:: DIFF_3 semantic presentation begin theorem Th1: :: DIFF_3:1 for h, x being Real for f being Function of REAL,REAL holds (cD (f,h)) . x = ((fD (f,(h / 2))) . x) - ((fD (f,(- (h / 2)))) . x) proof let h, x be Real; ::_thesis: for f being Function of REAL,REAL holds (cD (f,h)) . x = ((fD (f,(h / 2))) . x) - ((fD (f,(- (h / 2)))) . x) let f be Function of REAL,REAL; ::_thesis: (cD (f,h)) . x = ((fD (f,(h / 2))) . x) - ((fD (f,(- (h / 2)))) . x) (cD (f,h)) . x = (((f . (x + (h / 2))) - (f . x)) + (f . x)) - (f . (x - (h / 2))) by DIFF_1:5 .= ((f . (x + (h / 2))) - (f . x)) - ((f . (x - (h / 2))) - (f . x)) .= ((fD (f,(h / 2))) . x) - ((f . (x - (h / 2))) - (f . x)) by DIFF_1:3 .= ((fD (f,(h / 2))) . x) - ((fD (f,(- (h / 2)))) . x) by DIFF_1:3 ; hence (cD (f,h)) . x = ((fD (f,(h / 2))) . x) - ((fD (f,(- (h / 2)))) . x) ; ::_thesis: verum end; theorem Th2: :: DIFF_3:2 for h, x being Real for f being Function of REAL,REAL holds (fD (f,(- (h / 2)))) . x = - ((bD (f,(h / 2))) . x) proof let h, x be Real; ::_thesis: for f being Function of REAL,REAL holds (fD (f,(- (h / 2)))) . x = - ((bD (f,(h / 2))) . x) let f be Function of REAL,REAL; ::_thesis: (fD (f,(- (h / 2)))) . x = - ((bD (f,(h / 2))) . x) (fD (f,(- (h / 2)))) . x = (f . (x - (h / 2))) - (f . x) by DIFF_1:3 .= - ((f . x) - (f . (x - (h / 2)))) .= - ((bD (f,(h / 2))) . x) by DIFF_1:4 ; hence (fD (f,(- (h / 2)))) . x = - ((bD (f,(h / 2))) . x) ; ::_thesis: verum end; theorem :: DIFF_3:3 for h, x being Real for f being Function of REAL,REAL holds (cD (f,h)) . x = ((bD (f,(h / 2))) . x) - ((bD (f,(- (h / 2)))) . x) proof let h, x be Real; ::_thesis: for f being Function of REAL,REAL holds (cD (f,h)) . x = ((bD (f,(h / 2))) . x) - ((bD (f,(- (h / 2)))) . x) let f be Function of REAL,REAL; ::_thesis: (cD (f,h)) . x = ((bD (f,(h / 2))) . x) - ((bD (f,(- (h / 2)))) . x) (fD (f,(h / 2))) . x = - ((bD (f,(- (h / 2)))) . x) proof (fD (f,(h / 2))) . x = (f . (x + (h / 2))) - (f . x) by DIFF_1:3 .= - ((f . x) - (f . (x - (- (h / 2))))) .= - ((bD (f,(- (h / 2)))) . x) by DIFF_1:4 ; hence (fD (f,(h / 2))) . x = - ((bD (f,(- (h / 2)))) . x) ; ::_thesis: verum end; then (cD (f,h)) . x = (- ((bD (f,(- (h / 2)))) . x)) - ((fD (f,(- (h / 2)))) . x) by Th1 .= (- ((bD (f,(- (h / 2)))) . x)) - (- ((bD (f,(h / 2))) . x)) by Th2 .= ((bD (f,(h / 2))) . x) - ((bD (f,(- (h / 2)))) . x) ; hence (cD (f,h)) . x = ((bD (f,(h / 2))) . x) - ((bD (f,(- (h / 2)))) . x) ; ::_thesis: verum end; theorem :: DIFF_3:4 for n being Element of NAT for r, h, x being Real for f1, f2 being Function of REAL,REAL holds ((fdif (((r (#) f1) + f2),h)) . (n + 1)) . x = (r * (((fdif (f1,h)) . (n + 1)) . x)) + (((fdif (f2,h)) . (n + 1)) . x) proof let n be Element of NAT ; ::_thesis: for r, h, x being Real for f1, f2 being Function of REAL,REAL holds ((fdif (((r (#) f1) + f2),h)) . (n + 1)) . x = (r * (((fdif (f1,h)) . (n + 1)) . x)) + (((fdif (f2,h)) . (n + 1)) . x) let r, h, x be Real; ::_thesis: for f1, f2 being Function of REAL,REAL holds ((fdif (((r (#) f1) + f2),h)) . (n + 1)) . x = (r * (((fdif (f1,h)) . (n + 1)) . x)) + (((fdif (f2,h)) . (n + 1)) . x) let f1, f2 be Function of REAL,REAL; ::_thesis: ((fdif (((r (#) f1) + f2),h)) . (n + 1)) . x = (r * (((fdif (f1,h)) . (n + 1)) . x)) + (((fdif (f2,h)) . (n + 1)) . x) set g = r (#) f1; ((fdif (((r (#) f1) + f2),h)) . (n + 1)) . x = (((fdif ((r (#) f1),h)) . (n + 1)) . x) + (((fdif (f2,h)) . (n + 1)) . x) by DIFF_1:8 .= (r * (((fdif (f1,h)) . (n + 1)) . x)) + (((fdif (f2,h)) . (n + 1)) . x) by DIFF_1:7 ; hence ((fdif (((r (#) f1) + f2),h)) . (n + 1)) . x = (r * (((fdif (f1,h)) . (n + 1)) . x)) + (((fdif (f2,h)) . (n + 1)) . x) ; ::_thesis: verum end; theorem :: DIFF_3:5 for n being Element of NAT for r, h, x being Real for f1, f2 being Function of REAL,REAL holds ((fdif ((f1 + (r (#) f2)),h)) . (n + 1)) . x = (((fdif (f1,h)) . (n + 1)) . x) + (r * (((fdif (f2,h)) . (n + 1)) . x)) proof let n be Element of NAT ; ::_thesis: for r, h, x being Real for f1, f2 being Function of REAL,REAL holds ((fdif ((f1 + (r (#) f2)),h)) . (n + 1)) . x = (((fdif (f1,h)) . (n + 1)) . x) + (r * (((fdif (f2,h)) . (n + 1)) . x)) let r, h, x be Real; ::_thesis: for f1, f2 being Function of REAL,REAL holds ((fdif ((f1 + (r (#) f2)),h)) . (n + 1)) . x = (((fdif (f1,h)) . (n + 1)) . x) + (r * (((fdif (f2,h)) . (n + 1)) . x)) let f1, f2 be Function of REAL,REAL; ::_thesis: ((fdif ((f1 + (r (#) f2)),h)) . (n + 1)) . x = (((fdif (f1,h)) . (n + 1)) . x) + (r * (((fdif (f2,h)) . (n + 1)) . x)) set g = r (#) f2; ((fdif ((f1 + (r (#) f2)),h)) . (n + 1)) . x = (((fdif (f1,h)) . (n + 1)) . x) + (((fdif ((r (#) f2),h)) . (n + 1)) . x) by DIFF_1:8 .= (((fdif (f1,h)) . (n + 1)) . x) + (r * (((fdif (f2,h)) . (n + 1)) . x)) by DIFF_1:7 ; hence ((fdif ((f1 + (r (#) f2)),h)) . (n + 1)) . x = (((fdif (f1,h)) . (n + 1)) . x) + (r * (((fdif (f2,h)) . (n + 1)) . x)) ; ::_thesis: verum end; theorem :: DIFF_3:6 for n being Element of NAT for r1, r2, h, x being Real for f1, f2 being Function of REAL,REAL holds ((fdif (((r1 (#) f1) - (r2 (#) f2)),h)) . (n + 1)) . x = (r1 * (((fdif (f1,h)) . (n + 1)) . x)) - (r2 * (((fdif (f2,h)) . (n + 1)) . x)) proof let n be Element of NAT ; ::_thesis: for r1, r2, h, x being Real for f1, f2 being Function of REAL,REAL holds ((fdif (((r1 (#) f1) - (r2 (#) f2)),h)) . (n + 1)) . x = (r1 * (((fdif (f1,h)) . (n + 1)) . x)) - (r2 * (((fdif (f2,h)) . (n + 1)) . x)) let r1, r2, h, x be Real; ::_thesis: for f1, f2 being Function of REAL,REAL holds ((fdif (((r1 (#) f1) - (r2 (#) f2)),h)) . (n + 1)) . x = (r1 * (((fdif (f1,h)) . (n + 1)) . x)) - (r2 * (((fdif (f2,h)) . (n + 1)) . x)) let f1, f2 be Function of REAL,REAL; ::_thesis: ((fdif (((r1 (#) f1) - (r2 (#) f2)),h)) . (n + 1)) . x = (r1 * (((fdif (f1,h)) . (n + 1)) . x)) - (r2 * (((fdif (f2,h)) . (n + 1)) . x)) set g1 = r1 (#) f1; set g2 = r2 (#) f2; ((fdif (((r1 (#) f1) - (r2 (#) f2)),h)) . (n + 1)) . x = (((fdif ((r1 (#) f1),h)) . (n + 1)) . x) - (((fdif ((r2 (#) f2),h)) . (n + 1)) . x) by DIFF_1:9 .= (r1 * (((fdif (f1,h)) . (n + 1)) . x)) - (((fdif ((r2 (#) f2),h)) . (n + 1)) . x) by DIFF_1:7 .= (r1 * (((fdif (f1,h)) . (n + 1)) . x)) - (r2 * (((fdif (f2,h)) . (n + 1)) . x)) by DIFF_1:7 ; hence ((fdif (((r1 (#) f1) - (r2 (#) f2)),h)) . (n + 1)) . x = (r1 * (((fdif (f1,h)) . (n + 1)) . x)) - (r2 * (((fdif (f2,h)) . (n + 1)) . x)) ; ::_thesis: verum end; theorem :: DIFF_3:7 for h being Real for f being Function of REAL,REAL holds (fdif (f,h)) . 1 = fD (f,h) proof let h be Real; ::_thesis: for f being Function of REAL,REAL holds (fdif (f,h)) . 1 = fD (f,h) let f be Function of REAL,REAL; ::_thesis: (fdif (f,h)) . 1 = fD (f,h) (fdif (f,h)) . 1 = (fdif (f,h)) . (0 + 1) .= fD (((fdif (f,h)) . 0),h) by DIFF_1:def_6 .= fD (f,h) by DIFF_1:def_6 ; hence (fdif (f,h)) . 1 = fD (f,h) ; ::_thesis: verum end; theorem :: DIFF_3:8 for n being Element of NAT for r, h, x being Real for f1, f2 being Function of REAL,REAL holds ((bdif (((r (#) f1) + f2),h)) . (n + 1)) . x = (r * (((bdif (f1,h)) . (n + 1)) . x)) + (((bdif (f2,h)) . (n + 1)) . x) proof let n be Element of NAT ; ::_thesis: for r, h, x being Real for f1, f2 being Function of REAL,REAL holds ((bdif (((r (#) f1) + f2),h)) . (n + 1)) . x = (r * (((bdif (f1,h)) . (n + 1)) . x)) + (((bdif (f2,h)) . (n + 1)) . x) let r, h, x be Real; ::_thesis: for f1, f2 being Function of REAL,REAL holds ((bdif (((r (#) f1) + f2),h)) . (n + 1)) . x = (r * (((bdif (f1,h)) . (n + 1)) . x)) + (((bdif (f2,h)) . (n + 1)) . x) let f1, f2 be Function of REAL,REAL; ::_thesis: ((bdif (((r (#) f1) + f2),h)) . (n + 1)) . x = (r * (((bdif (f1,h)) . (n + 1)) . x)) + (((bdif (f2,h)) . (n + 1)) . x) set g = r (#) f1; ((bdif (((r (#) f1) + f2),h)) . (n + 1)) . x = (((bdif ((r (#) f1),h)) . (n + 1)) . x) + (((bdif (f2,h)) . (n + 1)) . x) by DIFF_1:15 .= (r * (((bdif (f1,h)) . (n + 1)) . x)) + (((bdif (f2,h)) . (n + 1)) . x) by DIFF_1:14 ; hence ((bdif (((r (#) f1) + f2),h)) . (n + 1)) . x = (r * (((bdif (f1,h)) . (n + 1)) . x)) + (((bdif (f2,h)) . (n + 1)) . x) ; ::_thesis: verum end; theorem :: DIFF_3:9 for n being Element of NAT for r, h, x being Real for f1, f2 being Function of REAL,REAL holds ((bdif ((f1 + (r (#) f2)),h)) . (n + 1)) . x = (((bdif (f1,h)) . (n + 1)) . x) + (r * (((bdif (f2,h)) . (n + 1)) . x)) proof let n be Element of NAT ; ::_thesis: for r, h, x being Real for f1, f2 being Function of REAL,REAL holds ((bdif ((f1 + (r (#) f2)),h)) . (n + 1)) . x = (((bdif (f1,h)) . (n + 1)) . x) + (r * (((bdif (f2,h)) . (n + 1)) . x)) let r, h, x be Real; ::_thesis: for f1, f2 being Function of REAL,REAL holds ((bdif ((f1 + (r (#) f2)),h)) . (n + 1)) . x = (((bdif (f1,h)) . (n + 1)) . x) + (r * (((bdif (f2,h)) . (n + 1)) . x)) let f1, f2 be Function of REAL,REAL; ::_thesis: ((bdif ((f1 + (r (#) f2)),h)) . (n + 1)) . x = (((bdif (f1,h)) . (n + 1)) . x) + (r * (((bdif (f2,h)) . (n + 1)) . x)) set g = r (#) f2; ((bdif ((f1 + (r (#) f2)),h)) . (n + 1)) . x = (((bdif (f1,h)) . (n + 1)) . x) + (((bdif ((r (#) f2),h)) . (n + 1)) . x) by DIFF_1:15 .= (((bdif (f1,h)) . (n + 1)) . x) + (r * (((bdif (f2,h)) . (n + 1)) . x)) by DIFF_1:14 ; hence ((bdif ((f1 + (r (#) f2)),h)) . (n + 1)) . x = (((bdif (f1,h)) . (n + 1)) . x) + (r * (((bdif (f2,h)) . (n + 1)) . x)) ; ::_thesis: verum end; theorem :: DIFF_3:10 for n being Element of NAT for r1, r2, h, x being Real for f1, f2 being Function of REAL,REAL holds ((bdif (((r1 (#) f1) - (r2 (#) f2)),h)) . (n + 1)) . x = (r1 * (((bdif (f1,h)) . (n + 1)) . x)) - (r2 * (((bdif (f2,h)) . (n + 1)) . x)) proof let n be Element of NAT ; ::_thesis: for r1, r2, h, x being Real for f1, f2 being Function of REAL,REAL holds ((bdif (((r1 (#) f1) - (r2 (#) f2)),h)) . (n + 1)) . x = (r1 * (((bdif (f1,h)) . (n + 1)) . x)) - (r2 * (((bdif (f2,h)) . (n + 1)) . x)) let r1, r2, h, x be Real; ::_thesis: for f1, f2 being Function of REAL,REAL holds ((bdif (((r1 (#) f1) - (r2 (#) f2)),h)) . (n + 1)) . x = (r1 * (((bdif (f1,h)) . (n + 1)) . x)) - (r2 * (((bdif (f2,h)) . (n + 1)) . x)) let f1, f2 be Function of REAL,REAL; ::_thesis: ((bdif (((r1 (#) f1) - (r2 (#) f2)),h)) . (n + 1)) . x = (r1 * (((bdif (f1,h)) . (n + 1)) . x)) - (r2 * (((bdif (f2,h)) . (n + 1)) . x)) set g1 = r1 (#) f1; set g2 = r2 (#) f2; ((bdif (((r1 (#) f1) - (r2 (#) f2)),h)) . (n + 1)) . x = (((bdif ((r1 (#) f1),h)) . (n + 1)) . x) - (((bdif ((r2 (#) f2),h)) . (n + 1)) . x) by DIFF_1:16 .= (r1 * (((bdif (f1,h)) . (n + 1)) . x)) - (((bdif ((r2 (#) f2),h)) . (n + 1)) . x) by DIFF_1:14 .= (r1 * (((bdif (f1,h)) . (n + 1)) . x)) - (r2 * (((bdif (f2,h)) . (n + 1)) . x)) by DIFF_1:14 ; hence ((bdif (((r1 (#) f1) - (r2 (#) f2)),h)) . (n + 1)) . x = (r1 * (((bdif (f1,h)) . (n + 1)) . x)) - (r2 * (((bdif (f2,h)) . (n + 1)) . x)) ; ::_thesis: verum end; theorem Th11: :: DIFF_3:11 for h being Real for f being Function of REAL,REAL holds (bdif (f,h)) . 1 = bD (f,h) proof let h be Real; ::_thesis: for f being Function of REAL,REAL holds (bdif (f,h)) . 1 = bD (f,h) let f be Function of REAL,REAL; ::_thesis: (bdif (f,h)) . 1 = bD (f,h) (bdif (f,h)) . 1 = (bdif (f,h)) . (0 + 1) .= bD (((bdif (f,h)) . 0),h) by DIFF_1:def_7 .= bD (f,h) by DIFF_1:def_7 ; hence (bdif (f,h)) . 1 = bD (f,h) ; ::_thesis: verum end; theorem :: DIFF_3:12 for m, n being Element of NAT for h, x being Real for f being Function of REAL,REAL holds ((bdif (((bdif (f,h)) . m),h)) . n) . x = ((bdif (f,h)) . (m + n)) . x proof let m, n be Element of NAT ; ::_thesis: for h, x being Real for f being Function of REAL,REAL holds ((bdif (((bdif (f,h)) . m),h)) . n) . x = ((bdif (f,h)) . (m + n)) . x let h, x be Real; ::_thesis: for f being Function of REAL,REAL holds ((bdif (((bdif (f,h)) . m),h)) . n) . x = ((bdif (f,h)) . (m + n)) . x let f be Function of REAL,REAL; ::_thesis: ((bdif (((bdif (f,h)) . m),h)) . n) . x = ((bdif (f,h)) . (m + n)) . x defpred S1[ Nat] means for x being Real holds ((bdif (((bdif (f,h)) . m),h)) . \$1) . x = ((bdif (f,h)) . (m + \$1)) . x; A1: S1[ 0 ] by DIFF_1:def_7; A2: for i being Nat st S1[i] holds S1[i + 1] proof let i be Nat; ::_thesis: ( S1[i] implies S1[i + 1] ) assume A3: for x being Real holds ((bdif (((bdif (f,h)) . m),h)) . i) . x = ((bdif (f,h)) . (m + i)) . x ; ::_thesis: S1[i + 1] let x be Real; ::_thesis: ((bdif (((bdif (f,h)) . m),h)) . (i + 1)) . x = ((bdif (f,h)) . (m + (i + 1))) . x (bdif (f,h)) . m is Function of REAL,REAL by DIFF_1:12; then A4: (bdif (((bdif (f,h)) . m),h)) . i is Function of REAL,REAL by DIFF_1:12; A5: (bdif (f,h)) . (m + i) is Function of REAL,REAL by DIFF_1:12; ((bdif (((bdif (f,h)) . m),h)) . (i + 1)) . x = (bD (((bdif (((bdif (f,h)) . m),h)) . i),h)) . x by DIFF_1:def_7 .= (((bdif (((bdif (f,h)) . m),h)) . i) . x) - (((bdif (((bdif (f,h)) . m),h)) . i) . (x - h)) by A4, DIFF_1:4 .= (((bdif (f,h)) . (m + i)) . x) - (((bdif (((bdif (f,h)) . m),h)) . i) . (x - h)) by A3 .= (((bdif (f,h)) . (m + i)) . x) - (((bdif (f,h)) . (m + i)) . (x - h)) by A3 .= (bD (((bdif (f,h)) . (m + i)),h)) . x by A5, DIFF_1:4 .= ((bdif (f,h)) . ((m + i) + 1)) . x by DIFF_1:def_7 ; hence ((bdif (((bdif (f,h)) . m),h)) . (i + 1)) . x = ((bdif (f,h)) . (m + (i + 1))) . x ; ::_thesis: verum end; for n being Nat holds S1[n] from NAT_1:sch_2(A1, A2); hence ((bdif (((bdif (f,h)) . m),h)) . n) . x = ((bdif (f,h)) . (m + n)) . x ; ::_thesis: verum end; theorem :: DIFF_3:13 for n being Element of NAT for r, h, x being Real for f1, f2 being Function of REAL,REAL holds ((cdif (((r (#) f1) + f2),h)) . (n + 1)) . x = (r * (((cdif (f1,h)) . (n + 1)) . x)) + (((cdif (f2,h)) . (n + 1)) . x) proof let n be Element of NAT ; ::_thesis: for r, h, x being Real for f1, f2 being Function of REAL,REAL holds ((cdif (((r (#) f1) + f2),h)) . (n + 1)) . x = (r * (((cdif (f1,h)) . (n + 1)) . x)) + (((cdif (f2,h)) . (n + 1)) . x) let r, h, x be Real; ::_thesis: for f1, f2 being Function of REAL,REAL holds ((cdif (((r (#) f1) + f2),h)) . (n + 1)) . x = (r * (((cdif (f1,h)) . (n + 1)) . x)) + (((cdif (f2,h)) . (n + 1)) . x) let f1, f2 be Function of REAL,REAL; ::_thesis: ((cdif (((r (#) f1) + f2),h)) . (n + 1)) . x = (r * (((cdif (f1,h)) . (n + 1)) . x)) + (((cdif (f2,h)) . (n + 1)) . x) set g = r (#) f1; ((cdif (((r (#) f1) + f2),h)) . (n + 1)) . x = (((cdif ((r (#) f1),h)) . (n + 1)) . x) + (((cdif (f2,h)) . (n + 1)) . x) by DIFF_1:22 .= (r * (((cdif (f1,h)) . (n + 1)) . x)) + (((cdif (f2,h)) . (n + 1)) . x) by DIFF_1:21 ; hence ((cdif (((r (#) f1) + f2),h)) . (n + 1)) . x = (r * (((cdif (f1,h)) . (n + 1)) . x)) + (((cdif (f2,h)) . (n + 1)) . x) ; ::_thesis: verum end; theorem :: DIFF_3:14 for n being Element of NAT for r, h, x being Real for f1, f2 being Function of REAL,REAL holds ((cdif ((f1 + (r (#) f2)),h)) . (n + 1)) . x = (((cdif (f1,h)) . (n + 1)) . x) + (r * (((cdif (f2,h)) . (n + 1)) . x)) proof let n be Element of NAT ; ::_thesis: for r, h, x being Real for f1, f2 being Function of REAL,REAL holds ((cdif ((f1 + (r (#) f2)),h)) . (n + 1)) . x = (((cdif (f1,h)) . (n + 1)) . x) + (r * (((cdif (f2,h)) . (n + 1)) . x)) let r, h, x be Real; ::_thesis: for f1, f2 being Function of REAL,REAL holds ((cdif ((f1 + (r (#) f2)),h)) . (n + 1)) . x = (((cdif (f1,h)) . (n + 1)) . x) + (r * (((cdif (f2,h)) . (n + 1)) . x)) let f1, f2 be Function of REAL,REAL; ::_thesis: ((cdif ((f1 + (r (#) f2)),h)) . (n + 1)) . x = (((cdif (f1,h)) . (n + 1)) . x) + (r * (((cdif (f2,h)) . (n + 1)) . x)) set g = r (#) f2; ((cdif ((f1 + (r (#) f2)),h)) . (n + 1)) . x = (((cdif (f1,h)) . (n + 1)) . x) + (((cdif ((r (#) f2),h)) . (n + 1)) . x) by DIFF_1:22 .= (((cdif (f1,h)) . (n + 1)) . x) + (r * (((cdif (f2,h)) . (n + 1)) . x)) by DIFF_1:21 ; hence ((cdif ((f1 + (r (#) f2)),h)) . (n + 1)) . x = (((cdif (f1,h)) . (n + 1)) . x) + (r * (((cdif (f2,h)) . (n + 1)) . x)) ; ::_thesis: verum end; theorem :: DIFF_3:15 for n being Element of NAT for r1, r2, h, x being Real for f1, f2 being Function of REAL,REAL holds ((cdif (((r1 (#) f1) - (r2 (#) f2)),h)) . (n + 1)) . x = (r1 * (((cdif (f1,h)) . (n + 1)) . x)) - (r2 * (((cdif (f2,h)) . (n + 1)) . x)) proof let n be Element of NAT ; ::_thesis: for r1, r2, h, x being Real for f1, f2 being Function of REAL,REAL holds ((cdif (((r1 (#) f1) - (r2 (#) f2)),h)) . (n + 1)) . x = (r1 * (((cdif (f1,h)) . (n + 1)) . x)) - (r2 * (((cdif (f2,h)) . (n + 1)) . x)) let r1, r2, h, x be Real; ::_thesis: for f1, f2 being Function of REAL,REAL holds ((cdif (((r1 (#) f1) - (r2 (#) f2)),h)) . (n + 1)) . x = (r1 * (((cdif (f1,h)) . (n + 1)) . x)) - (r2 * (((cdif (f2,h)) . (n + 1)) . x)) let f1, f2 be Function of REAL,REAL; ::_thesis: ((cdif (((r1 (#) f1) - (r2 (#) f2)),h)) . (n + 1)) . x = (r1 * (((cdif (f1,h)) . (n + 1)) . x)) - (r2 * (((cdif (f2,h)) . (n + 1)) . x)) set g1 = r1 (#) f1; set g2 = r2 (#) f2; ((cdif (((r1 (#) f1) - (r2 (#) f2)),h)) . (n + 1)) . x = (((cdif ((r1 (#) f1),h)) . (n + 1)) . x) - (((cdif ((r2 (#) f2),h)) . (n + 1)) . x) by DIFF_1:23 .= (r1 * (((cdif (f1,h)) . (n + 1)) . x)) - (((cdif ((r2 (#) f2),h)) . (n + 1)) . x) by DIFF_1:21 .= (r1 * (((cdif (f1,h)) . (n + 1)) . x)) - (r2 * (((cdif (f2,h)) . (n + 1)) . x)) by DIFF_1:21 ; hence ((cdif (((r1 (#) f1) - (r2 (#) f2)),h)) . (n + 1)) . x = (r1 * (((cdif (f1,h)) . (n + 1)) . x)) - (r2 * (((cdif (f2,h)) . (n + 1)) . x)) ; ::_thesis: verum end; theorem Th16: :: DIFF_3:16 for h being Real for f being Function of REAL,REAL holds (cdif (f,h)) . 1 = cD (f,h) proof let h be Real; ::_thesis: for f being Function of REAL,REAL holds (cdif (f,h)) . 1 = cD (f,h) let f be Function of REAL,REAL; ::_thesis: (cdif (f,h)) . 1 = cD (f,h) (cdif (f,h)) . 1 = (cdif (f,h)) . (0 + 1) .= cD (((cdif (f,h)) . 0),h) by DIFF_1:def_8 .= cD (f,h) by DIFF_1:def_8 ; hence (cdif (f,h)) . 1 = cD (f,h) ; ::_thesis: verum end; theorem :: DIFF_3:17 for m, n being Element of NAT for h, x being Real for f being Function of REAL,REAL holds ((cdif (((cdif (f,h)) . m),h)) . n) . x = ((cdif (f,h)) . (m + n)) . x proof let m, n be Element of NAT ; ::_thesis: for h, x being Real for f being Function of REAL,REAL holds ((cdif (((cdif (f,h)) . m),h)) . n) . x = ((cdif (f,h)) . (m + n)) . x let h, x be Real; ::_thesis: for f being Function of REAL,REAL holds ((cdif (((cdif (f,h)) . m),h)) . n) . x = ((cdif (f,h)) . (m + n)) . x let f be Function of REAL,REAL; ::_thesis: ((cdif (((cdif (f,h)) . m),h)) . n) . x = ((cdif (f,h)) . (m + n)) . x defpred S1[ Nat] means for x being Real holds ((cdif (((cdif (f,h)) . m),h)) . \$1) . x = ((cdif (f,h)) . (m + \$1)) . x; A1: S1[ 0 ] by DIFF_1:def_8; A2: for i being Nat st S1[i] holds S1[i + 1] proof let i be Nat; ::_thesis: ( S1[i] implies S1[i + 1] ) assume A3: for x being Real holds ((cdif (((cdif (f,h)) . m),h)) . i) . x = ((cdif (f,h)) . (m + i)) . x ; ::_thesis: S1[i + 1] let x be Real; ::_thesis: ((cdif (((cdif (f,h)) . m),h)) . (i + 1)) . x = ((cdif (f,h)) . (m + (i + 1))) . x (cdif (f,h)) . m is Function of REAL,REAL by DIFF_1:19; then A4: (cdif (((cdif (f,h)) . m),h)) . i is Function of REAL,REAL by DIFF_1:19; A5: (cdif (f,h)) . (m + i) is Function of REAL,REAL by DIFF_1:19; ((cdif (((cdif (f,h)) . m),h)) . (i + 1)) . x = (cD (((cdif (((cdif (f,h)) . m),h)) . i),h)) . x by DIFF_1:def_8 .= (((cdif (((cdif (f,h)) . m),h)) . i) . (x + (h / 2))) - (((cdif (((cdif (f,h)) . m),h)) . i) . (x - (h / 2))) by A4, DIFF_1:5 .= (((cdif (f,h)) . (m + i)) . (x + (h / 2))) - (((cdif (((cdif (f,h)) . m),h)) . i) . (x - (h / 2))) by A3 .= (((cdif (f,h)) . (m + i)) . (x + (h / 2))) - (((cdif (f,h)) . (m + i)) . (x - (h / 2))) by A3 .= (cD (((cdif (f,h)) . (m + i)),h)) . x by A5, DIFF_1:5 .= ((cdif (f,h)) . ((m + i) + 1)) . x by DIFF_1:def_8 ; hence ((cdif (((cdif (f,h)) . m),h)) . (i + 1)) . x = ((cdif (f,h)) . (m + (i + 1))) . x ; ::_thesis: verum end; for n being Nat holds S1[n] from NAT_1:sch_2(A1, A2); hence ((cdif (((cdif (f,h)) . m),h)) . n) . x = ((cdif (f,h)) . (m + n)) . x ; ::_thesis: verum end; theorem :: DIFF_3:18 for n being Element of NAT for h, x being Real for f being Function of REAL,REAL st ((fdif (f,h)) . n) . x = ((cdif (f,h)) . n) . (x + ((n / 2) * h)) holds ((bdif (f,h)) . n) . x = ((cdif (f,h)) . n) . (x - ((n / 2) * h)) proof let n be Element of NAT ; ::_thesis: for h, x being Real for f being Function of REAL,REAL st ((fdif (f,h)) . n) . x = ((cdif (f,h)) . n) . (x + ((n / 2) * h)) holds ((bdif (f,h)) . n) . x = ((cdif (f,h)) . n) . (x - ((n / 2) * h)) let h, x be Real; ::_thesis: for f being Function of REAL,REAL st ((fdif (f,h)) . n) . x = ((cdif (f,h)) . n) . (x + ((n / 2) * h)) holds ((bdif (f,h)) . n) . x = ((cdif (f,h)) . n) . (x - ((n / 2) * h)) let f be Function of REAL,REAL; ::_thesis: ( ((fdif (f,h)) . n) . x = ((cdif (f,h)) . n) . (x + ((n / 2) * h)) implies ((bdif (f,h)) . n) . x = ((cdif (f,h)) . n) . (x - ((n / 2) * h)) ) defpred S1[ Nat] means for x being Real holds ((bdif (f,h)) . \$1) . x = ((cdif (f,h)) . \$1) . (x - ((\$1 / 2) * h)); A1: S1[ 0 ] proof let x be Real; ::_thesis: ((bdif (f,h)) . 0) . x = ((cdif (f,h)) . 0) . (x - ((0 / 2) * h)) ((bdif (f,h)) . 0) . x = f . x by DIFF_1:def_7 .= ((cdif (f,h)) . 0) . (x - ((0 / 2) * h)) by DIFF_1:def_8 ; hence ((bdif (f,h)) . 0) . x = ((cdif (f,h)) . 0) . (x - ((0 / 2) * h)) ; ::_thesis: verum end; A2: for i being Nat st S1[i] holds S1[i + 1] proof let i be Nat; ::_thesis: ( S1[i] implies S1[i + 1] ) assume A3: for x being Real holds ((bdif (f,h)) . i) . x = ((cdif (f,h)) . i) . (x - ((i / 2) * h)) ; ::_thesis: S1[i + 1] let x be Real; ::_thesis: ((bdif (f,h)) . (i + 1)) . x = ((cdif (f,h)) . (i + 1)) . (x - (((i + 1) / 2) * h)) A4: (bdif (f,h)) . i is Function of REAL,REAL by DIFF_1:12; A5: (cdif (f,h)) . i is Function of REAL,REAL by DIFF_1:19; ((bdif (f,h)) . (i + 1)) . x = (bD (((bdif (f,h)) . i),h)) . x by DIFF_1:def_7 .= (((bdif (f,h)) . i) . x) - (((bdif (f,h)) . i) . (x - h)) by A4, DIFF_1:4 .= (((cdif (f,h)) . i) . (x - ((i / 2) * h))) - (((bdif (f,h)) . i) . (x - h)) by A3 .= (((cdif (f,h)) . i) . (x - ((i / 2) * h))) - (((cdif (f,h)) . i) . ((x - h) - ((i / 2) * h))) by A3 .= (((cdif (f,h)) . i) . ((x - (((i + 1) / 2) * h)) + (h / 2))) - (((cdif (f,h)) . i) . ((x - (((i + 1) / 2) * h)) - (h / 2))) .= (cD (((cdif (f,h)) . i),h)) . (x - (((i + 1) / 2) * h)) by A5, DIFF_1:5 .= ((cdif (f,h)) . (i + 1)) . (x - (((i + 1) / 2) * h)) by DIFF_1:def_8 ; hence ((bdif (f,h)) . (i + 1)) . x = ((cdif (f,h)) . (i + 1)) . (x - (((i + 1) / 2) * h)) ; ::_thesis: verum end; for n being Nat holds S1[n] from NAT_1:sch_2(A1, A2); hence ( ((fdif (f,h)) . n) . x = ((cdif (f,h)) . n) . (x + ((n / 2) * h)) implies ((bdif (f,h)) . n) . x = ((cdif (f,h)) . n) . (x - ((n / 2) * h)) ) ; ::_thesis: verum end; theorem :: DIFF_3:19 for n being Element of NAT for h, x being Real for f being Function of REAL,REAL st ((fdif (f,h)) . n) . x = ((cdif (f,h)) . n) . ((x + (((n - 1) / 2) * h)) + (h / 2)) holds ((bdif (f,h)) . n) . x = ((cdif (f,h)) . n) . ((x - (((n - 1) / 2) * h)) - (h / 2)) proof let n be Element of NAT ; ::_thesis: for h, x being Real for f being Function of REAL,REAL st ((fdif (f,h)) . n) . x = ((cdif (f,h)) . n) . ((x + (((n - 1) / 2) * h)) + (h / 2)) holds ((bdif (f,h)) . n) . x = ((cdif (f,h)) . n) . ((x - (((n - 1) / 2) * h)) - (h / 2)) let h, x be Real; ::_thesis: for f being Function of REAL,REAL st ((fdif (f,h)) . n) . x = ((cdif (f,h)) . n) . ((x + (((n - 1) / 2) * h)) + (h / 2)) holds ((bdif (f,h)) . n) . x = ((cdif (f,h)) . n) . ((x - (((n - 1) / 2) * h)) - (h / 2)) let f be Function of REAL,REAL; ::_thesis: ( ((fdif (f,h)) . n) . x = ((cdif (f,h)) . n) . ((x + (((n - 1) / 2) * h)) + (h / 2)) implies ((bdif (f,h)) . n) . x = ((cdif (f,h)) . n) . ((x - (((n - 1) / 2) * h)) - (h / 2)) ) defpred S1[ Nat] means for x being Real holds ((bdif (f,h)) . \$1) . x = ((cdif (f,h)) . \$1) . ((x - (((\$1 - 1) / 2) * h)) - (h / 2)); A1: S1[ 0 ] proof let x be Real; ::_thesis: ((bdif (f,h)) . 0) . x = ((cdif (f,h)) . 0) . ((x - (((0 - 1) / 2) * h)) - (h / 2)) ((bdif (f,h)) . 0) . x = f . x by DIFF_1:def_7 .= ((cdif (f,h)) . 0) . ((x - (((0 - 1) / 2) * h)) - (h / 2)) by DIFF_1:def_8 ; hence ((bdif (f,h)) . 0) . x = ((cdif (f,h)) . 0) . ((x - (((0 - 1) / 2) * h)) - (h / 2)) ; ::_thesis: verum end; A2: for i being Nat st S1[i] holds S1[i + 1] proof let i be Nat; ::_thesis: ( S1[i] implies S1[i + 1] ) assume A3: for x being Real holds ((bdif (f,h)) . i) . x = ((cdif (f,h)) . i) . ((x - (((i - 1) / 2) * h)) - (h / 2)) ; ::_thesis: S1[i + 1] let x be Real; ::_thesis: ((bdif (f,h)) . (i + 1)) . x = ((cdif (f,h)) . (i + 1)) . ((x - ((((i + 1) - 1) / 2) * h)) - (h / 2)) A4: (bdif (f,h)) . i is Function of REAL,REAL by DIFF_1:12; A5: (cdif (f,h)) . i is Function of REAL,REAL by DIFF_1:19; ((bdif (f,h)) . (i + 1)) . x = (bD (((bdif (f,h)) . i),h)) . x by DIFF_1:def_7 .= (((bdif (f,h)) . i) . x) - (((bdif (f,h)) . i) . (x - h)) by A4, DIFF_1:4 .= (((cdif (f,h)) . i) . ((x - (((i - 1) / 2) * h)) - (h / 2))) - (((bdif (f,h)) . i) . (x - h)) by A3 .= (((cdif (f,h)) . i) . ((x - (((i - 1) / 2) * h)) - (h / 2))) - (((cdif (f,h)) . i) . (((x - h) - (((i - 1) / 2) * h)) - (h / 2))) by A3 .= (((cdif (f,h)) . i) . (((x - ((i / 2) * h)) - (h / 2)) + (h / 2))) - (((cdif (f,h)) . i) . (((x - ((i / 2) * h)) - (h / 2)) - (h / 2))) .= (cD (((cdif (f,h)) . i),h)) . ((x - ((i / 2) * h)) - (h / 2)) by A5, DIFF_1:5 .= ((cdif (f,h)) . (i + 1)) . ((x - ((i / 2) * h)) - (h / 2)) by DIFF_1:def_8 ; hence ((bdif (f,h)) . (i + 1)) . x = ((cdif (f,h)) . (i + 1)) . ((x - ((((i + 1) - 1) / 2) * h)) - (h / 2)) ; ::_thesis: verum end; for n being Nat holds S1[n] from NAT_1:sch_2(A1, A2); hence ( ((fdif (f,h)) . n) . x = ((cdif (f,h)) . n) . ((x + (((n - 1) / 2) * h)) + (h / 2)) implies ((bdif (f,h)) . n) . x = ((cdif (f,h)) . n) . ((x - (((n - 1) / 2) * h)) - (h / 2)) ) ; ::_thesis: verum end; theorem :: DIFF_3:20 for x, h being Real for f being Function of REAL,REAL holds [!f,x,(x + h)!] = ((fD (f,h)) . x) / h proof let x, h be Real; ::_thesis: for f being Function of REAL,REAL holds [!f,x,(x + h)!] = ((fD (f,h)) . x) / h let f be Function of REAL,REAL; ::_thesis: [!f,x,(x + h)!] = ((fD (f,h)) . x) / h [!f,x,(x + h)!] = (- ((f . (x + h)) - (f . x))) / (- h) .= ((f . (x + h)) - (f . x)) / h by XCMPLX_1:191 .= ((fD (f,h)) . x) / h by DIFF_1:3 ; hence [!f,x,(x + h)!] = ((fD (f,h)) . x) / h ; ::_thesis: verum end; theorem :: DIFF_3:21 for x, h being Real for f being Function of REAL,REAL holds [!f,(x - h),x!] = ((bD (f,h)) . x) / h proof let x, h be Real; ::_thesis: for f being Function of REAL,REAL holds [!f,(x - h),x!] = ((bD (f,h)) . x) / h let f be Function of REAL,REAL; ::_thesis: [!f,(x - h),x!] = ((bD (f,h)) . x) / h [!f,(x - h),x!] = (((bdif (f,h)) . 1) . x) / h by DIFF_2:3 .= ((bD (f,h)) . x) / h by Th11 ; hence [!f,(x - h),x!] = ((bD (f,h)) . x) / h ; ::_thesis: verum end; theorem Th22: :: DIFF_3:22 for x, h being Real for f being Function of REAL,REAL holds [!f,(x - (h / 2)),(x + (h / 2))!] = ((cD (f,h)) . x) / h proof let x, h be Real; ::_thesis: for f being Function of REAL,REAL holds [!f,(x - (h / 2)),(x + (h / 2))!] = ((cD (f,h)) . x) / h let f be Function of REAL,REAL; ::_thesis: [!f,(x - (h / 2)),(x + (h / 2))!] = ((cD (f,h)) . x) / h [!f,(x - (h / 2)),(x + (h / 2))!] = [!f,(x + (h / 2)),(x - (h / 2))!] by DIFF_1:29 .= ((cD (f,h)) . x) / h by DIFF_1:5 ; hence [!f,(x - (h / 2)),(x + (h / 2))!] = ((cD (f,h)) . x) / h ; ::_thesis: verum end; theorem Th23: :: DIFF_3:23 for x, h being Real for f being Function of REAL,REAL holds [!f,(x - (h / 2)),(x + (h / 2))!] = (((cdif (f,h)) . 1) . x) / h proof let x, h be Real; ::_thesis: for f being Function of REAL,REAL holds [!f,(x - (h / 2)),(x + (h / 2))!] = (((cdif (f,h)) . 1) . x) / h let f be Function of REAL,REAL; ::_thesis: [!f,(x - (h / 2)),(x + (h / 2))!] = (((cdif (f,h)) . 1) . x) / h [!f,(x - (h / 2)),(x + (h / 2))!] = ((cD (f,h)) . x) / h by Th22 .= (((cdif (f,h)) . 1) . x) / h by Th16 ; hence [!f,(x - (h / 2)),(x + (h / 2))!] = (((cdif (f,h)) . 1) . x) / h ; ::_thesis: verum end; theorem :: DIFF_3:24 for h, x being Real for f being Function of REAL,REAL st h <> 0 holds [!f,(x - h),x,(x + h)!] = (((cdif (f,h)) . 2) . x) / ((2 * h) * h) proof let h, x be Real; ::_thesis: for f being Function of REAL,REAL st h <> 0 holds [!f,(x - h),x,(x + h)!] = (((cdif (f,h)) . 2) . x) / ((2 * h) * h) let f be Function of REAL,REAL; ::_thesis: ( h <> 0 implies [!f,(x - h),x,(x + h)!] = (((cdif (f,h)) . 2) . x) / ((2 * h) * h) ) assume h <> 0 ; ::_thesis: [!f,(x - h),x,(x + h)!] = (((cdif (f,h)) . 2) . x) / ((2 * h) * h) then ( x - h <> x & x - h <> x + h & x <> x + h ) ; then A1: x - h,x,x + h are_mutually_different by ZFMISC_1:def_5; A2: (cdif (f,h)) . 1 is Function of REAL,REAL by DIFF_1:19; [!f,(x - h),x,(x + h)!] = [!f,(x + h),x,(x - h)!] by A1, DIFF_1:34 .= ([!f,x,(x + h)!] - [!f,x,(x - h)!]) / ((x + h) - (x - h)) by DIFF_1:29 .= ([!f,((x + (h / 2)) - (h / 2)),((x + (h / 2)) + (h / 2))!] - [!f,((x - (h / 2)) - (h / 2)),((x - (h / 2)) + (h / 2))!]) / ((x + h) - (x - h)) by DIFF_1:29 .= (((((cdif (f,h)) . 1) . (x + (h / 2))) / h) - [!f,((x - (h / 2)) - (h / 2)),((x - (h / 2)) + (h / 2))!]) / ((x + h) - (x - h)) by Th23 .= (((((cdif (f,h)) . 1) . (x + (h / 2))) / h) - ((((cdif (f,h)) . 1) . (x - (h / 2))) / h)) / ((x + h) - (x - h)) by Th23 .= (((((cdif (f,h)) . 1) . (x + (h / 2))) - (((cdif (f,h)) . 1) . (x - (h / 2)))) / h) / ((x + h) - (x - h)) .= (((cD (((cdif (f,h)) . 1),h)) . x) / h) / (2 * h) by A2, DIFF_1:5 .= ((((cdif (f,h)) . (1 + 1)) . x) / h) / (2 * h) by DIFF_1:def_8 .= (((cdif (f,h)) . 2) . x) / ((2 * h) * h) by XCMPLX_1:78 ; hence [!f,(x - h),x,(x + h)!] = (((cdif (f,h)) . 2) . x) / ((2 * h) * h) ; ::_thesis: verum end; theorem Th25: :: DIFF_3:25 for x0, x1 being Real for f1, f2 being Function of REAL,REAL holds [!(f1 - f2),x0,x1!] = [!f1,x0,x1!] - [!f2,x0,x1!] proof let x0, x1 be Real; ::_thesis: for f1, f2 being Function of REAL,REAL holds [!(f1 - f2),x0,x1!] = [!f1,x0,x1!] - [!f2,x0,x1!] let f1, f2 be Function of REAL,REAL; ::_thesis: [!(f1 - f2),x0,x1!] = [!f1,x0,x1!] - [!f2,x0,x1!] [!(f1 - f2),x0,x1!] = (((f1 . x0) - (f2 . x0)) - ((f1 - f2) . x1)) / (x0 - x1) by VALUED_1:15 .= (((f1 . x0) - (f2 . x0)) - ((f1 . x1) - (f2 . x1))) / (x0 - x1) by VALUED_1:15 .= [!f1,x0,x1!] - [!f2,x0,x1!] ; hence [!(f1 - f2),x0,x1!] = [!f1,x0,x1!] - [!f2,x0,x1!] ; ::_thesis: verum end; theorem :: DIFF_3:26 for r, x0, x1 being Real for f1, f2 being Function of REAL,REAL holds [!((r (#) f1) + f2),x0,x1!] = (r * [!f1,x0,x1!]) + [!f2,x0,x1!] proof let r, x0, x1 be Real; ::_thesis: for f1, f2 being Function of REAL,REAL holds [!((r (#) f1) + f2),x0,x1!] = (r * [!f1,x0,x1!]) + [!f2,x0,x1!] let f1, f2 be Function of REAL,REAL; ::_thesis: [!((r (#) f1) + f2),x0,x1!] = (r * [!f1,x0,x1!]) + [!f2,x0,x1!] set g = r (#) f1; [!((r (#) f1) + f2),x0,x1!] = [!(r (#) f1),x0,x1!] + [!f2,x0,x1!] by DIFF_1:32 .= (r * [!f1,x0,x1!]) + [!f2,x0,x1!] by DIFF_1:31 ; hence [!((r (#) f1) + f2),x0,x1!] = (r * [!f1,x0,x1!]) + [!f2,x0,x1!] ; ::_thesis: verum end; theorem :: DIFF_3:27 for r, x0, x1 being Real for f1, f2 being Function of REAL,REAL holds [!((r (#) f1) - f2),x0,x1!] = (r * [!f1,x0,x1!]) - [!f2,x0,x1!] proof let r, x0, x1 be Real; ::_thesis: for f1, f2 being Function of REAL,REAL holds [!((r (#) f1) - f2),x0,x1!] = (r * [!f1,x0,x1!]) - [!f2,x0,x1!] let f1, f2 be Function of REAL,REAL; ::_thesis: [!((r (#) f1) - f2),x0,x1!] = (r * [!f1,x0,x1!]) - [!f2,x0,x1!] set g = r (#) f1; [!((r (#) f1) - f2),x0,x1!] = [!(r (#) f1),x0,x1!] - [!f2,x0,x1!] by Th25 .= (r * [!f1,x0,x1!]) - [!f2,x0,x1!] by DIFF_1:31 ; hence [!((r (#) f1) - f2),x0,x1!] = (r * [!f1,x0,x1!]) - [!f2,x0,x1!] ; ::_thesis: verum end; theorem :: DIFF_3:28 for r, x0, x1 being Real for f1, f2 being Function of REAL,REAL holds [!(f1 + (r (#) f2)),x0,x1!] = [!f1,x0,x1!] + (r * [!f2,x0,x1!]) proof let r, x0, x1 be Real; ::_thesis: for f1, f2 being Function of REAL,REAL holds [!(f1 + (r (#) f2)),x0,x1!] = [!f1,x0,x1!] + (r * [!f2,x0,x1!]) let f1, f2 be Function of REAL,REAL; ::_thesis: [!(f1 + (r (#) f2)),x0,x1!] = [!f1,x0,x1!] + (r * [!f2,x0,x1!]) set g = r (#) f2; [!(f1 + (r (#) f2)),x0,x1!] = [!f1,x0,x1!] + [!(r (#) f2),x0,x1!] by DIFF_1:32 .= [!f1,x0,x1!] + (r * [!f2,x0,x1!]) by DIFF_1:31 ; hence [!(f1 + (r (#) f2)),x0,x1!] = [!f1,x0,x1!] + (r * [!f2,x0,x1!]) ; ::_thesis: verum end; theorem :: DIFF_3:29 for r, x0, x1 being Real for f1, f2 being Function of REAL,REAL holds [!(f1 - (r (#) f2)),x0,x1!] = [!f1,x0,x1!] - (r * [!f2,x0,x1!]) proof let r, x0, x1 be Real; ::_thesis: for f1, f2 being Function of REAL,REAL holds [!(f1 - (r (#) f2)),x0,x1!] = [!f1,x0,x1!] - (r * [!f2,x0,x1!]) let f1, f2 be Function of REAL,REAL; ::_thesis: [!(f1 - (r (#) f2)),x0,x1!] = [!f1,x0,x1!] - (r * [!f2,x0,x1!]) set g = r (#) f2; [!(f1 - (r (#) f2)),x0,x1!] = [!f1,x0,x1!] - [!(r (#) f2),x0,x1!] by Th25 .= [!f1,x0,x1!] - (r * [!f2,x0,x1!]) by DIFF_1:31 ; hence [!(f1 - (r (#) f2)),x0,x1!] = [!f1,x0,x1!] - (r * [!f2,x0,x1!]) ; ::_thesis: verum end; theorem :: DIFF_3:30 for r1, r2, x0, x1 being Real for f1, f2 being Function of REAL,REAL holds [!((r1 (#) f1) - (r2 (#) f2)),x0,x1!] = (r1 * [!f1,x0,x1!]) - (r2 * [!f2,x0,x1!]) proof let r1, r2, x0, x1 be Real; ::_thesis: for f1, f2 being Function of REAL,REAL holds [!((r1 (#) f1) - (r2 (#) f2)),x0,x1!] = (r1 * [!f1,x0,x1!]) - (r2 * [!f2,x0,x1!]) let f1, f2 be Function of REAL,REAL; ::_thesis: [!((r1 (#) f1) - (r2 (#) f2)),x0,x1!] = (r1 * [!f1,x0,x1!]) - (r2 * [!f2,x0,x1!]) set g1 = r1 (#) f1; set g2 = r2 (#) f2; [!((r1 (#) f1) - (r2 (#) f2)),x0,x1!] = [!(r1 (#) f1),x0,x1!] - [!(r2 (#) f2),x0,x1!] by Th25 .= (r1 * [!f1,x0,x1!]) - [!(r2 (#) f2),x0,x1!] by DIFF_1:31 .= (r1 * [!f1,x0,x1!]) - (r2 * [!f2,x0,x1!]) by DIFF_1:31 ; hence [!((r1 (#) f1) - (r2 (#) f2)),x0,x1!] = (r1 * [!f1,x0,x1!]) - (r2 * [!f2,x0,x1!]) ; ::_thesis: verum end; theorem Th31: :: DIFF_3:31 for h, x being Real for f1, f2 being Function of REAL,REAL holds ((bdif ((f1 (#) f2),h)) . 1) . x = ((f1 . x) * (((bdif (f2,h)) . 1) . x)) + ((f2 . (x - h)) * (((bdif (f1,h)) . 1) . x)) proof let h, x be Real; ::_thesis: for f1, f2 being Function of REAL,REAL holds ((bdif ((f1 (#) f2),h)) . 1) . x = ((f1 . x) * (((bdif (f2,h)) . 1) . x)) + ((f2 . (x - h)) * (((bdif (f1,h)) . 1) . x)) let f1, f2 be Function of REAL,REAL; ::_thesis: ((bdif ((f1 (#) f2),h)) . 1) . x = ((f1 . x) * (((bdif (f2,h)) . 1) . x)) + ((f2 . (x - h)) * (((bdif (f1,h)) . 1) . x)) ((bdif ((f1 (#) f2),h)) . 1) . x = ((bdif ((f1 (#) f2),h)) . (0 + 1)) . x .= (bD (((bdif ((f1 (#) f2),h)) . 0),h)) . x by DIFF_1:def_7 .= (bD ((f1 (#) f2),h)) . x by DIFF_1:def_7 .= ((f1 (#) f2) . x) - ((f1 (#) f2) . (x - h)) by DIFF_1:4 .= ((f1 . x) * (f2 . x)) - ((f1 (#) f2) . (x - h)) by VALUED_1:5 .= ((f1 . x) * (f2 . x)) - ((f1 . (x - h)) * (f2 . (x - h))) by VALUED_1:5 .= ((f1 . x) * ((f2 . x) - (f2 . (x - h)))) + ((f2 . (x - h)) * ((f1 . x) - (f1 . (x - h)))) .= ((f1 . x) * ((bD (f2,h)) . x)) + ((f2 . (x - h)) * ((f1 . x) - (f1 . (x - h)))) by DIFF_1:4 .= ((f1 . x) * ((bD (f2,h)) . x)) + ((f2 . (x - h)) * ((bD (f1,h)) . x)) by DIFF_1:4 .= ((f1 . x) * ((bD (((bdif (f2,h)) . 0),h)) . x)) + ((f2 . (x - h)) * ((bD (f1,h)) . x)) by DIFF_1:def_7 .= ((f1 . x) * ((bD (((bdif (f2,h)) . 0),h)) . x)) + ((f2 . (x - h)) * ((bD (((bdif (f1,h)) . 0),h)) . x)) by DIFF_1:def_7 .= ((f1 . x) * (((bdif (f2,h)) . (0 + 1)) . x)) + ((f2 . (x - h)) * ((bD (((bdif (f1,h)) . 0),h)) . x)) by DIFF_1:def_7 .= ((f1 . x) * (((bdif (f2,h)) . 1) . x)) + ((f2 . (x - h)) * (((bdif (f1,h)) . 1) . x)) by DIFF_1:def_7 ; hence ((bdif ((f1 (#) f2),h)) . 1) . x = ((f1 . x) * (((bdif (f2,h)) . 1) . x)) + ((f2 . (x - h)) * (((bdif (f1,h)) . 1) . x)) ; ::_thesis: verum end; theorem :: DIFF_3:32 for x0, x1, x2 being Real for f being Function of REAL,REAL st x0,x1,x2 are_mutually_different holds [!f,x0,x1,x2!] = [!f,x0,x2,x1!] proof let x0, x1, x2 be Real; ::_thesis: for f being Function of REAL,REAL st x0,x1,x2 are_mutually_different holds [!f,x0,x1,x2!] = [!f,x0,x2,x1!] let f be Function of REAL,REAL; ::_thesis: ( x0,x1,x2 are_mutually_different implies [!f,x0,x1,x2!] = [!f,x0,x2,x1!] ) assume x0,x1,x2 are_mutually_different ; ::_thesis: [!f,x0,x1,x2!] = [!f,x0,x2,x1!] then A1: ( x2 - x1 <> 0 & x2 - x0 <> 0 & x1 - x0 <> 0 ) by ZFMISC_1:def_5; set x10 = x1 - x0; set x20 = x2 - x0; set x21 = x2 - x1; A2: [!f,x0,x2,x1!] = ((((f . x0) - (f . x2)) / (- (x2 - x0))) - (((f . x2) - (f . x1)) / (x2 - x1))) / (- (x1 - x0)) .= ((- (((f . x0) - (f . x2)) / (x2 - x0))) - (((f . x2) - (f . x1)) / (x2 - x1))) / (- (x1 - x0)) by XCMPLX_1:188 .= (- ((((f . x0) - (f . x2)) / (x2 - x0)) + (((f . x2) - (f . x1)) / (x2 - x1)))) / (- (x1 - x0)) .= ((((f . x0) - (f . x2)) / (x2 - x0)) + (((f . x2) - (f . x1)) / (x2 - x1))) / (x1 - x0) by XCMPLX_1:191 .= (((((f . x0) - (f . x2)) * (x2 - x1)) + (((f . x2) - (f . x1)) * (x2 - x0))) / ((x2 - x0) * (x2 - x1))) / (x1 - x0) by A1, XCMPLX_1:116 .= ((((f . x0) * (x2 - x1)) - ((f . x1) * (x2 - x0))) + ((f . x2) * (x1 - x0))) / (((x2 - x0) * (x2 - x1)) * (x1 - x0)) by XCMPLX_1:78 ; [!f,x0,x1,x2!] = ((((f . x0) - (f . x1)) / (- (x1 - x0))) - (((f . x1) - (f . x2)) / (- (x2 - x1)))) / (- (x2 - x0)) .= ((- (((f . x0) - (f . x1)) / (x1 - x0))) - (((f . x1) - (f . x2)) / (- (x2 - x1)))) / (- (x2 - x0)) by XCMPLX_1:188 .= ((- (((f . x0) - (f . x1)) / (x1 - x0))) - (- (((f . x1) - (f . x2)) / (x2 - x1)))) / (- (x2 - x0)) by XCMPLX_1:188 .= (- ((((f . x0) - (f . x1)) / (x1 - x0)) - (((f . x1) - (f . x2)) / (x2 - x1)))) / (- (x2 - x0)) .= ((((f . x0) - (f . x1)) / (x1 - x0)) - (((f . x1) - (f . x2)) / (x2 - x1))) / (x2 - x0) by XCMPLX_1:191 .= (((((f . x0) - (f . x1)) * (x2 - x1)) - (((f . x1) - (f . x2)) * (x1 - x0))) / ((x1 - x0) * (x2 - x1))) / (x2 - x0) by A1, XCMPLX_1:130 .= ((((f . x0) * (x2 - x1)) - ((f . x1) * (x2 - x0))) + ((f . x2) * (x1 - x0))) / (((x1 - x0) * (x2 - x1)) * (x2 - x0)) by XCMPLX_1:78 .= [!f,x0,x2,x1!] by A2 ; hence [!f,x0,x1,x2!] = [!f,x0,x2,x1!] ; ::_thesis: verum end; theorem :: DIFF_3:33 for h, x being Real for f1, f2 being Function of REAL,REAL for S being Seq_Sequence st ( for n, i being Nat st i <= n holds (S . n) . i = ((n choose i) * (((bdif (f1,h)) . i) . x)) * (((bdif (f2,h)) . (n -' i)) . (x - (i * h))) ) holds ( ((bdif ((f1 (#) f2),h)) . 1) . x = Sum ((S . 1),1) & ((bdif ((f1 (#) f2),h)) . 2) . x = Sum ((S . 2),2) ) proof let h, x be Real; ::_thesis: for f1, f2 being Function of REAL,REAL for S being Seq_Sequence st ( for n, i being Nat st i <= n holds (S . n) . i = ((n choose i) * (((bdif (f1,h)) . i) . x)) * (((bdif (f2,h)) . (n -' i)) . (x - (i * h))) ) holds ( ((bdif ((f1 (#) f2),h)) . 1) . x = Sum ((S . 1),1) & ((bdif ((f1 (#) f2),h)) . 2) . x = Sum ((S . 2),2) ) let f1, f2 be Function of REAL,REAL; ::_thesis: for S being Seq_Sequence st ( for n, i being Nat st i <= n holds (S . n) . i = ((n choose i) * (((bdif (f1,h)) . i) . x)) * (((bdif (f2,h)) . (n -' i)) . (x - (i * h))) ) holds ( ((bdif ((f1 (#) f2),h)) . 1) . x = Sum ((S . 1),1) & ((bdif ((f1 (#) f2),h)) . 2) . x = Sum ((S . 2),2) ) let S be Seq_Sequence; ::_thesis: ( ( for n, i being Nat st i <= n holds (S . n) . i = ((n choose i) * (((bdif (f1,h)) . i) . x)) * (((bdif (f2,h)) . (n -' i)) . (x - (i * h))) ) implies ( ((bdif ((f1 (#) f2),h)) . 1) . x = Sum ((S . 1),1) & ((bdif ((f1 (#) f2),h)) . 2) . x = Sum ((S . 2),2) ) ) assume A1: for n, i being Nat st i <= n holds (S . n) . i = ((n choose i) * (((bdif (f1,h)) . i) . x)) * (((bdif (f2,h)) . (n -' i)) . (x - (i * h))) ; ::_thesis: ( ((bdif ((f1 (#) f2),h)) . 1) . x = Sum ((S . 1),1) & ((bdif ((f1 (#) f2),h)) . 2) . x = Sum ((S . 2),2) ) A2: 1 -' 0 = 1 - 0 by XREAL_1:233 .= 1 ; A3: (S . 1) . 0 = ((1 choose 0) * (((bdif (f1,h)) . 0) . x)) * (((bdif (f2,h)) . (1 -' 0)) . (x - (0 * h))) by A1 .= (1 * (((bdif (f1,h)) . 0) . x)) * (((bdif (f2,h)) . (1 -' 0)) . (x - (0 * h))) by NEWTON:19 .= (f1 . x) * (((bdif (f2,h)) . 1) . x) by A2, DIFF_1:def_7 ; A4: 1 -' 1 = 1 - 1 by XREAL_1:233 .= 0 ; A5: (S . 1) . 1 = ((1 choose 1) * (((bdif (f1,h)) . 1) . x)) * (((bdif (f2,h)) . (1 -' 1)) . (x - (1 * h))) by A1 .= (1 * (((bdif (f1,h)) . 1) . x)) * (((bdif (f2,h)) . (1 -' 1)) . (x - (1 * h))) by NEWTON:21 .= (f2 . (x - h)) * (((bdif (f1,h)) . 1) . x) by A4, DIFF_1:def_7 ; A6: Sum ((S . 1),1) = (Partial_Sums (S . 1)) . (0 + 1) by SERIES_1:def_5 .= ((Partial_Sums (S . 1)) . 0) + ((S . 1) . 1) by SERIES_1:def_1 .= ((S . 1) . 0) + ((S . 1) . 1) by SERIES_1:def_1 .= ((bdif ((f1 (#) f2),h)) . 1) . x by A3, A5, Th31 ; A7: (bdif ((f1 (#) f2),h)) . 1 is Function of REAL,REAL by DIFF_1:12; A8: (bdif (f2,h)) . 1 is Function of REAL,REAL by DIFF_1:12; A9: (bdif (f1,h)) . 1 is Function of REAL,REAL by DIFF_1:12; A10: ((bdif ((f1 (#) f2),h)) . 2) . x = ((bdif ((f1 (#) f2),h)) . (1 + 1)) . x .= (bD (((bdif ((f1 (#) f2),h)) . 1),h)) . x by DIFF_1:def_7 .= (((bdif ((f1 (#) f2),h)) . 1) . x) - (((bdif ((f1 (#) f2),h)) . 1) . (x - h)) by A7, DIFF_1:4 .= (((f1 . x) * (((bdif (f2,h)) . 1) . x)) + ((((bdif (f1,h)) . 1) . x) * (f2 . (x - h)))) - (((bdif ((f1 (#) f2),h)) . 1) . (x - h)) by Th31 .= (((f1 . x) * (((bdif (f2,h)) . 1) . x)) + ((((bdif (f1,h)) . 1) . x) * (f2 . (x - h)))) - (((f1 . (x - h)) * (((bdif (f2,h)) . 1) . (x - h))) + ((((bdif (f1,h)) . 1) . (x - h)) * (f2 . ((x - h) - h)))) by Th31 .= ((((f1 . x) * ((((bdif (f2,h)) . 1) . x) - (((bdif (f2,h)) . 1) . (x - h)))) + (((f1 . x) - (f1 . (x - h))) * (((bdif (f2,h)) . 1) . (x - h)))) - (((((bdif (f1,h)) . 1) . (x - h)) - (((bdif (f1,h)) . 1) . x)) * (f2 . (x - (2 * h))))) - ((((bdif (f1,h)) . 1) . x) * ((f2 . (x - (2 * h))) - (f2 . (x - h)))) .= ((((f1 . x) * ((bD (((bdif (f2,h)) . 1),h)) . x)) + (((f1 . x) - (f1 . (x - h))) * (((bdif (f2,h)) . 1) . (x - h)))) - (((((bdif (f1,h)) . 1) . (x - h)) - (((bdif (f1,h)) . 1) . x)) * (f2 . (x - (2 * h))))) - ((((bdif (f1,h)) . 1) . x) * ((f2 . (x - (2 * h))) - (f2 . (x - h)))) by A8, DIFF_1:4 .= ((((f1 . x) * ((bD (((bdif (f2,h)) . 1),h)) . x)) + (((bD (f1,h)) . x) * (((bdif (f2,h)) . 1) . (x - h)))) + (((((bdif (f1,h)) . 1) . x) - (((bdif (f1,h)) . 1) . (x - h))) * (f2 . (x - (2 * h))))) - ((((bdif (f1,h)) . 1) . x) * ((f2 . (x - (2 * h))) - (f2 . (x - h)))) by DIFF_1:4 .= ((((f1 . x) * ((bD (((bdif (f2,h)) . 1),h)) . x)) + (((bD (f1,h)) . x) * (((bdif (f2,h)) . 1) . (x - h)))) + (((bD (((bdif (f1,h)) . 1),h)) . x) * (f2 . (x - (2 * h))))) + ((((bdif (f1,h)) . 1) . x) * ((f2 . (x - h)) - (f2 . ((x - h) - h)))) by A9, DIFF_1:4 .= ((((f1 . x) * ((bD (((bdif (f2,h)) . 1),h)) . x)) + (((bD (f1,h)) . x) * (((bdif (f2,h)) . 1) . (x - h)))) + (((bD (((bdif (f1,h)) . 1),h)) . x) * (f2 . (x - (2 * h))))) + ((((bdif (f1,h)) . 1) . x) * ((bD (f2,h)) . (x - h))) by DIFF_1:4 .= ((((f1 . x) * (((bdif (f2,h)) . (1 + 1)) . x)) + (((bD (f1,h)) . x) * (((bdif (f2,h)) . 1) . (x - h)))) + (((bD (((bdif (f1,h)) . 1),h)) . x) * (f2 . (x - (2 * h))))) + ((((bdif (f1,h)) . 1) . x) * ((bD (f2,h)) . (x - h))) by DIFF_1:def_7 .= ((((f1 . x) * (((bdif (f2,h)) . (1 + 1)) . x)) + (((bD (((bdif (f1,h)) . 0),h)) . x) * (((bdif (f2,h)) . 1) . (x - h)))) + (((bD (((bdif (f1,h)) . 1),h)) . x) * (f2 . (x - (2 * h))))) + ((((bdif (f1,h)) . 1) . x) * ((bD (f2,h)) . (x - h))) by DIFF_1:def_7 .= ((((f1 . x) * (((bdif (f2,h)) . 2) . x)) + (((bD (((bdif (f1,h)) . 0),h)) . x) * (((bdif (f2,h)) . 1) . (x - h)))) + ((((bdif (f1,h)) . 2) . x) * (f2 . (x - (2 * h))))) + ((((bdif (f1,h)) . 1) . x) * ((bD (f2,h)) . (x - h))) by DIFF_1:def_7 .= ((((f1 . x) * (((bdif (f2,h)) . 2) . x)) + ((((bdif (f1,h)) . (0 + 1)) . x) * (((bdif (f2,h)) . 1) . (x - h)))) + ((((bdif (f1,h)) . 2) . x) * (f2 . (x - (2 * h))))) + ((((bdif (f1,h)) . 1) . x) * ((bD (f2,h)) . (x - h))) by DIFF_1:def_7 .= ((((f1 . x) * (((bdif (f2,h)) . 2) . x)) + ((((bdif (f1,h)) . 1) . x) * (((bdif (f2,h)) . 1) . (x - h)))) + ((((bdif (f1,h)) . 2) . x) * (f2 . (x - (2 * h))))) + ((((bdif (f1,h)) . 1) . x) * ((bD (((bdif (f2,h)) . 0),h)) . (x - h))) by DIFF_1:def_7 .= ((((f1 . x) * (((bdif (f2,h)) . 2) . x)) + ((((bdif (f1,h)) . 1) . x) * (((bdif (f2,h)) . 1) . (x - h)))) + ((((bdif (f1,h)) . 2) . x) * (f2 . (x - (2 * h))))) + ((((bdif (f1,h)) . 1) . x) * (((bdif (f2,h)) . (0 + 1)) . (x - h))) by DIFF_1:def_7 .= (((f1 . x) * (((bdif (f2,h)) . 2) . x)) + (2 * ((((bdif (f1,h)) . 1) . x) * (((bdif (f2,h)) . 1) . (x - h))))) + ((((bdif (f1,h)) . 2) . x) * (f2 . (x - (2 * h)))) ; A11: 2 -' 0 = 2 - 0 by XREAL_1:233 .= 2 ; A12: (S . 2) . 0 = ((2 choose 0) * (((bdif (f1,h)) . 0) . x)) * (((bdif (f2,h)) . (2 -' 0)) . (x - (0 * h))) by A1 .= (1 * (((bdif (f1,h)) . 0) . x)) * (((bdif (f2,h)) . (2 -' 0)) . (x - (0 * h))) by NEWTON:19 .= (f1 . x) * (((bdif (f2,h)) . 2) . x) by A11, DIFF_1:def_7 ; A13: 2 -' 1 = 2 - 1 by XREAL_1:233 .= 1 ; A14: (S . 2) . 1 = ((2 choose 1) * (((bdif (f1,h)) . 1) . x)) * (((bdif (f2,h)) . (2 -' 1)) . (x - (1 * h))) by A1 .= (2 * (((bdif (f1,h)) . 1) . x)) * (((bdif (f2,h)) . 1) . (x - h)) by A13, NEWTON:23 ; A15: 2 -' 2 = 2 - 2 by XREAL_1:233 .= 0 ; A16: (S . 2) . 2 = ((2 choose 2) * (((bdif (f1,h)) . 2) . x)) * (((bdif (f2,h)) . (2 -' 2)) . (x - (2 * h))) by A1 .= (1 * (((bdif (f1,h)) . 2) . x)) * (((bdif (f2,h)) . (2 -' 2)) . (x - (2 * h))) by NEWTON:21 .= (((bdif (f1,h)) . 2) . x) * (f2 . (x - (2 * h))) by A15, DIFF_1:def_7 ; Sum ((S . 2),2) = (Partial_Sums (S . 2)) . (1 + 1) by SERIES_1:def_5 .= ((Partial_Sums (S . 2)) . (0 + 1)) + ((S . 2) . 2) by SERIES_1:def_1 .= (((Partial_Sums (S . 2)) . 0) + ((S . 2) . 1)) + ((S . 2) . 2) by SERIES_1:def_1 .= ((bdif ((f1 (#) f2),h)) . 2) . x by A10, A12, A14, A16, SERIES_1:def_1 ; hence ( ((bdif ((f1 (#) f2),h)) . 1) . x = Sum ((S . 1),1) & ((bdif ((f1 (#) f2),h)) . 2) . x = Sum ((S . 2),2) ) by A6; ::_thesis: verum end; theorem Th34: :: DIFF_3:34 for h, x being Real for f1, f2 being Function of REAL,REAL holds ((cdif ((f1 (#) f2),h)) . 1) . x = ((f1 . (x + (h / 2))) * (((cdif (f2,h)) . 1) . x)) + ((f2 . (x - (h / 2))) * (((cdif (f1,h)) . 1) . x)) proof let h, x be Real; ::_thesis: for f1, f2 being Function of REAL,REAL holds ((cdif ((f1 (#) f2),h)) . 1) . x = ((f1 . (x + (h / 2))) * (((cdif (f2,h)) . 1) . x)) + ((f2 . (x - (h / 2))) * (((cdif (f1,h)) . 1) . x)) let f1, f2 be Function of REAL,REAL; ::_thesis: ((cdif ((f1 (#) f2),h)) . 1) . x = ((f1 . (x + (h / 2))) * (((cdif (f2,h)) . 1) . x)) + ((f2 . (x - (h / 2))) * (((cdif (f1,h)) . 1) . x)) ((cdif ((f1 (#) f2),h)) . 1) . x = ((cdif ((f1 (#) f2),h)) . (0 + 1)) . x .= (cD (((cdif ((f1 (#) f2),h)) . 0),h)) . x by DIFF_1:def_8 .= (cD ((f1 (#) f2),h)) . x by DIFF_1:def_8 .= ((f1 (#) f2) . (x + (h / 2))) - ((f1 (#) f2) . (x - (h / 2))) by DIFF_1:5 .= ((f1 . (x + (h / 2))) * (f2 . (x + (h / 2)))) - ((f1 (#) f2) . (x - (h / 2))) by VALUED_1:5 .= ((f1 . (x + (h / 2))) * (f2 . (x + (h / 2)))) - ((f1 . (x - (h / 2))) * (f2 . (x - (h / 2)))) by VALUED_1:5 .= ((f1 . (x + (h / 2))) * ((f2 . (x + (h / 2))) - (f2 . (x - (h / 2))))) + ((f2 . (x - (h / 2))) * ((f1 . (x + (h / 2))) - (f1 . (x - (h / 2))))) .= ((f1 . (x + (h / 2))) * ((cD (f2,h)) . x)) + ((f2 . (x - (h / 2))) * ((f1 . (x + (h / 2))) - (f1 . (x - (h / 2))))) by DIFF_1:5 .= ((f1 . (x + (h / 2))) * ((cD (f2,h)) . x)) + ((f2 . (x - (h / 2))) * ((cD (f1,h)) . x)) by DIFF_1:5 .= ((f1 . (x + (h / 2))) * ((cD (((cdif (f2,h)) . 0),h)) . x)) + ((f2 . (x - (h / 2))) * ((cD (f1,h)) . x)) by DIFF_1:def_8 .= ((f1 . (x + (h / 2))) * ((cD (((cdif (f2,h)) . 0),h)) . x)) + ((f2 . (x - (h / 2))) * ((cD (((cdif (f1,h)) . 0),h)) . x)) by DIFF_1:def_8 .= ((f1 . (x + (h / 2))) * (((cdif (f2,h)) . (0 + 1)) . x)) + ((f2 . (x - (h / 2))) * ((cD (((cdif (f1,h)) . 0),h)) . x)) by DIFF_1:def_8 .= ((f1 . (x + (h / 2))) * (((cdif (f2,h)) . 1) . x)) + ((f2 . (x - (h / 2))) * (((cdif (f1,h)) . 1) . x)) by DIFF_1:def_8 ; hence ((cdif ((f1 (#) f2),h)) . 1) . x = ((f1 . (x + (h / 2))) * (((cdif (f2,h)) . 1) . x)) + ((f2 . (x - (h / 2))) * (((cdif (f1,h)) . 1) . x)) ; ::_thesis: verum end; theorem :: DIFF_3:35 for h, x being Real for f1, f2 being Function of REAL,REAL for S being Seq_Sequence st ( for n, i being Nat st i <= n holds (S . n) . i = ((n choose i) * (((cdif (f1,h)) . i) . (x + ((n -' i) * (h / 2))))) * (((cdif (f2,h)) . (n -' i)) . (x - (i * (h / 2)))) ) holds ( ((cdif ((f1 (#) f2),h)) . 1) . x = Sum ((S . 1),1) & ((cdif ((f1 (#) f2),h)) . 2) . x = Sum ((S . 2),2) ) proof let h, x be Real; ::_thesis: for f1, f2 being Function of REAL,REAL for S being Seq_Sequence st ( for n, i being Nat st i <= n holds (S . n) . i = ((n choose i) * (((cdif (f1,h)) . i) . (x + ((n -' i) * (h / 2))))) * (((cdif (f2,h)) . (n -' i)) . (x - (i * (h / 2)))) ) holds ( ((cdif ((f1 (#) f2),h)) . 1) . x = Sum ((S . 1),1) & ((cdif ((f1 (#) f2),h)) . 2) . x = Sum ((S . 2),2) ) let f1, f2 be Function of REAL,REAL; ::_thesis: for S being Seq_Sequence st ( for n, i being Nat st i <= n holds (S . n) . i = ((n choose i) * (((cdif (f1,h)) . i) . (x + ((n -' i) * (h / 2))))) * (((cdif (f2,h)) . (n -' i)) . (x - (i * (h / 2)))) ) holds ( ((cdif ((f1 (#) f2),h)) . 1) . x = Sum ((S . 1),1) & ((cdif ((f1 (#) f2),h)) . 2) . x = Sum ((S . 2),2) ) let S be Seq_Sequence; ::_thesis: ( ( for n, i being Nat st i <= n holds (S . n) . i = ((n choose i) * (((cdif (f1,h)) . i) . (x + ((n -' i) * (h / 2))))) * (((cdif (f2,h)) . (n -' i)) . (x - (i * (h / 2)))) ) implies ( ((cdif ((f1 (#) f2),h)) . 1) . x = Sum ((S . 1),1) & ((cdif ((f1 (#) f2),h)) . 2) . x = Sum ((S . 2),2) ) ) assume A1: for n, i being Nat st i <= n holds (S . n) . i = ((n choose i) * (((cdif (f1,h)) . i) . (x + ((n -' i) * (h / 2))))) * (((cdif (f2,h)) . (n -' i)) . (x - (i * (h / 2)))) ; ::_thesis: ( ((cdif ((f1 (#) f2),h)) . 1) . x = Sum ((S . 1),1) & ((cdif ((f1 (#) f2),h)) . 2) . x = Sum ((S . 2),2) ) A2: 1 -' 0 = 1 - 0 by XREAL_1:233 .= 1 ; A3: (S . 1) . 0 = ((1 choose 0) * (((cdif (f1,h)) . 0) . (x + ((1 -' 0) * (h / 2))))) * (((cdif (f2,h)) . (1 -' 0)) . (x - (0 * (h / 2)))) by A1 .= (1 * (((cdif (f1,h)) . 0) . (x + ((1 -' 0) * (h / 2))))) * (((cdif (f2,h)) . (1 -' 0)) . (x - (0 * (h / 2)))) by NEWTON:19 .= (f1 . (x + (h / 2))) * (((cdif (f2,h)) . 1) . x) by A2, DIFF_1:def_8 ; A4: 1 -' 1 = 1 - 1 by XREAL_1:233 .= 0 ; A5: (S . 1) . 1 = ((1 choose 1) * (((cdif (f1,h)) . 1) . (x + ((1 -' 1) * (h / 2))))) * (((cdif (f2,h)) . (1 -' 1)) . (x - (1 * (h / 2)))) by A1 .= (1 * (((cdif (f1,h)) . 1) . (x + ((1 -' 1) * (h / 2))))) * (((cdif (f2,h)) . (1 -' 1)) . (x - (1 * (h / 2)))) by NEWTON:21 .= (f2 . (x - (h / 2))) * (((cdif (f1,h)) . 1) . x) by A4, DIFF_1:def_8 ; A6: Sum ((S . 1),1) = (Partial_Sums (S . 1)) . (0 + 1) by SERIES_1:def_5 .= ((Partial_Sums (S . 1)) . 0) + ((S . 1) . 1) by SERIES_1:def_1 .= ((S . 1) . 0) + ((S . 1) . 1) by SERIES_1:def_1 .= ((cdif ((f1 (#) f2),h)) . 1) . x by A3, A5, Th34 ; A7: (cdif ((f1 (#) f2),h)) . 1 is Function of REAL,REAL by DIFF_1:19; A8: (cdif (f1,h)) . 1 is Function of REAL,REAL by DIFF_1:19; A9: (cdif (f2,h)) . 1 is Function of REAL,REAL by DIFF_1:19; A10: ((cdif ((f1 (#) f2),h)) . 2) . x = ((cdif ((f1 (#) f2),h)) . (1 + 1)) . x .= (cD (((cdif ((f1 (#) f2),h)) . 1),h)) . x by DIFF_1:def_8 .= (((cdif ((f1 (#) f2),h)) . 1) . (x + (h / 2))) - (((cdif ((f1 (#) f2),h)) . 1) . (x - (h / 2))) by A7, DIFF_1:5 .= (((f1 . ((x + (h / 2)) + (h / 2))) * (((cdif (f2,h)) . 1) . (x + (h / 2)))) + ((((cdif (f1,h)) . 1) . (x + (h / 2))) * (f2 . ((x + (h / 2)) - (h / 2))))) - (((cdif ((f1 (#) f2),h)) . 1) . (x - (h / 2))) by Th34 .= (((f1 . (x + h)) * (((cdif (f2,h)) . 1) . (x + (h / 2)))) + ((((cdif (f1,h)) . 1) . (x + (h / 2))) * (f2 . x))) - (((f1 . ((x - (h / 2)) + (h / 2))) * (((cdif (f2,h)) . 1) . (x - (h / 2)))) + ((((cdif (f1,h)) . 1) . (x - (h / 2))) * (f2 . ((x - (h / 2)) - (h / 2))))) by Th34 .= ((((f1 . (x + h)) * ((((cdif (f2,h)) . 1) . (x + (h / 2))) - (((cdif (f2,h)) . 1) . (x - (h / 2))))) + (((f1 . (x + h)) - (f1 . x)) * (((cdif (f2,h)) . 1) . (x - (h / 2))))) - (((((cdif (f1,h)) . 1) . (x - (h / 2))) - (((cdif (f1,h)) . 1) . (x + (h / 2)))) * (f2 . (x - h)))) - ((((cdif (f1,h)) . 1) . (x + (h / 2))) * ((f2 . (x - h)) - (f2 . x))) .= ((((f1 . (x + h)) * ((cD (((cdif (f2,h)) . 1),h)) . x)) + (((f1 . ((x + (h / 2)) + (h / 2))) - (f1 . ((x + (h / 2)) - (h / 2)))) * (((cdif (f2,h)) . 1) . (x - (h / 2))))) - (((((cdif (f1,h)) . 1) . (x - (h / 2))) - (((cdif (f1,h)) . 1) . (x + (h / 2)))) * (f2 . (x - h)))) - ((((cdif (f1,h)) . 1) . (x + (h / 2))) * ((f2 . (x - h)) - (f2 . x))) by A9, DIFF_1:5 .= ((((f1 . (x + h)) * ((cD (((cdif (f2,h)) . 1),h)) . x)) + (((cD (f1,h)) . (x + (h / 2))) * (((cdif (f2,h)) . 1) . (x - (h / 2))))) + (((((cdif (f1,h)) . 1) . (x + (h / 2))) - (((cdif (f1,h)) . 1) . (x - (h / 2)))) * (f2 . (x - h)))) - ((((cdif (f1,h)) . 1) . (x + (h / 2))) * ((f2 . (x - h)) - (f2 . x))) by DIFF_1:5 .= ((((f1 . (x + h)) * ((cD (((cdif (f2,h)) . 1),h)) . x)) + (((cD (f1,h)) . (x + (h / 2))) * (((cdif (f2,h)) . 1) . (x - (h / 2))))) + (((cD (((cdif (f1,h)) . 1),h)) . x) * (f2 . (x - h)))) + ((((cdif (f1,h)) . 1) . (x + (h / 2))) * ((f2 . ((x - (h / 2)) + (h / 2))) - (f2 . ((x - (h / 2)) - (h / 2))))) by A8, DIFF_1:5 .= ((((f1 . (x + h)) * ((cD (((cdif (f2,h)) . 1),h)) . x)) + (((cD (f1,h)) . (x + (h / 2))) * (((cdif (f2,h)) . 1) . (x - (h / 2))))) + (((cD (((cdif (f1,h)) . 1),h)) . x) * (f2 . (x - h)))) + ((((cdif (f1,h)) . 1) . (x + (h / 2))) * ((cD (f2,h)) . (x - (h / 2)))) by DIFF_1:5 .= ((((f1 . (x + h)) * (((cdif (f2,h)) . (1 + 1)) . x)) + (((cD (f1,h)) . (x + (h / 2))) * (((cdif (f2,h)) . 1) . (x - (h / 2))))) + (((cD (((cdif (f1,h)) . 1),h)) . x) * (f2 . (x - h)))) + ((((cdif (f1,h)) . 1) . (x + (h / 2))) * ((cD (f2,h)) . (x - (h / 2)))) by DIFF_1:def_8 .= ((((f1 . (x + h)) * (((cdif (f2,h)) . 2) . x)) + ((((cdif (f1,h)) . 1) . (x + (h / 2))) * (((cdif (f2,h)) . 1) . (x - (h / 2))))) + (((cD (((cdif (f1,h)) . 1),h)) . x) * (f2 . (x - h)))) + ((((cdif (f1,h)) . 1) . (x + (h / 2))) * ((cD (f2,h)) . (x - (h / 2)))) by Th16 .= ((((f1 . (x + h)) * (((cdif (f2,h)) . 2) . x)) + ((((cdif (f1,h)) . 1) . (x + (h / 2))) * (((cdif (f2,h)) . 1) . (x - (h / 2))))) + ((((cdif (f1,h)) . (1 + 1)) . x) * (f2 . (x - h)))) + ((((cdif (f1,h)) . 1) . (x + (h / 2))) * ((cD (f2,h)) . (x - (h / 2)))) by DIFF_1:def_8 .= ((((f1 . (x + h)) * (((cdif (f2,h)) . 2) . x)) + ((((cdif (f1,h)) . 1) . (x + (h / 2))) * (((cdif (f2,h)) . 1) . (x - (h / 2))))) + ((((cdif (f1,h)) . 2) . x) * (f2 . (x - h)))) + ((((cdif (f1,h)) . 1) . (x + (h / 2))) * (((cdif (f2,h)) . 1) . (x - (h / 2)))) by Th16 .= (((f1 . (x + h)) * (((cdif (f2,h)) . 2) . x)) + ((2 * (((cdif (f1,h)) . 1) . (x + (h / 2)))) * (((cdif (f2,h)) . 1) . (x - (h / 2))))) + ((((cdif (f1,h)) . 2) . x) * (f2 . (x - h))) ; A11: 2 -' 0 = 2 - 0 by XREAL_1:233 .= 2 ; A12: (S . 2) . 0 = ((2 choose 0) * (((cdif (f1,h)) . 0) . (x + ((2 -' 0) * (h / 2))))) * (((cdif (f2,h)) . (2 -' 0)) . (x - (0 * (h / 2)))) by A1 .= (1 * (((cdif (f1,h)) . 0) . (x + ((2 -' 0) * (h / 2))))) * (((cdif (f2,h)) . (2 -' 0)) . (x - (0 * (h / 2)))) by NEWTON:19 .= (f1 . (x + h)) * (((cdif (f2,h)) . 2) . x) by A11, DIFF_1:def_8 ; A13: 2 -' 1 = 2 - 1 by XREAL_1:233 .= 1 ; A14: (S . 2) . 1 = ((2 choose 1) * (((cdif (f1,h)) . 1) . (x + ((2 -' 1) * (h / 2))))) * (((cdif (f2,h)) . (2 -' 1)) . (x - (1 * (h / 2)))) by A1 .= (2 * (((cdif (f1,h)) . 1) . (x + (h / 2)))) * (((cdif (f2,h)) . 1) . (x - (h / 2))) by A13, NEWTON:23 ; A15: 2 -' 2 = 2 - 2 by XREAL_1:233 .= 0 ; A16: (S . 2) . 2 = ((2 choose 2) * (((cdif (f1,h)) . 2) . (x + ((2 -' 2) * (h / 2))))) * (((cdif (f2,h)) . (2 -' 2)) . (x - (2 * (h / 2)))) by A1 .= (1 * (((cdif (f1,h)) . 2) . (x + ((2 -' 2) * (h / 2))))) * (((cdif (f2,h)) . (2 -' 2)) . (x - (2 * (h / 2)))) by NEWTON:21 .= (((cdif (f1,h)) . 2) . x) * (f2 . (x - h)) by A15, DIFF_1:def_8 ; Sum ((S . 2),2) = (Partial_Sums (S . 2)) . (1 + 1) by SERIES_1:def_5 .= ((Partial_Sums (S . 2)) . (0 + 1)) + ((S . 2) . 2) by SERIES_1:def_1 .= (((Partial_Sums (S . 2)) . 0) + ((S . 2) . 1)) + ((S . 2) . 2) by SERIES_1:def_1 .= ((cdif ((f1 (#) f2),h)) . 2) . x by A10, A12, A14, A16, SERIES_1:def_1 ; hence ( ((cdif ((f1 (#) f2),h)) . 1) . x = Sum ((S . 1),1) & ((cdif ((f1 (#) f2),h)) . 2) . x = Sum ((S . 2),2) ) by A6; ::_thesis: verum end; theorem :: DIFF_3:36 for x0, x1 being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = sqrt x ) & x0 <> x1 & x0 > 0 & x1 > 0 holds [!f,x0,x1!] = 1 / ((sqrt x0) + (sqrt x1)) proof let x0, x1 be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = sqrt x ) & x0 <> x1 & x0 > 0 & x1 > 0 holds [!f,x0,x1!] = 1 / ((sqrt x0) + (sqrt x1)) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = sqrt x ) & x0 <> x1 & x0 > 0 & x1 > 0 implies [!f,x0,x1!] = 1 / ((sqrt x0) + (sqrt x1)) ) assume that A1: for x being Real holds f . x = sqrt x and A2: x0 <> x1 and A3: ( x0 > 0 & x1 > 0 ) ; ::_thesis: [!f,x0,x1!] = 1 / ((sqrt x0) + (sqrt x1)) [!f,x0,x1!] = ((sqrt x0) - (f . x1)) / (x0 - x1) by A1 .= ((sqrt x0) - (sqrt x1)) / (x0 - x1) by A1 .= 1 / ((sqrt x0) + (sqrt x1)) by A2, A3, SQUARE_1:36 ; hence [!f,x0,x1!] = 1 / ((sqrt x0) + (sqrt x1)) ; ::_thesis: verum end; theorem :: DIFF_3:37 for x0, x1, x2 being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = sqrt x ) & x0,x1,x2 are_mutually_different & x0 > 0 & x1 > 0 & x2 > 0 holds [!f,x0,x1,x2!] = - (1 / ((((sqrt x0) + (sqrt x1)) * ((sqrt x0) + (sqrt x2))) * ((sqrt x1) + (sqrt x2)))) proof let x0, x1, x2 be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = sqrt x ) & x0,x1,x2 are_mutually_different & x0 > 0 & x1 > 0 & x2 > 0 holds [!f,x0,x1,x2!] = - (1 / ((((sqrt x0) + (sqrt x1)) * ((sqrt x0) + (sqrt x2))) * ((sqrt x1) + (sqrt x2)))) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = sqrt x ) & x0,x1,x2 are_mutually_different & x0 > 0 & x1 > 0 & x2 > 0 implies [!f,x0,x1,x2!] = - (1 / ((((sqrt x0) + (sqrt x1)) * ((sqrt x0) + (sqrt x2))) * ((sqrt x1) + (sqrt x2)))) ) assume that A1: for x being Real holds f . x = sqrt x and A2: x0,x1,x2 are_mutually_different and A3: ( x0 > 0 & x1 > 0 & x2 > 0 ) ; ::_thesis: [!f,x0,x1,x2!] = - (1 / ((((sqrt x0) + (sqrt x1)) * ((sqrt x0) + (sqrt x2))) * ((sqrt x1) + (sqrt x2)))) A4: ( f . x0 = sqrt x0 & f . x1 = sqrt x1 & f . x2 = sqrt x2 ) by A1; ( sqrt x0 > 0 & sqrt x1 > 0 & sqrt x2 > 0 ) by A3, SQUARE_1:25; then A5: ( (sqrt x0) + (sqrt x1) > 0 & (sqrt x1) + (sqrt x2) > 0 ) by XREAL_1:34; A6: ( x0 <> x1 & x1 <> x2 & x2 <> x0 ) by A2, ZFMISC_1:def_5; then [!f,x0,x1,x2!] = ((1 / ((sqrt x0) + (sqrt x1))) - (((sqrt x1) - (sqrt x2)) / (x1 - x2))) / (x0 - x2) by A3, A4, SQUARE_1:36 .= ((1 / ((sqrt x0) + (sqrt x1))) - (1 / ((sqrt x1) + (sqrt x2)))) / (x0 - x2) by A3, A6, SQUARE_1:36 .= (((1 * ((sqrt x1) + (sqrt x2))) - (1 * ((sqrt x0) + (sqrt x1)))) / (((sqrt x0) + (sqrt x1)) * ((sqrt x1) + (sqrt x2)))) / (x0 - x2) by A5, XCMPLX_1:130 .= (((sqrt x2) - (sqrt x0)) / (((sqrt x0) + (sqrt x1)) * ((sqrt x1) + (sqrt x2)))) / (- (x2 - x0)) .= - ((((sqrt x2) - (sqrt x0)) / (((sqrt x0) + (sqrt x1)) * ((sqrt x1) + (sqrt x2)))) / (x2 - x0)) by XCMPLX_1:188 .= - ((1 / (((sqrt x0) + (sqrt x1)) * ((sqrt x1) + (sqrt x2)))) * (((sqrt x2) - (sqrt x0)) / (x2 - x0))) .= - ((1 / (((sqrt x0) + (sqrt x1)) * ((sqrt x1) + (sqrt x2)))) * (1 / ((sqrt x2) + (sqrt x0)))) by A3, A6, SQUARE_1:36 .= - (1 / ((((sqrt x0) + (sqrt x1)) * ((sqrt x1) + (sqrt x2))) * ((sqrt x2) + (sqrt x0)))) by XCMPLX_1:102 ; hence [!f,x0,x1,x2!] = - (1 / ((((sqrt x0) + (sqrt x1)) * ((sqrt x0) + (sqrt x2))) * ((sqrt x1) + (sqrt x2)))) ; ::_thesis: verum end; theorem :: DIFF_3:38 for x0, x1, x2, x3 being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = sqrt x ) & x0,x1,x2,x3 are_mutually_different & x0 > 0 & x1 > 0 & x2 > 0 & x3 > 0 holds [!f,x0,x1,x2,x3!] = ((((sqrt x0) + (sqrt x1)) + (sqrt x2)) + (sqrt x3)) / (((((((sqrt x0) + (sqrt x1)) * ((sqrt x0) + (sqrt x2))) * ((sqrt x0) + (sqrt x3))) * ((sqrt x1) + (sqrt x2))) * ((sqrt x1) + (sqrt x3))) * ((sqrt x2) + (sqrt x3))) proof let x0, x1, x2, x3 be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = sqrt x ) & x0,x1,x2,x3 are_mutually_different & x0 > 0 & x1 > 0 & x2 > 0 & x3 > 0 holds [!f,x0,x1,x2,x3!] = ((((sqrt x0) + (sqrt x1)) + (sqrt x2)) + (sqrt x3)) / (((((((sqrt x0) + (sqrt x1)) * ((sqrt x0) + (sqrt x2))) * ((sqrt x0) + (sqrt x3))) * ((sqrt x1) + (sqrt x2))) * ((sqrt x1) + (sqrt x3))) * ((sqrt x2) + (sqrt x3))) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = sqrt x ) & x0,x1,x2,x3 are_mutually_different & x0 > 0 & x1 > 0 & x2 > 0 & x3 > 0 implies [!f,x0,x1,x2,x3!] = ((((sqrt x0) + (sqrt x1)) + (sqrt x2)) + (sqrt x3)) / (((((((sqrt x0) + (sqrt x1)) * ((sqrt x0) + (sqrt x2))) * ((sqrt x0) + (sqrt x3))) * ((sqrt x1) + (sqrt x2))) * ((sqrt x1) + (sqrt x3))) * ((sqrt x2) + (sqrt x3))) ) assume that A1: for x being Real holds f . x = sqrt x and A2: x0,x1,x2,x3 are_mutually_different and A3: ( x0 > 0 & x1 > 0 & x2 > 0 & x3 > 0 ) ; ::_thesis: [!f,x0,x1,x2,x3!] = ((((sqrt x0) + (sqrt x1)) + (sqrt x2)) + (sqrt x3)) / (((((((sqrt x0) + (sqrt x1)) * ((sqrt x0) + (sqrt x2))) * ((sqrt x0) + (sqrt x3))) * ((sqrt x1) + (sqrt x2))) * ((sqrt x1) + (sqrt x3))) * ((sqrt x2) + (sqrt x3))) A4: ( f . x0 = sqrt x0 & f . x1 = sqrt x1 & f . x2 = sqrt x2 & f . x3 = sqrt x3 ) by A1; ( sqrt x0 > 0 & sqrt x1 > 0 & sqrt x2 > 0 & sqrt x3 > 0 ) by A3, SQUARE_1:25; then A5: ( (sqrt x0) + (sqrt x1) > 0 & (sqrt x1) + (sqrt x2) > 0 & (sqrt x2) + (sqrt x3) > 0 & (sqrt x2) + (sqrt x0) > 0 & (sqrt x3) + (sqrt x1) > 0 ) by XREAL_1:34; A6: ( x0 <> x1 & x0 <> x2 & x0 <> x3 & x1 <> x2 & x1 <> x3 & x2 <> x3 ) by A2, ZFMISC_1:def_6; then [!f,x0,x1,x2,x3!] = ((((1 / ((sqrt x0) + (sqrt x1))) - (((sqrt x1) - (sqrt x2)) / (x1 - x2))) / (x0 - x2)) - (((((sqrt x1) - (sqrt x2)) / (x1 - x2)) - (((sqrt x2) - (sqrt x3)) / (x2 - x3))) / (x1 - x3))) / (x0 - x3) by A3, A4, SQUARE_1:36 .= ((((1 / ((sqrt x0) + (sqrt x1))) - (1 / ((sqrt x1) + (sqrt x2)))) / (x0 - x2)) - (((((sqrt x1) - (sqrt x2)) / (x1 - x2)) - (((sqrt x2) - (sqrt x3)) / (x2 - x3))) / (x1 - x3))) / (x0 - x3) by A3, A6, SQUARE_1:36 .= ((((1 / ((sqrt x0) + (sqrt x1))) - (1 / ((sqrt x1) + (sqrt x2)))) / (x0 - x2)) - (((1 / ((sqrt x1) + (sqrt x2))) - (((sqrt x2) - (sqrt x3)) / (x2 - x3))) / (x1 - x3))) / (x0 - x3) by A3, A6, SQUARE_1:36 .= ((((1 / ((sqrt x0) + (sqrt x1))) - (1 / ((sqrt x1) + (sqrt x2)))) / (x0 - x2)) - (((1 / ((sqrt x1) + (sqrt x2))) - (1 / ((sqrt x2) + (sqrt x3)))) / (x1 - x3))) / (x0 - x3) by A3, A6, SQUARE_1:36 .= (((((1 * ((sqrt x1) + (sqrt x2))) - (1 * ((sqrt x0) + (sqrt x1)))) / (((sqrt x0) + (sqrt x1)) * ((sqrt x1) + (sqrt x2)))) / (x0 - x2)) - (((1 / ((sqrt x1) + (sqrt x2))) - (1 / ((sqrt x2) + (sqrt x3)))) / (x1 - x3))) / (x0 - x3) by A5, XCMPLX_1:130 .= (((((1 * ((sqrt x1) + (sqrt x2))) - (1 * ((sqrt x0) + (sqrt x1)))) / (((sqrt x0) + (sqrt x1)) * ((sqrt x1) + (sqrt x2)))) / (x0 - x2)) - ((((1 * ((sqrt x2) + (sqrt x3))) - (1 * ((sqrt x1) + (sqrt x2)))) / (((sqrt x1) + (sqrt x2)) * ((sqrt x2) + (sqrt x3)))) / (x1 - x3))) / (x0 - x3) by A5, XCMPLX_1:130 .= (((((sqrt x2) - (sqrt x0)) / (((sqrt x0) + (sqrt x1)) * ((sqrt x1) + (sqrt x2)))) / (- (x2 - x0))) - ((((sqrt x3) - (sqrt x1)) / (((sqrt x1) + (sqrt x2)) * ((sqrt x2) + (sqrt x3)))) / (- (x3 - x1)))) / (x0 - x3) .= ((- ((((sqrt x2) - (sqrt x0)) / (((sqrt x0) + (sqrt x1)) * ((sqrt x1) + (sqrt x2)))) / (x2 - x0))) - ((((sqrt x3) - (sqrt x1)) / (((sqrt x1) + (sqrt x2)) * ((sqrt x2) + (sqrt x3)))) / (- (x3 - x1)))) / (x0 - x3) by XCMPLX_1:188 .= ((- ((1 / (((sqrt x0) + (sqrt x1)) * ((sqrt x1) + (sqrt x2)))) * (((sqrt x2) - (sqrt x0)) / (x2 - x0)))) - (- ((((sqrt x3) - (sqrt x1)) / (((sqrt x1) + (sqrt x2)) * ((sqrt x2) + (sqrt x3)))) / (x3 - x1)))) / (x0 - x3) by XCMPLX_1:188 .= ((- ((1 / (((sqrt x0) + (sqrt x1)) * ((sqrt x1) + (sqrt x2)))) * (1 / ((sqrt x2) + (sqrt x0))))) - (- ((((sqrt x3) - (sqrt x1)) / (((sqrt x1) + (sqrt x2)) * ((sqrt x2) + (sqrt x3)))) / (x3 - x1)))) / (x0 - x3) by A3, A6, SQUARE_1:36 .= ((- (1 / ((((sqrt x0) + (sqrt x1)) * ((sqrt x1) + (sqrt x2))) * ((sqrt x2) + (sqrt x0))))) + ((1 / (((sqrt x1) + (sqrt x2)) * ((sqrt x2) + (sqrt x3)))) * (((sqrt x3) - (sqrt x1)) / (x3 - x1)))) / (x0 - x3) by XCMPLX_1:102 .= ((- (1 / ((((sqrt x0) + (sqrt x1)) * ((sqrt x1) + (sqrt x2))) * ((sqrt x2) + (sqrt x0))))) + ((1 / (((sqrt x1) + (sqrt x2)) * ((sqrt x2) + (sqrt x3)))) * (1 / ((sqrt x3) + (sqrt x1))))) / (x0 - x3) by A3, A6, SQUARE_1:36 .= ((1 / ((((sqrt x1) + (sqrt x2)) * ((sqrt x2) + (sqrt x3))) * ((sqrt x3) + (sqrt x1)))) - (1 / ((((sqrt x0) + (sqrt x1)) * ((sqrt x1) + (sqrt x2))) * ((sqrt x2) + (sqrt x0))))) / (x0 - x3) by XCMPLX_1:102 .= (((1 * ((((sqrt x0) + (sqrt x1)) * ((sqrt x1) + (sqrt x2))) * ((sqrt x2) + (sqrt x0)))) - (1 * ((((sqrt x1) + (sqrt x2)) * ((sqrt x2) + (sqrt x3))) * ((sqrt x3) + (sqrt x1))))) / (((((sqrt x1) + (sqrt x2)) * ((sqrt x2) + (sqrt x3))) * ((sqrt x3) + (sqrt x1))) * ((((sqrt x0) + (sqrt x1)) * ((sqrt x1) + (sqrt x2))) * ((sqrt x2) + (sqrt x0))))) / (x0 - x3) by A5, XCMPLX_1:130 .= ((((((sqrt x0) + (sqrt x1)) * ((sqrt x2) + (sqrt x0))) - (((sqrt x2) + (sqrt x3)) * ((sqrt x3) + (sqrt x1)))) * ((sqrt x1) + (sqrt x2))) / (((((((sqrt x2) + (sqrt x3)) * ((sqrt x3) + (sqrt x1))) * ((sqrt x0) + (sqrt x1))) * ((sqrt x1) + (sqrt x2))) * ((sqrt x2) + (sqrt x0))) * ((sqrt x1) + (sqrt x2)))) / (x0 - x3) .= (((((((sqrt x0) * (sqrt x2)) + ((sqrt x0) * (sqrt x0))) + ((sqrt x1) * (sqrt x2))) + ((sqrt x1) * (sqrt x0))) - (((((sqrt x2) * (sqrt x3)) + ((sqrt x2) * (sqrt x1))) + ((sqrt x3) * (sqrt x3))) + ((sqrt x3) * (sqrt x1)))) / ((((((sqrt x2) + (sqrt x3)) * ((sqrt x3) + (sqrt x1))) * ((sqrt x0) + (sqrt x1))) * ((sqrt x1) + (sqrt x2))) * ((sqrt x2) + (sqrt x0)))) / (x0 - x3) by A5, XCMPLX_1:91 .= (((((((sqrt x0) * (sqrt x2)) + (sqrt (x0 ^2))) + ((sqrt x1) * (sqrt x2))) + ((sqrt x1) * (sqrt x0))) - (((((sqrt x2) * (sqrt x3)) + ((sqrt x2) * (sqrt x1))) + ((sqrt x3) * (sqrt x3))) + ((sqrt x3) * (sqrt x1)))) / ((((((sqrt x2) + (sqrt x3)) * ((sqrt x3) + (sqrt x1))) * ((sqrt x0) + (sqrt x1))) * ((sqrt x1) + (sqrt x2))) * ((sqrt x2) + (sqrt x0)))) / (x0 - x3) by A3, SQUARE_1:29 .= (((((((sqrt x0) * (sqrt x2)) + x0) + ((sqrt x1) * (sqrt x2))) + ((sqrt x1) * (sqrt x0))) - (((((sqrt x2) * (sqrt x3)) + ((sqrt x2) * (sqrt x1))) + ((sqrt x3) * (sqrt x3))) + ((sqrt x3) * (sqrt x1)))) / ((((((sqrt x2) + (sqrt x3)) * ((sqrt x3) + (sqrt x1))) * ((sqrt x0) + (sqrt x1))) * ((sqrt x1) + (sqrt x2))) * ((sqrt x2) + (sqrt x0)))) / (x0 - x3) by A3, SQUARE_1:22 .= (((((((sqrt x0) * (sqrt x2)) + x0) + ((sqrt x1) * (sqrt x2))) + ((sqrt x1) * (sqrt x0))) - (((((sqrt x2) * (sqrt x3)) + ((sqrt x2) * (sqrt x1))) + (sqrt (x3 ^2))) + ((sqrt x3) * (sqrt x1)))) / ((((((sqrt x2) + (sqrt x3)) * ((sqrt x3) + (sqrt x1))) * ((sqrt x0) + (sqrt x1))) * ((sqrt x1) + (sqrt x2))) * ((sqrt x2) + (sqrt x0)))) / (x0 - x3) by A3, SQUARE_1:29 .= (((((((sqrt x0) * (sqrt x2)) + x0) + ((sqrt x1) * (sqrt x2))) + ((sqrt x1) * (sqrt x0))) - (((((sqrt x2) * (sqrt x3)) + ((sqrt x2) * (sqrt x1))) + x3) + ((sqrt x3) * (sqrt x1)))) / ((((((sqrt x2) + (sqrt x3)) * ((sqrt x3) + (sqrt x1))) * ((sqrt x0) + (sqrt x1))) * ((sqrt x1) + (sqrt x2))) * ((sqrt x2) + (sqrt x0)))) / (x0 - x3) by A3, SQUARE_1:22 .= ((((sqrt x2) + (sqrt x1)) * ((sqrt x0) - (sqrt x3))) + (x0 - x3)) / (((((((sqrt x2) + (sqrt x3)) * ((sqrt x3) + (sqrt x1))) * ((sqrt x0) + (sqrt x1))) * ((sqrt x1) + (sqrt x2))) * ((sqrt x2) + (sqrt x0))) * (x0 - x3)) by XCMPLX_1:78 .= ((((sqrt x0) - (sqrt x3)) * ((sqrt x2) + (sqrt x1))) + (((sqrt x0) - (sqrt x3)) * ((sqrt x0) + (sqrt x3)))) / (((((((sqrt x2) + (sqrt x3)) * ((sqrt x3) + (sqrt x1))) * ((sqrt x0) + (sqrt x1))) * ((sqrt x1) + (sqrt x2))) * ((sqrt x2) + (sqrt x0))) * (x0 - x3)) by A3, SQUARE_1:35 .= (((sqrt x0) - (sqrt x3)) * ((((sqrt x2) + (sqrt x1)) + (sqrt x0)) + (sqrt x3))) / (((((((sqrt x2) + (sqrt x3)) * ((sqrt x3) + (sqrt x1))) * ((sqrt x0) + (sqrt x1))) * ((sqrt x1) + (sqrt x2))) * ((sqrt x2) + (sqrt x0))) * (x0 - x3)) .= (((sqrt x0) - (sqrt x3)) / (x0 - x3)) * (((((sqrt x2) + (sqrt x1)) + (sqrt x0)) + (sqrt x3)) / ((((((sqrt x2) + (sqrt x3)) * ((sqrt x3) + (sqrt x1))) * ((sqrt x0) + (sqrt x1))) * ((sqrt x1) + (sqrt x2))) * ((sqrt x2) + (sqrt x0)))) by XCMPLX_1:76 .= (1 / ((sqrt x0) + (sqrt x3))) * (((((sqrt x2) + (sqrt x1)) + (sqrt x0)) + (sqrt x3)) / ((((((sqrt x2) + (sqrt x3)) * ((sqrt x3) + (sqrt x1))) * ((sqrt x0) + (sqrt x1))) * ((sqrt x1) + (sqrt x2))) * ((sqrt x2) + (sqrt x0)))) by A3, A6, SQUARE_1:36 .= ((((sqrt x2) + (sqrt x1)) + (sqrt x0)) + (sqrt x3)) / (((((((sqrt x2) + (sqrt x3)) * ((sqrt x3) + (sqrt x1))) * ((sqrt x0) + (sqrt x1))) * ((sqrt x1) + (sqrt x2))) * ((sqrt x2) + (sqrt x0))) * ((sqrt x0) + (sqrt x3))) by XCMPLX_1:103 .= ((((sqrt x0) + (sqrt x1)) + (sqrt x2)) + (sqrt x3)) / (((((((sqrt x0) + (sqrt x1)) * ((sqrt x0) + (sqrt x2))) * ((sqrt x0) + (sqrt x3))) * ((sqrt x1) + (sqrt x2))) * ((sqrt x1) + (sqrt x3))) * ((sqrt x2) + (sqrt x3))) ; hence [!f,x0,x1,x2,x3!] = ((((sqrt x0) + (sqrt x1)) + (sqrt x2)) + (sqrt x3)) / (((((((sqrt x0) + (sqrt x1)) * ((sqrt x0) + (sqrt x2))) * ((sqrt x0) + (sqrt x3))) * ((sqrt x1) + (sqrt x2))) * ((sqrt x1) + (sqrt x3))) * ((sqrt x2) + (sqrt x3))) ; ::_thesis: verum end; theorem :: DIFF_3:39 for x, h being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = sqrt x ) & x > 0 & x + h > 0 holds (fD (f,h)) . x = (sqrt (x + h)) - (sqrt x) proof let x, h be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = sqrt x ) & x > 0 & x + h > 0 holds (fD (f,h)) . x = (sqrt (x + h)) - (sqrt x) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = sqrt x ) & x > 0 & x + h > 0 implies (fD (f,h)) . x = (sqrt (x + h)) - (sqrt x) ) assume A1: for x being Real holds f . x = sqrt x ; ::_thesis: ( not x > 0 or not x + h > 0 or (fD (f,h)) . x = (sqrt (x + h)) - (sqrt x) ) (fD (f,h)) . x = (f . (x + h)) - (f . x) by DIFF_1:3 .= (sqrt (x + h)) - (f . x) by A1 .= (sqrt (x + h)) - (sqrt x) by A1 ; hence ( not x > 0 or not x + h > 0 or (fD (f,h)) . x = (sqrt (x + h)) - (sqrt x) ) ; ::_thesis: verum end; theorem :: DIFF_3:40 for x, h being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = sqrt x ) & x > 0 & x - h > 0 holds (bD (f,h)) . x = (sqrt x) - (sqrt (x - h)) proof let x, h be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = sqrt x ) & x > 0 & x - h > 0 holds (bD (f,h)) . x = (sqrt x) - (sqrt (x - h)) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = sqrt x ) & x > 0 & x - h > 0 implies (bD (f,h)) . x = (sqrt x) - (sqrt (x - h)) ) assume A1: for x being Real holds f . x = sqrt x ; ::_thesis: ( not x > 0 or not x - h > 0 or (bD (f,h)) . x = (sqrt x) - (sqrt (x - h)) ) (bD (f,h)) . x = (f . x) - (f . (x - h)) by DIFF_1:4 .= (sqrt x) - (f . (x - h)) by A1 .= (sqrt x) - (sqrt (x - h)) by A1 ; hence ( not x > 0 or not x - h > 0 or (bD (f,h)) . x = (sqrt x) - (sqrt (x - h)) ) ; ::_thesis: verum end; theorem :: DIFF_3:41 for x, h being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = sqrt x ) & x + (h / 2) > 0 & x - (h / 2) > 0 holds (cD (f,h)) . x = (sqrt (x + (h / 2))) - (sqrt (x - (h / 2))) proof let x, h be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = sqrt x ) & x + (h / 2) > 0 & x - (h / 2) > 0 holds (cD (f,h)) . x = (sqrt (x + (h / 2))) - (sqrt (x - (h / 2))) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = sqrt x ) & x + (h / 2) > 0 & x - (h / 2) > 0 implies (cD (f,h)) . x = (sqrt (x + (h / 2))) - (sqrt (x - (h / 2))) ) assume A1: for x being Real holds f . x = sqrt x ; ::_thesis: ( not x + (h / 2) > 0 or not x - (h / 2) > 0 or (cD (f,h)) . x = (sqrt (x + (h / 2))) - (sqrt (x - (h / 2))) ) (cD (f,h)) . x = (f . (x + (h / 2))) - (f . (x - (h / 2))) by DIFF_1:5 .= (sqrt (x + (h / 2))) - (f . (x - (h / 2))) by A1 .= (sqrt (x + (h / 2))) - (sqrt (x - (h / 2))) by A1 ; hence ( not x + (h / 2) > 0 or not x - (h / 2) > 0 or (cD (f,h)) . x = (sqrt (x + (h / 2))) - (sqrt (x - (h / 2))) ) ; ::_thesis: verum end; theorem :: DIFF_3:42 for x0, x1 being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = x ^2 ) & x0 <> x1 holds [!f,x0,x1!] = x0 + x1 proof let x0, x1 be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = x ^2 ) & x0 <> x1 holds [!f,x0,x1!] = x0 + x1 let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = x ^2 ) & x0 <> x1 implies [!f,x0,x1!] = x0 + x1 ) assume that A1: for x being Real holds f . x = x ^2 and A2: x0 <> x1 ; ::_thesis: [!f,x0,x1!] = x0 + x1 A3: x0 - x1 <> 0 by A2; [!f,x0,x1!] = ((x0 ^2) - (f . x1)) / (x0 - x1) by A1 .= ((x0 ^2) - (x1 ^2)) / (x0 - x1) by A1 .= ((x0 - x1) * (x0 + x1)) / (x0 - x1) .= x0 + x1 by A3, XCMPLX_1:89 ; hence [!f,x0,x1!] = x0 + x1 ; ::_thesis: verum end; theorem :: DIFF_3:43 for x0, x1, x2 being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = x ^2 ) & x0,x1,x2 are_mutually_different holds [!f,x0,x1,x2!] = 1 proof let x0, x1, x2 be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = x ^2 ) & x0,x1,x2 are_mutually_different holds [!f,x0,x1,x2!] = 1 let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = x ^2 ) & x0,x1,x2 are_mutually_different implies [!f,x0,x1,x2!] = 1 ) assume that A1: for x being Real holds f . x = x ^2 and A2: x0,x1,x2 are_mutually_different ; ::_thesis: [!f,x0,x1,x2!] = 1 A3: ( f . x0 = x0 ^2 & f . x1 = x1 ^2 & f . x2 = x2 ^2 ) by A1; A4: ( x0 - x1 <> 0 & x1 - x2 <> 0 & x0 - x2 <> 0 ) by A2, ZFMISC_1:def_5; [!f,x0,x1,x2!] = ((((x0 - x1) * (x0 + x1)) / (x0 - x1)) - (((x1 - x2) * (x1 + x2)) / (x1 - x2))) / (x0 - x2) by A3 .= ((x0 + x1) - (((x1 - x2) * (x1 + x2)) / (x1 - x2))) / (x0 - x2) by A4, XCMPLX_1:89 .= ((x0 + x1) - (x1 + x2)) / (x0 - x2) by A4, XCMPLX_1:89 .= 1 by A4, XCMPLX_1:60 ; hence [!f,x0,x1,x2!] = 1 ; ::_thesis: verum end; theorem :: DIFF_3:44 for x0, x1, x2, x3 being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = x ^2 ) & x0,x1,x2,x3 are_mutually_different holds [!f,x0,x1,x2,x3!] = 0 proof let x0, x1, x2, x3 be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = x ^2 ) & x0,x1,x2,x3 are_mutually_different holds [!f,x0,x1,x2,x3!] = 0 let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = x ^2 ) & x0,x1,x2,x3 are_mutually_different implies [!f,x0,x1,x2,x3!] = 0 ) assume that A1: for x being Real holds f . x = x ^2 and A2: x0,x1,x2,x3 are_mutually_different ; ::_thesis: [!f,x0,x1,x2,x3!] = 0 A3: ( f . x0 = x0 ^2 & f . x1 = x1 ^2 & f . x2 = x2 ^2 & f . x3 = x3 ^2 ) by A1; A4: ( x0 - x1 <> 0 & x1 - x2 <> 0 & x2 - x3 <> 0 & x0 - x2 <> 0 & x1 - x3 <> 0 & x0 - x3 <> 0 ) by A2, ZFMISC_1:def_6; [!f,x0,x1,x2,x3!] = ((((((x0 - x1) * (x0 + x1)) / (x0 - x1)) - (((x1 - x2) * (x1 + x2)) / (x1 - x2))) / (x0 - x2)) - (((((x1 - x2) * (x1 + x2)) / (x1 - x2)) - (((x2 - x3) * (x2 + x3)) / (x2 - x3))) / (x1 - x3))) / (x0 - x3) by A3 .= ((((x0 + x1) - (((x1 - x2) * (x1 + x2)) / (x1 - x2))) / (x0 - x2)) - (((((x1 - x2) * (x1 + x2)) / (x1 - x2)) - (((x2 - x3) * (x2 + x3)) / (x2 - x3))) / (x1 - x3))) / (x0 - x3) by A4, XCMPLX_1:89 .= ((((x0 + x1) - (x1 + x2)) / (x0 - x2)) - (((((x1 - x2) * (x1 + x2)) / (x1 - x2)) - (((x2 - x3) * (x2 + x3)) / (x2 - x3))) / (x1 - x3))) / (x0 - x3) by A4, XCMPLX_1:89 .= ((((x0 + x1) - (x1 + x2)) / (x0 - x2)) - (((x1 + x2) - (((x2 - x3) * (x2 + x3)) / (x2 - x3))) / (x1 - x3))) / (x0 - x3) by A4, XCMPLX_1:89 .= ((((x0 + x1) - (x1 + x2)) / (x0 - x2)) - (((x1 + x2) - (x2 + x3)) / (x1 - x3))) / (x0 - x3) by A4, XCMPLX_1:89 .= (1 - ((x1 - x3) / (x1 - x3))) / (x0 - x3) by A4, XCMPLX_1:60 .= (1 - 1) / (x0 - x3) by A4, XCMPLX_1:60 .= 0 ; hence [!f,x0,x1,x2,x3!] = 0 ; ::_thesis: verum end; theorem :: DIFF_3:45 for h, x being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = x ^2 ) holds (fD (f,h)) . x = ((2 * x) * h) + (h ^2) proof let h, x be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = x ^2 ) holds (fD (f,h)) . x = ((2 * x) * h) + (h ^2) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = x ^2 ) implies (fD (f,h)) . x = ((2 * x) * h) + (h ^2) ) assume A1: for x being Real holds f . x = x ^2 ; ::_thesis: (fD (f,h)) . x = ((2 * x) * h) + (h ^2) then f . (x + h) = (x + h) ^2 ; then (fD (f,h)) . x = ((x + h) ^2) - (f . x) by DIFF_1:3 .= (((x ^2) + ((2 * x) * h)) + (h ^2)) - (x ^2) by A1 .= ((2 * x) * h) + (h ^2) ; hence (fD (f,h)) . x = ((2 * x) * h) + (h ^2) ; ::_thesis: verum end; theorem :: DIFF_3:46 for h, x being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = x ^2 ) holds (bD (f,h)) . x = h * ((2 * x) - h) proof let h, x be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = x ^2 ) holds (bD (f,h)) . x = h * ((2 * x) - h) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = x ^2 ) implies (bD (f,h)) . x = h * ((2 * x) - h) ) assume A1: for x being Real holds f . x = x ^2 ; ::_thesis: (bD (f,h)) . x = h * ((2 * x) - h) then A2: f . (x - h) = (x - h) ^2 ; (bD (f,h)) . x = (f . x) - (f . (x - h)) by DIFF_1:4 .= (x ^2) - ((x - h) ^2) by A1, A2 .= h * ((2 * x) - h) ; hence (bD (f,h)) . x = h * ((2 * x) - h) ; ::_thesis: verum end; theorem :: DIFF_3:47 for h, x being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = x ^2 ) holds (cD (f,h)) . x = (2 * h) * x proof let h, x be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = x ^2 ) holds (cD (f,h)) . x = (2 * h) * x let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = x ^2 ) implies (cD (f,h)) . x = (2 * h) * x ) assume A1: for x being Real holds f . x = x ^2 ; ::_thesis: (cD (f,h)) . x = (2 * h) * x (cD (f,h)) . x = (f . (x + (h / 2))) - (f . (x - (h / 2))) by DIFF_1:5 .= ((x + (h / 2)) ^2) - (f . (x - (h / 2))) by A1 .= ((x + (h / 2)) ^2) - ((x - (h / 2)) ^2) by A1 .= h * (2 * x) ; hence (cD (f,h)) . x = (2 * h) * x ; ::_thesis: verum end; theorem :: DIFF_3:48 for k, x0, x1 being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = k / (x ^2) ) & x0 <> x1 & x0 <> 0 & x1 <> 0 holds [!f,x0,x1!] = - ((k / (x0 * x1)) * ((1 / x0) + (1 / x1))) proof let k, x0, x1 be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = k / (x ^2) ) & x0 <> x1 & x0 <> 0 & x1 <> 0 holds [!f,x0,x1!] = - ((k / (x0 * x1)) * ((1 / x0) + (1 / x1))) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = k / (x ^2) ) & x0 <> x1 & x0 <> 0 & x1 <> 0 implies [!f,x0,x1!] = - ((k / (x0 * x1)) * ((1 / x0) + (1 / x1))) ) assume that A1: for x being Real holds f . x = k / (x ^2) and A2: ( x0 <> x1 & x0 <> 0 & x1 <> 0 ) ; ::_thesis: [!f,x0,x1!] = - ((k / (x0 * x1)) * ((1 / x0) + (1 / x1))) A3: x1 - x0 <> 0 by A2; ( f . x0 = k / (x0 ^2) & f . x1 = k / (x1 ^2) ) by A1; then [!f,x0,x1!] = (k * ((1 / (x0 ^2)) - (1 / (x1 ^2)))) / (x0 - x1) .= (k * (((1 * (x1 ^2)) - (1 * (x0 ^2))) / ((x0 ^2) * (x1 ^2)))) / (x0 - x1) by A2, XCMPLX_1:130 .= k * ((((x1 - x0) * (x1 + x0)) / ((x0 ^2) * (x1 ^2))) / (- (x1 - x0))) .= k * (- ((((x1 - x0) * (x1 + x0)) / ((x0 ^2) * (x1 ^2))) / (x1 - x0))) by XCMPLX_1:188 .= - (k * ((((x1 - x0) * (x1 + x0)) / (x1 - x0)) / ((x0 ^2) * (x1 ^2)))) .= - (k * ((x1 + x0) / (((x0 * x0) * x1) * x1))) by A3, XCMPLX_1:89 .= - (k * ((x1 / (x1 * ((x0 * x0) * x1))) + (x0 / (x0 * ((x0 * x1) * x1))))) .= - (k * (((1 / ((x0 * x0) * x1)) * (x1 / x1)) + (x0 / (x0 * ((x0 * x1) * x1))))) by XCMPLX_1:103 .= - (k * (((1 / ((x0 * x0) * x1)) * (x1 / x1)) + ((1 / ((x0 * x1) * x1)) * (x0 / x0)))) by XCMPLX_1:103 .= - (k * (((1 / ((x0 * x0) * x1)) * 1) + ((1 / ((x0 * x1) * x1)) * (x0 / x0)))) by A2, XCMPLX_1:60 .= - (k * ((1 / (x0 * (x0 * x1))) + (1 / ((x0 * x1) * x1)))) by A2, XCMPLX_1:60 .= - (k * (((1 / x0) * (1 / (x0 * x1))) + (1 / ((x0 * x1) * x1)))) by XCMPLX_1:102 .= - (k * (((1 / x0) * (1 / (x0 * x1))) + ((1 / (x0 * x1)) * (1 / x1)))) by XCMPLX_1:102 .= - ((k / (x0 * x1)) * ((1 / x0) + (1 / x1))) ; hence [!f,x0,x1!] = - ((k / (x0 * x1)) * ((1 / x0) + (1 / x1))) ; ::_thesis: verum end; theorem :: DIFF_3:49 for k, x0, x1, x2 being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = k / (x ^2) ) & x0 <> 0 & x1 <> 0 & x2 <> 0 & x0,x1,x2 are_mutually_different holds [!f,x0,x1,x2!] = (k / ((x0 * x1) * x2)) * (((1 / x0) + (1 / x1)) + (1 / x2)) proof let k, x0, x1, x2 be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = k / (x ^2) ) & x0 <> 0 & x1 <> 0 & x2 <> 0 & x0,x1,x2 are_mutually_different holds [!f,x0,x1,x2!] = (k / ((x0 * x1) * x2)) * (((1 / x0) + (1 / x1)) + (1 / x2)) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = k / (x ^2) ) & x0 <> 0 & x1 <> 0 & x2 <> 0 & x0,x1,x2 are_mutually_different implies [!f,x0,x1,x2!] = (k / ((x0 * x1) * x2)) * (((1 / x0) + (1 / x1)) + (1 / x2)) ) assume that A1: for x being Real holds f . x = k / (x ^2) and A2: ( x0 <> 0 & x1 <> 0 & x2 <> 0 ) and A3: x0,x1,x2 are_mutually_different ; ::_thesis: [!f,x0,x1,x2!] = (k / ((x0 * x1) * x2)) * (((1 / x0) + (1 / x1)) + (1 / x2)) A4: ( f . x0 = k / (x0 ^2) & f . x1 = k / (x1 ^2) & f . x2 = k / (x2 ^2) ) by A1; A5: ( x1 - x0 <> 0 & x2 - x1 <> 0 & x0 - x2 <> 0 ) by A3, ZFMISC_1:def_5; [!f,x0,x1,x2!] = (((k * ((1 / (x0 ^2)) - (1 / (x1 ^2)))) / (x0 - x1)) - ((k * ((1 / (x1 ^2)) - (1 / (x2 ^2)))) / (x1 - x2))) / (x0 - x2) by A4 .= (((k * (((1 * (x1 ^2)) - (1 * (x0 ^2))) / ((x0 ^2) * (x1 ^2)))) / (x0 - x1)) - ((k * ((1 / (x1 ^2)) - (1 / (x2 ^2)))) / (x1 - x2))) / (x0 - x2) by A2, XCMPLX_1:130 .= (((k * (((1 * (x1 ^2)) - (1 * (x0 ^2))) / ((x0 ^2) * (x1 ^2)))) / (x0 - x1)) - ((k * (((1 * (x2 ^2)) - (1 * (x1 ^2))) / ((x1 ^2) * (x2 ^2)))) / (x1 - x2))) / (x0 - x2) by A2, XCMPLX_1:130 .= ((k * ((((x1 - x0) * (x1 + x0)) / ((x0 ^2) * (x1 ^2))) / (- (x1 - x0)))) - (k * ((((x2 - x1) * (x2 + x1)) / ((x1 ^2) * (x2 ^2))) / (- (x2 - x1))))) / (x0 - x2) .= ((k * (- ((((x1 - x0) * (x1 + x0)) / ((x0 ^2) * (x1 ^2))) / (x1 - x0)))) - (k * ((((x2 - x1) * (x2 + x1)) / ((x1 ^2) * (x2 ^2))) / (- (x2 - x1))))) / (x0 - x2) by XCMPLX_1:188 .= ((- (k * ((((x1 - x0) * (x1 + x0)) / ((x0 ^2) * (x1 ^2))) / (x1 - x0)))) - (k * (- ((((x2 - x1) * (x2 + x1)) / ((x1 ^2) * (x2 ^2))) / (x2 - x1))))) / (x0 - x2) by XCMPLX_1:188 .= ((- (k * ((((x1 - x0) * (x1 + x0)) / (x1 - x0)) / ((x0 ^2) * (x1 ^2))))) + (k * ((((x2 - x1) * (x2 + x1)) / (x2 - x1)) / ((x1 ^2) * (x2 ^2))))) / (x0 - x2) .= ((- (k * ((x1 + x0) / ((x0 ^2) * (x1 ^2))))) + (k * ((((x2 - x1) * (x2 + x1)) / (x2 - x1)) / ((x1 ^2) * (x2 ^2))))) / (x0 - x2) by A5, XCMPLX_1:89 .= ((- (k * ((x1 / (((x0 * x0) * x1) * x1)) + (x0 / (((x0 * x0) * x1) * x1))))) + (k * ((x2 + x1) / (((x1 * x1) * x2) * x2)))) / (x0 - x2) by A5, XCMPLX_1:89 .= ((- (k * (((1 / ((x0 * x0) * x1)) * (x1 / x1)) + (x0 / (x0 * ((x0 * x1) * x1)))))) + (k * ((x2 / (x2 * ((x1 * x1) * x2))) + (x1 / (x1 * ((x1 * x2) * x2)))))) / (x0 - x2) by XCMPLX_1:103 .= ((- (k * (((1 / ((x0 * x0) * x1)) * (x1 / x1)) + (x0 / (x0 * ((x0 * x1) * x1)))))) + (k * (((1 / ((x1 * x1) * x2)) * (x2 / x2)) + (x1 / (x1 * ((x1 * x2) * x2)))))) / (x0 - x2) by XCMPLX_1:103 .= ((- (k * (((1 / ((x0 * x0) * x1)) * (x1 / x1)) + ((1 / ((x0 * x1) * x1)) * (x0 / x0))))) + (k * (((1 / ((x1 * x1) * x2)) * (x2 / x2)) + (x1 / (x1 * ((x1 * x2) * x2)))))) / (x0 - x2) by XCMPLX_1:103 .= ((- (k * (((1 / ((x0 * x0) * x1)) * (x1 / x1)) + ((1 / ((x0 * x1) * x1)) * (x0 / x0))))) + (k * (((1 / ((x1 * x1) * x2)) * (x2 / x2)) + ((1 / ((x1 * x2) * x2)) * (x1 / x1))))) / (x0 - x2) by XCMPLX_1:103 .= ((- (k * (((1 / ((x0 * x0) * x1)) * 1) + ((1 / ((x0 * x1) * x1)) * (x0 / x0))))) + (k * (((1 / ((x1 * x1) * x2)) * (x2 / x2)) + ((1 / ((x1 * x2) * x2)) * (x1 / x1))))) / (x0 - x2) by A2, XCMPLX_1:60 .= ((- (k * (((1 / ((x0 * x0) * x1)) * 1) + ((1 / ((x0 * x1) * x1)) * (x0 / x0))))) + (k * (((1 / ((x1 * x1) * x2)) * 1) + ((1 / ((x1 * x2) * x2)) * (x1 / x1))))) / (x0 - x2) by A2, XCMPLX_1:60 .= ((- (k * (((1 / ((x0 * x0) * x1)) * 1) + ((1 / ((x0 * x1) * x1)) * 1)))) + (k * (((1 / ((x1 * x1) * x2)) * 1) + ((1 / ((x1 * x2) * x2)) * (x1 / x1))))) / (x0 - x2) by A2, XCMPLX_1:60 .= ((- (k * ((1 / (x0 * (x0 * x1))) + (1 / ((x0 * x1) * x1))))) + (k * ((1 / (x1 * (x1 * x2))) + (1 / ((x1 * x2) * x2))))) / (x0 - x2) by A2, XCMPLX_1:60 .= ((- (k * (((1 / x0) * (1 / (x0 * x1))) + (1 / ((x0 * x1) * x1))))) + (k * ((1 / (x1 * (x1 * x2))) + (1 / ((x1 * x2) * x2))))) / (x0 - x2) by XCMPLX_1:102 .= ((- (k * (((1 / x0) * (1 / (x0 * x1))) + (1 / ((x0 * x1) * x1))))) + (k * (((1 / x1) * (1 / (x1 * x2))) + (1 / ((x1 * x2) * x2))))) / (x0 - x2) by XCMPLX_1:102 .= ((- (k * (((1 / x0) * (1 / (x0 * x1))) + ((1 / (x0 * x1)) * (1 / x1))))) + (k * (((1 / x1) * (1 / (x1 * x2))) + (1 / ((x1 * x2) * x2))))) / (x0 - x2) by XCMPLX_1:102 .= ((- ((k * (1 / (x0 * x1))) * ((1 / x0) + (1 / x1)))) + (k * (((1 / x1) * (1 / (x1 * x2))) + ((1 / (x1 * x2)) * (1 / x2))))) / (x0 - x2) by XCMPLX_1:102 .= k * ((((1 / (x1 * x2)) / (x0 - x2)) * ((1 / x1) + (1 / x2))) - (((1 / (x0 * x1)) / (x0 - x2)) * ((1 / x0) + (1 / x1)))) .= k * (((1 / ((x1 * x2) * (x0 - x2))) * ((1 / x1) + (1 / x2))) - (((1 / (x0 * x1)) / (x0 - x2)) * ((1 / x0) + (1 / x1)))) by XCMPLX_1:78 .= k * (((1 / ((x1 * x2) * (x0 - x2))) * ((1 / x1) + (1 / x2))) - ((1 / ((x0 * x1) * (x0 - x2))) * ((1 / x0) + (1 / x1)))) by XCMPLX_1:78 .= k * (((1 / ((x1 * x2) * (x0 - x2))) * (((1 * x2) + (1 * x1)) / (x1 * x2))) - ((1 / ((x0 * x1) * (x0 - x2))) * ((1 / x0) + (1 / x1)))) by A2, XCMPLX_1:116 .= k * (((1 / ((x1 * x2) * (x0 - x2))) * (((1 * x2) + (1 * x1)) / (x1 * x2))) - ((1 / ((x0 * x1) * (x0 - x2))) * (((1 * x1) + (1 * x0)) / (x0 * x1)))) by A2, XCMPLX_1:116 .= k * (((1 / ((x1 * x2) * (x0 - x2))) / ((x1 * x2) / (x2 + x1))) - ((1 / ((x0 * x1) * (x0 - x2))) * ((x1 + x0) / (x0 * x1)))) by XCMPLX_1:79 .= k * (((1 / ((x1 * x2) * (x0 - x2))) / ((x1 * x2) / (x2 + x1))) - ((1 / ((x0 * x1) * (x0 - x2))) / ((x0 * x1) / (x1 + x0)))) by XCMPLX_1:79 .= k * ((((1 / ((x1 * x2) * (x0 - x2))) / (x1 * x2)) * (x2 + x1)) - ((1 / ((x0 * x1) * (x0 - x2))) / ((x0 * x1) / (x1 + x0)))) by XCMPLX_1:82 .= k * ((((1 / ((x1 * x2) * (x0 - x2))) / (x1 * x2)) * (x2 + x1)) - (((1 / ((x0 * x1) * (x0 - x2))) / (x0 * x1)) * (x1 + x0))) by XCMPLX_1:82 .= k * ((((x2 + x1) / ((x1 * x2) * (x0 - x2))) / (x1 * x2)) - (((x1 + x0) / ((x0 * x1) * (x0 - x2))) / (x0 * x1))) .= k * (((x2 + x1) / (((x1 * x2) * (x0 - x2)) * (x1 * x2))) - (((x1 + x0) / ((x0 * x1) * (x0 - x2))) / (x0 * x1))) by XCMPLX_1:78 .= k * (((x2 + x1) / (((x1 * x2) * (x0 - x2)) * (x1 * x2))) - ((x1 + x0) / (((x0 * x1) * (x0 - x2)) * (x0 * x1)))) by XCMPLX_1:78 .= k * (((x2 + x1) / (((x1 ^2) * (x0 - x2)) * (x2 ^2))) - ((x1 + x0) / (((x1 ^2) * (x0 - x2)) * (x0 ^2)))) .= k * ((((x2 + x1) / ((x1 ^2) * (x0 - x2))) / (x2 ^2)) - ((x1 + x0) / (((x1 ^2) * (x0 - x2)) * (x0 ^2)))) by XCMPLX_1:78 .= k * (((1 / ((x1 ^2) * (x0 - x2))) * ((x2 + x1) / (x2 ^2))) - (((x1 + x0) / ((x1 ^2) * (x0 - x2))) / (x0 ^2))) by XCMPLX_1:78 .= k * ((1 / ((x1 ^2) * (x0 - x2))) * (((x2 + x1) / (x2 ^2)) - ((x1 + x0) / (x0 ^2)))) .= k * ((1 / ((x1 ^2) * (x0 - x2))) * ((((x2 + x1) * (x0 ^2)) - ((x1 + x0) * (x2 ^2))) / ((x2 ^2) * (x0 ^2)))) by A2, XCMPLX_1:130 .= k * ((((1 * (x0 - x2)) / ((x1 ^2) * (x0 - x2))) * ((x1 * (x0 + x2)) + (x0 * x2))) / ((x2 ^2) * (x0 ^2))) .= k * (((1 / (x1 ^2)) * (((x1 * x0) + (x1 * x2)) + (x0 * x2))) / ((x2 ^2) * (x0 ^2))) by A5, XCMPLX_1:91 .= k * (((((x1 * x0) + (x1 * x2)) + (x0 * x2)) / (x1 ^2)) / ((x2 ^2) * (x0 ^2))) .= k * ((((x1 * x0) + (x1 * x2)) + (x0 * x2)) / ((x1 ^2) * ((x2 ^2) * (x0 ^2)))) by XCMPLX_1:78 .= k * ((((1 * (x1 * x0)) / (((x1 * (x2 ^2)) * x0) * (x1 * x0))) + ((1 * (x1 * x2)) / (((x1 * x2) * (x0 ^2)) * (x1 * x2)))) + ((1 * (x0 * x2)) / ((((x1 ^2) * x2) * x0) * (x0 * x2)))) .= k * (((1 / ((x1 * (x2 ^2)) * x0)) + ((1 * (x1 * x2)) / (((x1 * x2) * (x0 ^2)) * (x1 * x2)))) + ((1 * (x0 * x2)) / ((((x1 ^2) * x2) * x0) * (x0 * x2)))) by A2, XCMPLX_1:91 .= k * (((1 / ((x1 * (x2 ^2)) * x0)) + (1 / ((x1 * x2) * (x0 ^2)))) + ((1 * (x0 * x2)) / ((((x1 ^2) * x2) * x0) * (x0 * x2)))) by A2, XCMPLX_1:91 .= k * (((1 / (((x1 * x2) * x0) * x2)) + (1 / (((x1 * x2) * x0) * x0))) + (1 / (((x1 * x2) * x0) * x1))) by A2, XCMPLX_1:91 .= k * ((((1 / ((x1 * x2) * x0)) * (1 / x2)) + (1 / (((x1 * x2) * x0) * x0))) + (1 / (((x1 * x2) * x0) * x1))) by XCMPLX_1:102 .= k * ((((1 / ((x1 * x2) * x0)) * (1 / x2)) + ((1 / ((x1 * x2) * x0)) * (1 / x0))) + (1 / (((x1 * x2) * x0) * x1))) by XCMPLX_1:102 .= k * ((((1 / ((x1 * x2) * x0)) * (1 / x2)) + ((1 / ((x1 * x2) * x0)) * (1 / x0))) + ((1 / ((x1 * x2) * x0)) * (1 / x1))) by XCMPLX_1:102 .= (k / ((x0 * x1) * x2)) * (((1 / x0) + (1 / x1)) + (1 / x2)) ; hence [!f,x0,x1,x2!] = (k / ((x0 * x1) * x2)) * (((1 / x0) + (1 / x1)) + (1 / x2)) ; ::_thesis: verum end; theorem :: DIFF_3:50 for k, x, h being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = k / (x ^2) ) & x <> 0 & x + h <> 0 holds (fD (f,h)) . x = (((- k) * h) * ((2 * x) + h)) / (((x ^2) + (h * x)) ^2) proof let k, x, h be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = k / (x ^2) ) & x <> 0 & x + h <> 0 holds (fD (f,h)) . x = (((- k) * h) * ((2 * x) + h)) / (((x ^2) + (h * x)) ^2) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = k / (x ^2) ) & x <> 0 & x + h <> 0 implies (fD (f,h)) . x = (((- k) * h) * ((2 * x) + h)) / (((x ^2) + (h * x)) ^2) ) assume that A1: for x being Real holds f . x = k / (x ^2) and A2: ( x <> 0 & x + h <> 0 ) ; ::_thesis: (fD (f,h)) . x = (((- k) * h) * ((2 * x) + h)) / (((x ^2) + (h * x)) ^2) (fD (f,h)) . x = (f . (x + h)) - (f . x) by DIFF_1:3 .= (k / ((x + h) ^2)) - (f . x) by A1 .= (k / ((x + h) ^2)) - (k / (x ^2)) by A1 .= ((k * (x ^2)) - (k * ((x + h) ^2))) / (((x + h) ^2) * (x ^2)) by A2, XCMPLX_1:130 .= (((- k) * h) * ((2 * x) + h)) / (((x ^2) + (h * x)) ^2) ; hence (fD (f,h)) . x = (((- k) * h) * ((2 * x) + h)) / (((x ^2) + (h * x)) ^2) ; ::_thesis: verum end; theorem :: DIFF_3:51 for k, x, h being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = k / (x ^2) ) & x <> 0 & x - h <> 0 holds (bD (f,h)) . x = (((- k) * h) * ((2 * x) - h)) / (((x ^2) - (x * h)) ^2) proof let k, x, h be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = k / (x ^2) ) & x <> 0 & x - h <> 0 holds (bD (f,h)) . x = (((- k) * h) * ((2 * x) - h)) / (((x ^2) - (x * h)) ^2) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = k / (x ^2) ) & x <> 0 & x - h <> 0 implies (bD (f,h)) . x = (((- k) * h) * ((2 * x) - h)) / (((x ^2) - (x * h)) ^2) ) assume that A1: for x being Real holds f . x = k / (x ^2) and A2: ( x <> 0 & x - h <> 0 ) ; ::_thesis: (bD (f,h)) . x = (((- k) * h) * ((2 * x) - h)) / (((x ^2) - (x * h)) ^2) A3: f . (x - h) = k / ((x - h) ^2) by A1; (bD (f,h)) . x = (f . x) - (f . (x - h)) by DIFF_1:4 .= (k / (x ^2)) - (k / ((x - h) ^2)) by A1, A3 .= ((k * ((x - h) ^2)) - (k * (x ^2))) / ((x ^2) * ((x - h) ^2)) by A2, XCMPLX_1:130 .= (((- k) * h) * ((2 * x) - h)) / (((x ^2) - (x * h)) ^2) ; hence (bD (f,h)) . x = (((- k) * h) * ((2 * x) - h)) / (((x ^2) - (x * h)) ^2) ; ::_thesis: verum end; theorem :: DIFF_3:52 for k, x, h being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = k / (x ^2) ) & x + (h / 2) <> 0 & x - (h / 2) <> 0 holds (cD (f,h)) . x = (- (((2 * h) * k) * x)) / (((x ^2) - ((h / 2) ^2)) ^2) proof let k, x, h be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = k / (x ^2) ) & x + (h / 2) <> 0 & x - (h / 2) <> 0 holds (cD (f,h)) . x = (- (((2 * h) * k) * x)) / (((x ^2) - ((h / 2) ^2)) ^2) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = k / (x ^2) ) & x + (h / 2) <> 0 & x - (h / 2) <> 0 implies (cD (f,h)) . x = (- (((2 * h) * k) * x)) / (((x ^2) - ((h / 2) ^2)) ^2) ) assume that A1: for x being Real holds f . x = k / (x ^2) and A2: ( x + (h / 2) <> 0 & x - (h / 2) <> 0 ) ; ::_thesis: (cD (f,h)) . x = (- (((2 * h) * k) * x)) / (((x ^2) - ((h / 2) ^2)) ^2) (cD (f,h)) . x = (f . (x + (h / 2))) - (f . (x - (h / 2))) by DIFF_1:5 .= (k / ((x + (h / 2)) ^2)) - (f . (x - (h / 2))) by A1 .= (k / ((x + (h / 2)) ^2)) - (k / ((x - (h / 2)) ^2)) by A1 .= ((k * ((x - (h / 2)) ^2)) - (k * ((x + (h / 2)) ^2))) / (((x + (h / 2)) ^2) * ((x - (h / 2)) ^2)) by A2, XCMPLX_1:130 .= (- (((2 * h) * k) * x)) / (((x ^2) - ((h / 2) ^2)) ^2) ; hence (cD (f,h)) . x = (- (((2 * h) * k) * x)) / (((x ^2) - ((h / 2) ^2)) ^2) ; ::_thesis: verum end; theorem :: DIFF_3:53 for x0, x1 being Real holds [!((sin (#) sin) (#) sin),x0,x1!] = ((1 / 2) * (((3 * (cos ((x0 + x1) / 2))) * (sin ((x0 - x1) / 2))) - ((cos ((3 * (x0 + x1)) / 2)) * (sin ((3 * (x0 - x1)) / 2))))) / (x0 - x1) proof let x0, x1 be Real; ::_thesis: [!((sin (#) sin) (#) sin),x0,x1!] = ((1 / 2) * (((3 * (cos ((x0 + x1) / 2))) * (sin ((x0 - x1) / 2))) - ((cos ((3 * (x0 + x1)) / 2)) * (sin ((3 * (x0 - x1)) / 2))))) / (x0 - x1) set y = 3 * x0; set z = 3 * x1; [!((sin (#) sin) (#) sin),x0,x1!] = ((((sin (#) sin) . x0) * (sin . x0)) - (((sin (#) sin) (#) sin) . x1)) / (x0 - x1) by VALUED_1:5 .= ((((sin . x0) * (sin . x0)) * (sin . x0)) - (((sin (#) sin) (#) sin) . x1)) / (x0 - x1) by VALUED_1:5 .= ((((sin . x0) * (sin . x0)) * (sin . x0)) - (((sin (#) sin) . x1) * (sin . x1))) / (x0 - x1) by VALUED_1:5 .= ((((sin x0) * (sin x0)) * (sin x0)) - (((sin x1) * (sin x1)) * (sin x1))) / (x0 - x1) by VALUED_1:5 .= (((1 / 4) * ((((sin ((x0 + x0) - x0)) + (sin ((x0 + x0) - x0))) + (sin ((x0 + x0) - x0))) - (sin ((x0 + x0) + x0)))) - (((sin x1) * (sin x1)) * (sin x1))) / (x0 - x1) by SIN_COS4:33 .= (((1 / 4) * ((3 * (sin x0)) - (sin (3 * x0)))) - ((1 / 4) * ((((sin ((x1 + x1) - x1)) + (sin ((x1 + x1) - x1))) + (sin ((x1 + x1) - x1))) - (sin ((x1 + x1) + x1))))) / (x0 - x1) by SIN_COS4:33 .= ((1 / 4) * ((3 * ((sin x0) - (sin x1))) - ((sin (3 * x0)) - (sin (3 * x1))))) / (x0 - x1) .= ((1 / 4) * ((3 * (2 * ((cos ((x0 + x1) / 2)) * (sin ((x0 - x1) / 2))))) - ((sin (3 * x0)) - (sin (3 * x1))))) / (x0 - x1) by SIN_COS4:16 .= ((1 / 4) * ((3 * (2 * ((cos ((x0 + x1) / 2)) * (sin ((x0 - x1) / 2))))) - (2 * ((cos (((3 * x0) + (3 * x1)) / 2)) * (sin (((3 * x0) - (3 * x1)) / 2)))))) / (x0 - x1) by SIN_COS4:16 .= ((1 / 2) * (((3 * (cos ((x0 + x1) / 2))) * (sin ((x0 - x1) / 2))) - ((cos ((3 * (x0 + x1)) / 2)) * (sin ((3 * (x0 - x1)) / 2))))) / (x0 - x1) ; hence [!((sin (#) sin) (#) sin),x0,x1!] = ((1 / 2) * (((3 * (cos ((x0 + x1) / 2))) * (sin ((x0 - x1) / 2))) - ((cos ((3 * (x0 + x1)) / 2)) * (sin ((3 * (x0 - x1)) / 2))))) / (x0 - x1) ; ::_thesis: verum end; theorem :: DIFF_3:54 for h, x being Real holds (fD (((sin (#) sin) (#) sin),h)) . x = (1 / 2) * (((3 * (cos (((2 * x) + h) / 2))) * (sin (h / 2))) - ((cos ((3 * ((2 * x) + h)) / 2)) * (sin ((3 * h) / 2)))) proof let h, x be Real; ::_thesis: (fD (((sin (#) sin) (#) sin),h)) . x = (1 / 2) * (((3 * (cos (((2 * x) + h) / 2))) * (sin (h / 2))) - ((cos ((3 * ((2 * x) + h)) / 2)) * (sin ((3 * h) / 2)))) (fD (((sin (#) sin) (#) sin),h)) . x = (((sin (#) sin) (#) sin) . (x + h)) - (((sin (#) sin) (#) sin) . x) by DIFF_1:3 .= (((sin (#) sin) . (x + h)) * (sin . (x + h))) - (((sin (#) sin) (#) sin) . x) by VALUED_1:5 .= (((sin . (x + h)) * (sin . (x + h))) * (sin . (x + h))) - (((sin (#) sin) (#) sin) . x) by VALUED_1:5 .= (((sin . (x + h)) * (sin . (x + h))) * (sin . (x + h))) - (((sin (#) sin) . x) * (sin . x)) by VALUED_1:5 .= (((sin (x + h)) * (sin (x + h))) * (sin (x + h))) - (((sin x) * (sin x)) * (sin x)) by VALUED_1:5 .= ((1 / 4) * ((((sin (((x + h) + (x + h)) - (x + h))) + (sin (((x + h) + (x + h)) - (x + h)))) + (sin (((x + h) + (x + h)) - (x + h)))) - (sin (((x + h) + (x + h)) + (x + h))))) - (((sin x) * (sin x)) * (sin x)) by SIN_COS4:33 .= ((1 / 4) * ((((sin (x + h)) + (sin (x + h))) + (sin (x + h))) - (sin (3 * (x + h))))) - ((1 / 4) * ((((sin ((x + x) - x)) + (sin ((x + x) - x))) + (sin ((x + x) - x))) - (sin ((x + x) + x)))) by SIN_COS4:33 .= (1 / 4) * ((3 * ((sin (x + h)) - (sin x))) - ((sin (3 * (x + h))) - (sin (3 * x)))) .= (1 / 4) * ((3 * (2 * ((cos (((x + h) + x) / 2)) * (sin (((x + h) - x) / 2))))) - ((sin (3 * (x + h))) - (sin (3 * x)))) by SIN_COS4:16 .= (1 / 4) * ((3 * (2 * ((cos (((x + h) + x) / 2)) * (sin (((x + h) - x) / 2))))) - (2 * ((cos (((3 * (x + h)) + (3 * x)) / 2)) * (sin (((3 * (x + h)) - (3 * x)) / 2))))) by SIN_COS4:16 .= (1 / 2) * (((3 * (cos (((2 * x) + h) / 2))) * (sin (h / 2))) - ((cos ((3 * ((2 * x) + h)) / 2)) * (sin ((3 * h) / 2)))) ; hence (fD (((sin (#) sin) (#) sin),h)) . x = (1 / 2) * (((3 * (cos (((2 * x) + h) / 2))) * (sin (h / 2))) - ((cos ((3 * ((2 * x) + h)) / 2)) * (sin ((3 * h) / 2)))) ; ::_thesis: verum end; theorem :: DIFF_3:55 for h, x being Real holds (bD (((sin (#) sin) (#) sin),h)) . x = (1 / 2) * (((3 * (cos (((2 * x) - h) / 2))) * (sin (h / 2))) - ((cos ((3 * ((2 * x) - h)) / 2)) * (sin ((3 * h) / 2)))) proof let h, x be Real; ::_thesis: (bD (((sin (#) sin) (#) sin),h)) . x = (1 / 2) * (((3 * (cos (((2 * x) - h) / 2))) * (sin (h / 2))) - ((cos ((3 * ((2 * x) - h)) / 2)) * (sin ((3 * h) / 2)))) (bD (((sin (#) sin) (#) sin),h)) . x = (((sin (#) sin) (#) sin) . x) - (((sin (#) sin) (#) sin) . (x - h)) by DIFF_1:4 .= (((sin (#) sin) . x) * (sin . x)) - (((sin (#) sin) (#) sin) . (x - h)) by VALUED_1:5 .= (((sin . x) * (sin . x)) * (sin . x)) - (((sin (#) sin) (#) sin) . (x - h)) by VALUED_1:5 .= (((sin . x) * (sin . x)) * (sin . x)) - (((sin (#) sin) . (x - h)) * (sin . (x - h))) by VALUED_1:5 .= (((sin x) * (sin x)) * (sin x)) - (((sin (x - h)) * (sin (x - h))) * (sin (x - h))) by VALUED_1:5 .= ((1 / 4) * ((((sin ((x + x) - x)) + (sin ((x + x) - x))) + (sin ((x + x) - x))) - (sin ((x + x) + x)))) - (((sin (x - h)) * (sin (x - h))) * (sin (x - h))) by SIN_COS4:33 .= ((1 / 4) * ((((sin x) + (sin x)) + (sin x)) - (sin (3 * x)))) - ((1 / 4) * ((((sin (((x - h) + (x - h)) - (x - h))) + (sin (((x - h) + (x - h)) - (x - h)))) + (sin (((x - h) + (x - h)) - (x - h)))) - (sin (((x - h) + (x - h)) + (x - h))))) by SIN_COS4:33 .= (1 / 4) * ((3 * ((sin x) - (sin (x - h)))) - ((sin (3 * x)) - (sin (3 * (x - h))))) .= (1 / 4) * ((3 * (2 * ((cos ((x + (x - h)) / 2)) * (sin ((x - (x - h)) / 2))))) - ((sin (3 * x)) - (sin (3 * (x - h))))) by SIN_COS4:16 .= (1 / 4) * ((3 * (2 * ((cos (((2 * x) - h) / 2)) * (sin (h / 2))))) - (2 * ((cos (((3 * x) + (3 * (x - h))) / 2)) * (sin (((3 * x) - (3 * (x - h))) / 2))))) by SIN_COS4:16 .= (1 / 2) * (((3 * (cos (((2 * x) - h) / 2))) * (sin (h / 2))) - ((cos ((3 * ((2 * x) - h)) / 2)) * (sin ((3 * h) / 2)))) ; hence (bD (((sin (#) sin) (#) sin),h)) . x = (1 / 2) * (((3 * (cos (((2 * x) - h) / 2))) * (sin (h / 2))) - ((cos ((3 * ((2 * x) - h)) / 2)) * (sin ((3 * h) / 2)))) ; ::_thesis: verum end; theorem :: DIFF_3:56 for h, x being Real holds (cD (((sin (#) sin) (#) sin),h)) . x = (1 / 2) * (((3 * (cos x)) * (sin (h / 2))) - ((cos (3 * x)) * (sin ((3 * h) / 2)))) proof let h, x be Real; ::_thesis: (cD (((sin (#) sin) (#) sin),h)) . x = (1 / 2) * (((3 * (cos x)) * (sin (h / 2))) - ((cos (3 * x)) * (sin ((3 * h) / 2)))) (cD (((sin (#) sin) (#) sin),h)) . x = (((sin (#) sin) (#) sin) . (x + (h / 2))) - (((sin (#) sin) (#) sin) . (x - (h / 2))) by DIFF_1:5 .= (((sin (#) sin) . (x + (h / 2))) * (sin . (x + (h / 2)))) - (((sin (#) sin) (#) sin) . (x - (h / 2))) by VALUED_1:5 .= (((sin . (x + (h / 2))) * (sin . (x + (h / 2)))) * (sin . (x + (h / 2)))) - (((sin (#) sin) (#) sin) . (x - (h / 2))) by VALUED_1:5 .= (((sin . (x + (h / 2))) * (sin . (x + (h / 2)))) * (sin . (x + (h / 2)))) - (((sin (#) sin) . (x - (h / 2))) * (sin . (x - (h / 2)))) by VALUED_1:5 .= (((sin (x + (h / 2))) * (sin (x + (h / 2)))) * (sin (x + (h / 2)))) - (((sin (x - (h / 2))) * (sin (x - (h / 2)))) * (sin (x - (h / 2)))) by VALUED_1:5 .= ((1 / 4) * ((((sin (((x + (h / 2)) + (x + (h / 2))) - (x + (h / 2)))) + (sin (((x + (h / 2)) + (x + (h / 2))) - (x + (h / 2))))) + (sin (((x + (h / 2)) + (x + (h / 2))) - (x + (h / 2))))) - (sin (((x + (h / 2)) + (x + (h / 2))) + (x + (h / 2)))))) - (((sin (x - (h / 2))) * (sin (x - (h / 2)))) * (sin (x - (h / 2)))) by SIN_COS4:33 .= ((1 / 4) * ((((sin (x + (h / 2))) + (sin (x + (h / 2)))) + (sin (x + (h / 2)))) - (sin (3 * (x + (h / 2)))))) - ((1 / 4) * ((((sin (((x - (h / 2)) + (x - (h / 2))) - (x - (h / 2)))) + (sin (((x - (h / 2)) + (x - (h / 2))) - (x - (h / 2))))) + (sin (((x - (h / 2)) + (x - (h / 2))) - (x - (h / 2))))) - (sin (((x - (h / 2)) + (x - (h / 2))) + (x - (h / 2)))))) by SIN_COS4:33 .= (1 / 4) * ((3 * ((sin (x + (h / 2))) - (sin (x - (h / 2))))) - ((sin (3 * (x + (h / 2)))) - (sin (3 * (x - (h / 2)))))) .= (1 / 4) * ((3 * (2 * ((cos (((x + (h / 2)) + (x - (h / 2))) / 2)) * (sin (((x + (h / 2)) - (x - (h / 2))) / 2))))) - ((sin (3 * (x + (h / 2)))) - (sin (3 * (x - (h / 2)))))) by SIN_COS4:16 .= (1 / 4) * ((3 * (2 * ((cos ((2 * x) / 2)) * (sin (h / 2))))) - (2 * ((cos (((3 * (x + (h / 2))) + (3 * (x - (h / 2)))) / 2)) * (sin (((3 * (x + (h / 2))) - (3 * (x - (h / 2)))) / 2))))) by SIN_COS4:16 .= (1 / 2) * (((3 * (cos x)) * (sin (h / 2))) - ((cos (3 * x)) * (sin ((3 * h) / 2)))) ; hence (cD (((sin (#) sin) (#) sin),h)) . x = (1 / 2) * (((3 * (cos x)) * (sin (h / 2))) - ((cos (3 * x)) * (sin ((3 * h) / 2)))) ; ::_thesis: verum end; theorem :: DIFF_3:57 for x0, x1 being Real holds [!((cos (#) cos) (#) cos),x0,x1!] = - (((1 / 2) * (((3 * (sin ((x0 + x1) / 2))) * (sin ((x0 - x1) / 2))) + ((sin (((3 * x0) + (3 * x1)) / 2)) * (sin (((3 * x0) - (3 * x1)) / 2))))) / (x0 - x1)) proof let x0, x1 be Real; ::_thesis: [!((cos (#) cos) (#) cos),x0,x1!] = - (((1 / 2) * (((3 * (sin ((x0 + x1) / 2))) * (sin ((x0 - x1) / 2))) + ((sin (((3 * x0) + (3 * x1)) / 2)) * (sin (((3 * x0) - (3 * x1)) / 2))))) / (x0 - x1)) [!((cos (#) cos) (#) cos),x0,x1!] = ((((cos (#) cos) . x0) * (cos . x0)) - (((cos (#) cos) (#) cos) . x1)) / (x0 - x1) by VALUED_1:5 .= ((((cos . x0) * (cos . x0)) * (cos . x0)) - (((cos (#) cos) (#) cos) . x1)) / (x0 - x1) by VALUED_1:5 .= ((((cos . x0) * (cos . x0)) * (cos . x0)) - (((cos (#) cos) . x1) * (cos . x1))) / (x0 - x1) by VALUED_1:5 .= ((((cos x0) * (cos x0)) * (cos x0)) - (((cos x1) * (cos x1)) * (cos x1))) / (x0 - x1) by VALUED_1:5 .= (((1 / 4) * ((((cos ((x0 + x0) - x0)) + (cos ((x0 + x0) - x0))) + (cos ((x0 + x0) - x0))) + (cos ((x0 + x0) + x0)))) - (((cos x1) * (cos x1)) * (cos x1))) / (x0 - x1) by SIN_COS4:36 .= (((1 / 4) * ((((cos x0) + (cos x0)) + (cos x0)) + (cos (3 * x0)))) - ((1 / 4) * ((((cos ((x1 + x1) - x1)) + (cos ((x1 + x1) - x1))) + (cos ((x1 + x1) - x1))) + (cos ((x1 + x1) + x1))))) / (x0 - x1) by SIN_COS4:36 .= ((1 / 4) * ((3 * ((cos x0) - (cos x1))) + ((cos (3 * x0)) - (cos (3 * x1))))) / (x0 - x1) .= ((1 / 4) * ((3 * (- (2 * ((sin ((x0 + x1) / 2)) * (sin ((x0 - x1) / 2)))))) + ((cos (3 * x0)) - (cos (3 * x1))))) / (x0 - x1) by SIN_COS4:18 .= ((1 / 4) * (((3 * (- 2)) * ((sin ((x0 + x1) / 2)) * (sin ((x0 - x1) / 2)))) + (- (2 * ((sin (((3 * x0) + (3 * x1)) / 2)) * (sin (((3 * x0) - (3 * x1)) / 2))))))) / (x0 - x1) by SIN_COS4:18 .= (- ((1 / 2) * (((3 * (sin ((x0 + x1) / 2))) * (sin ((x0 - x1) / 2))) + ((sin (((3 * x0) + (3 * x1)) / 2)) * (sin (((3 * x0) - (3 * x1)) / 2)))))) / (x0 - x1) ; hence [!((cos (#) cos) (#) cos),x0,x1!] = - (((1 / 2) * (((3 * (sin ((x0 + x1) / 2))) * (sin ((x0 - x1) / 2))) + ((sin (((3 * x0) + (3 * x1)) / 2)) * (sin (((3 * x0) - (3 * x1)) / 2))))) / (x0 - x1)) ; ::_thesis: verum end; theorem :: DIFF_3:58 for h, x being Real holds (fD (((cos (#) cos) (#) cos),h)) . x = - ((1 / 2) * (((3 * (sin (((2 * x) + h) / 2))) * (sin (h / 2))) + ((sin ((3 * ((2 * x) + h)) / 2)) * (sin ((3 * h) / 2))))) proof let h, x be Real; ::_thesis: (fD (((cos (#) cos) (#) cos),h)) . x = - ((1 / 2) * (((3 * (sin (((2 * x) + h) / 2))) * (sin (h / 2))) + ((sin ((3 * ((2 * x) + h)) / 2)) * (sin ((3 * h) / 2))))) (fD (((cos (#) cos) (#) cos),h)) . x = (((cos (#) cos) (#) cos) . (x + h)) - (((cos (#) cos) (#) cos) . x) by DIFF_1:3 .= (((cos (#) cos) . (x + h)) * (cos . (x + h))) - (((cos (#) cos) (#) cos) . x) by VALUED_1:5 .= (((cos . (x + h)) * (cos . (x + h))) * (cos . (x + h))) - (((cos (#) cos) (#) cos) . x) by VALUED_1:5 .= (((cos . (x + h)) * (cos . (x + h))) * (cos . (x + h))) - (((cos (#) cos) . x) * (cos . x)) by VALUED_1:5 .= (((cos (x + h)) * (cos (x + h))) * (cos (x + h))) - (((cos x) * (cos x)) * (cos x)) by VALUED_1:5 .= ((1 / 4) * ((((cos (((x + h) + (x + h)) - (x + h))) + (cos (((x + h) + (x + h)) - (x + h)))) + (cos (((x + h) + (x + h)) - (x + h)))) + (cos (((x + h) + (x + h)) + (x + h))))) - (((cos x) * (cos x)) * (cos x)) by SIN_COS4:36 .= ((1 / 4) * ((((cos (x + h)) + (cos (x + h))) + (cos (x + h))) + (cos (3 * (x + h))))) - ((1 / 4) * ((((cos ((x + x) - x)) + (cos ((x + x) - x))) + (cos ((x + x) - x))) + (cos ((x + x) + x)))) by SIN_COS4:36 .= (1 / 4) * ((3 * ((cos (x + h)) - (cos x))) + ((cos (3 * (x + h))) - (cos (3 * x)))) .= (1 / 4) * ((3 * (- (2 * ((sin (((x + h) + x) / 2)) * (sin (((x + h) - x) / 2)))))) + ((cos (3 * (x + h))) - (cos (3 * x)))) by SIN_COS4:18 .= (1 / 4) * ((3 * (- (2 * ((sin (((2 * x) + h) / 2)) * (sin (h / 2)))))) + (- (2 * ((sin (((3 * (x + h)) + (3 * x)) / 2)) * (sin (((3 * (x + h)) - (3 * x)) / 2)))))) by SIN_COS4:18 .= (- (1 / 2)) * (((3 * (sin (((2 * x) + h) / 2))) * (sin (h / 2))) + ((sin ((3 * ((2 * x) + h)) / 2)) * (sin ((3 * h) / 2)))) ; hence (fD (((cos (#) cos) (#) cos),h)) . x = - ((1 / 2) * (((3 * (sin (((2 * x) + h) / 2))) * (sin (h / 2))) + ((sin ((3 * ((2 * x) + h)) / 2)) * (sin ((3 * h) / 2))))) ; ::_thesis: verum end; theorem :: DIFF_3:59 for h, x being Real holds (bD (((cos (#) cos) (#) cos),h)) . x = - ((1 / 2) * (((3 * (sin (((2 * x) - h) / 2))) * (sin (h / 2))) + ((sin ((3 * ((2 * x) - h)) / 2)) * (sin ((3 * h) / 2))))) proof let h, x be Real; ::_thesis: (bD (((cos (#) cos) (#) cos),h)) . x = - ((1 / 2) * (((3 * (sin (((2 * x) - h) / 2))) * (sin (h / 2))) + ((sin ((3 * ((2 * x) - h)) / 2)) * (sin ((3 * h) / 2))))) (bD (((cos (#) cos) (#) cos),h)) . x = (((cos (#) cos) (#) cos) . x) - (((cos (#) cos) (#) cos) . (x - h)) by DIFF_1:4 .= (((cos (#) cos) . x) * (cos . x)) - (((cos (#) cos) (#) cos) . (x - h)) by VALUED_1:5 .= (((cos . x) * (cos . x)) * (cos . x)) - (((cos (#) cos) (#) cos) . (x - h)) by VALUED_1:5 .= (((cos . x) * (cos . x)) * (cos . x)) - (((cos (#) cos) . (x - h)) * (cos . (x - h))) by VALUED_1:5 .= (((cos x) * (cos x)) * (cos x)) - (((cos (x - h)) * (cos (x - h))) * (cos (x - h))) by VALUED_1:5 .= ((1 / 4) * ((((cos ((x + x) - x)) + (cos ((x + x) - x))) + (cos ((x + x) - x))) + (cos ((x + x) + x)))) - (((cos (x - h)) * (cos (x - h))) * (cos (x - h))) by SIN_COS4:36 .= ((1 / 4) * ((((cos x) + (cos x)) + (cos x)) + (cos (3 * x)))) - ((1 / 4) * ((((cos (((x - h) + (x - h)) - (x - h))) + (cos (((x - h) + (x - h)) - (x - h)))) + (cos (((x - h) + (x - h)) - (x - h)))) + (cos (((x - h) + (x - h)) + (x - h))))) by SIN_COS4:36 .= (1 / 4) * ((3 * ((cos x) - (cos (x - h)))) + ((cos (3 * x)) - (cos (3 * (x - h))))) .= (1 / 4) * ((3 * (- (2 * ((sin ((x + (x - h)) / 2)) * (sin ((x - (x - h)) / 2)))))) + ((cos (3 * x)) - (cos (3 * (x - h))))) by SIN_COS4:18 .= (1 / 4) * ((3 * (- (2 * ((sin (((2 * x) - h) / 2)) * (sin (h / 2)))))) + (- (2 * ((sin (((3 * x) + (3 * (x - h))) / 2)) * (sin (((3 * x) - (3 * (x - h))) / 2)))))) by SIN_COS4:18 .= (- (1 / 2)) * ((3 * ((sin (((2 * x) - h) / 2)) * (sin (h / 2)))) + ((sin ((3 * ((2 * x) - h)) / 2)) * (sin ((3 * h) / 2)))) ; hence (bD (((cos (#) cos) (#) cos),h)) . x = - ((1 / 2) * (((3 * (sin (((2 * x) - h) / 2))) * (sin (h / 2))) + ((sin ((3 * ((2 * x) - h)) / 2)) * (sin ((3 * h) / 2))))) ; ::_thesis: verum end; theorem :: DIFF_3:60 for h, x being Real holds (cD (((cos (#) cos) (#) cos),h)) . x = - ((1 / 2) * (((3 * (sin x)) * (sin (h / 2))) + ((sin (3 * x)) * (sin ((3 * h) / 2))))) proof let h, x be Real; ::_thesis: (cD (((cos (#) cos) (#) cos),h)) . x = - ((1 / 2) * (((3 * (sin x)) * (sin (h / 2))) + ((sin (3 * x)) * (sin ((3 * h) / 2))))) (cD (((cos (#) cos) (#) cos),h)) . x = (((cos (#) cos) (#) cos) . (x + (h / 2))) - (((cos (#) cos) (#) cos) . (x - (h / 2))) by DIFF_1:5 .= (((cos (#) cos) . (x + (h / 2))) * (cos . (x + (h / 2)))) - (((cos (#) cos) (#) cos) . (x - (h / 2))) by VALUED_1:5 .= (((cos . (x + (h / 2))) * (cos . (x + (h / 2)))) * (cos . (x + (h / 2)))) - (((cos (#) cos) (#) cos) . (x - (h / 2))) by VALUED_1:5 .= (((cos . (x + (h / 2))) * (cos . (x + (h / 2)))) * (cos . (x + (h / 2)))) - (((cos (#) cos) . (x - (h / 2))) * (cos . (x - (h / 2)))) by VALUED_1:5 .= (((cos (x + (h / 2))) * (cos (x + (h / 2)))) * (cos (x + (h / 2)))) - (((cos (x - (h / 2))) * (cos (x - (h / 2)))) * (cos (x - (h / 2)))) by VALUED_1:5 .= ((1 / 4) * ((((cos (((x + (h / 2)) + (x + (h / 2))) - (x + (h / 2)))) + (cos (((x + (h / 2)) + (x + (h / 2))) - (x + (h / 2))))) + (cos (((x + (h / 2)) + (x + (h / 2))) - (x + (h / 2))))) + (cos (((x + (h / 2)) + (x + (h / 2))) + (x + (h / 2)))))) - (((cos (x - (h / 2))) * (cos (x - (h / 2)))) * (cos (x - (h / 2)))) by SIN_COS4:36 .= ((1 / 4) * ((((cos (x + (h / 2))) + (cos (x + (h / 2)))) + (cos (x + (h / 2)))) + (cos (3 * (x + (h / 2)))))) - ((1 / 4) * ((((cos (((x - (h / 2)) + (x - (h / 2))) - (x - (h / 2)))) + (cos (((x - (h / 2)) + (x - (h / 2))) - (x - (h / 2))))) + (cos (((x - (h / 2)) + (x - (h / 2))) - (x - (h / 2))))) + (cos (((x - (h / 2)) + (x - (h / 2))) + (x - (h / 2)))))) by SIN_COS4:36 .= (1 / 4) * ((3 * ((cos (x + (h / 2))) - (cos (x - (h / 2))))) + ((cos (3 * (x + (h / 2)))) - (cos (3 * (x - (h / 2)))))) .= (1 / 4) * ((3 * (- (2 * ((sin (((x + (h / 2)) + (x - (h / 2))) / 2)) * (sin (((x + (h / 2)) - (x - (h / 2))) / 2)))))) + ((cos (3 * (x + (h / 2)))) - (cos (3 * (x - (h / 2)))))) by SIN_COS4:18 .= (1 / 4) * ((3 * (- (2 * ((sin ((2 * x) / 2)) * (sin (h / 2)))))) + (- (2 * ((sin (((3 * (x + (h / 2))) + (3 * (x - (h / 2)))) / 2)) * (sin (((3 * (x + (h / 2))) - (3 * (x - (h / 2)))) / 2)))))) by SIN_COS4:18 .= - ((1 / 2) * ((3 * ((sin x) * (sin (h / 2)))) + ((sin (3 * x)) * (sin ((3 * h) / 2))))) ; hence (cD (((cos (#) cos) (#) cos),h)) . x = - ((1 / 2) * (((3 * (sin x)) * (sin (h / 2))) + ((sin (3 * x)) * (sin ((3 * h) / 2))))) ; ::_thesis: verum end; theorem :: DIFF_3:61 for x0, x1 being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = 1 / (sin x) ) & sin x0 <> 0 & sin x1 <> 0 holds [!f,x0,x1!] = - (((2 * ((sin x1) - (sin x0))) / ((cos (x0 + x1)) - (cos (x0 - x1)))) / (x0 - x1)) proof let x0, x1 be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = 1 / (sin x) ) & sin x0 <> 0 & sin x1 <> 0 holds [!f,x0,x1!] = - (((2 * ((sin x1) - (sin x0))) / ((cos (x0 + x1)) - (cos (x0 - x1)))) / (x0 - x1)) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = 1 / (sin x) ) & sin x0 <> 0 & sin x1 <> 0 implies [!f,x0,x1!] = - (((2 * ((sin x1) - (sin x0))) / ((cos (x0 + x1)) - (cos (x0 - x1)))) / (x0 - x1)) ) assume that A1: for x being Real holds f . x = 1 / (sin x) and A2: ( sin x0 <> 0 & sin x1 <> 0 ) ; ::_thesis: [!f,x0,x1!] = - (((2 * ((sin x1) - (sin x0))) / ((cos (x0 + x1)) - (cos (x0 - x1)))) / (x0 - x1)) ( f . x0 = 1 / (sin x0) & f . x1 = 1 / (sin x1) ) by A1; then [!f,x0,x1!] = (((1 * (sin x1)) - (1 * (sin x0))) / ((sin x0) * (sin x1))) / (x0 - x1) by A2, XCMPLX_1:130 .= (((sin x1) - (sin x0)) / (- ((1 / 2) * ((cos (x0 + x1)) - (cos (x0 - x1)))))) / (x0 - x1) by SIN_COS4:29 .= (((sin x1) - (sin x0)) / ((- (1 / 2)) * ((cos (x0 + x1)) - (cos (x0 - x1))))) / (x0 - x1) .= ((((sin x1) - (sin x0)) / (- (1 / 2))) / ((cos (x0 + x1)) - (cos (x0 - x1)))) / (x0 - x1) by XCMPLX_1:78 .= (((- 2) * ((sin x1) - (sin x0))) / ((cos (x0 + x1)) - (cos (x0 - x1)))) / (x0 - x1) ; hence [!f,x0,x1!] = - (((2 * ((sin x1) - (sin x0))) / ((cos (x0 + x1)) - (cos (x0 - x1)))) / (x0 - x1)) ; ::_thesis: verum end; theorem :: DIFF_3:62 for x, h being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = 1 / (sin x) ) & sin x <> 0 & sin (x + h) <> 0 holds (fD (f,h)) . x = - ((2 * ((sin x) - (sin (x + h)))) / ((cos ((2 * x) + h)) - (cos h))) proof let x, h be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = 1 / (sin x) ) & sin x <> 0 & sin (x + h) <> 0 holds (fD (f,h)) . x = - ((2 * ((sin x) - (sin (x + h)))) / ((cos ((2 * x) + h)) - (cos h))) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = 1 / (sin x) ) & sin x <> 0 & sin (x + h) <> 0 implies (fD (f,h)) . x = - ((2 * ((sin x) - (sin (x + h)))) / ((cos ((2 * x) + h)) - (cos h))) ) assume that A1: for x being Real holds f . x = 1 / (sin x) and A2: ( sin x <> 0 & sin (x + h) <> 0 ) ; ::_thesis: (fD (f,h)) . x = - ((2 * ((sin x) - (sin (x + h)))) / ((cos ((2 * x) + h)) - (cos h))) f . (x + h) = 1 / (sin (x + h)) by A1; then (fD (f,h)) . x = (1 / (sin (x + h))) - (f . x) by DIFF_1:3 .= (1 / (sin (x + h))) - (1 / (sin x)) by A1 .= ((1 * (sin x)) - (1 * (sin (x + h)))) / ((sin (x + h)) * (sin x)) by A2, XCMPLX_1:130 .= ((sin x) - (sin (x + h))) / (- ((1 / 2) * ((cos ((x + h) + x)) - (cos ((x + h) - x))))) by SIN_COS4:29 .= ((sin x) - (sin (x + h))) / ((- (1 / 2)) * ((cos ((x + h) + x)) - (cos ((x + h) - x)))) .= (((sin x) - (sin (x + h))) / (- (1 / 2))) / ((cos ((2 * x) + h)) - (cos h)) by XCMPLX_1:78 .= (- 2) * (((sin x) - (sin (x + h))) / ((cos ((2 * x) + h)) - (cos h))) ; hence (fD (f,h)) . x = - ((2 * ((sin x) - (sin (x + h)))) / ((cos ((2 * x) + h)) - (cos h))) ; ::_thesis: verum end; theorem :: DIFF_3:63 for x, h being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = 1 / (sin x) ) & sin x <> 0 & sin (x - h) <> 0 holds (bD (f,h)) . x = ((- 2) * ((sin (x - h)) - (sin x))) / ((cos ((2 * x) - h)) - (cos h)) proof let x, h be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = 1 / (sin x) ) & sin x <> 0 & sin (x - h) <> 0 holds (bD (f,h)) . x = ((- 2) * ((sin (x - h)) - (sin x))) / ((cos ((2 * x) - h)) - (cos h)) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = 1 / (sin x) ) & sin x <> 0 & sin (x - h) <> 0 implies (bD (f,h)) . x = ((- 2) * ((sin (x - h)) - (sin x))) / ((cos ((2 * x) - h)) - (cos h)) ) assume that A1: for x being Real holds f . x = 1 / (sin x) and A2: ( sin x <> 0 & sin (x - h) <> 0 ) ; ::_thesis: (bD (f,h)) . x = ((- 2) * ((sin (x - h)) - (sin x))) / ((cos ((2 * x) - h)) - (cos h)) (bD (f,h)) . x = (f . x) - (f . (x - h)) by DIFF_1:4 .= (1 / (sin x)) - (f . (x - h)) by A1 .= (1 / (sin x)) - (1 / (sin (x - h))) by A1 .= ((1 * (sin (x - h))) - (1 * (sin x))) / ((sin x) * (sin (x - h))) by A2, XCMPLX_1:130 .= ((sin (x - h)) - (sin x)) / (- ((1 / 2) * ((cos (x + (x - h))) - (cos (x - (x - h)))))) by SIN_COS4:29 .= ((sin (x - h)) - (sin x)) / ((- (1 / 2)) * ((cos ((2 * x) - h)) - (cos h))) .= (((sin (x - h)) - (sin x)) / (- (1 / 2))) / ((cos ((2 * x) - h)) - (cos h)) by XCMPLX_1:78 .= (- 2) * (((sin (x - h)) - (sin x)) / ((cos ((2 * x) - h)) - (cos h))) ; hence (bD (f,h)) . x = ((- 2) * ((sin (x - h)) - (sin x))) / ((cos ((2 * x) - h)) - (cos h)) ; ::_thesis: verum end; theorem :: DIFF_3:64 for x, h being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = 1 / (sin x) ) & sin (x + (h / 2)) <> 0 & sin (x - (h / 2)) <> 0 holds (cD (f,h)) . x = - ((2 * ((sin (x - (h / 2))) - (sin (x + (h / 2))))) / ((cos (2 * x)) - (cos h))) proof let x, h be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = 1 / (sin x) ) & sin (x + (h / 2)) <> 0 & sin (x - (h / 2)) <> 0 holds (cD (f,h)) . x = - ((2 * ((sin (x - (h / 2))) - (sin (x + (h / 2))))) / ((cos (2 * x)) - (cos h))) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = 1 / (sin x) ) & sin (x + (h / 2)) <> 0 & sin (x - (h / 2)) <> 0 implies (cD (f,h)) . x = - ((2 * ((sin (x - (h / 2))) - (sin (x + (h / 2))))) / ((cos (2 * x)) - (cos h))) ) assume that A1: for x being Real holds f . x = 1 / (sin x) and A2: ( sin (x + (h / 2)) <> 0 & sin (x - (h / 2)) <> 0 ) ; ::_thesis: (cD (f,h)) . x = - ((2 * ((sin (x - (h / 2))) - (sin (x + (h / 2))))) / ((cos (2 * x)) - (cos h))) (cD (f,h)) . x = (f . (x + (h / 2))) - (f . (x - (h / 2))) by DIFF_1:5 .= (1 / (sin (x + (h / 2)))) - (f . (x - (h / 2))) by A1 .= (1 / (sin (x + (h / 2)))) - (1 / (sin (x - (h / 2)))) by A1 .= ((1 * (sin (x - (h / 2)))) - (1 * (sin (x + (h / 2))))) / ((sin (x + (h / 2))) * (sin (x - (h / 2)))) by A2, XCMPLX_1:130 .= ((sin (x - (h / 2))) - (sin (x + (h / 2)))) / (- ((1 / 2) * ((cos ((x + (h / 2)) + (x - (h / 2)))) - (cos ((x + (h / 2)) - (x - (h / 2))))))) by SIN_COS4:29 .= ((sin (x - (h / 2))) - (sin (x + (h / 2)))) / ((- (1 / 2)) * ((cos (2 * x)) - (cos h))) .= (((sin (x - (h / 2))) - (sin (x + (h / 2)))) / (- (1 / 2))) / ((cos (2 * x)) - (cos h)) by XCMPLX_1:78 .= (- 2) * (((sin (x - (h / 2))) - (sin (x + (h / 2)))) / ((cos (2 * x)) - (cos h))) ; hence (cD (f,h)) . x = - ((2 * ((sin (x - (h / 2))) - (sin (x + (h / 2))))) / ((cos (2 * x)) - (cos h))) ; ::_thesis: verum end; theorem :: DIFF_3:65 for x0, x1 being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = 1 / (cos x) ) & x0 <> x1 & cos x0 <> 0 & cos x1 <> 0 holds [!f,x0,x1!] = ((2 * ((cos x1) - (cos x0))) / ((cos (x0 + x1)) + (cos (x0 - x1)))) / (x0 - x1) proof let x0, x1 be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = 1 / (cos x) ) & x0 <> x1 & cos x0 <> 0 & cos x1 <> 0 holds [!f,x0,x1!] = ((2 * ((cos x1) - (cos x0))) / ((cos (x0 + x1)) + (cos (x0 - x1)))) / (x0 - x1) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = 1 / (cos x) ) & x0 <> x1 & cos x0 <> 0 & cos x1 <> 0 implies [!f,x0,x1!] = ((2 * ((cos x1) - (cos x0))) / ((cos (x0 + x1)) + (cos (x0 - x1)))) / (x0 - x1) ) assume that A1: for x being Real holds f . x = 1 / (cos x) and x0 <> x1 and A2: ( cos x0 <> 0 & cos x1 <> 0 ) ; ::_thesis: [!f,x0,x1!] = ((2 * ((cos x1) - (cos x0))) / ((cos (x0 + x1)) + (cos (x0 - x1)))) / (x0 - x1) ( f . x0 = 1 / (cos x0) & f . x1 = 1 / (cos x1) ) by A1; then [!f,x0,x1!] = (((1 * (cos x1)) - (1 * (cos x0))) / ((cos x0) * (cos x1))) / (x0 - x1) by A2, XCMPLX_1:130 .= (((cos x1) - (cos x0)) / ((1 / 2) * ((cos (x0 + x1)) + (cos (x0 - x1))))) / (x0 - x1) by SIN_COS4:32 .= ((((cos x1) - (cos x0)) / (1 / 2)) / ((cos (x0 + x1)) + (cos (x0 - x1)))) / (x0 - x1) by XCMPLX_1:78 .= (2 * (((cos x1) - (cos x0)) / ((cos (x0 + x1)) + (cos (x0 - x1))))) / (x0 - x1) ; hence [!f,x0,x1!] = ((2 * ((cos x1) - (cos x0))) / ((cos (x0 + x1)) + (cos (x0 - x1)))) / (x0 - x1) ; ::_thesis: verum end; theorem :: DIFF_3:66 for x, h being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = 1 / (cos x) ) & cos x <> 0 & cos (x + h) <> 0 holds (fD (f,h)) . x = (2 * ((cos x) - (cos (x + h)))) / ((cos ((2 * x) + h)) + (cos h)) proof let x, h be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = 1 / (cos x) ) & cos x <> 0 & cos (x + h) <> 0 holds (fD (f,h)) . x = (2 * ((cos x) - (cos (x + h)))) / ((cos ((2 * x) + h)) + (cos h)) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = 1 / (cos x) ) & cos x <> 0 & cos (x + h) <> 0 implies (fD (f,h)) . x = (2 * ((cos x) - (cos (x + h)))) / ((cos ((2 * x) + h)) + (cos h)) ) assume that A1: for x being Real holds f . x = 1 / (cos x) and A2: ( cos x <> 0 & cos (x + h) <> 0 ) ; ::_thesis: (fD (f,h)) . x = (2 * ((cos x) - (cos (x + h)))) / ((cos ((2 * x) + h)) + (cos h)) f . (x + h) = 1 / (cos (x + h)) by A1; then (fD (f,h)) . x = (1 / (cos (x + h))) - (f . x) by DIFF_1:3 .= (1 / (cos (x + h))) - (1 / (cos x)) by A1 .= ((1 * (cos x)) - (1 * (cos (x + h)))) / ((cos (x + h)) * (cos x)) by A2, XCMPLX_1:130 .= ((cos x) - (cos (x + h))) / ((1 / 2) * ((cos ((x + h) + x)) + (cos ((x + h) - x)))) by SIN_COS4:32 .= (((cos x) - (cos (x + h))) / (1 / 2)) / ((cos ((2 * x) + h)) + (cos h)) by XCMPLX_1:78 .= 2 * (((cos x) - (cos (x + h))) / ((cos ((2 * x) + h)) + (cos h))) ; hence (fD (f,h)) . x = (2 * ((cos x) - (cos (x + h)))) / ((cos ((2 * x) + h)) + (cos h)) ; ::_thesis: verum end; theorem :: DIFF_3:67 for x, h being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = 1 / (cos x) ) & cos x <> 0 & cos (x - h) <> 0 holds (bD (f,h)) . x = (2 * ((cos (x - h)) - (cos x))) / ((cos ((2 * x) - h)) + (cos h)) proof let x, h be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = 1 / (cos x) ) & cos x <> 0 & cos (x - h) <> 0 holds (bD (f,h)) . x = (2 * ((cos (x - h)) - (cos x))) / ((cos ((2 * x) - h)) + (cos h)) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = 1 / (cos x) ) & cos x <> 0 & cos (x - h) <> 0 implies (bD (f,h)) . x = (2 * ((cos (x - h)) - (cos x))) / ((cos ((2 * x) - h)) + (cos h)) ) assume that A1: for x being Real holds f . x = 1 / (cos x) and A2: ( cos x <> 0 & cos (x - h) <> 0 ) ; ::_thesis: (bD (f,h)) . x = (2 * ((cos (x - h)) - (cos x))) / ((cos ((2 * x) - h)) + (cos h)) (bD (f,h)) . x = (f . x) - (f . (x - h)) by DIFF_1:4 .= (1 / (cos x)) - (f . (x - h)) by A1 .= (1 / (cos x)) - (1 / (cos (x - h))) by A1 .= ((1 * (cos (x - h))) - (1 * (cos x))) / ((cos x) * (cos (x - h))) by A2, XCMPLX_1:130 .= ((cos (x - h)) - (cos x)) / ((1 / 2) * ((cos (x + (x - h))) + (cos (x - (x - h))))) by SIN_COS4:32 .= (((cos (x - h)) - (cos x)) / (1 / 2)) / ((cos ((2 * x) - h)) + (cos h)) by XCMPLX_1:78 .= 2 * (((cos (x - h)) - (cos x)) / ((cos ((2 * x) - h)) + (cos h))) ; hence (bD (f,h)) . x = (2 * ((cos (x - h)) - (cos x))) / ((cos ((2 * x) - h)) + (cos h)) ; ::_thesis: verum end; theorem :: DIFF_3:68 for x, h being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = 1 / (cos x) ) & cos (x + (h / 2)) <> 0 & cos (x - (h / 2)) <> 0 holds (cD (f,h)) . x = (2 * ((cos (x - (h / 2))) - (cos (x + (h / 2))))) / ((cos (2 * x)) + (cos h)) proof let x, h be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = 1 / (cos x) ) & cos (x + (h / 2)) <> 0 & cos (x - (h / 2)) <> 0 holds (cD (f,h)) . x = (2 * ((cos (x - (h / 2))) - (cos (x + (h / 2))))) / ((cos (2 * x)) + (cos h)) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = 1 / (cos x) ) & cos (x + (h / 2)) <> 0 & cos (x - (h / 2)) <> 0 implies (cD (f,h)) . x = (2 * ((cos (x - (h / 2))) - (cos (x + (h / 2))))) / ((cos (2 * x)) + (cos h)) ) assume that A1: for x being Real holds f . x = 1 / (cos x) and A2: ( cos (x + (h / 2)) <> 0 & cos (x - (h / 2)) <> 0 ) ; ::_thesis: (cD (f,h)) . x = (2 * ((cos (x - (h / 2))) - (cos (x + (h / 2))))) / ((cos (2 * x)) + (cos h)) (cD (f,h)) . x = (f . (x + (h / 2))) - (f . (x - (h / 2))) by DIFF_1:5 .= (1 / (cos (x + (h / 2)))) - (f . (x - (h / 2))) by A1 .= (1 / (cos (x + (h / 2)))) - (1 / (cos (x - (h / 2)))) by A1 .= ((1 * (cos (x - (h / 2)))) - (1 * (cos (x + (h / 2))))) / ((cos (x + (h / 2))) * (cos (x - (h / 2)))) by A2, XCMPLX_1:130 .= ((cos (x - (h / 2))) - (cos (x + (h / 2)))) / ((1 / 2) * ((cos ((x + (h / 2)) + (x - (h / 2)))) + (cos ((x + (h / 2)) - (x - (h / 2)))))) by SIN_COS4:32 .= (((cos (x - (h / 2))) - (cos (x + (h / 2)))) / (1 / 2)) / ((cos (2 * x)) + (cos h)) by XCMPLX_1:78 .= 2 * (((cos (x - (h / 2))) - (cos (x + (h / 2)))) / ((cos (2 * x)) + (cos h))) ; hence (cD (f,h)) . x = (2 * ((cos (x - (h / 2))) - (cos (x + (h / 2))))) / ((cos (2 * x)) + (cos h)) ; ::_thesis: verum end; theorem :: DIFF_3:69 for x0, x1 being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = 1 / ((sin x) ^2) ) & x0 <> x1 & sin x0 <> 0 & sin x1 <> 0 holds [!f,x0,x1!] = ((((16 * (cos ((x1 + x0) / 2))) * (sin ((x1 - x0) / 2))) * (cos ((x1 - x0) / 2))) * (sin ((x1 + x0) / 2))) / ((((cos (x0 + x1)) - (cos (x0 - x1))) ^2) * (x0 - x1)) proof let x0, x1 be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = 1 / ((sin x) ^2) ) & x0 <> x1 & sin x0 <> 0 & sin x1 <> 0 holds [!f,x0,x1!] = ((((16 * (cos ((x1 + x0) / 2))) * (sin ((x1 - x0) / 2))) * (cos ((x1 - x0) / 2))) * (sin ((x1 + x0) / 2))) / ((((cos (x0 + x1)) - (cos (x0 - x1))) ^2) * (x0 - x1)) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = 1 / ((sin x) ^2) ) & x0 <> x1 & sin x0 <> 0 & sin x1 <> 0 implies [!f,x0,x1!] = ((((16 * (cos ((x1 + x0) / 2))) * (sin ((x1 - x0) / 2))) * (cos ((x1 - x0) / 2))) * (sin ((x1 + x0) / 2))) / ((((cos (x0 + x1)) - (cos (x0 - x1))) ^2) * (x0 - x1)) ) assume that A1: for x being Real holds f . x = 1 / ((sin x) ^2) and x0 <> x1 and A2: ( sin x0 <> 0 & sin x1 <> 0 ) ; ::_thesis: [!f,x0,x1!] = ((((16 * (cos ((x1 + x0) / 2))) * (sin ((x1 - x0) / 2))) * (cos ((x1 - x0) / 2))) * (sin ((x1 + x0) / 2))) / ((((cos (x0 + x1)) - (cos (x0 - x1))) ^2) * (x0 - x1)) ( f . x0 = 1 / ((sin x0) ^2) & f . x1 = 1 / ((sin x1) ^2) ) by A1; then [!f,x0,x1!] = (((1 * ((sin x1) ^2)) - (1 * ((sin x0) ^2))) / (((sin x0) ^2) * ((sin x1) ^2))) / (x0 - x1) by A2, XCMPLX_1:130 .= ((((sin x1) ^2) - ((sin x0) ^2)) / (((sin x0) * (sin x1)) ^2)) / (x0 - x1) .= ((((sin x1) ^2) - ((sin x0) ^2)) / ((- ((1 / 2) * ((cos (x0 + x1)) - (cos (x0 - x1))))) ^2)) / (x0 - x1) by SIN_COS4:29 .= ((((sin x1) ^2) - ((sin x0) ^2)) / ((1 / 4) * (((cos (x0 + x1)) - (cos (x0 - x1))) ^2))) / (x0 - x1) .= (((((sin x1) ^2) - ((sin x0) ^2)) / (1 / 4)) / (((cos (x0 + x1)) - (cos (x0 - x1))) ^2)) / (x0 - x1) by XCMPLX_1:78 .= ((4 * (((sin x1) - (sin x0)) * ((sin x1) + (sin x0)))) / (((cos (x0 + x1)) - (cos (x0 - x1))) ^2)) / (x0 - x1) .= ((4 * ((2 * ((cos ((x1 + x0) / 2)) * (sin ((x1 - x0) / 2)))) * ((sin x1) + (sin x0)))) / (((cos (x0 + x1)) - (cos (x0 - x1))) ^2)) / (x0 - x1) by SIN_COS4:16 .= ((4 * ((2 * ((cos ((x1 + x0) / 2)) * (sin ((x1 - x0) / 2)))) * (2 * ((cos ((x1 - x0) / 2)) * (sin ((x1 + x0) / 2)))))) / (((cos (x0 + x1)) - (cos (x0 - x1))) ^2)) / (x0 - x1) by SIN_COS4:15 .= ((((16 * (cos ((x1 + x0) / 2))) * (sin ((x1 - x0) / 2))) * (cos ((x1 - x0) / 2))) * (sin ((x1 + x0) / 2))) / ((((cos (x0 + x1)) - (cos (x0 - x1))) ^2) * (x0 - x1)) by XCMPLX_1:78 ; hence [!f,x0,x1!] = ((((16 * (cos ((x1 + x0) / 2))) * (sin ((x1 - x0) / 2))) * (cos ((x1 - x0) / 2))) * (sin ((x1 + x0) / 2))) / ((((cos (x0 + x1)) - (cos (x0 - x1))) ^2) * (x0 - x1)) ; ::_thesis: verum end; theorem :: DIFF_3:70 for x, h being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = 1 / ((sin x) ^2) ) & sin x <> 0 & sin (x + h) <> 0 holds (fD (f,h)) . x = ((((16 * (cos (((2 * x) + h) / 2))) * (sin ((- h) / 2))) * (cos ((- h) / 2))) * (sin (((2 * x) + h) / 2))) / (((cos ((2 * x) + h)) - (cos h)) ^2) proof let x, h be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = 1 / ((sin x) ^2) ) & sin x <> 0 & sin (x + h) <> 0 holds (fD (f,h)) . x = ((((16 * (cos (((2 * x) + h) / 2))) * (sin ((- h) / 2))) * (cos ((- h) / 2))) * (sin (((2 * x) + h) / 2))) / (((cos ((2 * x) + h)) - (cos h)) ^2) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = 1 / ((sin x) ^2) ) & sin x <> 0 & sin (x + h) <> 0 implies (fD (f,h)) . x = ((((16 * (cos (((2 * x) + h) / 2))) * (sin ((- h) / 2))) * (cos ((- h) / 2))) * (sin (((2 * x) + h) / 2))) / (((cos ((2 * x) + h)) - (cos h)) ^2) ) assume that A1: for x being Real holds f . x = 1 / ((sin x) ^2) and A2: ( sin x <> 0 & sin (x + h) <> 0 ) ; ::_thesis: (fD (f,h)) . x = ((((16 * (cos (((2 * x) + h) / 2))) * (sin ((- h) / 2))) * (cos ((- h) / 2))) * (sin (((2 * x) + h) / 2))) / (((cos ((2 * x) + h)) - (cos h)) ^2) (fD (f,h)) . x = (f . (x + h)) - (f . x) by DIFF_1:3 .= (1 / ((sin (x + h)) ^2)) - (f . x) by A1 .= (1 / ((sin (x + h)) ^2)) - (1 / ((sin x) ^2)) by A1 .= ((1 * ((sin x) ^2)) - (1 * ((sin (x + h)) ^2))) / (((sin (x + h)) ^2) * ((sin x) ^2)) by A2, XCMPLX_1:130 .= (((sin x) ^2) - ((sin (x + h)) ^2)) / (((sin (x + h)) * (sin x)) ^2) .= (((sin x) ^2) - ((sin (x + h)) ^2)) / ((- ((1 / 2) * ((cos ((x + h) + x)) - (cos ((x + h) - x))))) ^2) by SIN_COS4:29 .= (((sin x) ^2) - ((sin (x + h)) ^2)) / ((1 / 4) * (((cos ((2 * x) + h)) - (cos h)) ^2)) .= ((((sin x) ^2) - ((sin (x + h)) ^2)) / (1 / 4)) / (((cos ((2 * x) + h)) - (cos h)) ^2) by XCMPLX_1:78 .= 4 * ((((sin x) - (sin (x + h))) * ((sin x) + (sin (x + h)))) / (((cos ((2 * x) + h)) - (cos h)) ^2)) .= 4 * (((2 * ((cos ((x + (x + h)) / 2)) * (sin ((x - (x + h)) / 2)))) * ((sin x) + (sin (x + h)))) / (((cos ((2 * x) + h)) - (cos h)) ^2)) by SIN_COS4:16 .= 4 * (((2 * ((cos (((2 * x) + h) / 2)) * (sin ((- h) / 2)))) * (2 * ((cos ((- h) / 2)) * (sin (((2 * x) + h) / 2))))) / (((cos ((2 * x) + h)) - (cos h)) ^2)) by SIN_COS4:15 .= ((((16 * (cos (((2 * x) + h) / 2))) * (sin ((- h) / 2))) * (cos ((- h) / 2))) * (sin (((2 * x) + h) / 2))) / (((cos ((2 * x) + h)) - (cos h)) ^2) ; hence (fD (f,h)) . x = ((((16 * (cos (((2 * x) + h) / 2))) * (sin ((- h) / 2))) * (cos ((- h) / 2))) * (sin (((2 * x) + h) / 2))) / (((cos ((2 * x) + h)) - (cos h)) ^2) ; ::_thesis: verum end; theorem :: DIFF_3:71 for x, h being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = 1 / ((sin x) ^2) ) & sin x <> 0 & sin (x - h) <> 0 holds (bD (f,h)) . x = ((((16 * (cos (((2 * x) - h) / 2))) * (sin ((- h) / 2))) * (cos ((- h) / 2))) * (sin (((2 * x) - h) / 2))) / (((cos ((2 * x) - h)) - (cos h)) ^2) proof let x, h be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = 1 / ((sin x) ^2) ) & sin x <> 0 & sin (x - h) <> 0 holds (bD (f,h)) . x = ((((16 * (cos (((2 * x) - h) / 2))) * (sin ((- h) / 2))) * (cos ((- h) / 2))) * (sin (((2 * x) - h) / 2))) / (((cos ((2 * x) - h)) - (cos h)) ^2) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = 1 / ((sin x) ^2) ) & sin x <> 0 & sin (x - h) <> 0 implies (bD (f,h)) . x = ((((16 * (cos (((2 * x) - h) / 2))) * (sin ((- h) / 2))) * (cos ((- h) / 2))) * (sin (((2 * x) - h) / 2))) / (((cos ((2 * x) - h)) - (cos h)) ^2) ) assume that A1: for x being Real holds f . x = 1 / ((sin x) ^2) and A2: ( sin x <> 0 & sin (x - h) <> 0 ) ; ::_thesis: (bD (f,h)) . x = ((((16 * (cos (((2 * x) - h) / 2))) * (sin ((- h) / 2))) * (cos ((- h) / 2))) * (sin (((2 * x) - h) / 2))) / (((cos ((2 * x) - h)) - (cos h)) ^2) (bD (f,h)) . x = (f . x) - (f . (x - h)) by DIFF_1:4 .= (1 / ((sin x) ^2)) - (f . (x - h)) by A1 .= (1 / ((sin x) ^2)) - (1 / ((sin (x - h)) ^2)) by A1 .= ((1 * ((sin (x - h)) ^2)) - (1 * ((sin x) ^2))) / (((sin x) ^2) * ((sin (x - h)) ^2)) by A2, XCMPLX_1:130 .= (((sin (x - h)) ^2) - ((sin x) ^2)) / (((sin x) * (sin (x - h))) ^2) .= (((sin (x - h)) ^2) - ((sin x) ^2)) / ((- ((1 / 2) * ((cos (x + (x - h))) - (cos (x - (x - h)))))) ^2) by SIN_COS4:29 .= (((sin (x - h)) ^2) - ((sin x) ^2)) / ((1 / 4) * (((cos ((2 * x) - h)) - (cos h)) ^2)) .= ((((sin (x - h)) ^2) - ((sin x) ^2)) / (1 / 4)) / (((cos ((2 * x) - h)) - (cos h)) ^2) by XCMPLX_1:78 .= 4 * ((((sin (x - h)) - (sin x)) * ((sin (x - h)) + (sin x))) / (((cos ((2 * x) - h)) - (cos h)) ^2)) .= 4 * (((2 * ((cos (((x - h) + x) / 2)) * (sin (((x - h) - x) / 2)))) * ((sin (x - h)) + (sin x))) / (((cos ((2 * x) - h)) - (cos h)) ^2)) by SIN_COS4:16 .= 4 * (((2 * ((cos (((2 * x) - h) / 2)) * (sin ((- h) / 2)))) * (2 * ((cos ((- h) / 2)) * (sin (((2 * x) - h) / 2))))) / (((cos ((2 * x) - h)) - (cos h)) ^2)) by SIN_COS4:15 .= ((((16 * (cos (((2 * x) - h) / 2))) * (sin ((- h) / 2))) * (cos ((- h) / 2))) * (sin (((2 * x) - h) / 2))) / (((cos ((2 * x) - h)) - (cos h)) ^2) ; hence (bD (f,h)) . x = ((((16 * (cos (((2 * x) - h) / 2))) * (sin ((- h) / 2))) * (cos ((- h) / 2))) * (sin (((2 * x) - h) / 2))) / (((cos ((2 * x) - h)) - (cos h)) ^2) ; ::_thesis: verum end; theorem :: DIFF_3:72 for x, h being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = 1 / ((sin x) ^2) ) & sin (x + (h / 2)) <> 0 & sin (x - (h / 2)) <> 0 holds (cD (f,h)) . x = ((((16 * (cos x)) * (sin ((- h) / 2))) * (cos ((- h) / 2))) * (sin x)) / (((cos (2 * x)) - (cos h)) ^2) proof let x, h be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = 1 / ((sin x) ^2) ) & sin (x + (h / 2)) <> 0 & sin (x - (h / 2)) <> 0 holds (cD (f,h)) . x = ((((16 * (cos x)) * (sin ((- h) / 2))) * (cos ((- h) / 2))) * (sin x)) / (((cos (2 * x)) - (cos h)) ^2) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = 1 / ((sin x) ^2) ) & sin (x + (h / 2)) <> 0 & sin (x - (h / 2)) <> 0 implies (cD (f,h)) . x = ((((16 * (cos x)) * (sin ((- h) / 2))) * (cos ((- h) / 2))) * (sin x)) / (((cos (2 * x)) - (cos h)) ^2) ) assume that A1: for x being Real holds f . x = 1 / ((sin x) ^2) and A2: ( sin (x + (h / 2)) <> 0 & sin (x - (h / 2)) <> 0 ) ; ::_thesis: (cD (f,h)) . x = ((((16 * (cos x)) * (sin ((- h) / 2))) * (cos ((- h) / 2))) * (sin x)) / (((cos (2 * x)) - (cos h)) ^2) (cD (f,h)) . x = (f . (x + (h / 2))) - (f . (x - (h / 2))) by DIFF_1:5 .= (1 / ((sin (x + (h / 2))) ^2)) - (f . (x - (h / 2))) by A1 .= (1 / ((sin (x + (h / 2))) ^2)) - (1 / ((sin (x - (h / 2))) ^2)) by A1 .= ((1 * ((sin (x - (h / 2))) ^2)) - (1 * ((sin (x + (h / 2))) ^2))) / (((sin (x + (h / 2))) ^2) * ((sin (x - (h / 2))) ^2)) by A2, XCMPLX_1:130 .= (((sin (x - (h / 2))) ^2) - ((sin (x + (h / 2))) ^2)) / (((sin (x + (h / 2))) * (sin (x - (h / 2)))) ^2) .= (((sin (x - (h / 2))) ^2) - ((sin (x + (h / 2))) ^2)) / ((- ((1 / 2) * ((cos ((x + (h / 2)) + (x - (h / 2)))) - (cos ((x + (h / 2)) - (x - (h / 2))))))) ^2) by SIN_COS4:29 .= (((sin (x - (h / 2))) ^2) - ((sin (x + (h / 2))) ^2)) / ((1 / 4) * (((cos (2 * x)) - (cos h)) ^2)) .= ((((sin (x - (h / 2))) ^2) - ((sin (x + (h / 2))) ^2)) / (1 / 4)) / (((cos (2 * x)) - (cos h)) ^2) by XCMPLX_1:78 .= 4 * ((((sin (x - (h / 2))) - (sin (x + (h / 2)))) * ((sin (x - (h / 2))) + (sin (x + (h / 2))))) / (((cos (2 * x)) - (cos h)) ^2)) .= 4 * (((2 * ((cos (((x - (h / 2)) + (x + (h / 2))) / 2)) * (sin (((x - (h / 2)) - (x + (h / 2))) / 2)))) * ((sin (x - (h / 2))) + (sin (x + (h / 2))))) / (((cos (2 * x)) - (cos h)) ^2)) by SIN_COS4:16 .= 4 * (((2 * ((cos ((2 * x) / 2)) * (sin ((- h) / 2)))) * (2 * ((cos ((- h) / 2)) * (sin ((2 * x) / 2))))) / (((cos (2 * x)) - (cos h)) ^2)) by SIN_COS4:15 .= ((((16 * (cos x)) * (sin ((- h) / 2))) * (cos ((- h) / 2))) * (sin x)) / (((cos (2 * x)) - (cos h)) ^2) ; hence (cD (f,h)) . x = ((((16 * (cos x)) * (sin ((- h) / 2))) * (cos ((- h) / 2))) * (sin x)) / (((cos (2 * x)) - (cos h)) ^2) ; ::_thesis: verum end; theorem :: DIFF_3:73 for x0, x1 being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = 1 / ((cos x) ^2) ) & x0 <> x1 & cos x0 <> 0 & cos x1 <> 0 holds [!f,x0,x1!] = ((((((- 16) * (sin ((x1 + x0) / 2))) * (sin ((x1 - x0) / 2))) * (cos ((x1 + x0) / 2))) * (cos ((x1 - x0) / 2))) / (((cos (x0 + x1)) + (cos (x0 - x1))) ^2)) / (x0 - x1) proof let x0, x1 be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = 1 / ((cos x) ^2) ) & x0 <> x1 & cos x0 <> 0 & cos x1 <> 0 holds [!f,x0,x1!] = ((((((- 16) * (sin ((x1 + x0) / 2))) * (sin ((x1 - x0) / 2))) * (cos ((x1 + x0) / 2))) * (cos ((x1 - x0) / 2))) / (((cos (x0 + x1)) + (cos (x0 - x1))) ^2)) / (x0 - x1) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = 1 / ((cos x) ^2) ) & x0 <> x1 & cos x0 <> 0 & cos x1 <> 0 implies [!f,x0,x1!] = ((((((- 16) * (sin ((x1 + x0) / 2))) * (sin ((x1 - x0) / 2))) * (cos ((x1 + x0) / 2))) * (cos ((x1 - x0) / 2))) / (((cos (x0 + x1)) + (cos (x0 - x1))) ^2)) / (x0 - x1) ) assume that A1: for x being Real holds f . x = 1 / ((cos x) ^2) and x0 <> x1 and A2: ( cos x0 <> 0 & cos x1 <> 0 ) ; ::_thesis: [!f,x0,x1!] = ((((((- 16) * (sin ((x1 + x0) / 2))) * (sin ((x1 - x0) / 2))) * (cos ((x1 + x0) / 2))) * (cos ((x1 - x0) / 2))) / (((cos (x0 + x1)) + (cos (x0 - x1))) ^2)) / (x0 - x1) ( f . x0 = 1 / ((cos x0) ^2) & f . x1 = 1 / ((cos x1) ^2) ) by A1; then [!f,x0,x1!] = (((1 * ((cos x1) ^2)) - (1 * ((cos x0) ^2))) / (((cos x0) ^2) * ((cos x1) ^2))) / (x0 - x1) by A2, XCMPLX_1:130 .= ((((cos x1) ^2) - ((cos x0) ^2)) / (((cos x0) * (cos x1)) ^2)) / (x0 - x1) .= ((((cos x1) ^2) - ((cos x0) ^2)) / (((1 / 2) * ((cos (x0 + x1)) + (cos (x0 - x1)))) ^2)) / (x0 - x1) by SIN_COS4:32 .= ((((cos x1) ^2) - ((cos x0) ^2)) / ((1 / 4) * (((cos (x0 + x1)) + (cos (x0 - x1))) ^2))) / (x0 - x1) .= (((((cos x1) ^2) - ((cos x0) ^2)) / (1 / 4)) / (((cos (x0 + x1)) + (cos (x0 - x1))) ^2)) / (x0 - x1) by XCMPLX_1:78 .= ((4 * (((cos x1) - (cos x0)) * ((cos x1) + (cos x0)))) / (((cos (x0 + x1)) + (cos (x0 - x1))) ^2)) / (x0 - x1) .= ((4 * ((- (2 * ((sin ((x1 + x0) / 2)) * (sin ((x1 - x0) / 2))))) * ((cos x1) + (cos x0)))) / (((cos (x0 + x1)) + (cos (x0 - x1))) ^2)) / (x0 - x1) by SIN_COS4:18 .= ((4 * ((- (2 * ((sin ((x1 + x0) / 2)) * (sin ((x1 - x0) / 2))))) * (2 * ((cos ((x1 + x0) / 2)) * (cos ((x1 - x0) / 2)))))) / (((cos (x0 + x1)) + (cos (x0 - x1))) ^2)) / (x0 - x1) by SIN_COS4:17 .= ((((((- 16) * (sin ((x1 + x0) / 2))) * (sin ((x1 - x0) / 2))) * (cos ((x1 + x0) / 2))) * (cos ((x1 - x0) / 2))) / (((cos (x0 + x1)) + (cos (x0 - x1))) ^2)) / (x0 - x1) ; hence [!f,x0,x1!] = ((((((- 16) * (sin ((x1 + x0) / 2))) * (sin ((x1 - x0) / 2))) * (cos ((x1 + x0) / 2))) * (cos ((x1 - x0) / 2))) / (((cos (x0 + x1)) + (cos (x0 - x1))) ^2)) / (x0 - x1) ; ::_thesis: verum end; theorem :: DIFF_3:74 for x, h being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = 1 / ((cos x) ^2) ) & cos x <> 0 & cos (x + h) <> 0 holds (fD (f,h)) . x = (((((- 16) * (sin (((2 * x) + h) / 2))) * (sin ((- h) / 2))) * (cos (((2 * x) + h) / 2))) * (cos ((- h) / 2))) / (((cos ((2 * x) + h)) + (cos h)) ^2) proof let x, h be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = 1 / ((cos x) ^2) ) & cos x <> 0 & cos (x + h) <> 0 holds (fD (f,h)) . x = (((((- 16) * (sin (((2 * x) + h) / 2))) * (sin ((- h) / 2))) * (cos (((2 * x) + h) / 2))) * (cos ((- h) / 2))) / (((cos ((2 * x) + h)) + (cos h)) ^2) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = 1 / ((cos x) ^2) ) & cos x <> 0 & cos (x + h) <> 0 implies (fD (f,h)) . x = (((((- 16) * (sin (((2 * x) + h) / 2))) * (sin ((- h) / 2))) * (cos (((2 * x) + h) / 2))) * (cos ((- h) / 2))) / (((cos ((2 * x) + h)) + (cos h)) ^2) ) assume that A1: for x being Real holds f . x = 1 / ((cos x) ^2) and A2: ( cos x <> 0 & cos (x + h) <> 0 ) ; ::_thesis: (fD (f,h)) . x = (((((- 16) * (sin (((2 * x) + h) / 2))) * (sin ((- h) / 2))) * (cos (((2 * x) + h) / 2))) * (cos ((- h) / 2))) / (((cos ((2 * x) + h)) + (cos h)) ^2) (fD (f,h)) . x = (f . (x + h)) - (f . x) by DIFF_1:3 .= (1 / ((cos (x + h)) ^2)) - (f . x) by A1 .= (1 / ((cos (x + h)) ^2)) - (1 / ((cos x) ^2)) by A1 .= ((1 * ((cos x) ^2)) - (1 * ((cos (x + h)) ^2))) / (((cos (x + h)) ^2) * ((cos x) ^2)) by A2, XCMPLX_1:130 .= (((cos x) ^2) - ((cos (x + h)) ^2)) / (((cos (x + h)) * (cos x)) ^2) .= (((cos x) ^2) - ((cos (x + h)) ^2)) / (((1 / 2) * ((cos ((x + h) + x)) + (cos ((x + h) - x)))) ^2) by SIN_COS4:32 .= (((cos x) ^2) - ((cos (x + h)) ^2)) / ((1 / 4) * (((cos ((2 * x) + h)) + (cos h)) ^2)) .= ((((cos x) ^2) - ((cos (x + h)) ^2)) / (1 / 4)) / (((cos ((2 * x) + h)) + (cos h)) ^2) by XCMPLX_1:78 .= 4 * ((((cos x) - (cos (x + h))) * ((cos x) + (cos (x + h)))) / (((cos ((2 * x) + h)) + (cos h)) ^2)) .= 4 * (((- (2 * ((sin ((x + (x + h)) / 2)) * (sin ((x - (x + h)) / 2))))) * ((cos x) + (cos (x + h)))) / (((cos ((2 * x) + h)) + (cos h)) ^2)) by SIN_COS4:18 .= 4 * (((- (2 * ((sin (((2 * x) + h) / 2)) * (sin ((- h) / 2))))) * (2 * ((cos (((2 * x) + h) / 2)) * (cos ((- h) / 2))))) / (((cos ((2 * x) + h)) + (cos h)) ^2)) by SIN_COS4:17 .= (((((- 16) * (sin (((2 * x) + h) / 2))) * (sin ((- h) / 2))) * (cos (((2 * x) + h) / 2))) * (cos ((- h) / 2))) / (((cos ((2 * x) + h)) + (cos h)) ^2) ; hence (fD (f,h)) . x = (((((- 16) * (sin (((2 * x) + h) / 2))) * (sin ((- h) / 2))) * (cos (((2 * x) + h) / 2))) * (cos ((- h) / 2))) / (((cos ((2 * x) + h)) + (cos h)) ^2) ; ::_thesis: verum end; theorem :: DIFF_3:75 for x, h being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = 1 / ((cos x) ^2) ) & cos x <> 0 & cos (x - h) <> 0 holds (bD (f,h)) . x = (((((- 16) * (sin (((2 * x) - h) / 2))) * (sin ((- h) / 2))) * (cos (((2 * x) - h) / 2))) * (cos ((- h) / 2))) / (((cos ((2 * x) - h)) + (cos h)) ^2) proof let x, h be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = 1 / ((cos x) ^2) ) & cos x <> 0 & cos (x - h) <> 0 holds (bD (f,h)) . x = (((((- 16) * (sin (((2 * x) - h) / 2))) * (sin ((- h) / 2))) * (cos (((2 * x) - h) / 2))) * (cos ((- h) / 2))) / (((cos ((2 * x) - h)) + (cos h)) ^2) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = 1 / ((cos x) ^2) ) & cos x <> 0 & cos (x - h) <> 0 implies (bD (f,h)) . x = (((((- 16) * (sin (((2 * x) - h) / 2))) * (sin ((- h) / 2))) * (cos (((2 * x) - h) / 2))) * (cos ((- h) / 2))) / (((cos ((2 * x) - h)) + (cos h)) ^2) ) assume that A1: for x being Real holds f . x = 1 / ((cos x) ^2) and A2: ( cos x <> 0 & cos (x - h) <> 0 ) ; ::_thesis: (bD (f,h)) . x = (((((- 16) * (sin (((2 * x) - h) / 2))) * (sin ((- h) / 2))) * (cos (((2 * x) - h) / 2))) * (cos ((- h) / 2))) / (((cos ((2 * x) - h)) + (cos h)) ^2) (bD (f,h)) . x = (f . x) - (f . (x - h)) by DIFF_1:4 .= (1 / ((cos x) ^2)) - (f . (x - h)) by A1 .= (1 / ((cos x) ^2)) - (1 / ((cos (x - h)) ^2)) by A1 .= ((1 * ((cos (x - h)) ^2)) - (1 * ((cos x) ^2))) / (((cos x) ^2) * ((cos (x - h)) ^2)) by A2, XCMPLX_1:130 .= (((cos (x - h)) ^2) - ((cos x) ^2)) / (((cos x) * (cos (x - h))) ^2) .= (((cos (x - h)) ^2) - ((cos x) ^2)) / (((1 / 2) * ((cos (x + (x - h))) + (cos (x - (x - h))))) ^2) by SIN_COS4:32 .= (((cos (x - h)) ^2) - ((cos x) ^2)) / ((1 / 4) * (((cos ((2 * x) - h)) + (cos h)) ^2)) .= ((((cos (x - h)) ^2) - ((cos x) ^2)) / (1 / 4)) / (((cos ((2 * x) - h)) + (cos h)) ^2) by XCMPLX_1:78 .= 4 * ((((cos (x - h)) - (cos x)) * ((cos (x - h)) + (cos x))) / (((cos ((2 * x) - h)) + (cos h)) ^2)) .= 4 * (((- (2 * ((sin (((x - h) + x) / 2)) * (sin (((x - h) - x) / 2))))) * ((cos (x - h)) + (cos x))) / (((cos ((2 * x) - h)) + (cos h)) ^2)) by SIN_COS4:18 .= 4 * (((- (2 * ((sin (((2 * x) - h) / 2)) * (sin ((- h) / 2))))) * (2 * ((cos (((2 * x) - h) / 2)) * (cos ((- h) / 2))))) / (((cos ((2 * x) - h)) + (cos h)) ^2)) by SIN_COS4:17 .= (((((- 16) * (sin (((2 * x) - h) / 2))) * (sin ((- h) / 2))) * (cos (((2 * x) - h) / 2))) * (cos ((- h) / 2))) / (((cos ((2 * x) - h)) + (cos h)) ^2) ; hence (bD (f,h)) . x = (((((- 16) * (sin (((2 * x) - h) / 2))) * (sin ((- h) / 2))) * (cos (((2 * x) - h) / 2))) * (cos ((- h) / 2))) / (((cos ((2 * x) - h)) + (cos h)) ^2) ; ::_thesis: verum end; theorem :: DIFF_3:76 for x, h being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = 1 / ((cos x) ^2) ) & cos (x + (h / 2)) <> 0 & cos (x - (h / 2)) <> 0 holds (cD (f,h)) . x = (((((- 16) * (sin x)) * (sin ((- h) / 2))) * (cos x)) * (cos ((- h) / 2))) / (((cos (2 * x)) + (cos h)) ^2) proof let x, h be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = 1 / ((cos x) ^2) ) & cos (x + (h / 2)) <> 0 & cos (x - (h / 2)) <> 0 holds (cD (f,h)) . x = (((((- 16) * (sin x)) * (sin ((- h) / 2))) * (cos x)) * (cos ((- h) / 2))) / (((cos (2 * x)) + (cos h)) ^2) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = 1 / ((cos x) ^2) ) & cos (x + (h / 2)) <> 0 & cos (x - (h / 2)) <> 0 implies (cD (f,h)) . x = (((((- 16) * (sin x)) * (sin ((- h) / 2))) * (cos x)) * (cos ((- h) / 2))) / (((cos (2 * x)) + (cos h)) ^2) ) assume that A1: for x being Real holds f . x = 1 / ((cos x) ^2) and A2: ( cos (x + (h / 2)) <> 0 & cos (x - (h / 2)) <> 0 ) ; ::_thesis: (cD (f,h)) . x = (((((- 16) * (sin x)) * (sin ((- h) / 2))) * (cos x)) * (cos ((- h) / 2))) / (((cos (2 * x)) + (cos h)) ^2) (cD (f,h)) . x = (f . (x + (h / 2))) - (f . (x - (h / 2))) by DIFF_1:5 .= (1 / ((cos (x + (h / 2))) ^2)) - (f . (x - (h / 2))) by A1 .= (1 / ((cos (x + (h / 2))) ^2)) - (1 / ((cos (x - (h / 2))) ^2)) by A1 .= ((1 * ((cos (x - (h / 2))) ^2)) - (1 * ((cos (x + (h / 2))) ^2))) / (((cos (x + (h / 2))) ^2) * ((cos (x - (h / 2))) ^2)) by A2, XCMPLX_1:130 .= (((cos (x - (h / 2))) ^2) - ((cos (x + (h / 2))) ^2)) / (((cos (x + (h / 2))) * (cos (x - (h / 2)))) ^2) .= (((cos (x - (h / 2))) ^2) - ((cos (x + (h / 2))) ^2)) / (((1 / 2) * ((cos ((x + (h / 2)) + (x - (h / 2)))) + (cos ((x + (h / 2)) - (x - (h / 2)))))) ^2) by SIN_COS4:32 .= (((cos (x - (h / 2))) ^2) - ((cos (x + (h / 2))) ^2)) / ((1 / 4) * (((cos (2 * x)) + (cos h)) ^2)) .= ((((cos (x - (h / 2))) ^2) - ((cos (x + (h / 2))) ^2)) / (1 / 4)) / (((cos (2 * x)) + (cos h)) ^2) by XCMPLX_1:78 .= 4 * ((((cos (x - (h / 2))) - (cos (x + (h / 2)))) * ((cos (x - (h / 2))) + (cos (x + (h / 2))))) / (((cos (2 * x)) + (cos h)) ^2)) .= 4 * (((- (2 * ((sin (((x - (h / 2)) + (x + (h / 2))) / 2)) * (sin (((x - (h / 2)) - (x + (h / 2))) / 2))))) * ((cos (x - (h / 2))) + (cos (x + (h / 2))))) / (((cos (2 * x)) + (cos h)) ^2)) by SIN_COS4:18 .= 4 * (((- (2 * ((sin ((2 * x) / 2)) * (sin ((- h) / 2))))) * (2 * ((cos ((2 * x) / 2)) * (cos ((- h) / 2))))) / (((cos (2 * x)) + (cos h)) ^2)) by SIN_COS4:17 .= (((((- 16) * (sin x)) * (sin ((- h) / 2))) * (cos x)) * (cos ((- h) / 2))) / (((cos (2 * x)) + (cos h)) ^2) ; hence (cD (f,h)) . x = (((((- 16) * (sin x)) * (sin ((- h) / 2))) * (cos x)) * (cos ((- h) / 2))) / (((cos (2 * x)) + (cos h)) ^2) ; ::_thesis: verum end; theorem :: DIFF_3:77 for x0, x1 being Real st x0 in dom tan & x1 in dom tan holds [!(tan (#) sin),x0,x1!] = ((((1 / (cos x0)) - (cos x0)) - (1 / (cos x1))) + (cos x1)) / (x0 - x1) proof let x0, x1 be Real; ::_thesis: ( x0 in dom tan & x1 in dom tan implies [!(tan (#) sin),x0,x1!] = ((((1 / (cos x0)) - (cos x0)) - (1 / (cos x1))) + (cos x1)) / (x0 - x1) ) assume A1: ( x0 in dom tan & x1 in dom tan ) ; ::_thesis: [!(tan (#) sin),x0,x1!] = ((((1 / (cos x0)) - (cos x0)) - (1 / (cos x1))) + (cos x1)) / (x0 - x1) [!(tan (#) sin),x0,x1!] = (((tan . x0) * (sin . x0)) - ((tan (#) sin) . x1)) / (x0 - x1) by VALUED_1:5 .= (((tan . x0) * (sin . x0)) - ((tan . x1) * (sin . x1))) / (x0 - x1) by VALUED_1:5 .= ((((sin . x0) * ((cos . x0) ")) * (sin . x0)) - ((tan . x1) * (sin . x1))) / (x0 - x1) by A1, RFUNCT_1:def_1 .= ((((sin x0) / (cos x0)) * (sin x0)) - (((sin x1) / (cos x1)) * (sin x1))) / (x0 - x1) by A1, RFUNCT_1:def_1 .= (((sin x0) / ((cos x0) / (sin x0))) - (((sin x1) / (cos x1)) * (sin x1))) / (x0 - x1) by XCMPLX_1:82 .= (((sin x0) / ((cos x0) / (sin x0))) - ((sin x1) / ((cos x1) / (sin x1)))) / (x0 - x1) by XCMPLX_1:82 .= ((((sin x0) * (sin x0)) / (cos x0)) - ((sin x1) / ((cos x1) / (sin x1)))) / (x0 - x1) by XCMPLX_1:77 .= ((((sin x0) * (sin x0)) / (cos x0)) - (((sin x1) * (sin x1)) / (cos x1))) / (x0 - x1) by XCMPLX_1:77 .= (((1 - ((cos x0) * (cos x0))) / (cos x0)) - (((sin x1) * (sin x1)) / (cos x1))) / (x0 - x1) by SIN_COS4:4 .= (((1 / (cos x0)) - (((cos x0) * (cos x0)) / (cos x0))) - ((1 - ((cos x1) * (cos x1))) / (cos x1))) / (x0 - x1) by SIN_COS4:4 .= (((1 / (cos x0)) - (cos x0)) - ((1 / (cos x1)) - (((cos x1) * (cos x1)) / (cos x1)))) / (x0 - x1) by A1, FDIFF_8:1, XCMPLX_1:89 .= (((1 / (cos x0)) - (cos x0)) - ((1 / (cos x1)) - (cos x1))) / (x0 - x1) by A1, FDIFF_8:1, XCMPLX_1:89 .= ((((1 / (cos x0)) - (cos x0)) - (1 / (cos x1))) + (cos x1)) / (x0 - x1) ; hence [!(tan (#) sin),x0,x1!] = ((((1 / (cos x0)) - (cos x0)) - (1 / (cos x1))) + (cos x1)) / (x0 - x1) ; ::_thesis: verum end; theorem :: DIFF_3:78 for x, h being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = (tan (#) sin) . x ) & x in dom tan & x + h in dom tan holds (fD (f,h)) . x = (((1 / (cos (x + h))) - (cos (x + h))) - (1 / (cos x))) + (cos x) proof let x, h be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = (tan (#) sin) . x ) & x in dom tan & x + h in dom tan holds (fD (f,h)) . x = (((1 / (cos (x + h))) - (cos (x + h))) - (1 / (cos x))) + (cos x) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = (tan (#) sin) . x ) & x in dom tan & x + h in dom tan implies (fD (f,h)) . x = (((1 / (cos (x + h))) - (cos (x + h))) - (1 / (cos x))) + (cos x) ) assume that A1: for x being Real holds f . x = (tan (#) sin) . x and A2: ( x in dom tan & x + h in dom tan ) ; ::_thesis: (fD (f,h)) . x = (((1 / (cos (x + h))) - (cos (x + h))) - (1 / (cos x))) + (cos x) (fD (f,h)) . x = (f . (x + h)) - (f . x) by DIFF_1:3 .= ((tan (#) sin) . (x + h)) - (f . x) by A1 .= ((tan (#) sin) . (x + h)) - ((tan (#) sin) . x) by A1 .= ((tan . (x + h)) * (sin . (x + h))) - ((tan (#) sin) . x) by VALUED_1:5 .= ((tan . (x + h)) * (sin . (x + h))) - ((tan . x) * (sin . x)) by VALUED_1:5 .= (((sin . (x + h)) * ((cos . (x + h)) ")) * (sin . (x + h))) - ((tan . x) * (sin . x)) by A2, RFUNCT_1:def_1 .= (((sin (x + h)) / (cos (x + h))) * (sin (x + h))) - (((sin x) / (cos x)) * (sin x)) by A2, RFUNCT_1:def_1 .= ((sin (x + h)) / ((cos (x + h)) / (sin (x + h)))) - (((sin x) / (cos x)) * (sin x)) by XCMPLX_1:82 .= ((sin (x + h)) / ((cos (x + h)) / (sin (x + h)))) - ((sin x) / ((cos x) / (sin x))) by XCMPLX_1:82 .= (((sin (x + h)) * (sin (x + h))) / (cos (x + h))) - ((sin x) / ((cos x) / (sin x))) by XCMPLX_1:77 .= (((sin (x + h)) * (sin (x + h))) / (cos (x + h))) - (((sin x) * (sin x)) / (cos x)) by XCMPLX_1:77 .= ((1 - ((cos (x + h)) * (cos (x + h)))) / (cos (x + h))) - (((sin x) * (sin x)) / (cos x)) by SIN_COS4:4 .= ((1 / (cos (x + h))) - (((cos (x + h)) * (cos (x + h))) / (cos (x + h)))) - ((1 - ((cos x) * (cos x))) / (cos x)) by SIN_COS4:4 .= ((1 / (cos (x + h))) - (cos (x + h))) - ((1 / (cos x)) - (((cos x) * (cos x)) / (cos x))) by A2, FDIFF_8:1, XCMPLX_1:89 .= ((1 / (cos (x + h))) - (cos (x + h))) - ((1 / (cos x)) - (cos x)) by A2, FDIFF_8:1, XCMPLX_1:89 .= (((1 / (cos (x + h))) - (cos (x + h))) - (1 / (cos x))) + (cos x) ; hence (fD (f,h)) . x = (((1 / (cos (x + h))) - (cos (x + h))) - (1 / (cos x))) + (cos x) ; ::_thesis: verum end; theorem :: DIFF_3:79 for x, h being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = (tan (#) sin) . x ) & x in dom tan & x - h in dom tan holds (bD (f,h)) . x = (((1 / (cos x)) - (cos x)) - (1 / (cos (x - h)))) + (cos (x - h)) proof let x, h be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = (tan (#) sin) . x ) & x in dom tan & x - h in dom tan holds (bD (f,h)) . x = (((1 / (cos x)) - (cos x)) - (1 / (cos (x - h)))) + (cos (x - h)) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = (tan (#) sin) . x ) & x in dom tan & x - h in dom tan implies (bD (f,h)) . x = (((1 / (cos x)) - (cos x)) - (1 / (cos (x - h)))) + (cos (x - h)) ) assume that A1: for x being Real holds f . x = (tan (#) sin) . x and A2: ( x in dom tan & x - h in dom tan ) ; ::_thesis: (bD (f,h)) . x = (((1 / (cos x)) - (cos x)) - (1 / (cos (x - h)))) + (cos (x - h)) (bD (f,h)) . x = (f . x) - (f . (x - h)) by DIFF_1:4 .= ((tan (#) sin) . x) - (f . (x - h)) by A1 .= ((tan (#) sin) . x) - ((tan (#) sin) . (x - h)) by A1 .= ((tan . x) * (sin . x)) - ((tan (#) sin) . (x - h)) by VALUED_1:5 .= ((tan . x) * (sin . x)) - ((tan . (x - h)) * (sin . (x - h))) by VALUED_1:5 .= (((sin . x) * ((cos . x) ")) * (sin . x)) - ((tan . (x - h)) * (sin . (x - h))) by A2, RFUNCT_1:def_1 .= (((sin x) / (cos x)) * (sin x)) - (((sin (x - h)) / (cos (x - h))) * (sin (x - h))) by A2, RFUNCT_1:def_1 .= ((sin x) / ((cos x) / (sin x))) - (((sin (x - h)) / (cos (x - h))) * (sin (x - h))) by XCMPLX_1:82 .= ((sin x) / ((cos x) / (sin x))) - ((sin (x - h)) / ((cos (x - h)) / (sin (x - h)))) by XCMPLX_1:82 .= (((sin x) * (sin x)) / (cos x)) - ((sin (x - h)) / ((cos (x - h)) / (sin (x - h)))) by XCMPLX_1:77 .= (((sin x) * (sin x)) / (cos x)) - (((sin (x - h)) * (sin (x - h))) / (cos (x - h))) by XCMPLX_1:77 .= ((1 - ((cos x) * (cos x))) / (cos x)) - (((sin (x - h)) * (sin (x - h))) / (cos (x - h))) by SIN_COS4:4 .= ((1 / (cos x)) - (((cos x) * (cos x)) / (cos x))) - ((1 - ((cos (x - h)) * (cos (x - h)))) / (cos (x - h))) by SIN_COS4:4 .= ((1 / (cos x)) - (cos x)) - ((1 / (cos (x - h))) - (((cos (x - h)) * (cos (x - h))) / (cos (x - h)))) by A2, FDIFF_8:1, XCMPLX_1:89 .= ((1 / (cos x)) - (cos x)) - ((1 / (cos (x - h))) - (cos (x - h))) by A2, FDIFF_8:1, XCMPLX_1:89 .= (((1 / (cos x)) - (cos x)) - (1 / (cos (x - h)))) + (cos (x - h)) ; hence (bD (f,h)) . x = (((1 / (cos x)) - (cos x)) - (1 / (cos (x - h)))) + (cos (x - h)) ; ::_thesis: verum end; theorem :: DIFF_3:80 for x, h being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = (tan (#) sin) . x ) & x + (h / 2) in dom tan & x - (h / 2) in dom tan holds (cD (f,h)) . x = (((1 / (cos (x + (h / 2)))) - (cos (x + (h / 2)))) - (1 / (cos (x - (h / 2))))) + (cos (x - (h / 2))) proof let x, h be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = (tan (#) sin) . x ) & x + (h / 2) in dom tan & x - (h / 2) in dom tan holds (cD (f,h)) . x = (((1 / (cos (x + (h / 2)))) - (cos (x + (h / 2)))) - (1 / (cos (x - (h / 2))))) + (cos (x - (h / 2))) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = (tan (#) sin) . x ) & x + (h / 2) in dom tan & x - (h / 2) in dom tan implies (cD (f,h)) . x = (((1 / (cos (x + (h / 2)))) - (cos (x + (h / 2)))) - (1 / (cos (x - (h / 2))))) + (cos (x - (h / 2))) ) assume that A1: for x being Real holds f . x = (tan (#) sin) . x and A2: ( x + (h / 2) in dom tan & x - (h / 2) in dom tan ) ; ::_thesis: (cD (f,h)) . x = (((1 / (cos (x + (h / 2)))) - (cos (x + (h / 2)))) - (1 / (cos (x - (h / 2))))) + (cos (x - (h / 2))) (cD (f,h)) . x = (f . (x + (h / 2))) - (f . (x - (h / 2))) by DIFF_1:5 .= ((tan (#) sin) . (x + (h / 2))) - (f . (x - (h / 2))) by A1 .= ((tan (#) sin) . (x + (h / 2))) - ((tan (#) sin) . (x - (h / 2))) by A1 .= ((tan . (x + (h / 2))) * (sin . (x + (h / 2)))) - ((tan (#) sin) . (x - (h / 2))) by VALUED_1:5 .= ((tan . (x + (h / 2))) * (sin . (x + (h / 2)))) - ((tan . (x - (h / 2))) * (sin . (x - (h / 2)))) by VALUED_1:5 .= (((sin . (x + (h / 2))) * ((cos . (x + (h / 2))) ")) * (sin . (x + (h / 2)))) - ((tan . (x - (h / 2))) * (sin . (x - (h / 2)))) by A2, RFUNCT_1:def_1 .= (((sin (x + (h / 2))) / (cos (x + (h / 2)))) * (sin (x + (h / 2)))) - (((sin (x - (h / 2))) / (cos (x - (h / 2)))) * (sin (x - (h / 2)))) by A2, RFUNCT_1:def_1 .= ((sin (x + (h / 2))) / ((cos (x + (h / 2))) / (sin (x + (h / 2))))) - (((sin (x - (h / 2))) / (cos (x - (h / 2)))) * (sin (x - (h / 2)))) by XCMPLX_1:82 .= ((sin (x + (h / 2))) / ((cos (x + (h / 2))) / (sin (x + (h / 2))))) - ((sin (x - (h / 2))) / ((cos (x - (h / 2))) / (sin (x - (h / 2))))) by XCMPLX_1:82 .= (((sin (x + (h / 2))) * (sin (x + (h / 2)))) / (cos (x + (h / 2)))) - ((sin (x - (h / 2))) / ((cos (x - (h / 2))) / (sin (x - (h / 2))))) by XCMPLX_1:77 .= (((sin (x + (h / 2))) * (sin (x + (h / 2)))) / (cos (x + (h / 2)))) - (((sin (x - (h / 2))) * (sin (x - (h / 2)))) / (cos (x - (h / 2)))) by XCMPLX_1:77 .= ((1 - ((cos (x + (h / 2))) * (cos (x + (h / 2))))) / (cos (x + (h / 2)))) - (((sin (x - (h / 2))) * (sin (x - (h / 2)))) / (cos (x - (h / 2)))) by SIN_COS4:4 .= ((1 / (cos (x + (h / 2)))) - (((cos (x + (h / 2))) * (cos (x + (h / 2)))) / (cos (x + (h / 2))))) - ((1 - ((cos (x - (h / 2))) * (cos (x - (h / 2))))) / (cos (x - (h / 2)))) by SIN_COS4:4 .= ((1 / (cos (x + (h / 2)))) - (cos (x + (h / 2)))) - ((1 / (cos (x - (h / 2)))) - (((cos (x - (h / 2))) * (cos (x - (h / 2)))) / (cos (x - (h / 2))))) by A2, FDIFF_8:1, XCMPLX_1:89 .= ((1 / (cos (x + (h / 2)))) - (cos (x + (h / 2)))) - ((1 / (cos (x - (h / 2)))) - (cos (x - (h / 2)))) by A2, FDIFF_8:1, XCMPLX_1:89 .= (((1 / (cos (x + (h / 2)))) - (cos (x + (h / 2)))) - (1 / (cos (x - (h / 2))))) + (cos (x - (h / 2))) ; hence (cD (f,h)) . x = (((1 / (cos (x + (h / 2)))) - (cos (x + (h / 2)))) - (1 / (cos (x - (h / 2))))) + (cos (x - (h / 2))) ; ::_thesis: verum end; theorem :: DIFF_3:81 for x0, x1 being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = (tan (#) cos) . x ) & x0 in dom tan & x1 in dom tan holds [!f,x0,x1!] = ((sin x0) - (sin x1)) / (x0 - x1) proof let x0, x1 be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = (tan (#) cos) . x ) & x0 in dom tan & x1 in dom tan holds [!f,x0,x1!] = ((sin x0) - (sin x1)) / (x0 - x1) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = (tan (#) cos) . x ) & x0 in dom tan & x1 in dom tan implies [!f,x0,x1!] = ((sin x0) - (sin x1)) / (x0 - x1) ) assume that A1: for x being Real holds f . x = (tan (#) cos) . x and A2: ( x0 in dom tan & x1 in dom tan ) ; ::_thesis: [!f,x0,x1!] = ((sin x0) - (sin x1)) / (x0 - x1) A3: f . x0 = (tan (#) cos) . x0 by A1; f . x1 = (tan (#) cos) . x1 by A1; then [!f,x0,x1!] = (((tan . x0) * (cos . x0)) - ((tan (#) cos) . x1)) / (x0 - x1) by A3, VALUED_1:5 .= (((tan . x0) * (cos . x0)) - ((tan . x1) * (cos . x1))) / (x0 - x1) by VALUED_1:5 .= ((((sin . x0) * ((cos . x0) ")) * (cos . x0)) - ((tan . x1) * (cos . x1))) / (x0 - x1) by A2, RFUNCT_1:def_1 .= ((((sin x0) / (cos x0)) * (cos x0)) - (((sin x1) / (cos x1)) * (cos x1))) / (x0 - x1) by A2, RFUNCT_1:def_1 .= (((sin x0) / ((cos x0) / (cos x0))) - (((sin x1) / (cos x1)) * (cos x1))) / (x0 - x1) by XCMPLX_1:82 .= (((sin x0) / ((cos x0) * (1 / (cos x0)))) - ((sin x1) / ((cos x1) / (cos x1)))) / (x0 - x1) by XCMPLX_1:82 .= (((sin x0) / 1) - ((sin x1) / ((cos x1) * (1 / (cos x1))))) / (x0 - x1) by A2, FDIFF_8:1, XCMPLX_1:106 .= (((sin x0) / 1) - ((sin x1) / 1)) / (x0 - x1) by A2, FDIFF_8:1, XCMPLX_1:106 .= ((sin x0) - (sin x1)) / (x0 - x1) ; hence [!f,x0,x1!] = ((sin x0) - (sin x1)) / (x0 - x1) ; ::_thesis: verum end; theorem :: DIFF_3:82 for x, h being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = (tan (#) cos) . x ) & x in dom tan & x + h in dom tan holds (fD (f,h)) . x = (sin (x + h)) - (sin x) proof let x, h be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = (tan (#) cos) . x ) & x in dom tan & x + h in dom tan holds (fD (f,h)) . x = (sin (x + h)) - (sin x) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = (tan (#) cos) . x ) & x in dom tan & x + h in dom tan implies (fD (f,h)) . x = (sin (x + h)) - (sin x) ) assume that A1: for x being Real holds f . x = (tan (#) cos) . x and A2: ( x in dom tan & x + h in dom tan ) ; ::_thesis: (fD (f,h)) . x = (sin (x + h)) - (sin x) (fD (f,h)) . x = (f . (x + h)) - (f . x) by DIFF_1:3 .= ((tan (#) cos) . (x + h)) - (f . x) by A1 .= ((tan (#) cos) . (x + h)) - ((tan (#) cos) . x) by A1 .= ((tan . (x + h)) * (cos . (x + h))) - ((tan (#) cos) . x) by VALUED_1:5 .= ((tan . (x + h)) * (cos . (x + h))) - ((tan . x) * (cos . x)) by VALUED_1:5 .= (((sin . (x + h)) * ((cos . (x + h)) ")) * (cos . (x + h))) - ((tan . x) * (cos . x)) by A2, RFUNCT_1:def_1 .= (((sin (x + h)) / (cos (x + h))) * (cos (x + h))) - (((sin x) / (cos x)) * (cos x)) by A2, RFUNCT_1:def_1 .= ((sin (x + h)) / ((cos (x + h)) / (cos (x + h)))) - (((sin x) / (cos x)) * (cos x)) by XCMPLX_1:82 .= ((sin (x + h)) / ((cos (x + h)) * (1 / (cos (x + h))))) - ((sin x) / ((cos x) / (cos x))) by XCMPLX_1:82 .= ((sin (x + h)) / 1) - ((sin x) / ((cos x) * (1 / (cos x)))) by A2, FDIFF_8:1, XCMPLX_1:106 .= ((sin (x + h)) / 1) - ((sin x) / 1) by A2, FDIFF_8:1, XCMPLX_1:106 .= (sin (x + h)) - (sin x) ; hence (fD (f,h)) . x = (sin (x + h)) - (sin x) ; ::_thesis: verum end; theorem :: DIFF_3:83 for x, h being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = (tan (#) cos) . x ) & x in dom tan & x - h in dom tan holds (bD (f,h)) . x = (sin x) - (sin (x - h)) proof let x, h be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = (tan (#) cos) . x ) & x in dom tan & x - h in dom tan holds (bD (f,h)) . x = (sin x) - (sin (x - h)) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = (tan (#) cos) . x ) & x in dom tan & x - h in dom tan implies (bD (f,h)) . x = (sin x) - (sin (x - h)) ) assume that A1: for x being Real holds f . x = (tan (#) cos) . x and A2: ( x in dom tan & x - h in dom tan ) ; ::_thesis: (bD (f,h)) . x = (sin x) - (sin (x - h)) (bD (f,h)) . x = (f . x) - (f . (x - h)) by DIFF_1:4 .= ((tan (#) cos) . x) - (f . (x - h)) by A1 .= ((tan (#) cos) . x) - ((tan (#) cos) . (x - h)) by A1 .= ((tan . x) * (cos . x)) - ((tan (#) cos) . (x - h)) by VALUED_1:5 .= ((tan . x) * (cos . x)) - ((tan . (x - h)) * (cos . (x - h))) by VALUED_1:5 .= (((sin . x) * ((cos . x) ")) * (cos . x)) - ((tan . (x - h)) * (cos . (x - h))) by A2, RFUNCT_1:def_1 .= (((sin x) / (cos x)) * (cos x)) - (((sin (x - h)) / (cos (x - h))) * (cos (x - h))) by A2, RFUNCT_1:def_1 .= ((sin x) / ((cos x) / (cos x))) - (((sin (x - h)) / (cos (x - h))) * (cos (x - h))) by XCMPLX_1:82 .= ((sin x) / ((cos x) * (1 / (cos x)))) - ((sin (x - h)) / ((cos (x - h)) / (cos (x - h)))) by XCMPLX_1:82 .= ((sin x) / 1) - ((sin (x - h)) / ((cos (x - h)) * (1 / (cos (x - h))))) by A2, FDIFF_8:1, XCMPLX_1:106 .= ((sin x) / 1) - ((sin (x - h)) / 1) by A2, FDIFF_8:1, XCMPLX_1:106 .= (sin x) - (sin (x - h)) ; hence (bD (f,h)) . x = (sin x) - (sin (x - h)) ; ::_thesis: verum end; theorem :: DIFF_3:84 for x, h being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = (tan (#) cos) . x ) & x + (h / 2) in dom tan & x - (h / 2) in dom tan holds (cD (f,h)) . x = (sin (x + (h / 2))) - (sin (x - (h / 2))) proof let x, h be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = (tan (#) cos) . x ) & x + (h / 2) in dom tan & x - (h / 2) in dom tan holds (cD (f,h)) . x = (sin (x + (h / 2))) - (sin (x - (h / 2))) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = (tan (#) cos) . x ) & x + (h / 2) in dom tan & x - (h / 2) in dom tan implies (cD (f,h)) . x = (sin (x + (h / 2))) - (sin (x - (h / 2))) ) assume that A1: for x being Real holds f . x = (tan (#) cos) . x and A2: ( x + (h / 2) in dom tan & x - (h / 2) in dom tan ) ; ::_thesis: (cD (f,h)) . x = (sin (x + (h / 2))) - (sin (x - (h / 2))) (cD (f,h)) . x = (f . (x + (h / 2))) - (f . (x - (h / 2))) by DIFF_1:5 .= ((tan (#) cos) . (x + (h / 2))) - (f . (x - (h / 2))) by A1 .= ((tan (#) cos) . (x + (h / 2))) - ((tan (#) cos) . (x - (h / 2))) by A1 .= ((tan . (x + (h / 2))) * (cos . (x + (h / 2)))) - ((tan (#) cos) . (x - (h / 2))) by VALUED_1:5 .= ((tan . (x + (h / 2))) * (cos . (x + (h / 2)))) - ((tan . (x - (h / 2))) * (cos . (x - (h / 2)))) by VALUED_1:5 .= (((sin . (x + (h / 2))) * ((cos . (x + (h / 2))) ")) * (cos . (x + (h / 2)))) - ((tan . (x - (h / 2))) * (cos . (x - (h / 2)))) by A2, RFUNCT_1:def_1 .= (((sin (x + (h / 2))) / (cos (x + (h / 2)))) * (cos (x + (h / 2)))) - (((sin (x - (h / 2))) / (cos (x - (h / 2)))) * (cos (x - (h / 2)))) by A2, RFUNCT_1:def_1 .= ((sin (x + (h / 2))) / ((cos (x + (h / 2))) / (cos (x + (h / 2))))) - (((sin (x - (h / 2))) / (cos (x - (h / 2)))) * (cos (x - (h / 2)))) by XCMPLX_1:82 .= ((sin (x + (h / 2))) / ((cos (x + (h / 2))) * (1 / (cos (x + (h / 2)))))) - ((sin (x - (h / 2))) / ((cos (x - (h / 2))) / (cos (x - (h / 2))))) by XCMPLX_1:82 .= ((sin (x + (h / 2))) / 1) - ((sin (x - (h / 2))) / ((cos (x - (h / 2))) * (1 / (cos (x - (h / 2)))))) by A2, FDIFF_8:1, XCMPLX_1:106 .= ((sin (x + (h / 2))) / 1) - ((sin (x - (h / 2))) / 1) by A2, FDIFF_8:1, XCMPLX_1:106 .= (sin (x + (h / 2))) - (sin (x - (h / 2))) ; hence (cD (f,h)) . x = (sin (x + (h / 2))) - (sin (x - (h / 2))) ; ::_thesis: verum end; theorem :: DIFF_3:85 for x0, x1 being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = (cot (#) cos) . x ) & x0 in dom cot & x1 in dom cot holds [!f,x0,x1!] = ((((1 / (sin x0)) - (sin x0)) - (1 / (sin x1))) + (sin x1)) / (x0 - x1) proof let x0, x1 be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = (cot (#) cos) . x ) & x0 in dom cot & x1 in dom cot holds [!f,x0,x1!] = ((((1 / (sin x0)) - (sin x0)) - (1 / (sin x1))) + (sin x1)) / (x0 - x1) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = (cot (#) cos) . x ) & x0 in dom cot & x1 in dom cot implies [!f,x0,x1!] = ((((1 / (sin x0)) - (sin x0)) - (1 / (sin x1))) + (sin x1)) / (x0 - x1) ) assume that A1: for x being Real holds f . x = (cot (#) cos) . x and A2: ( x0 in dom cot & x1 in dom cot ) ; ::_thesis: [!f,x0,x1!] = ((((1 / (sin x0)) - (sin x0)) - (1 / (sin x1))) + (sin x1)) / (x0 - x1) A3: f . x0 = (cot (#) cos) . x0 by A1; f . x1 = (cot (#) cos) . x1 by A1; then [!f,x0,x1!] = (((cot . x0) * (cos . x0)) - ((cot (#) cos) . x1)) / (x0 - x1) by A3, VALUED_1:5 .= (((cot . x0) * (cos . x0)) - ((cot . x1) * (cos . x1))) / (x0 - x1) by VALUED_1:5 .= ((((cos . x0) * ((sin . x0) ")) * (cos . x0)) - ((cot . x1) * (cos . x1))) / (x0 - x1) by A2, RFUNCT_1:def_1 .= ((((cos x0) / (sin x0)) * (cos x0)) - (((cos x1) / (sin x1)) * (cos x1))) / (x0 - x1) by A2, RFUNCT_1:def_1 .= (((cos x0) / ((sin x0) / (cos x0))) - (((cos x1) / (sin x1)) * (cos x1))) / (x0 - x1) by XCMPLX_1:82 .= (((cos x0) / ((sin x0) / (cos x0))) - ((cos x1) / ((sin x1) / (cos x1)))) / (x0 - x1) by XCMPLX_1:82 .= ((((cos x0) * (cos x0)) / (sin x0)) - ((cos x1) / ((sin x1) / (cos x1)))) / (x0 - x1) by XCMPLX_1:77 .= ((((cos x0) * (cos x0)) / (sin x0)) - (((cos x1) * (cos x1)) / (sin x1))) / (x0 - x1) by XCMPLX_1:77 .= (((1 - ((sin x0) * (sin x0))) / (sin x0)) - (((cos x1) * (cos x1)) / (sin x1))) / (x0 - x1) by SIN_COS4:5 .= (((1 / (sin x0)) - (((sin x0) * (sin x0)) / (sin x0))) - ((1 - ((sin x1) * (sin x1))) / (sin x1))) / (x0 - x1) by SIN_COS4:5 .= (((1 / (sin x0)) - (sin x0)) - ((1 / (sin x1)) - (((sin x1) * (sin x1)) / (sin x1)))) / (x0 - x1) by A2, FDIFF_8:2, XCMPLX_1:89 .= (((1 / (sin x0)) - (sin x0)) - ((1 / (sin x1)) - (sin x1))) / (x0 - x1) by A2, FDIFF_8:2, XCMPLX_1:89 .= ((((1 / (sin x0)) - (sin x0)) - (1 / (sin x1))) + (sin x1)) / (x0 - x1) ; hence [!f,x0,x1!] = ((((1 / (sin x0)) - (sin x0)) - (1 / (sin x1))) + (sin x1)) / (x0 - x1) ; ::_thesis: verum end; theorem :: DIFF_3:86 for x, h being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = (cot (#) cos) . x ) & x in dom cot & x + h in dom cot holds (fD (f,h)) . x = (((1 / (sin (x + h))) - (sin (x + h))) - (1 / (sin x))) + (sin x) proof let x, h be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = (cot (#) cos) . x ) & x in dom cot & x + h in dom cot holds (fD (f,h)) . x = (((1 / (sin (x + h))) - (sin (x + h))) - (1 / (sin x))) + (sin x) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = (cot (#) cos) . x ) & x in dom cot & x + h in dom cot implies (fD (f,h)) . x = (((1 / (sin (x + h))) - (sin (x + h))) - (1 / (sin x))) + (sin x) ) assume that A1: for x being Real holds f . x = (cot (#) cos) . x and A2: ( x in dom cot & x + h in dom cot ) ; ::_thesis: (fD (f,h)) . x = (((1 / (sin (x + h))) - (sin (x + h))) - (1 / (sin x))) + (sin x) (fD (f,h)) . x = (f . (x + h)) - (f . x) by DIFF_1:3 .= ((cot (#) cos) . (x + h)) - (f . x) by A1 .= ((cot (#) cos) . (x + h)) - ((cot (#) cos) . x) by A1 .= ((cot . (x + h)) * (cos . (x + h))) - ((cot (#) cos) . x) by VALUED_1:5 .= ((cot . (x + h)) * (cos . (x + h))) - ((cot . x) * (cos . x)) by VALUED_1:5 .= (((cos . (x + h)) * ((sin . (x + h)) ")) * (cos . (x + h))) - ((cot . x) * (cos . x)) by A2, RFUNCT_1:def_1 .= (((cos (x + h)) / (sin (x + h))) * (cos (x + h))) - (((cos x) / (sin x)) * (cos x)) by A2, RFUNCT_1:def_1 .= ((cos (x + h)) / ((sin (x + h)) / (cos (x + h)))) - (((cos x) / (sin x)) * (cos x)) by XCMPLX_1:82 .= ((cos (x + h)) / ((sin (x + h)) / (cos (x + h)))) - ((cos x) / ((sin x) / (cos x))) by XCMPLX_1:82 .= (((cos (x + h)) * (cos (x + h))) / (sin (x + h))) - ((cos x) / ((sin x) / (cos x))) by XCMPLX_1:77 .= (((cos (x + h)) * (cos (x + h))) / (sin (x + h))) - (((cos x) * (cos x)) / (sin x)) by XCMPLX_1:77 .= ((1 - ((sin (x + h)) * (sin (x + h)))) / (sin (x + h))) - (((cos x) * (cos x)) / (sin x)) by SIN_COS4:5 .= ((1 / (sin (x + h))) - (((sin (x + h)) * (sin (x + h))) / (sin (x + h)))) - ((1 - ((sin x) * (sin x))) / (sin x)) by SIN_COS4:5 .= ((1 / (sin (x + h))) - (sin (x + h))) - ((1 / (sin x)) - (((sin x) * (sin x)) / (sin x))) by A2, FDIFF_8:2, XCMPLX_1:89 .= ((1 / (sin (x + h))) - (sin (x + h))) - ((1 / (sin x)) - (sin x)) by A2, FDIFF_8:2, XCMPLX_1:89 .= (((1 / (sin (x + h))) - (sin (x + h))) - (1 / (sin x))) + (sin x) ; hence (fD (f,h)) . x = (((1 / (sin (x + h))) - (sin (x + h))) - (1 / (sin x))) + (sin x) ; ::_thesis: verum end; theorem :: DIFF_3:87 for x, h being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = (cot (#) cos) . x ) & x in dom cot & x - h in dom cot holds (bD (f,h)) . x = (((1 / (sin x)) - (sin x)) - (1 / (sin (x - h)))) + (sin (x - h)) proof let x, h be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = (cot (#) cos) . x ) & x in dom cot & x - h in dom cot holds (bD (f,h)) . x = (((1 / (sin x)) - (sin x)) - (1 / (sin (x - h)))) + (sin (x - h)) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = (cot (#) cos) . x ) & x in dom cot & x - h in dom cot implies (bD (f,h)) . x = (((1 / (sin x)) - (sin x)) - (1 / (sin (x - h)))) + (sin (x - h)) ) assume that A1: for x being Real holds f . x = (cot (#) cos) . x and A2: ( x in dom cot & x - h in dom cot ) ; ::_thesis: (bD (f,h)) . x = (((1 / (sin x)) - (sin x)) - (1 / (sin (x - h)))) + (sin (x - h)) (bD (f,h)) . x = (f . x) - (f . (x - h)) by DIFF_1:4 .= ((cot (#) cos) . x) - (f . (x - h)) by A1 .= ((cot (#) cos) . x) - ((cot (#) cos) . (x - h)) by A1 .= ((cot . x) * (cos . x)) - ((cot (#) cos) . (x - h)) by VALUED_1:5 .= ((cot . x) * (cos . x)) - ((cot . (x - h)) * (cos . (x - h))) by VALUED_1:5 .= (((cos . x) * ((sin . x) ")) * (cos . x)) - ((cot . (x - h)) * (cos . (x - h))) by A2, RFUNCT_1:def_1 .= (((cos x) / (sin x)) * (cos x)) - (((cos (x - h)) / (sin (x - h))) * (cos (x - h))) by A2, RFUNCT_1:def_1 .= ((cos x) / ((sin x) / (cos x))) - (((cos (x - h)) / (sin (x - h))) * (cos (x - h))) by XCMPLX_1:82 .= ((cos x) / ((sin x) / (cos x))) - ((cos (x - h)) / ((sin (x - h)) / (cos (x - h)))) by XCMPLX_1:82 .= (((cos x) * (cos x)) / (sin x)) - ((cos (x - h)) / ((sin (x - h)) / (cos (x - h)))) by XCMPLX_1:77 .= (((cos x) * (cos x)) / (sin x)) - (((cos (x - h)) * (cos (x - h))) / (sin (x - h))) by XCMPLX_1:77 .= ((1 - ((sin x) * (sin x))) / (sin x)) - (((cos (x - h)) * (cos (x - h))) / (sin (x - h))) by SIN_COS4:5 .= ((1 / (sin x)) - (((sin x) * (sin x)) / (sin x))) - ((1 - ((sin (x - h)) * (sin (x - h)))) / (sin (x - h))) by SIN_COS4:5 .= ((1 / (sin x)) - (sin x)) - ((1 / (sin (x - h))) - (((sin (x - h)) * (sin (x - h))) / (sin (x - h)))) by A2, FDIFF_8:2, XCMPLX_1:89 .= ((1 / (sin x)) - (sin x)) - ((1 / (sin (x - h))) - (sin (x - h))) by A2, FDIFF_8:2, XCMPLX_1:89 .= (((1 / (sin x)) - (sin x)) - (1 / (sin (x - h)))) + (sin (x - h)) ; hence (bD (f,h)) . x = (((1 / (sin x)) - (sin x)) - (1 / (sin (x - h)))) + (sin (x - h)) ; ::_thesis: verum end; theorem :: DIFF_3:88 for x, h being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = (cot (#) cos) . x ) & x + (h / 2) in dom cot & x - (h / 2) in dom cot holds (cD (f,h)) . x = (((1 / (sin (x + (h / 2)))) - (sin (x + (h / 2)))) - (1 / (sin (x - (h / 2))))) + (sin (x - (h / 2))) proof let x, h be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = (cot (#) cos) . x ) & x + (h / 2) in dom cot & x - (h / 2) in dom cot holds (cD (f,h)) . x = (((1 / (sin (x + (h / 2)))) - (sin (x + (h / 2)))) - (1 / (sin (x - (h / 2))))) + (sin (x - (h / 2))) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = (cot (#) cos) . x ) & x + (h / 2) in dom cot & x - (h / 2) in dom cot implies (cD (f,h)) . x = (((1 / (sin (x + (h / 2)))) - (sin (x + (h / 2)))) - (1 / (sin (x - (h / 2))))) + (sin (x - (h / 2))) ) assume that A1: for x being Real holds f . x = (cot (#) cos) . x and A2: ( x + (h / 2) in dom cot & x - (h / 2) in dom cot ) ; ::_thesis: (cD (f,h)) . x = (((1 / (sin (x + (h / 2)))) - (sin (x + (h / 2)))) - (1 / (sin (x - (h / 2))))) + (sin (x - (h / 2))) (cD (f,h)) . x = (f . (x + (h / 2))) - (f . (x - (h / 2))) by DIFF_1:5 .= ((cot (#) cos) . (x + (h / 2))) - (f . (x - (h / 2))) by A1 .= ((cot (#) cos) . (x + (h / 2))) - ((cot (#) cos) . (x - (h / 2))) by A1 .= ((cot . (x + (h / 2))) * (cos . (x + (h / 2)))) - ((cot (#) cos) . (x - (h / 2))) by VALUED_1:5 .= ((cot . (x + (h / 2))) * (cos . (x + (h / 2)))) - ((cot . (x - (h / 2))) * (cos . (x - (h / 2)))) by VALUED_1:5 .= (((cos . (x + (h / 2))) * ((sin . (x + (h / 2))) ")) * (cos . (x + (h / 2)))) - ((cot . (x - (h / 2))) * (cos . (x - (h / 2)))) by A2, RFUNCT_1:def_1 .= (((cos (x + (h / 2))) / (sin (x + (h / 2)))) * (cos (x + (h / 2)))) - (((cos (x - (h / 2))) / (sin (x - (h / 2)))) * (cos (x - (h / 2)))) by A2, RFUNCT_1:def_1 .= ((cos (x + (h / 2))) / ((sin (x + (h / 2))) / (cos (x + (h / 2))))) - (((cos (x - (h / 2))) / (sin (x - (h / 2)))) * (cos (x - (h / 2)))) by XCMPLX_1:82 .= ((cos (x + (h / 2))) / ((sin (x + (h / 2))) / (cos (x + (h / 2))))) - ((cos (x - (h / 2))) / ((sin (x - (h / 2))) / (cos (x - (h / 2))))) by XCMPLX_1:82 .= (((cos (x + (h / 2))) * (cos (x + (h / 2)))) / (sin (x + (h / 2)))) - ((cos (x - (h / 2))) / ((sin (x - (h / 2))) / (cos (x - (h / 2))))) by XCMPLX_1:77 .= (((cos (x + (h / 2))) * (cos (x + (h / 2)))) / (sin (x + (h / 2)))) - (((cos (x - (h / 2))) * (cos (x - (h / 2)))) / (sin (x - (h / 2)))) by XCMPLX_1:77 .= ((1 - ((sin (x + (h / 2))) * (sin (x + (h / 2))))) / (sin (x + (h / 2)))) - (((cos (x - (h / 2))) * (cos (x - (h / 2)))) / (sin (x - (h / 2)))) by SIN_COS4:5 .= ((1 / (sin (x + (h / 2)))) - (((sin (x + (h / 2))) * (sin (x + (h / 2)))) / (sin (x + (h / 2))))) - ((1 - ((sin (x - (h / 2))) * (sin (x - (h / 2))))) / (sin (x - (h / 2)))) by SIN_COS4:5 .= ((1 / (sin (x + (h / 2)))) - (sin (x + (h / 2)))) - ((1 / (sin (x - (h / 2)))) - (((sin (x - (h / 2))) * (sin (x - (h / 2)))) / (sin (x - (h / 2))))) by A2, FDIFF_8:2, XCMPLX_1:89 .= ((1 / (sin (x + (h / 2)))) - (sin (x + (h / 2)))) - ((1 / (sin (x - (h / 2)))) - (sin (x - (h / 2)))) by A2, FDIFF_8:2, XCMPLX_1:89 .= (((1 / (sin (x + (h / 2)))) - (sin (x + (h / 2)))) - (1 / (sin (x - (h / 2))))) + (sin (x - (h / 2))) ; hence (cD (f,h)) . x = (((1 / (sin (x + (h / 2)))) - (sin (x + (h / 2)))) - (1 / (sin (x - (h / 2))))) + (sin (x - (h / 2))) ; ::_thesis: verum end; theorem :: DIFF_3:89 for x0, x1 being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = (cot (#) sin) . x ) & x0 in dom cot & x1 in dom cot holds [!f,x0,x1!] = ((cos x0) - (cos x1)) / (x0 - x1) proof let x0, x1 be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = (cot (#) sin) . x ) & x0 in dom cot & x1 in dom cot holds [!f,x0,x1!] = ((cos x0) - (cos x1)) / (x0 - x1) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = (cot (#) sin) . x ) & x0 in dom cot & x1 in dom cot implies [!f,x0,x1!] = ((cos x0) - (cos x1)) / (x0 - x1) ) assume that A1: for x being Real holds f . x = (cot (#) sin) . x and A2: ( x0 in dom cot & x1 in dom cot ) ; ::_thesis: [!f,x0,x1!] = ((cos x0) - (cos x1)) / (x0 - x1) A3: f . x0 = (cot (#) sin) . x0 by A1; f . x1 = (cot (#) sin) . x1 by A1; then [!f,x0,x1!] = (((cot . x0) * (sin . x0)) - ((cot (#) sin) . x1)) / (x0 - x1) by A3, VALUED_1:5 .= (((cot . x0) * (sin . x0)) - ((cot . x1) * (sin . x1))) / (x0 - x1) by VALUED_1:5 .= ((((cos . x0) * ((sin . x0) ")) * (sin . x0)) - ((cot . x1) * (sin . x1))) / (x0 - x1) by A2, RFUNCT_1:def_1 .= ((((cos x0) / (sin x0)) * (sin x0)) - (((cos x1) / (sin x1)) * (sin x1))) / (x0 - x1) by A2, RFUNCT_1:def_1 .= (((cos x0) / ((sin x0) / (sin x0))) - (((cos x1) / (sin x1)) * (sin x1))) / (x0 - x1) by XCMPLX_1:82 .= (((cos x0) / ((sin x0) * (1 / (sin x0)))) - ((cos x1) / ((sin x1) / (sin x1)))) / (x0 - x1) by XCMPLX_1:82 .= (((cos x0) / 1) - ((cos x1) / ((sin x1) * (1 / (sin x1))))) / (x0 - x1) by A2, FDIFF_8:2, XCMPLX_1:106 .= (((cos x0) / 1) - ((cos x1) / 1)) / (x0 - x1) by A2, FDIFF_8:2, XCMPLX_1:106 .= ((cos x0) - (cos x1)) / (x0 - x1) ; hence [!f,x0,x1!] = ((cos x0) - (cos x1)) / (x0 - x1) ; ::_thesis: verum end; theorem :: DIFF_3:90 for x, h being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = (cot (#) sin) . x ) & x in dom cot & x + h in dom cot holds (fD (f,h)) . x = (cos (x + h)) - (cos x) proof let x, h be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = (cot (#) sin) . x ) & x in dom cot & x + h in dom cot holds (fD (f,h)) . x = (cos (x + h)) - (cos x) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = (cot (#) sin) . x ) & x in dom cot & x + h in dom cot implies (fD (f,h)) . x = (cos (x + h)) - (cos x) ) assume that A1: for x being Real holds f . x = (cot (#) sin) . x and A2: ( x in dom cot & x + h in dom cot ) ; ::_thesis: (fD (f,h)) . x = (cos (x + h)) - (cos x) (fD (f,h)) . x = (f . (x + h)) - (f . x) by DIFF_1:3 .= ((cot (#) sin) . (x + h)) - (f . x) by A1 .= ((cot (#) sin) . (x + h)) - ((cot (#) sin) . x) by A1 .= ((cot . (x + h)) * (sin . (x + h))) - ((cot (#) sin) . x) by VALUED_1:5 .= ((cot . (x + h)) * (sin . (x + h))) - ((cot . x) * (sin . x)) by VALUED_1:5 .= (((cos . (x + h)) * ((sin . (x + h)) ")) * (sin . (x + h))) - ((cot . x) * (sin . x)) by A2, RFUNCT_1:def_1 .= (((cos (x + h)) / (sin (x + h))) * (sin (x + h))) - (((cos x) / (sin x)) * (sin x)) by A2, RFUNCT_1:def_1 .= ((cos (x + h)) / ((sin (x + h)) / (sin (x + h)))) - (((cos x) / (sin x)) * (sin x)) by XCMPLX_1:82 .= ((cos (x + h)) / ((sin (x + h)) * (1 / (sin (x + h))))) - ((cos x) / ((sin x) / (sin x))) by XCMPLX_1:82 .= ((cos (x + h)) / 1) - ((cos x) / ((sin x) * (1 / (sin x)))) by A2, FDIFF_8:2, XCMPLX_1:106 .= ((cos (x + h)) / 1) - ((cos x) / 1) by A2, FDIFF_8:2, XCMPLX_1:106 .= (cos (x + h)) - (cos x) ; hence (fD (f,h)) . x = (cos (x + h)) - (cos x) ; ::_thesis: verum end; theorem :: DIFF_3:91 for x, h being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = (cot (#) sin) . x ) & x in dom cot & x - h in dom cot holds (bD (f,h)) . x = (cos x) - (cos (x - h)) proof let x, h be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = (cot (#) sin) . x ) & x in dom cot & x - h in dom cot holds (bD (f,h)) . x = (cos x) - (cos (x - h)) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = (cot (#) sin) . x ) & x in dom cot & x - h in dom cot implies (bD (f,h)) . x = (cos x) - (cos (x - h)) ) assume that A1: for x being Real holds f . x = (cot (#) sin) . x and A2: ( x in dom cot & x - h in dom cot ) ; ::_thesis: (bD (f,h)) . x = (cos x) - (cos (x - h)) (bD (f,h)) . x = (f . x) - (f . (x - h)) by DIFF_1:4 .= ((cot (#) sin) . x) - (f . (x - h)) by A1 .= ((cot (#) sin) . x) - ((cot (#) sin) . (x - h)) by A1 .= ((cot . x) * (sin . x)) - ((cot (#) sin) . (x - h)) by VALUED_1:5 .= ((cot . x) * (sin . x)) - ((cot . (x - h)) * (sin . (x - h))) by VALUED_1:5 .= (((cos . x) * ((sin . x) ")) * (sin . x)) - ((cot . (x - h)) * (sin . (x - h))) by A2, RFUNCT_1:def_1 .= (((cos x) / (sin x)) * (sin x)) - (((cos (x - h)) / (sin (x - h))) * (sin (x - h))) by A2, RFUNCT_1:def_1 .= ((cos x) / ((sin x) / (sin x))) - (((cos (x - h)) / (sin (x - h))) * (sin (x - h))) by XCMPLX_1:82 .= ((cos x) / ((sin x) * (1 / (sin x)))) - ((cos (x - h)) / ((sin (x - h)) / (sin (x - h)))) by XCMPLX_1:82 .= ((cos x) / 1) - ((cos (x - h)) / ((sin (x - h)) * (1 / (sin (x - h))))) by A2, FDIFF_8:2, XCMPLX_1:106 .= ((cos x) / 1) - ((cos (x - h)) / 1) by A2, FDIFF_8:2, XCMPLX_1:106 .= (cos x) - (cos (x - h)) ; hence (bD (f,h)) . x = (cos x) - (cos (x - h)) ; ::_thesis: verum end; theorem :: DIFF_3:92 for x, h being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = (cot (#) sin) . x ) & x + (h / 2) in dom cot & x - (h / 2) in dom cot holds (cD (f,h)) . x = (cos (x + (h / 2))) - (cos (x - (h / 2))) proof let x, h be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = (cot (#) sin) . x ) & x + (h / 2) in dom cot & x - (h / 2) in dom cot holds (cD (f,h)) . x = (cos (x + (h / 2))) - (cos (x - (h / 2))) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = (cot (#) sin) . x ) & x + (h / 2) in dom cot & x - (h / 2) in dom cot implies (cD (f,h)) . x = (cos (x + (h / 2))) - (cos (x - (h / 2))) ) assume that A1: for x being Real holds f . x = (cot (#) sin) . x and A2: ( x + (h / 2) in dom cot & x - (h / 2) in dom cot ) ; ::_thesis: (cD (f,h)) . x = (cos (x + (h / 2))) - (cos (x - (h / 2))) (cD (f,h)) . x = (f . (x + (h / 2))) - (f . (x - (h / 2))) by DIFF_1:5 .= ((cot (#) sin) . (x + (h / 2))) - (f . (x - (h / 2))) by A1 .= ((cot (#) sin) . (x + (h / 2))) - ((cot (#) sin) . (x - (h / 2))) by A1 .= ((cot . (x + (h / 2))) * (sin . (x + (h / 2)))) - ((cot (#) sin) . (x - (h / 2))) by VALUED_1:5 .= ((cot . (x + (h / 2))) * (sin . (x + (h / 2)))) - ((cot . (x - (h / 2))) * (sin . (x - (h / 2)))) by VALUED_1:5 .= (((cos . (x + (h / 2))) * ((sin . (x + (h / 2))) ")) * (sin . (x + (h / 2)))) - ((cot . (x - (h / 2))) * (sin . (x - (h / 2)))) by A2, RFUNCT_1:def_1 .= (((cos (x + (h / 2))) / (sin (x + (h / 2)))) * (sin (x + (h / 2)))) - (((cos (x - (h / 2))) / (sin (x - (h / 2)))) * (sin (x - (h / 2)))) by A2, RFUNCT_1:def_1 .= ((cos (x + (h / 2))) / ((sin (x + (h / 2))) / (sin (x + (h / 2))))) - (((cos (x - (h / 2))) / (sin (x - (h / 2)))) * (sin (x - (h / 2)))) by XCMPLX_1:82 .= ((cos (x + (h / 2))) / ((sin (x + (h / 2))) * (1 / (sin (x + (h / 2)))))) - ((cos (x - (h / 2))) / ((sin (x - (h / 2))) / (sin (x - (h / 2))))) by XCMPLX_1:82 .= ((cos (x + (h / 2))) / 1) - ((cos (x - (h / 2))) / ((sin (x - (h / 2))) * (1 / (sin (x - (h / 2)))))) by A2, FDIFF_8:2, XCMPLX_1:106 .= ((cos (x + (h / 2))) / 1) - ((cos (x - (h / 2))) / 1) by A2, FDIFF_8:2, XCMPLX_1:106 .= (cos (x + (h / 2))) - (cos (x - (h / 2))) ; hence (cD (f,h)) . x = (cos (x + (h / 2))) - (cos (x - (h / 2))) ; ::_thesis: verum end; theorem :: DIFF_3:93 for x0, x1 being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = (tan (#) tan) . x ) & x0 in dom tan & x1 in dom tan holds [!f,x0,x1!] = (((cos x1) ^2) - ((cos x0) ^2)) / ((((cos x0) * (cos x1)) ^2) * (x0 - x1)) proof let x0, x1 be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = (tan (#) tan) . x ) & x0 in dom tan & x1 in dom tan holds [!f,x0,x1!] = (((cos x1) ^2) - ((cos x0) ^2)) / ((((cos x0) * (cos x1)) ^2) * (x0 - x1)) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = (tan (#) tan) . x ) & x0 in dom tan & x1 in dom tan implies [!f,x0,x1!] = (((cos x1) ^2) - ((cos x0) ^2)) / ((((cos x0) * (cos x1)) ^2) * (x0 - x1)) ) assume that A1: for x being Real holds f . x = (tan (#) tan) . x and A2: ( x0 in dom tan & x1 in dom tan ) ; ::_thesis: [!f,x0,x1!] = (((cos x1) ^2) - ((cos x0) ^2)) / ((((cos x0) * (cos x1)) ^2) * (x0 - x1)) A3: ( cos x0 <> 0 & cos x1 <> 0 ) by A2, FDIFF_8:1; A4: f . x0 = (tan (#) tan) . x0 by A1; f . x1 = (tan (#) tan) . x1 by A1; then [!f,x0,x1!] = (((tan . x0) * (tan . x0)) - ((tan (#) tan) . x1)) / (x0 - x1) by A4, VALUED_1:5 .= (((tan . x0) * (tan . x0)) - ((tan . x1) * (tan . x1))) / (x0 - x1) by VALUED_1:5 .= ((((sin . x0) * ((cos . x0) ")) * (tan . x0)) - ((tan . x1) * (tan . x1))) / (x0 - x1) by A2, RFUNCT_1:def_1 .= ((((sin . x0) * ((cos . x0) ")) * ((sin . x0) * ((cos . x0) "))) - ((tan . x1) * (tan . x1))) / (x0 - x1) by A2, RFUNCT_1:def_1 .= ((((sin . x0) * ((cos . x0) ")) * ((sin . x0) * ((cos . x0) "))) - (((sin . x1) * ((cos . x1) ")) * (tan . x1))) / (x0 - x1) by A2, RFUNCT_1:def_1 .= (((tan x0) ^2) - ((tan x1) ^2)) / (x0 - x1) by A2, RFUNCT_1:def_1 .= (((tan x0) - (tan x1)) * ((tan x0) + (tan x1))) / (x0 - x1) .= (((sin (x0 - x1)) / ((cos x0) * (cos x1))) * ((tan x0) + (tan x1))) / (x0 - x1) by A3, SIN_COS4:20 .= (((sin (x0 - x1)) / ((cos x0) * (cos x1))) * ((sin (x0 + x1)) / ((cos x0) * (cos x1)))) / (x0 - x1) by A3, SIN_COS4:19 .= (((sin (x0 + x1)) * (sin (x0 - x1))) / (((cos x0) * (cos x1)) ^2)) / (x0 - x1) by XCMPLX_1:76 .= ((((cos x1) ^2) - ((cos x0) ^2)) / (((cos x0) * (cos x1)) ^2)) / (x0 - x1) by SIN_COS4:38 .= (((cos x1) ^2) - ((cos x0) ^2)) / ((((cos x0) * (cos x1)) ^2) * (x0 - x1)) by XCMPLX_1:78 ; hence [!f,x0,x1!] = (((cos x1) ^2) - ((cos x0) ^2)) / ((((cos x0) * (cos x1)) ^2) * (x0 - x1)) ; ::_thesis: verum end; theorem :: DIFF_3:94 for x, h being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = (tan (#) tan) . x ) & x in dom tan & x + h in dom tan holds (fD (f,h)) . x = - (((1 / 2) * ((cos (2 * (x + h))) - (cos (2 * x)))) / (((cos (x + h)) * (cos x)) ^2)) proof let x, h be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = (tan (#) tan) . x ) & x in dom tan & x + h in dom tan holds (fD (f,h)) . x = - (((1 / 2) * ((cos (2 * (x + h))) - (cos (2 * x)))) / (((cos (x + h)) * (cos x)) ^2)) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = (tan (#) tan) . x ) & x in dom tan & x + h in dom tan implies (fD (f,h)) . x = - (((1 / 2) * ((cos (2 * (x + h))) - (cos (2 * x)))) / (((cos (x + h)) * (cos x)) ^2)) ) assume that A1: for x being Real holds f . x = (tan (#) tan) . x and A2: ( x in dom tan & x + h in dom tan ) ; ::_thesis: (fD (f,h)) . x = - (((1 / 2) * ((cos (2 * (x + h))) - (cos (2 * x)))) / (((cos (x + h)) * (cos x)) ^2)) A3: ( cos x <> 0 & cos (x + h) <> 0 ) by A2, FDIFF_8:1; (fD (f,h)) . x = (f . (x + h)) - (f . x) by DIFF_1:3 .= ((tan (#) tan) . (x + h)) - (f . x) by A1 .= ((tan (#) tan) . (x + h)) - ((tan (#) tan) . x) by A1 .= ((tan . (x + h)) * (tan . (x + h))) - ((tan (#) tan) . x) by VALUED_1:5 .= ((tan . (x + h)) * (tan . (x + h))) - ((tan . x) * (tan . x)) by VALUED_1:5 .= (((sin . (x + h)) * ((cos . (x + h)) ")) * (tan . (x + h))) - ((tan . x) * (tan . x)) by A2, RFUNCT_1:def_1 .= (((sin . (x + h)) * ((cos . (x + h)) ")) * ((sin . (x + h)) * ((cos . (x + h)) "))) - ((tan . x) * (tan . x)) by A2, RFUNCT_1:def_1 .= (((sin . (x + h)) * ((cos . (x + h)) ")) * ((sin . (x + h)) * ((cos . (x + h)) "))) - (((sin . x) * ((cos . x) ")) * (tan . x)) by A2, RFUNCT_1:def_1 .= ((tan (x + h)) ^2) - ((tan x) ^2) by A2, RFUNCT_1:def_1 .= ((tan (x + h)) - (tan x)) * ((tan (x + h)) + (tan x)) .= ((sin ((x + h) - x)) / ((cos (x + h)) * (cos x))) * ((tan (x + h)) + (tan x)) by A3, SIN_COS4:20 .= ((sin ((x + h) - x)) / ((cos (x + h)) * (cos x))) * ((sin ((x + h) + x)) / ((cos (x + h)) * (cos x))) by A3, SIN_COS4:19 .= ((sin ((2 * x) + h)) * (sin h)) / (((cos (x + h)) * (cos x)) ^2) by XCMPLX_1:76 .= (- ((1 / 2) * ((cos (((2 * x) + h) + h)) - (cos (((2 * x) + h) - h))))) / (((cos (x + h)) * (cos x)) ^2) by SIN_COS4:29 .= - (((1 / 2) * ((cos (2 * (x + h))) - (cos (2 * x)))) / (((cos (x + h)) * (cos x)) ^2)) ; hence (fD (f,h)) . x = - (((1 / 2) * ((cos (2 * (x + h))) - (cos (2 * x)))) / (((cos (x + h)) * (cos x)) ^2)) ; ::_thesis: verum end; theorem :: DIFF_3:95 for x, h being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = (tan (#) tan) . x ) & x in dom tan & x - h in dom tan holds (bD (f,h)) . x = - (((1 / 2) * ((cos (2 * x)) - (cos (2 * (h - x))))) / (((cos x) * (cos (x - h))) ^2)) proof let x, h be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = (tan (#) tan) . x ) & x in dom tan & x - h in dom tan holds (bD (f,h)) . x = - (((1 / 2) * ((cos (2 * x)) - (cos (2 * (h - x))))) / (((cos x) * (cos (x - h))) ^2)) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = (tan (#) tan) . x ) & x in dom tan & x - h in dom tan implies (bD (f,h)) . x = - (((1 / 2) * ((cos (2 * x)) - (cos (2 * (h - x))))) / (((cos x) * (cos (x - h))) ^2)) ) assume that A1: for x being Real holds f . x = (tan (#) tan) . x and A2: ( x in dom tan & x - h in dom tan ) ; ::_thesis: (bD (f,h)) . x = - (((1 / 2) * ((cos (2 * x)) - (cos (2 * (h - x))))) / (((cos x) * (cos (x - h))) ^2)) A3: ( cos x <> 0 & cos (x - h) <> 0 ) by A2, FDIFF_8:1; (bD (f,h)) . x = (f . x) - (f . (x - h)) by DIFF_1:4 .= ((tan (#) tan) . x) - (f . (x - h)) by A1 .= ((tan (#) tan) . x) - ((tan (#) tan) . (x - h)) by A1 .= ((tan . x) * (tan . x)) - ((tan (#) tan) . (x - h)) by VALUED_1:5 .= ((tan . x) * (tan . x)) - ((tan . (x - h)) * (tan . (x - h))) by VALUED_1:5 .= (((sin . x) * ((cos . x) ")) * (tan . x)) - ((tan . (x - h)) * (tan . (x - h))) by A2, RFUNCT_1:def_1 .= (((sin . x) * ((cos . x) ")) * ((sin . x) * ((cos . x) "))) - ((tan . (x - h)) * (tan . (x - h))) by A2, RFUNCT_1:def_1 .= (((sin . x) * ((cos . x) ")) * ((sin . x) * ((cos . x) "))) - (((sin . (x - h)) * ((cos . (x - h)) ")) * (tan . (x - h))) by A2, RFUNCT_1:def_1 .= ((tan x) ^2) - ((tan (x - h)) ^2) by A2, RFUNCT_1:def_1 .= ((tan x) - (tan (x - h))) * ((tan x) + (tan (x - h))) .= ((sin (x - (x - h))) / ((cos x) * (cos (x - h)))) * ((tan x) + (tan (x - h))) by A3, SIN_COS4:20 .= ((sin h) / ((cos x) * (cos (x - h)))) * ((sin (x + (x - h))) / ((cos x) * (cos (x - h)))) by A3, SIN_COS4:19 .= ((sin h) * (sin ((2 * x) - h))) / (((cos x) * (cos (x - h))) ^2) by XCMPLX_1:76 .= (- ((1 / 2) * ((cos (h + ((2 * x) - h))) - (cos (h - ((2 * x) - h)))))) / (((cos x) * (cos (x - h))) ^2) by SIN_COS4:29 .= - (((1 / 2) * ((cos (2 * x)) - (cos (2 * (h - x))))) / (((cos x) * (cos (x - h))) ^2)) ; hence (bD (f,h)) . x = - (((1 / 2) * ((cos (2 * x)) - (cos (2 * (h - x))))) / (((cos x) * (cos (x - h))) ^2)) ; ::_thesis: verum end; theorem :: DIFF_3:96 for x, h being Real for f being Function of REAL,REAL st ( for x being Real holds f . x = (tan (#) tan) . x ) & x + (h / 2) in dom tan & x - (h / 2) in dom tan holds (cD (f,h)) . x = - (((1 / 2) * ((cos (h + (2 * x))) - (cos (h - (2 * x))))) / (((cos (x + (h / 2))) * (cos (x - (h / 2)))) ^2)) proof let x, h be Real; ::_thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = (tan (#) tan) . x ) & x + (h / 2) in dom tan & x - (h / 2) in dom tan holds (cD (f,h)) . x = - (((1 / 2) * ((cos (h + (2 * x))) - (cos (h - (2 * x))))) / (((cos (x + (h / 2))) * (cos (x - (h / 2)))) ^2)) let f be Function of REAL,REAL; ::_thesis: ( ( for x being Real holds f . x = (tan (#) tan) . x ) & x + (h / 2) in dom tan & x - (h / 2) in dom tan implies (cD (f,h)) . x = - (((1 / 2) * ((cos (h + (2 * x))) - (cos (h - (2 * x))))) / (((cos (x + (h / 2))) * (cos (x - (h / 2)))) ^2)) ) assume that A1: for x being Real holds f . x = (tan (#) tan) . x and A2: ( x + (h / 2) in dom tan & x - (h / 2) in dom tan ) ; ::_thesis: (cD (f,h)) . x = - (((1 / 2) * ((cos (h + (2 * x))) - (cos (h - (2 * x))))) / (((cos (x + (h / 2))) * (cos (x - (h / 2)))) ^2)) A3: ( cos (x + (h / 2)) <> 0 & cos (x - (h / 2)) <> 0 ) by A2, FDIFF_8:1; (cD (f,h)) . x = (f . (x + (h / 2))) - (f . (x - (h / 2))) by DIFF_1:5 .= ((tan (#) tan) . (x + (h / 2))) - (f . (x - (h / 2))) by A1 .= ((tan (#) tan) . (x + (h / 2))) - ((tan (#) tan) . (x - (h / 2))) by A1 .= ((tan . (x + (h / 2))) * (tan . (x + (h / 2)))) - ((tan (#) tan) . (x - (h / 2))) by VALUED_1:5 .= ((tan . (x + (h / 2))) * (tan . (x + (h / 2)))) - ((tan . (x - (h / 2))) * (tan . (x - (h / 2)))) by VALUED_1:5 .= (((sin . (x + (h / 2))) * ((cos . (x + (h / 2))) ")) * (tan . (x + (h / 2)))) - ((tan . (x - (h / 2))) * (tan . (x - (h / 2)))) by A2, RFUNCT_1:def_1 .= (((sin . (x + (h / 2))) * ((cos . (x + (h / 2))) ")) * ((sin . (x + (h / 2))) * ((cos . (x + (h / 2))) "))) - ((tan . (x - (h / 2))) * (tan . (x - (h / 2)))) by A2, RFUNCT_1:def_1 .= (((sin . (x + (h / 2))) * ((cos . (x + (h / 2))) ")) * ((sin . (x + (h / 2))) * ((cos . (x + (h / 2))) "))) - (((sin . (x - (h / 2))) * ((cos . (x - (h / 2))) ")) * (tan . (x - (h / 2)))) by A2, RFUNCT_1:def_1 .= ((tan (x + (h / 2))) ^2) - ((tan (x - (h / 2))) ^2) by A2, RFUNCT_1:def_1 .= ((tan (x + (h / 2))) - (tan (x - (h / 2)))) * ((tan (x + (h / 2))) + (tan (x - (h / 2)))) .= ((sin ((x + (h / 2)) - (x - (h / 2)))) / ((cos (x + (h / 2))) * (cos (x - (h / 2))))) * ((tan (x + (h / 2))) + (tan (x - (h / 2)))) by A3, SIN_COS4:20 .= ((sin h) / ((cos (x + (h / 2))) * (cos (x - (h / 2))))) * ((sin ((x + (h / 2)) + (x - (h / 2)))) / ((cos (x + (h / 2))) * (cos (x - (h / 2))))) by A3, SIN_COS4:19 .= ((sin h) * (sin (2 * x))) / (((cos (x + (h / 2))) * (cos (x - (h / 2)))) ^2) by XCMPLX_1:76 .= (- ((1 / 2) * ((cos (h + (2 * x))) - (cos (h - (2 * x)))))) / (((cos (x + (h / 2))) * (cos (x - (h / 2)))) ^2) by SIN_COS4:29 .= - (((1 / 2) * ((cos (h + (2 * x))) - (cos (h - (2 * x))))) / (((cos (x + (h / 2))) * (cos (x - (h / 2)))) ^2)) ; hence (cD (f,h)) . x = - (((1 / 2) * ((cos (h + (2 * x))) - (cos (h - (2 * x))))) / (((cos (x + (h / 2))) * (cos (x - (h / 2)))) ^2)) ; ::_thesis: verum end;