = a ^2 & f . x > 0 & f3 . x = x / a & f3 . x > - 1 & f3 . x < 1 & x <> 0 & a > 0 ) ) & dom (arcsin * f3) = Z & Z c= dom (((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) & (arcsin * f3) | A is continuous holds integral ((arcsin * f3),A) = ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) . (upper_bound A)) - ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) . (lower_bound A)) proof let a be Real; ::_thesis: for A being non empty closed_interval Subset of REAL for Z being open Subset of REAL for f, f1, f2, f3 being PartFunc of REAL,REAL st A c= Z & f = f1 - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds ( f1 . x = a ^2 & f . x > 0 & f3 . x = x / a & f3 . x > - 1 & f3 . x < 1 & x <> 0 & a > 0 ) ) & dom (arcsin * f3) = Z & Z c= dom (((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) & (arcsin * f3) | A is continuous holds integral ((arcsin * f3),A) = ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) . (upper_bound A)) - ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) . (lower_bound A)) let A be non empty closed_interval Subset of REAL; ::_thesis: for Z being open Subset of REAL for f, f1, f2, f3 being PartFunc of REAL,REAL st A c= Z & f = f1 - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds ( f1 . x = a ^2 & f . x > 0 & f3 . x = x / a & f3 . x > - 1 & f3 . x < 1 & x <> 0 & a > 0 ) ) & dom (arcsin * f3) = Z & Z c= dom (((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) & (arcsin * f3) | A is continuous holds integral ((arcsin * f3),A) = ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) . (upper_bound A)) - ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) . (lower_bound A)) let Z be open Subset of REAL; ::_thesis: for f, f1, f2, f3 being PartFunc of REAL,REAL st A c= Z & f = f1 - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds ( f1 . x = a ^2 & f . x > 0 & f3 . x = x / a & f3 . x > - 1 & f3 . x < 1 & x <> 0 & a > 0 ) ) & dom (arcsin * f3) = Z & Z c= dom (((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) & (arcsin * f3) | A is continuous holds integral ((arcsin * f3),A) = ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) . (upper_bound A)) - ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) . (lower_bound A)) let f, f1, f2, f3 be PartFunc of REAL,REAL; ::_thesis: ( A c= Z & f = f1 - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds ( f1 . x = a ^2 & f . x > 0 & f3 . x = x / a & f3 . x > - 1 & f3 . x < 1 & x <> 0 & a > 0 ) ) & dom (arcsin * f3) = Z & Z c= dom (((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) & (arcsin * f3) | A is continuous implies integral ((arcsin * f3),A) = ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) . (upper_bound A)) - ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) . (lower_bound A)) ) assume that A1: A c= Z and A2: ( f = f1 - f2 & f2 = #Z 2 ) and A3: for x being Real st x in Z holds ( f1 . x = a ^2 & f . x > 0 & f3 . x = x / a & f3 . x > - 1 & f3 . x < 1 & x <> 0 & a > 0 ) and A4: dom (arcsin * f3) = Z and A5: Z c= dom (((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) and A6: (arcsin * f3) | A is continuous ; ::_thesis: integral ((arcsin * f3),A) = ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) . (upper_bound A)) - ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) . (lower_bound A)) A7: arcsin * f3 is_integrable_on A by A1, A4, A6, INTEGRA5:11; A8: ((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f) is_differentiable_on Z by A2, A3, A5, FDIFF_7:28; A9: for x being Real st x in dom ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) `| Z) holds ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) `| Z) . x = (arcsin * f3) . x proof let x be Real; ::_thesis: ( x in dom ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) `| Z) implies ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) `| Z) . x = (arcsin * f3) . x ) assume x in dom ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) `| Z) ; ::_thesis: ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) `| Z) . x = (arcsin * f3) . x then A10: x in Z by A8, FDIFF_1:def_7; then ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) `| Z) . x = arcsin . (x / a) by A2, A3, A5, FDIFF_7:28 .= arcsin . (f3 . x) by A3, A10 .= (arcsin * f3) . x by A4, A10, FUNCT_1:12 ; hence ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) `| Z) . x = (arcsin * f3) . x ; ::_thesis: verum end; dom ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) `| Z) = dom (arcsin * f3) by A4, A8, FDIFF_1:def_7; then (((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) `| Z = arcsin * f3 by A9, PARTFUN1:5; hence integral ((arcsin * f3),A) = ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) . (upper_bound A)) - ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) . (lower_bound A)) by A1, A4, A6, A7, A8, INTEGRA5:10, INTEGRA5:13; ::_thesis: verum end; theorem :: INTEGR11:70 for A being non empty closed_interval Subset of REAL for Z being open Subset of REAL for f, f1, f2 being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & f = f1 - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds ( f1 . x = 1 & f . x > 0 & x <> 0 ) ) & dom arccos = Z & Z c= dom (((id Z) (#) arccos) - ((#R (1 / 2)) * f)) holds integral (arccos,A) = ((((id Z) (#) arccos) - ((#R (1 / 2)) * f)) . (upper_bound A)) - ((((id Z) (#) arccos) - ((#R (1 / 2)) * f)) . (lower_bound A)) proof let A be non empty closed_interval Subset of REAL; ::_thesis: for Z being open Subset of REAL for f, f1, f2 being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & f = f1 - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds ( f1 . x = 1 & f . x > 0 & x <> 0 ) ) & dom arccos = Z & Z c= dom (((id Z) (#) arccos) - ((#R (1 / 2)) * f)) holds integral (arccos,A) = ((((id Z) (#) arccos) - ((#R (1 / 2)) * f)) . (upper_bound A)) - ((((id Z) (#) arccos) - ((#R (1 / 2)) * f)) . (lower_bound A)) let Z be open Subset of REAL; ::_thesis: for f, f1, f2 being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & f = f1 - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds ( f1 . x = 1 & f . x > 0 & x <> 0 ) ) & dom arccos = Z & Z c= dom (((id Z) (#) arccos) - ((#R (1 / 2)) * f)) holds integral (arccos,A) = ((((id Z) (#) arccos) - ((#R (1 / 2)) * f)) . (upper_bound A)) - ((((id Z) (#) arccos) - ((#R (1 / 2)) * f)) . (lower_bound A)) let f, f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( A c= Z & Z c= ].(- 1),1.[ & f = f1 - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds ( f1 . x = 1 & f . x > 0 & x <> 0 ) ) & dom arccos = Z & Z c= dom (((id Z) (#) arccos) - ((#R (1 / 2)) * f)) implies integral (arccos,A) = ((((id Z) (#) arccos) - ((#R (1 / 2)) * f)) . (upper_bound A)) - ((((id Z) (#) arccos) - ((#R (1 / 2)) * f)) . (lower_bound A)) ) assume that A1: A c= Z and A2: ( Z c= ].(- 1),1.[ & f = f1 - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds ( f1 . x = 1 & f . x > 0 & x <> 0 ) ) ) and A3: dom arccos = Z and A4: Z c= dom (((id Z) (#) arccos) - ((#R (1 / 2)) * f)) ; ::_thesis: integral (arccos,A) = ((((id Z) (#) arccos) - ((#R (1 / 2)) * f)) . (upper_bound A)) - ((((id Z) (#) arccos) - ((#R (1 / 2)) * f)) . (lower_bound A)) A5: arccos | A is bounded by A1, A3, INTEGRA5:10; A6: ((id Z) (#) arccos) - ((#R (1 / 2)) * f) is_differentiable_on Z by A2, A4, FDIFF_7:24; A7: for x being Real st x in dom ((((id Z) (#) arccos) - ((#R (1 / 2)) * f)) `| Z) holds ((((id Z) (#) arccos) - ((#R (1 / 2)) * f)) `| Z) . x = arccos . x proof let x be Real; ::_thesis: ( x in dom ((((id Z) (#) arccos) - ((#R (1 / 2)) * f)) `| Z) implies ((((id Z) (#) arccos) - ((#R (1 / 2)) * f)) `| Z) . x = arccos . x ) assume x in dom ((((id Z) (#) arccos) - ((#R (1 / 2)) * f)) `| Z) ; ::_thesis: ((((id Z) (#) arccos) - ((#R (1 / 2)) * f)) `| Z) . x = arccos . x then x in Z by A6, FDIFF_1:def_7; hence ((((id Z) (#) arccos) - ((#R (1 / 2)) * f)) `| Z) . x = arccos . x by A2, A4, FDIFF_7:24; ::_thesis: verum end; dom ((((id Z) (#) arccos) - ((#R (1 / 2)) * f)) `| Z) = dom arccos by A3, A6, FDIFF_1:def_7; then (((id Z) (#) arccos) - ((#R (1 / 2)) * f)) `| Z = arccos by A7, PARTFUN1:5; hence integral (arccos,A) = ((((id Z) (#) arccos) - ((#R (1 / 2)) * f)) . (upper_bound A)) - ((((id Z) (#) arccos) - ((#R (1 / 2)) * f)) . (lower_bound A)) by A1, A3, A5, A6, INTEGRA5:11, INTEGRA5:13; ::_thesis: verum end; theorem :: INTEGR11:71 for a being Real for A being non empty closed_interval Subset of REAL for Z being open Subset of REAL for f, f1, f2, f3 being PartFunc of REAL,REAL st A c= Z & f = f1 - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds ( f1 . x = a ^2 & f . x > 0 & f3 . x = x / a & f3 . x > - 1 & f3 . x < 1 & x <> 0 & a > 0 ) ) & dom (arccos * f3) = Z & Z = dom (((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) & (arccos * f3) | A is continuous holds integral ((arccos * f3),A) = ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) . (upper_bound A)) - ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) . (lower_bound A)) proof let a be Real; ::_thesis: for A being non empty closed_interval Subset of REAL for Z being open Subset of REAL for f, f1, f2, f3 being PartFunc of REAL,REAL st A c= Z & f = f1 - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds ( f1 . x = a ^2 & f . x > 0 & f3 . x = x / a & f3 . x > - 1 & f3 . x < 1 & x <> 0 & a > 0 ) ) & dom (arccos * f3) = Z & Z = dom (((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) & (arccos * f3) | A is continuous holds integral ((arccos * f3),A) = ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) . (upper_bound A)) - ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) . (lower_bound A)) let A be non empty closed_interval Subset of REAL; ::_thesis: for Z being open Subset of REAL for f, f1, f2, f3 being PartFunc of REAL,REAL st A c= Z & f = f1 - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds ( f1 . x = a ^2 & f . x > 0 & f3 . x = x / a & f3 . x > - 1 & f3 . x < 1 & x <> 0 & a > 0 ) ) & dom (arccos * f3) = Z & Z = dom (((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) & (arccos * f3) | A is continuous holds integral ((arccos * f3),A) = ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) . (upper_bound A)) - ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) . (lower_bound A)) let Z be open Subset of REAL; ::_thesis: for f, f1, f2, f3 being PartFunc of REAL,REAL st A c= Z & f = f1 - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds ( f1 . x = a ^2 & f . x > 0 & f3 . x = x / a & f3 . x > - 1 & f3 . x < 1 & x <> 0 & a > 0 ) ) & dom (arccos * f3) = Z & Z = dom (((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) & (arccos * f3) | A is continuous holds integral ((arccos * f3),A) = ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) . (upper_bound A)) - ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) . (lower_bound A)) let f, f1, f2, f3 be PartFunc of REAL,REAL; ::_thesis: ( A c= Z & f = f1 - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds ( f1 . x = a ^2 & f . x > 0 & f3 . x = x / a & f3 . x > - 1 & f3 . x < 1 & x <> 0 & a > 0 ) ) & dom (arccos * f3) = Z & Z = dom (((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) & (arccos * f3) | A is continuous implies integral ((arccos * f3),A) = ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) . (upper_bound A)) - ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) . (lower_bound A)) ) assume that A1: A c= Z and A2: ( f = f1 - f2 & f2 = #Z 2 ) and A3: for x being Real st x in Z holds ( f1 . x = a ^2 & f . x > 0 & f3 . x = x / a & f3 . x > - 1 & f3 . x < 1 & x <> 0 & a > 0 ) and A4: dom (arccos * f3) = Z and A5: Z = dom (((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) and A6: (arccos * f3) | A is continuous ; ::_thesis: integral ((arccos * f3),A) = ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) . (upper_bound A)) - ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) . (lower_bound A)) A7: arccos * f3 is_integrable_on A by A1, A4, A6, INTEGRA5:11; A8: ((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f) is_differentiable_on Z by A2, A3, A5, FDIFF_7:29; A9: for x being Real st x in dom ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) `| Z) holds ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) `| Z) . x = (arccos * f3) . x proof let x be Real; ::_thesis: ( x in dom ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) `| Z) implies ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) `| Z) . x = (arccos * f3) . x ) assume x in dom ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) `| Z) ; ::_thesis: ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) `| Z) . x = (arccos * f3) . x then A10: x in Z by A8, FDIFF_1:def_7; then ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) `| Z) . x = arccos . (x / a) by A2, A3, A5, FDIFF_7:29 .= arccos . (f3 . x) by A3, A10 .= (arccos * f3) . x by A4, A10, FUNCT_1:12 ; hence ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) `| Z) . x = (arccos * f3) . x ; ::_thesis: verum end; dom ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) `| Z) = dom (arccos * f3) by A4, A8, FDIFF_1:def_7; then (((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) `| Z = arccos * f3 by A9, PARTFUN1:5; hence integral ((arccos * f3),A) = ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) . (upper_bound A)) - ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) . (lower_bound A)) by A1, A4, A6, A7, A8, INTEGRA5:10, INTEGRA5:13; ::_thesis: verum end; theorem :: INTEGR11:72 for A being non empty closed_interval Subset of REAL for Z being open Subset of REAL for f2, f1 being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & f2 = #Z 2 & ( for x being Real st x in Z holds f1 . x = 1 ) & Z = dom arctan & Z = dom (((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) holds integral (arctan,A) = ((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) . (upper_bound A)) - ((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) . (lower_bound A)) proof let A be non empty closed_interval Subset of REAL; ::_thesis: for Z being open Subset of REAL for f2, f1 being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & f2 = #Z 2 & ( for x being Real st x in Z holds f1 . x = 1 ) & Z = dom arctan & Z = dom (((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) holds integral (arctan,A) = ((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) . (upper_bound A)) - ((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) . (lower_bound A)) let Z be open Subset of REAL; ::_thesis: for f2, f1 being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & f2 = #Z 2 & ( for x being Real st x in Z holds f1 . x = 1 ) & Z = dom arctan & Z = dom (((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) holds integral (arctan,A) = ((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) . (upper_bound A)) - ((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) . (lower_bound A)) let f2, f1 be PartFunc of REAL,REAL; ::_thesis: ( A c= Z & Z c= ].(- 1),1.[ & f2 = #Z 2 & ( for x being Real st x in Z holds f1 . x = 1 ) & Z = dom arctan & Z = dom (((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) implies integral (arctan,A) = ((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) . (upper_bound A)) - ((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) . (lower_bound A)) ) assume that A1: A c= Z and A2: Z c= ].(- 1),1.[ and A3: ( f2 = #Z 2 & ( for x being Real st x in Z holds f1 . x = 1 ) ) and A4: Z = dom arctan and A5: Z = dom (((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) ; ::_thesis: integral (arctan,A) = ((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) . (upper_bound A)) - ((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) . (lower_bound A)) ( ].(- 1),1.[ c= [.(- 1),1.] & A c= ].(- 1),1.[ ) by A1, A2, XBOOLE_1:1, XXREAL_1:25; then arctan | A is continuous by FCONT_1:16, SIN_COS9:53, XBOOLE_1:1; then A6: ( arctan is_integrable_on A & arctan | A is bounded ) by A1, A4, INTEGRA5:10, INTEGRA5:11; A7: ((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z by A2, A3, A5, SIN_COS9:103; A8: for x being Real st x in dom ((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) holds ((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . x proof let x be Real; ::_thesis: ( x in dom ((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) implies ((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . x ) assume x in dom ((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) ; ::_thesis: ((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . x then x in Z by A7, FDIFF_1:def_7; hence ((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . x by A2, A3, A5, SIN_COS9:103; ::_thesis: verum end; dom ((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) = dom arctan by A4, A7, FDIFF_1:def_7; then (((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z = arctan by A8, PARTFUN1:5; hence integral (arctan,A) = ((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) . (upper_bound A)) - ((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) . (lower_bound A)) by A1, A2, A3, A5, A6, INTEGRA5:13, SIN_COS9:103; ::_thesis: verum end; theorem :: INTEGR11:73 for A being non empty closed_interval Subset of REAL for Z being open Subset of REAL for f2, f1 being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & f2 = #Z 2 & ( for x being Real st x in Z holds f1 . x = 1 ) & dom arccot = Z & Z = dom (((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) holds integral (arccot,A) = ((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) . (upper_bound A)) - ((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) . (lower_bound A)) proof let A be non empty closed_interval Subset of REAL; ::_thesis: for Z being open Subset of REAL for f2, f1 being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & f2 = #Z 2 & ( for x being Real st x in Z holds f1 . x = 1 ) & dom arccot = Z & Z = dom (((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) holds integral (arccot,A) = ((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) . (upper_bound A)) - ((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) . (lower_bound A)) let Z be open Subset of REAL; ::_thesis: for f2, f1 being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & f2 = #Z 2 & ( for x being Real st x in Z holds f1 . x = 1 ) & dom arccot = Z & Z = dom (((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) holds integral (arccot,A) = ((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) . (upper_bound A)) - ((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) . (lower_bound A)) let f2, f1 be PartFunc of REAL,REAL; ::_thesis: ( A c= Z & Z c= ].(- 1),1.[ & f2 = #Z 2 & ( for x being Real st x in Z holds f1 . x = 1 ) & dom arccot = Z & Z = dom (((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) implies integral (arccot,A) = ((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) . (upper_bound A)) - ((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) . (lower_bound A)) ) assume that A1: A c= Z and A2: Z c= ].(- 1),1.[ and A3: ( f2 = #Z 2 & ( for x being Real st x in Z holds f1 . x = 1 ) ) and A4: dom arccot = Z and A5: Z = dom (((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) ; ::_thesis: integral (arccot,A) = ((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) . (upper_bound A)) - ((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) . (lower_bound A)) ( ].(- 1),1.[ c= [.(- 1),1.] & A c= ].(- 1),1.[ ) by A1, A2, XBOOLE_1:1, XXREAL_1:25; then arccot | A is continuous by FCONT_1:16, SIN_COS9:54, XBOOLE_1:1; then A6: ( arccot is_integrable_on A & arccot | A is bounded ) by A1, A4, INTEGRA5:10, INTEGRA5:11; A7: ((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z by A2, A3, A5, SIN_COS9:104; A8: for x being Real st x in dom ((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) holds ((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . x proof let x be Real; ::_thesis: ( x in dom ((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) implies ((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . x ) assume x in dom ((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) ; ::_thesis: ((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . x then x in Z by A7, FDIFF_1:def_7; hence ((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . x by A2, A3, A5, SIN_COS9:104; ::_thesis: verum end; dom ((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) = dom arccot by A4, A7, FDIFF_1:def_7; then (((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z = arccot by A8, PARTFUN1:5; hence integral (arccot,A) = ((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) . (upper_bound A)) - ((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) . (lower_bound A)) by A1, A2, A3, A5, A6, INTEGRA5:13, SIN_COS9:104; ::_thesis: verum end;