:: BOR_CANT semantic presentation begin definition let D be set ; let x, y be ext-real number ; let a, b be Element of D; :: original: IFGT redefine func IFGT (x,y,a,b) -> Element of D; coherence IFGT (x,y,a,b) is Element of D by XXREAL_0:def_11; end; theorem Th1: :: BOR_CANT:1 for k being Element of NAT for x being Element of REAL st k is odd & x > 0 & x <= 1 holds (((- x) rExpSeq) . (k + 1)) + (((- x) rExpSeq) . (k + 2)) >= 0 proof let k be Element of NAT ; ::_thesis: for x being Element of REAL st k is odd & x > 0 & x <= 1 holds (((- x) rExpSeq) . (k + 1)) + (((- x) rExpSeq) . (k + 2)) >= 0 let x be Element of REAL ; ::_thesis: ( k is odd & x > 0 & x <= 1 implies (((- x) rExpSeq) . (k + 1)) + (((- x) rExpSeq) . (k + 2)) >= 0 ) assume that A1: k is odd and A2: x > 0 and A3: x <= 1 ; ::_thesis: (((- x) rExpSeq) . (k + 1)) + (((- x) rExpSeq) . (k + 2)) >= 0 consider m being Element of NAT such that A4: k = (2 * m) + 1 by A1, ABIAN:9; set q = m + 1; A5: k + 2 = (2 * (m + 1)) + 1 by A4; consider m being Element of NAT such that A6: k = (2 * m) + 1 by A1, ABIAN:9; A7: for k being Element of NAT st k is even holds (- x) |^ k > 0 proof let k be Element of NAT ; ::_thesis: ( k is even implies (- x) |^ k > 0 ) assume k is even ; ::_thesis: (- x) |^ k > 0 then consider m being Element of NAT such that A8: k = 2 * m by ABIAN:def_2; defpred S1[ Element of NAT ] means (- x) |^ (2 * \$1) > 0 ; A9: S1[ 0 ] by NEWTON:4; A10: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A11: S1[k] ; ::_thesis: S1[k + 1] (- x) |^ (2 * (k + 1)) = (- x) |^ ((2 * k) + 2) ; then A12: (- x) |^ (2 * (k + 1)) = ((- x) |^ (2 * k)) * ((- x) |^ 2) by NEWTON:8; (- x) * (- x) > 0 by A2; then (- x) |^ 2 > 0 by NEWTON:81; hence S1[k + 1] by A11, A12; ::_thesis: verum end; for k being Element of NAT holds S1[k] from NAT_1:sch_1(A9, A10); hence (- x) |^ k > 0 by A8; ::_thesis: verum end; A13: (x |^ (k + 2)) / ((- x) |^ (k + 1)) = x proof x |^ (k + 2) = x |^ ((k + 1) + 1) ; then A14: x |^ (k + 2) = (x |^ (k + 1)) * x by NEWTON:6; x |^ (k + 2) = x * ((- x) |^ (k + 1)) by A6, A14, POWER:1; then (x |^ (k + 2)) / ((- x) |^ (k + 1)) = (x * ((- x) |^ (k + 1))) * (((- x) |^ (k + 1)) ") by XCMPLX_0:def_9; then A15: (x |^ (k + 2)) / ((- x) |^ (k + 1)) = x * (((- x) |^ (k + 1)) * (((- x) |^ (k + 1)) ")) ; ((- x) |^ (k + 1)) * (((- x) |^ (k + 1)) ") = 1 proof A16: 0 < (- x) |^ (k + 1) by A6, A7; A17: 1 <= ((- x) |^ (k + 1)) / ((- x) |^ (k + 1)) by A6, A7, XREAL_1:181; A18: ((- x) |^ (k + 1)) / ((- x) |^ (k + 1)) <= 1 by A16, XREAL_1:185; ((- x) |^ (k + 1)) / ((- x) |^ (k + 1)) = 1 by A17, A18, XXREAL_0:1; hence ((- x) |^ (k + 1)) * (((- x) |^ (k + 1)) ") = 1 by XCMPLX_0:def_9; ::_thesis: verum end; hence (x |^ (k + 2)) / ((- x) |^ (k + 1)) = x by A15; ::_thesis: verum end; A19: 1 <= ((k + 2) !) / ((k + 1) !) proof (k + 2) ! = ((k + 1) + 1) * ((k + 1) !) by NEWTON:15; then A20: ((k + 2) !) * (((k + 1) !) ") = ((k + 1) + 1) * (((k + 1) !) * (((k + 1) !) ")) ; A21: 1 <= ((k + 1) !) / ((k + 1) !) by XREAL_1:181; A22: ((k + 1) !) / ((k + 1) !) <= 1 by XREAL_1:183; ((k + 1) !) / ((k + 1) !) = 1 by A21, A22, XXREAL_0:1; then ((k + 2) !) * (((k + 1) !) ") = ((k + 1) + 1) * 1 by A20, XCMPLX_0:def_9; then ((k + 2) !) * (((k + 1) !) ") >= 1 by NAT_1:11; hence 1 <= ((k + 2) !) / ((k + 1) !) by XCMPLX_0:def_9; ::_thesis: verum end; ( (x |^ (k + 2)) / ((- x) |^ (k + 1)) <= ((k + 2) !) / ((k + 1) !) implies (((- x) rExpSeq) . (k + 1)) + (((- x) rExpSeq) . (k + 2)) >= 0 ) proof assume A23: (x |^ (k + 2)) / ((- x) |^ (k + 1)) <= ((k + 2) !) / ((k + 1) !) ; ::_thesis: (((- x) rExpSeq) . (k + 1)) + (((- x) rExpSeq) . (k + 2)) >= 0 (x |^ (k + 2)) * (((- x) |^ (k + 1)) ") <= ((k + 2) !) / ((k + 1) !) by A23, XCMPLX_0:def_9; then A24: (x |^ (k + 2)) * (((- x) |^ (k + 1)) ") <= (((k + 1) !) ") * ((k + 2) !) by XCMPLX_0:def_9; (- x) |^ (k + 1) > 0 by A6, A7; then A25: (x |^ (k + 2)) / ((k + 2) !) <= (((k + 1) !) ") / (((- x) |^ (k + 1)) ") by A24, XREAL_1:102; A26: (((k + 1) !) ") * 1 = 1 / ((k + 1) !) by XCMPLX_0:def_9; ( (((k + 1) !) ") / (((- x) |^ (k + 1)) ") = (1 / ((k + 1) !)) * ((((- x) |^ (k + 1)) ") ") & 1 * (((k + 1) !) ") = 1 / ((k + 1) !) ) by A26, XCMPLX_0:def_9; then A27: (((k + 1) !) ") / (((- x) |^ (k + 1)) ") = ((- x) |^ (k + 1)) / ((k + 1) !) by XCMPLX_0:def_9; (x rExpSeq) . (k + 2) <= ((- x) |^ (k + 1)) / ((k + 1) !) by A25, A27, SIN_COS:def_5; then (x rExpSeq) . (k + 2) <= ((- x) rExpSeq) . (k + 1) by SIN_COS:def_5; then ((x rExpSeq) . (k + 2)) - (((- x) rExpSeq) . (k + 1)) <= (((- x) rExpSeq) . (k + 1)) - (((- x) rExpSeq) . (k + 1)) by XREAL_1:9; then A28: ( ((x rExpSeq) . (k + 2)) - (((- x) rExpSeq) . (k + 1)) <= 0 & - (((x rExpSeq) . (k + 2)) - (((- x) rExpSeq) . (k + 1))) >= 0 ) ; - ((x rExpSeq) . (k + 2)) = ((- x) rExpSeq) . (k + 2) proof defpred S1[ Element of NAT ] means - (x |^ ((2 * \$1) + 1)) = (- x) |^ ((2 * \$1) + 1); A29: (- x) |^ ((2 * 0) + 1) = - x by NEWTON:5; A30: S1[ 0 ] by A29, NEWTON:5; A31: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A32: S1[k] ; ::_thesis: S1[k + 1] - (x |^ ((2 * (k + 1)) + 1)) = - ((x |^ (((2 * k) + 1) + 1)) * x) by NEWTON:6; then - (x |^ ((2 * (k + 1)) + 1)) = - (((x |^ ((2 * k) + 1)) * x) * x) by NEWTON:6; then - (x |^ ((2 * (k + 1)) + 1)) = (((- x) |^ ((2 * k) + 1)) * (- x)) * (- x) by A32; then - (x |^ ((2 * (k + 1)) + 1)) = ((- x) |^ (((2 * k) + 1) + 1)) * (- x) by NEWTON:6; hence S1[k + 1] by NEWTON:6; ::_thesis: verum end; A33: for k being Element of NAT holds S1[k] from NAT_1:sch_1(A30, A31); consider m being Element of NAT such that A34: k + 2 = (2 * m) + 1 by A5; A35: - (x |^ (k + 2)) = (- x) |^ (k + 2) by A33, A34; - ((x rExpSeq) . (k + 2)) = - ((x |^ (k + 2)) / ((k + 2) !)) by SIN_COS:def_5; then - ((x rExpSeq) . (k + 2)) = - ((x |^ (k + 2)) * (((k + 2) !) ")) by XCMPLX_0:def_9; then - ((x rExpSeq) . (k + 2)) = (- (x |^ (k + 2))) * (((k + 2) !) ") ; then - ((x rExpSeq) . (k + 2)) = (- (x |^ (k + 2))) / ((k + 2) !) by XCMPLX_0:def_9; hence - ((x rExpSeq) . (k + 2)) = ((- x) rExpSeq) . (k + 2) by A35, SIN_COS:def_5; ::_thesis: verum end; hence (((- x) rExpSeq) . (k + 1)) + (((- x) rExpSeq) . (k + 2)) >= 0 by A28; ::_thesis: verum end; hence (((- x) rExpSeq) . (k + 1)) + (((- x) rExpSeq) . (k + 2)) >= 0 by A3, A19, A13, XXREAL_0:2; ::_thesis: verum end; theorem Th2: :: BOR_CANT:2 for x being Element of REAL holds 1 + x <= exp_R . x proof let x be Element of REAL ; ::_thesis: 1 + x <= exp_R . x percases ( x > 0 or x = 0 or x < 0 ) ; supposeA1: x > 0 ; ::_thesis: 1 + x <= exp_R . x set B2 = NAT --> (1 + x); A2: for n being Element of NAT st x > 0 holds (NAT --> (1 + x)) . n <= (Partial_Sums (x rExpSeq)) . (n + 1) proof let n be Element of NAT ; ::_thesis: ( x > 0 implies (NAT --> (1 + x)) . n <= (Partial_Sums (x rExpSeq)) . (n + 1) ) defpred S1[ Nat] means (NAT --> (1 + x)) . \$1 <= (Partial_Sums (x rExpSeq)) . (1 + \$1); (Partial_Sums (x rExpSeq)) . 1 = ((Partial_Sums (x rExpSeq)) . 0) + ((x rExpSeq) . (0 + 1)) by SERIES_1:def_1; then A3: (Partial_Sums (x rExpSeq)) . 1 = ((x rExpSeq) . 0) + ((x rExpSeq) . 1) by SERIES_1:def_1; A4: (x rExpSeq) . 0 = (x |^ 0) / (0 !) by SIN_COS:def_5 .= 1 by NEWTON:4, NEWTON:12 ; (x rExpSeq) . 1 = (x |^ 1) / (1 !) by SIN_COS:def_5; then (x rExpSeq) . 1 = x by NEWTON:5, NEWTON:13; then A5: S1[ 0 ] by A4, A3, FUNCOP_1:7; A6: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A7: S1[k] ; ::_thesis: S1[k + 1] A8: (Partial_Sums (x rExpSeq)) . (1 + (k + 1)) = ((Partial_Sums (x rExpSeq)) . (k + 1)) + ((x rExpSeq) . ((k + 1) + 1)) by SERIES_1:def_1; A9: (x rExpSeq) . ((k + 1) + 1) > 0 proof ( x |^ ((k + 1) + 1) > 0 & ((k + 1) + 1) ! > 0 ) by A1, NEWTON:83; then (x |^ ((k + 1) + 1)) / (((k + 1) + 1) !) > 0 ; hence (x rExpSeq) . ((k + 1) + 1) > 0 by SIN_COS:def_5; ::_thesis: verum end; A10: 1 + x <= (Partial_Sums (x rExpSeq)) . (k + 1) by A7, FUNCOP_1:7; (Partial_Sums (x rExpSeq)) . (k + 1) <= ((x rExpSeq) . ((k + 1) + 1)) + ((Partial_Sums (x rExpSeq)) . (k + 1)) by A9, XREAL_1:31; then 1 + x <= ((Partial_Sums (x rExpSeq)) . (k + 1)) + ((x rExpSeq) . ((k + 1) + 1)) by A10, XXREAL_0:2; hence S1[k + 1] by A8, FUNCOP_1:7; ::_thesis: verum end; for k being Element of NAT holds S1[k] from NAT_1:sch_1(A5, A6); hence ( x > 0 implies (NAT --> (1 + x)) . n <= (Partial_Sums (x rExpSeq)) . (n + 1) ) ; ::_thesis: verum end; A11: for n being Element of NAT holds (NAT --> (1 + x)) . n <= ((Partial_Sums (x rExpSeq)) ^\ 1) . n proof let n be Element of NAT ; ::_thesis: (NAT --> (1 + x)) . n <= ((Partial_Sums (x rExpSeq)) ^\ 1) . n (NAT --> (1 + x)) . n <= (Partial_Sums (x rExpSeq)) . (n + 1) by A1, A2; hence (NAT --> (1 + x)) . n <= ((Partial_Sums (x rExpSeq)) ^\ 1) . n by NAT_1:def_3; ::_thesis: verum end; A12: lim (NAT --> (1 + x)) = (NAT --> (1 + x)) . 1 by SEQ_4:26 .= 1 + x by FUNCOP_1:7 ; x rExpSeq is summable by SIN_COS:45; then A13: Partial_Sums (x rExpSeq) is convergent by SERIES_1:def_2; then A14: ( lim ((Partial_Sums (x rExpSeq)) ^\ 1) = lim (Partial_Sums (x rExpSeq)) & (Partial_Sums (x rExpSeq)) ^\ 1 is convergent ) by SEQ_4:20; lim (NAT --> (1 + x)) <= lim ((Partial_Sums (x rExpSeq)) ^\ 1) by A13, A11, SEQ_2:18; then lim (NAT --> (1 + x)) <= Sum (x rExpSeq) by A14, SERIES_1:def_3; hence 1 + x <= exp_R . x by A12, SIN_COS:def_22; ::_thesis: verum end; suppose x = 0 ; ::_thesis: 1 + x <= exp_R . x hence 1 + x <= exp_R . x by SIN_COS:51; ::_thesis: verum end; supposeA15: x < 0 ; ::_thesis: 1 + x <= exp_R . x set y = - x; 1 - (- x) <= exp_R . (- (- x)) proof percases ( - x <= 1 or - x > 1 ) ; supposeA16: - x <= 1 ; ::_thesis: 1 - (- x) <= exp_R . (- (- x)) for x being Element of REAL st x > 0 & x <= 1 holds 1 - x <= exp_R . (- x) proof let x be Element of REAL ; ::_thesis: ( x > 0 & x <= 1 implies 1 - x <= exp_R . (- x) ) assume that A17: x > 0 and A18: x <= 1 ; ::_thesis: 1 - x <= exp_R . (- x) set B2 = NAT --> (1 - x); A19: for n being Element of NAT holds (NAT --> (1 - x)) . n <= (Partial_Sums ((- x) rExpSeq)) . (n + 1) proof let n be Element of NAT ; ::_thesis: (NAT --> (1 - x)) . n <= (Partial_Sums ((- x) rExpSeq)) . (n + 1) defpred S1[ Element of NAT ] means (NAT --> (1 - x)) . \$1 <= (Partial_Sums ((- x) rExpSeq)) . (\$1 + 1); (Partial_Sums ((- x) rExpSeq)) . (0 + 1) = ((Partial_Sums ((- x) rExpSeq)) . 0) + (((- x) rExpSeq) . 1) by SERIES_1:def_1; then A20: (Partial_Sums ((- x) rExpSeq)) . (0 + 1) = (((- x) rExpSeq) . 0) + (((- x) rExpSeq) . 1) by SERIES_1:def_1; ((- x) rExpSeq) . 1 = ((- x) |^ 1) / (1 !) by SIN_COS:def_5; then A21: ((- x) rExpSeq) . 1 = - x by NEWTON:5, NEWTON:13; ((- x) rExpSeq) . 0 = ((- x) |^ 0) / (0 !) by SIN_COS:def_5 .= 1 by NEWTON:4, NEWTON:12 ; then A22: S1[ 0 ] by A21, A20, FUNCOP_1:7; A23: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A24: S1[k] ; ::_thesis: S1[k + 1] percases ( k is even or k is odd ) ; suppose k is even ; ::_thesis: S1[k + 1] then consider m being Element of NAT such that A25: k = 2 * m by ABIAN:def_2; A26: 1 - x <= (Partial_Sums ((- x) rExpSeq)) . (k + 1) by A24, FUNCOP_1:7; A27: for k being Element of NAT st k is even & k > 0 holds for y being Real holds (y rExpSeq) . k >= 0 proof let k be Element of NAT ; ::_thesis: ( k is even & k > 0 implies for y being Real holds (y rExpSeq) . k >= 0 ) assume that A28: k is even and A29: k > 0 ; ::_thesis: for y being Real holds (y rExpSeq) . k >= 0 let y be Real; ::_thesis: (y rExpSeq) . k >= 0 percases ( y > 0 or y = 0 or y < 0 ) ; suppose y > 0 ; ::_thesis: (y rExpSeq) . k >= 0 then y |^ k > 0 by NEWTON:83; then (y |^ k) / (k !) > 0 ; hence (y rExpSeq) . k >= 0 by SIN_COS:def_5; ::_thesis: verum end; suppose y = 0 ; ::_thesis: (y rExpSeq) . k >= 0 then A30: y |^ k = 0 by A29, NEWTON:84; (y |^ k) / (k !) = 0 by A30; hence (y rExpSeq) . k >= 0 by SIN_COS:def_5; ::_thesis: verum end; supposeA31: y < 0 ; ::_thesis: (y rExpSeq) . k >= 0 consider m being Element of NAT such that A32: k = 2 * m by A28, ABIAN:def_2; y |^ k = y |^ (m + m) by A32; then y |^ k = (y |^ m) * (y |^ m) by NEWTON:8; then A33: y |^ k = (y * y) |^ m by NEWTON:7; y |^ k >= 0 by A31, A33, NEWTON:83; then (y |^ k) / (k !) >= 0 ; hence (y rExpSeq) . k >= 0 by SIN_COS:def_5; ::_thesis: verum end; end; end; A34: ((- x) rExpSeq) . (k + 2) >= 0 by A25, A27; A35: (Partial_Sums ((- x) rExpSeq)) . (k + 1) <= ((Partial_Sums ((- x) rExpSeq)) . (k + 1)) + (((- x) rExpSeq) . (k + 2)) by A34, XREAL_1:31; 1 - x <= ((Partial_Sums ((- x) rExpSeq)) . (k + 1)) + (((- x) rExpSeq) . ((k + 1) + 1)) by A26, A35, XXREAL_0:2; then 1 - x <= (Partial_Sums ((- x) rExpSeq)) . (k + 2) by SERIES_1:def_1; hence S1[k + 1] by FUNCOP_1:7; ::_thesis: verum end; suppose k is odd ; ::_thesis: S1[k + 1] then consider m being Element of NAT such that A36: k = (2 * m) + 1 by ABIAN:9; A37: for k being Element of NAT for x being Element of REAL st k is odd & x > 0 & x <= 1 holds 1 - x <= (Partial_Sums ((- x) rExpSeq)) . k proof let k be Element of NAT ; ::_thesis: for x being Element of REAL st k is odd & x > 0 & x <= 1 holds 1 - x <= (Partial_Sums ((- x) rExpSeq)) . k let x be Element of REAL ; ::_thesis: ( k is odd & x > 0 & x <= 1 implies 1 - x <= (Partial_Sums ((- x) rExpSeq)) . k ) assume A38: k is odd ; ::_thesis: ( not x > 0 or not x <= 1 or 1 - x <= (Partial_Sums ((- x) rExpSeq)) . k ) assume A39: x > 0 ; ::_thesis: ( not x <= 1 or 1 - x <= (Partial_Sums ((- x) rExpSeq)) . k ) assume A40: x <= 1 ; ::_thesis: 1 - x <= (Partial_Sums ((- x) rExpSeq)) . k defpred S2[ Element of NAT ] means 1 - x <= (Partial_Sums ((- x) rExpSeq)) . ((2 * \$1) + 1); (Partial_Sums ((- x) rExpSeq)) . ((2 * 0) + 1) = ((Partial_Sums ((- x) rExpSeq)) . 0) + (((- x) rExpSeq) . 1) by SERIES_1:def_1; then A41: (Partial_Sums ((- x) rExpSeq)) . ((2 * 0) + 1) = (((- x) rExpSeq) . 0) + (((- x) rExpSeq) . 1) by SERIES_1:def_1; A42: ((- x) rExpSeq) . 0 = ((- x) |^ 0) / (0 !) by SIN_COS:def_5 .= 1 by NEWTON:4, NEWTON:12 ; ((- x) rExpSeq) . 1 = ((- x) |^ 1) / (1 !) by SIN_COS:def_5; then A43: ((- x) rExpSeq) . (1 + 0) = - x by NEWTON:5, NEWTON:13; A44: S2[ 0 ] by A43, A41, A42; A45: for k being Element of NAT st S2[k] holds S2[k + 1] proof let k be Element of NAT ; ::_thesis: ( S2[k] implies S2[k + 1] ) assume A46: S2[k] ; ::_thesis: S2[k + 1] (Partial_Sums ((- x) rExpSeq)) . ((2 * k) + 1) <= (Partial_Sums ((- x) rExpSeq)) . ((2 * k) + 3) proof (Partial_Sums ((- x) rExpSeq)) . ((2 * k) + 3) = ((Partial_Sums ((- x) rExpSeq)) . ((2 * k) + 2)) + (((- x) rExpSeq) . (((2 * k) + 2) + 1)) by SERIES_1:def_1; then (Partial_Sums ((- x) rExpSeq)) . ((2 * k) + 3) = ((Partial_Sums ((- x) rExpSeq)) . (((2 * k) + 1) + 1)) + (((- x) rExpSeq) . ((2 * k) + 3)) ; then (Partial_Sums ((- x) rExpSeq)) . ((2 * k) + 3) = (((Partial_Sums ((- x) rExpSeq)) . ((2 * k) + 1)) + (((- x) rExpSeq) . ((2 * k) + 2))) + (((- x) rExpSeq) . ((2 * k) + 3)) by SERIES_1:def_1; then A47: (Partial_Sums ((- x) rExpSeq)) . ((2 * k) + 3) = ((Partial_Sums ((- x) rExpSeq)) . ((2 * k) + 1)) + ((((- x) rExpSeq) . ((2 * k) + 2)) + (((- x) rExpSeq) . ((2 * k) + 3))) ; (((- x) rExpSeq) . (((2 * k) + 1) + 1)) + (((- x) rExpSeq) . (((2 * k) + 1) + 2)) >= 0 by A39, A40, Th1; hence (Partial_Sums ((- x) rExpSeq)) . ((2 * k) + 1) <= (Partial_Sums ((- x) rExpSeq)) . ((2 * k) + 3) by A47, XREAL_1:31; ::_thesis: verum end; hence S2[k + 1] by A46, XXREAL_0:2; ::_thesis: verum end; A48: for k being Element of NAT holds S2[k] from NAT_1:sch_1(A44, A45); consider m being Element of NAT such that A49: k = (2 * m) + 1 by A38, ABIAN:9; thus 1 - x <= (Partial_Sums ((- x) rExpSeq)) . k by A48, A49; ::_thesis: verum end; 1 - x <= (Partial_Sums ((- x) rExpSeq)) . (k + 2) by A36, A17, A18, A37; hence S1[k + 1] by FUNCOP_1:7; ::_thesis: verum end; end; end; for k being Element of NAT holds S1[k] from NAT_1:sch_1(A22, A23); hence (NAT --> (1 - x)) . n <= (Partial_Sums ((- x) rExpSeq)) . (n + 1) ; ::_thesis: verum end; A50: for n being Element of NAT holds (NAT --> (1 - x)) . n <= ((Partial_Sums ((- x) rExpSeq)) ^\ 1) . n proof let n be Element of NAT ; ::_thesis: (NAT --> (1 - x)) . n <= ((Partial_Sums ((- x) rExpSeq)) ^\ 1) . n (NAT --> (1 - x)) . n <= (Partial_Sums ((- x) rExpSeq)) . (n + 1) by A19; hence (NAT --> (1 - x)) . n <= ((Partial_Sums ((- x) rExpSeq)) ^\ 1) . n by NAT_1:def_3; ::_thesis: verum end; A51: lim (NAT --> (1 - x)) = (NAT --> (1 - x)) . 1 by SEQ_4:26 .= 1 - x by FUNCOP_1:7 ; (- x) rExpSeq is summable by SIN_COS:45; then A52: Partial_Sums ((- x) rExpSeq) is convergent by SERIES_1:def_2; then A53: ( lim ((Partial_Sums ((- x) rExpSeq)) ^\ 1) = lim (Partial_Sums ((- x) rExpSeq)) & (Partial_Sums ((- x) rExpSeq)) ^\ 1 is convergent ) by SEQ_4:20; lim (NAT --> (1 - x)) <= lim ((Partial_Sums ((- x) rExpSeq)) ^\ 1) by A52, A50, SEQ_2:18; then lim (NAT --> (1 - x)) <= Sum ((- x) rExpSeq) by A53, SERIES_1:def_3; hence 1 - x <= exp_R . (- x) by A51, SIN_COS:def_22; ::_thesis: verum end; hence 1 - (- x) <= exp_R . (- (- x)) by A15, A16; ::_thesis: verum end; supposeA54: - x > 1 ; ::_thesis: 1 - (- x) <= exp_R . (- (- x)) 0 < exp_R . (- (- x)) by A54, SIN_COS:53; hence 1 - (- x) <= exp_R . (- (- x)) by A54, XREAL_1:49; ::_thesis: verum end; end; end; hence 1 + x <= exp_R . x ; ::_thesis: verum end; end; end; definition let s be Real_Sequence; func JSum s -> Real_Sequence means :Def1: :: BOR_CANT:def 1 for d being Nat holds it . d = Sum ((- (s . d)) rExpSeq); existence ex b1 being Real_Sequence st for d being Nat holds b1 . d = Sum ((- (s . d)) rExpSeq) proof deffunc H1( Element of NAT ) -> Element of REAL = Sum ((- (s . \$1)) rExpSeq); consider f being Real_Sequence such that A1: for d being Element of NAT holds f . d = H1(d) from FUNCT_2:sch_4(); take f ; ::_thesis: for d being Nat holds f . d = Sum ((- (s . d)) rExpSeq) let d be Nat; ::_thesis: f . d = Sum ((- (s . d)) rExpSeq) d in NAT by ORDINAL1:def_12; hence f . d = Sum ((- (s . d)) rExpSeq) by A1; ::_thesis: verum end; uniqueness for b1, b2 being Real_Sequence st ( for d being Nat holds b1 . d = Sum ((- (s . d)) rExpSeq) ) & ( for d being Nat holds b2 . d = Sum ((- (s . d)) rExpSeq) ) holds b1 = b2 proof let f1, f2 be Real_Sequence; ::_thesis: ( ( for d being Nat holds f1 . d = Sum ((- (s . d)) rExpSeq) ) & ( for d being Nat holds f2 . d = Sum ((- (s . d)) rExpSeq) ) implies f1 = f2 ) assume that A2: for d being Nat holds f1 . d = Sum ((- (s . d)) rExpSeq) and A3: for d being Nat holds f2 . d = Sum ((- (s . d)) rExpSeq) ; ::_thesis: f1 = f2 let d be Element of NAT ; :: according to FUNCT_2:def_8 ::_thesis: f1 . d = f2 . d f1 . d = Sum ((- (s . d)) rExpSeq) by A2; hence f1 . d = f2 . d by A3; ::_thesis: verum end; end; :: deftheorem Def1 defines JSum BOR_CANT:def_1_:_ for s, b2 being Real_Sequence holds ( b2 = JSum s iff for d being Nat holds b2 . d = Sum ((- (s . d)) rExpSeq) ); theorem Th3: :: BOR_CANT:3 for Omega being non empty set for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma for n being Element of NAT holds (Partial_Product (JSum (Prob * A))) . n = exp_R . (- ((Partial_Sums (Prob * A)) . n)) proof let Omega be non empty set ; ::_thesis: for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma for n being Element of NAT holds (Partial_Product (JSum (Prob * A))) . n = exp_R . (- ((Partial_Sums (Prob * A)) . n)) let Sigma be SigmaField of Omega; ::_thesis: for Prob being Probability of Sigma for A being SetSequence of Sigma for n being Element of NAT holds (Partial_Product (JSum (Prob * A))) . n = exp_R . (- ((Partial_Sums (Prob * A)) . n)) let Prob be Probability of Sigma; ::_thesis: for A being SetSequence of Sigma for n being Element of NAT holds (Partial_Product (JSum (Prob * A))) . n = exp_R . (- ((Partial_Sums (Prob * A)) . n)) let A be SetSequence of Sigma; ::_thesis: for n being Element of NAT holds (Partial_Product (JSum (Prob * A))) . n = exp_R . (- ((Partial_Sums (Prob * A)) . n)) let n be Element of NAT ; ::_thesis: (Partial_Product (JSum (Prob * A))) . n = exp_R . (- ((Partial_Sums (Prob * A)) . n)) defpred S1[ Element of NAT ] means exp_R . (- ((Partial_Sums (Prob * A)) . \$1)) = (Partial_Product (JSum (Prob * A))) . \$1; A1: exp_R . (- ((Partial_Sums (Prob * A)) . 0)) = exp_R . (- ((Prob * A) . 0)) by SERIES_1:def_1; (Partial_Product (JSum (Prob * A))) . 0 = (JSum (Prob * A)) . 0 by SERIES_3:def_1; then (Partial_Product (JSum (Prob * A))) . 0 = Sum ((- ((Prob * A) . 0)) rExpSeq) by Def1; then A2: S1[ 0 ] by A1, SIN_COS:def_22; A3: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A4: S1[k] ; ::_thesis: S1[k + 1] A5: (Partial_Product (JSum (Prob * A))) . (k + 1) = ((Partial_Product (JSum (Prob * A))) . k) * ((JSum (Prob * A)) . (k + 1)) by SERIES_3:def_1; A6: (Partial_Product (JSum (Prob * A))) . (k + 1) = (exp_R . (- ((Partial_Sums (Prob * A)) . k))) * (Sum ((- ((Prob * A) . (k + 1))) rExpSeq)) by A4, A5, Def1; A7: (exp_R (- ((Partial_Sums (Prob * A)) . k))) * (exp_R (- ((Prob * A) . (k + 1)))) = exp_R ((- ((Partial_Sums (Prob * A)) . k)) + (- ((Prob * A) . (k + 1)))) by SIN_COS:50; - (((Partial_Sums (Prob * A)) . k) + ((Prob * A) . (k + 1))) = - ((Partial_Sums (Prob * A)) . (k + 1)) by SERIES_1:def_1; hence S1[k + 1] by A7, A6, SIN_COS:def_22; ::_thesis: verum end; for k being Element of NAT holds S1[k] from NAT_1:sch_1(A2, A3); hence (Partial_Product (JSum (Prob * A))) . n = exp_R . (- ((Partial_Sums (Prob * A)) . n)) ; ::_thesis: verum end; theorem Th4: :: BOR_CANT:4 for Omega being non empty set for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma for n being Element of NAT holds (Partial_Product (Prob * (Complement A))) . n <= (Partial_Product (JSum (Prob * A))) . n proof let Omega be non empty set ; ::_thesis: for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma for n being Element of NAT holds (Partial_Product (Prob * (Complement A))) . n <= (Partial_Product (JSum (Prob * A))) . n let Sigma be SigmaField of Omega; ::_thesis: for Prob being Probability of Sigma for A being SetSequence of Sigma for n being Element of NAT holds (Partial_Product (Prob * (Complement A))) . n <= (Partial_Product (JSum (Prob * A))) . n let Prob be Probability of Sigma; ::_thesis: for A being SetSequence of Sigma for n being Element of NAT holds (Partial_Product (Prob * (Complement A))) . n <= (Partial_Product (JSum (Prob * A))) . n let A be SetSequence of Sigma; ::_thesis: for n being Element of NAT holds (Partial_Product (Prob * (Complement A))) . n <= (Partial_Product (JSum (Prob * A))) . n let n be Element of NAT ; ::_thesis: (Partial_Product (Prob * (Complement A))) . n <= (Partial_Product (JSum (Prob * A))) . n defpred S1[ Element of NAT ] means (Partial_Product (Prob * (Complement A))) . \$1 <= (Partial_Product (JSum (Prob * A))) . \$1; A1: (Partial_Product (Prob * (Complement A))) . 0 = (Prob * (Complement A)) . 0 by SERIES_3:def_1; dom (Prob * (Complement A)) = NAT by FUNCT_2:def_1; then A2: (Prob * (Complement A)) . 0 = Prob . ((Complement A) . 0) by FUNCT_1:12; A3: (Partial_Product (Prob * (Complement A))) . 0 = Prob . ((A . 0) `) by A2, A1, PROB_1:def_2; Prob . ((A . 0) `) = Prob . (([#] Sigma) \ (A . 0)) by SUBSET_1:def_4; then A4: (Partial_Product (Prob * (Complement A))) . 0 = 1 - (Prob . (A . 0)) by A3, PROB_1:32; (Partial_Product (JSum (Prob * A))) . 0 = (JSum (Prob * A)) . 0 by SERIES_3:def_1; then (Partial_Product (JSum (Prob * A))) . 0 = Sum ((- ((Prob * A) . 0)) rExpSeq) by Def1; then A5: (Partial_Product (JSum (Prob * A))) . 0 = exp_R . (- ((Prob * A) . 0)) by SIN_COS:def_22; A6: dom (Prob * A) = NAT by FUNCT_2:def_1; 1 + (- (Prob . (A . 0))) <= exp_R . (- (Prob . (A . 0))) by Th2; then A7: S1[ 0 ] by A4, A6, A5, FUNCT_1:12; A8: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A9: S1[k] ; ::_thesis: S1[k + 1] ( Prob . ((Complement A) . (k + 1)) = Prob . ((A . (k + 1)) `) & (A . (k + 1)) ` = ([#] Sigma) \ (A . (k + 1)) ) by PROB_1:def_2, SUBSET_1:def_4; then A10: Prob . ((Complement A) . (k + 1)) = 1 - (Prob . (A . (k + 1))) by PROB_1:32; A11: 1 + (- (Prob . (A . (k + 1)))) <= exp_R . (- (Prob . (A . (k + 1)))) by Th2; dom (Prob * (Complement A)) = NAT by FUNCT_2:def_1; then A12: (Prob * (Complement A)) . (k + 1) <= exp_R . (- (Prob . (A . (k + 1)))) by A11, A10, FUNCT_1:12; A13: ((Prob * (Complement A)) . (k + 1)) * ((Partial_Product (JSum (Prob * A))) . k) <= (exp_R . (- (Prob . (A . (k + 1))))) * ((Partial_Product (JSum (Prob * A))) . k) proof for n being Element of NAT holds (JSum (Prob * A)) . n > 0 proof let n be Element of NAT ; ::_thesis: (JSum (Prob * A)) . n > 0 A14: exp_R . (- ((Prob * A) . n)) > 0 by SIN_COS:54; (JSum (Prob * A)) . n = Sum ((- ((Prob * A) . n)) rExpSeq) by Def1; hence (JSum (Prob * A)) . n > 0 by A14, SIN_COS:def_22; ::_thesis: verum end; then (Partial_Product (JSum (Prob * A))) . k > 0 by SERIES_3:43; hence ((Prob * (Complement A)) . (k + 1)) * ((Partial_Product (JSum (Prob * A))) . k) <= (exp_R . (- (Prob . (A . (k + 1))))) * ((Partial_Product (JSum (Prob * A))) . k) by A12, XREAL_1:64; ::_thesis: verum end; A15: ((Partial_Product (Prob * (Complement A))) . k) * ((Prob * (Complement A)) . (k + 1)) <= ((Partial_Product (JSum (Prob * A))) . k) * ((Prob * (Complement A)) . (k + 1)) proof for n being Element of NAT holds (Prob * (Complement A)) . n >= 0 proof let n be Element of NAT ; ::_thesis: (Prob * (Complement A)) . n >= 0 A16: Prob . ((Complement A) . n) >= 0 by PROB_1:def_8; dom (Prob * (Complement A)) = NAT by FUNCT_2:def_1; hence (Prob * (Complement A)) . n >= 0 by A16, FUNCT_1:12; ::_thesis: verum end; then (Prob * (Complement A)) . (k + 1) >= 0 ; hence ((Partial_Product (Prob * (Complement A))) . k) * ((Prob * (Complement A)) . (k + 1)) <= ((Partial_Product (JSum (Prob * A))) . k) * ((Prob * (Complement A)) . (k + 1)) by A9, XREAL_1:64; ::_thesis: verum end; ((Partial_Product (Prob * (Complement A))) . k) * ((Prob * (Complement A)) . (k + 1)) <= (exp_R . (- (Prob . (A . (k + 1))))) * ((Partial_Product (JSum (Prob * A))) . k) by A15, A13, XXREAL_0:2; then (Partial_Product (Prob * (Complement A))) . (k + 1) <= (exp_R . (- (Prob . (A . (k + 1))))) * ((Partial_Product (JSum (Prob * A))) . k) by SERIES_3:def_1; then (Partial_Product (Prob * (Complement A))) . (k + 1) <= (Sum ((- (Prob . (A . (k + 1)))) rExpSeq)) * ((Partial_Product (JSum (Prob * A))) . k) by SIN_COS:def_22; then (Partial_Product (Prob * (Complement A))) . (k + 1) <= (Sum ((- (Prob . (A . (k + 1)))) rExpSeq)) * (exp_R . (- ((Partial_Sums (Prob * A)) . k))) by Th3; then (Partial_Product (Prob * (Complement A))) . (k + 1) <= (exp_R (- (Prob . (A . (k + 1))))) * (exp_R (- ((Partial_Sums (Prob * A)) . k))) by SIN_COS:def_22; then A17: (Partial_Product (Prob * (Complement A))) . (k + 1) <= exp_R ((- (Prob . (A . (k + 1)))) + (- ((Partial_Sums (Prob * A)) . k))) by SIN_COS:50; dom (Prob * A) = NAT by FUNCT_2:def_1; then (Prob * A) . (k + 1) = Prob . (A . (k + 1)) by FUNCT_1:12; then (- (Prob . (A . (k + 1)))) + (- ((Partial_Sums (Prob * A)) . k)) = - (((Prob * A) . (k + 1)) + ((Partial_Sums (Prob * A)) . k)) ; then (Partial_Product (Prob * (Complement A))) . (k + 1) <= exp_R . (- ((Partial_Sums (Prob * A)) . (k + 1))) by A17, SERIES_1:def_1; hence S1[k + 1] by Th3; ::_thesis: verum end; for k being Element of NAT holds S1[k] from NAT_1:sch_1(A7, A8); hence (Partial_Product (Prob * (Complement A))) . n <= (Partial_Product (JSum (Prob * A))) . n ; ::_thesis: verum end; definition let n1, n2 be Element of NAT ; func Special_Function (n1,n2) -> sequence of NAT means :Def2: :: BOR_CANT:def 2 for n being Element of NAT holds it . n = IFGT (n,n1,(n + n2),n); existence ex b1 being sequence of NAT st for n being Element of NAT holds b1 . n = IFGT (n,n1,(n + n2),n) proof deffunc H1( Element of NAT ) -> Element of NAT = IFGT (\$1,n1,(\$1 + n2),\$1); ex f being sequence of NAT st for n being Element of NAT holds f . n = H1(n) from FUNCT_2:sch_4(); hence ex b1 being sequence of NAT st for n being Element of NAT holds b1 . n = IFGT (n,n1,(n + n2),n) ; ::_thesis: verum end; uniqueness for b1, b2 being sequence of NAT st ( for n being Element of NAT holds b1 . n = IFGT (n,n1,(n + n2),n) ) & ( for n being Element of NAT holds b2 . n = IFGT (n,n1,(n + n2),n) ) holds b1 = b2 proof let s1, s2 be sequence of NAT; ::_thesis: ( ( for n being Element of NAT holds s1 . n = IFGT (n,n1,(n + n2),n) ) & ( for n being Element of NAT holds s2 . n = IFGT (n,n1,(n + n2),n) ) implies s1 = s2 ) assume that A1: for n being Element of NAT holds s1 . n = IFGT (n,n1,(n + n2),n) and A2: for n being Element of NAT holds s2 . n = IFGT (n,n1,(n + n2),n) ; ::_thesis: s1 = s2 let n be Element of NAT ; :: according to FUNCT_2:def_8 ::_thesis: s1 . n = s2 . n ( s1 . n = IFGT (n,n1,(n + n2),n) & s2 . n = IFGT (n,n1,(n + n2),n) ) by A1, A2; hence s1 . n = s2 . n ; ::_thesis: verum end; end; :: deftheorem Def2 defines Special_Function BOR_CANT:def_2_:_ for n1, n2 being Element of NAT for b3 being sequence of NAT holds ( b3 = Special_Function (n1,n2) iff for n being Element of NAT holds b3 . n = IFGT (n,n1,(n + n2),n) ); definition let k be Element of NAT ; func Special_Function2 k -> sequence of NAT means :Def3: :: BOR_CANT:def 3 for n being Element of NAT holds it . n = n + k; existence ex b1 being sequence of NAT st for n being Element of NAT holds b1 . n = n + k proof deffunc H1( Element of NAT ) -> Element of NAT = \$1 + k; consider f being sequence of NAT such that A1: for n being Element of NAT holds f . n = H1(n) from FUNCT_2:sch_4(); take f ; ::_thesis: for n being Element of NAT holds f . n = n + k let n be Nat; ::_thesis: ( n is Element of REAL & n is Element of NAT implies f . n = n + k ) thus ( n is Element of REAL & n is Element of NAT implies f . n = n + k ) by A1; ::_thesis: verum end; uniqueness for b1, b2 being sequence of NAT st ( for n being Element of NAT holds b1 . n = n + k ) & ( for n being Element of NAT holds b2 . n = n + k ) holds b1 = b2 proof let s1, s2 be sequence of NAT; ::_thesis: ( ( for n being Element of NAT holds s1 . n = n + k ) & ( for n being Element of NAT holds s2 . n = n + k ) implies s1 = s2 ) assume that A2: for n being Element of NAT holds s1 . n = n + k and A3: for n being Element of NAT holds s2 . n = n + k ; ::_thesis: s1 = s2 let n be Element of NAT ; :: according to FUNCT_2:def_8 ::_thesis: s1 . n = s2 . n ( s1 . n = n + k & s2 . n = n + k ) by A2, A3; hence s1 . n = s2 . n ; ::_thesis: verum end; end; :: deftheorem Def3 defines Special_Function2 BOR_CANT:def_3_:_ for k being Element of NAT for b2 being sequence of NAT holds ( b2 = Special_Function2 k iff for n being Element of NAT holds b2 . n = n + k ); definition let k be Element of NAT ; func Special_Function3 k -> sequence of NAT means :Def4: :: BOR_CANT:def 4 for n being Element of NAT holds it . n = IFGT (n,k,0,1); existence ex b1 being sequence of NAT st for n being Element of NAT holds b1 . n = IFGT (n,k,0,1) proof deffunc H1( Element of NAT ) -> Element of NAT = IFGT (\$1,k,0,1); ex f being sequence of NAT st for n being Element of NAT holds f . n = H1(n) from FUNCT_2:sch_4(); hence ex b1 being sequence of NAT st for n being Element of NAT holds b1 . n = IFGT (n,k,0,1) ; ::_thesis: verum end; uniqueness for b1, b2 being sequence of NAT st ( for n being Element of NAT holds b1 . n = IFGT (n,k,0,1) ) & ( for n being Element of NAT holds b2 . n = IFGT (n,k,0,1) ) holds b1 = b2 proof let s1, s2 be sequence of NAT; ::_thesis: ( ( for n being Element of NAT holds s1 . n = IFGT (n,k,0,1) ) & ( for n being Element of NAT holds s2 . n = IFGT (n,k,0,1) ) implies s1 = s2 ) assume that A1: for n being Element of NAT holds s1 . n = IFGT (n,k,0,1) and A2: for n being Element of NAT holds s2 . n = IFGT (n,k,0,1) ; ::_thesis: s1 = s2 let n be Element of NAT ; :: according to FUNCT_2:def_8 ::_thesis: s1 . n = s2 . n ( s1 . n = IFGT (n,k,0,1) & s2 . n = IFGT (n,k,0,1) ) by A1, A2; hence s1 . n = s2 . n ; ::_thesis: verum end; end; :: deftheorem Def4 defines Special_Function3 BOR_CANT:def_4_:_ for k being Element of NAT for b2 being sequence of NAT holds ( b2 = Special_Function3 k iff for n being Element of NAT holds b2 . n = IFGT (n,k,0,1) ); definition let n1, n2 be Element of NAT ; func Special_Function4 (n1,n2) -> sequence of NAT means :Def5: :: BOR_CANT:def 5 for n being Element of NAT holds it . n = IFGT (n,(n1 + 1),(n + n2),n); existence ex b1 being sequence of NAT st for n being Element of NAT holds b1 . n = IFGT (n,(n1 + 1),(n + n2),n) proof deffunc H1( Element of NAT ) -> Element of NAT = IFGT (\$1,(n1 + 1),(\$1 + n2),\$1); ex f being sequence of NAT st for n being Element of NAT holds f . n = H1(n) from FUNCT_2:sch_4(); hence ex b1 being sequence of NAT st for n being Element of NAT holds b1 . n = IFGT (n,(n1 + 1),(n + n2),n) ; ::_thesis: verum end; uniqueness for b1, b2 being sequence of NAT st ( for n being Element of NAT holds b1 . n = IFGT (n,(n1 + 1),(n + n2),n) ) & ( for n being Element of NAT holds b2 . n = IFGT (n,(n1 + 1),(n + n2),n) ) holds b1 = b2 proof let s1, s2 be sequence of NAT; ::_thesis: ( ( for n being Element of NAT holds s1 . n = IFGT (n,(n1 + 1),(n + n2),n) ) & ( for n being Element of NAT holds s2 . n = IFGT (n,(n1 + 1),(n + n2),n) ) implies s1 = s2 ) assume that A1: for n being Element of NAT holds s1 . n = IFGT (n,(n1 + 1),(n + n2),n) and A2: for n being Element of NAT holds s2 . n = IFGT (n,(n1 + 1),(n + n2),n) ; ::_thesis: s1 = s2 let n be Element of NAT ; :: according to FUNCT_2:def_8 ::_thesis: s1 . n = s2 . n ( s1 . n = IFGT (n,(n1 + 1),(n + n2),n) & s2 . n = IFGT (n,(n1 + 1),(n + n2),n) ) by A1, A2; hence s1 . n = s2 . n ; ::_thesis: verum end; end; :: deftheorem Def5 defines Special_Function4 BOR_CANT:def_5_:_ for n1, n2 being Element of NAT for b3 being sequence of NAT holds ( b3 = Special_Function4 (n1,n2) iff for n being Element of NAT holds b3 . n = IFGT (n,(n1 + 1),(n + n2),n) ); registration let n1, n2 be Element of NAT ; cluster Special_Function (n1,n2) -> one-to-one ; coherence Special_Function (n1,n2) is one-to-one proof let x1, x2 be set ; :: according to FUNCT_1:def_4 ::_thesis: ( not x1 in dom (Special_Function (n1,n2)) or not x2 in dom (Special_Function (n1,n2)) or not (Special_Function (n1,n2)) . x1 = (Special_Function (n1,n2)) . x2 or x1 = x2 ) assume that A1: x1 in dom (Special_Function (n1,n2)) and A2: x2 in dom (Special_Function (n1,n2)) ; ::_thesis: ( not (Special_Function (n1,n2)) . x1 = (Special_Function (n1,n2)) . x2 or x1 = x2 ) assume A3: (Special_Function (n1,n2)) . x1 = (Special_Function (n1,n2)) . x2 ; ::_thesis: x1 = x2 reconsider x1 = x1 as Element of NAT by A1; reconsider x2 = x2 as Element of NAT by A2; A4: ( (Special_Function (n1,n2)) . x2 = IFGT (x2,n1,(x2 + n2),x2) & (Special_Function (n1,n2)) . x1 = IFGT (x1,n1,(x1 + n2),x1) ) by Def2; percases ( ( x1 <= n1 & x2 <= n1 ) or ( x1 <= n1 & x2 > n1 ) or ( x2 <= n1 & x1 > n1 ) or ( x1 > n1 & x2 > n1 ) ) ; supposeA5: ( x1 <= n1 & x2 <= n1 ) ; ::_thesis: x1 = x2 ( IFGT (x2,n1,(x2 + n2),x2) = x2 & IFGT (x1,n1,(x1 + n2),x1) = x1 ) by A5, XXREAL_0:def_11; hence x1 = x2 by A4, A3; ::_thesis: verum end; supposeA6: ( x1 <= n1 & x2 > n1 ) ; ::_thesis: x1 = x2 then IFGT (x2,n1,(x2 + n2),x2) = x2 + n2 by XXREAL_0:def_11; then A7: (Special_Function (n1,n2)) . x2 = x2 + n2 by Def2; A8: IFGT (x1,n1,(x1 + n2),x1) = x1 by A6, XXREAL_0:def_11; A9: (Special_Function (n1,n2)) . x1 = x1 by Def2, A8; ( x1 <> x2 implies (Special_Function (n1,n2)) . x1 <> (Special_Function (n1,n2)) . x2 ) proof assume x1 <> x2 ; ::_thesis: (Special_Function (n1,n2)) . x1 <> (Special_Function (n1,n2)) . x2 ( x1 < x2 & 0 <= n2 ) by A6, XXREAL_0:2; hence (Special_Function (n1,n2)) . x1 <> (Special_Function (n1,n2)) . x2 by A9, A7, XREAL_1:31; ::_thesis: verum end; hence x1 = x2 by A3; ::_thesis: verum end; supposeA10: ( x2 <= n1 & x1 > n1 ) ; ::_thesis: x1 = x2 A11: (Special_Function (n1,n2)) . x1 = IFGT (x1,n1,(x1 + n2),x1) by Def2; A12: (Special_Function (n1,n2)) . x1 = x1 + n2 by A11, A10, XXREAL_0:def_11; A13: (Special_Function (n1,n2)) . x2 = IFGT (x2,n1,(x2 + n2),x2) by Def2; A14: (Special_Function (n1,n2)) . x2 = x2 by A13, A10, XXREAL_0:def_11; ( x2 <> x1 implies (Special_Function (n1,n2)) . x2 <> (Special_Function (n1,n2)) . x1 ) proof assume x2 <> x1 ; ::_thesis: (Special_Function (n1,n2)) . x2 <> (Special_Function (n1,n2)) . x1 ( x2 < x1 & 0 <= n2 ) by A10, XXREAL_0:2; hence (Special_Function (n1,n2)) . x2 <> (Special_Function (n1,n2)) . x1 by A14, A12, XREAL_1:31; ::_thesis: verum end; hence x1 = x2 by A3; ::_thesis: verum end; supposeA15: ( x1 > n1 & x2 > n1 ) ; ::_thesis: x1 = x2 ( IFGT (x2,n1,(x2 + n2),x2) = x2 + n2 & IFGT (x1,n1,(x1 + n2),x1) = x1 + n2 ) by A15, XXREAL_0:def_11; then x2 + n2 = x1 + n2 by A4, A3; hence x1 = x2 ; ::_thesis: verum end; end; end; cluster Special_Function4 (n1,n2) -> one-to-one ; coherence Special_Function4 (n1,n2) is one-to-one proof let x1, x2 be set ; :: according to FUNCT_1:def_4 ::_thesis: ( not x1 in dom (Special_Function4 (n1,n2)) or not x2 in dom (Special_Function4 (n1,n2)) or not (Special_Function4 (n1,n2)) . x1 = (Special_Function4 (n1,n2)) . x2 or x1 = x2 ) assume that A16: x1 in dom (Special_Function4 (n1,n2)) and A17: x2 in dom (Special_Function4 (n1,n2)) ; ::_thesis: ( not (Special_Function4 (n1,n2)) . x1 = (Special_Function4 (n1,n2)) . x2 or x1 = x2 ) assume A18: (Special_Function4 (n1,n2)) . x1 = (Special_Function4 (n1,n2)) . x2 ; ::_thesis: x1 = x2 reconsider x1 = x1 as Element of NAT by A16; reconsider x2 = x2 as Element of NAT by A17; percases ( ( x1 <= n1 + 1 & x2 <= n1 + 1 ) or ( x1 > n1 + 1 & x2 <= n1 + 1 ) or ( x1 <= n1 + 1 & x2 > n1 + 1 ) or ( x1 > n1 + 1 & x2 > n1 + 1 ) ) ; supposeA19: ( x1 <= n1 + 1 & x2 <= n1 + 1 ) ; ::_thesis: x1 = x2 A20: ( (Special_Function4 (n1,n2)) . x2 = IFGT (x2,(n1 + 1),(x2 + n2),x2) & (Special_Function4 (n1,n2)) . x1 = IFGT (x1,(n1 + 1),(x1 + n2),x1) ) by Def5; ( IFGT (x2,(n1 + 1),(x2 + n2),x2) = x2 & IFGT (x1,(n1 + 1),(x1 + n2),x1) = x1 ) by A19, XXREAL_0:def_11; hence x1 = x2 by A20, A18; ::_thesis: verum end; supposeA21: ( x1 > n1 + 1 & x2 <= n1 + 1 ) ; ::_thesis: x1 = x2 A22: ( (Special_Function4 (n1,n2)) . x2 = IFGT (x2,(n1 + 1),(x2 + n2),x2) & (Special_Function4 (n1,n2)) . x1 = IFGT (x1,(n1 + 1),(x1 + n2),x1) ) by Def5; A23: ( (Special_Function4 (n1,n2)) . x2 = x2 & (Special_Function4 (n1,n2)) . x1 = x1 + n2 ) by A22, A21, XXREAL_0:def_11; ( x1 <> x2 implies (Special_Function4 (n1,n2)) . x2 <> (Special_Function4 (n1,n2)) . x1 ) proof assume x1 <> x2 ; ::_thesis: (Special_Function4 (n1,n2)) . x2 <> (Special_Function4 (n1,n2)) . x1 ( (Special_Function4 (n1,n2)) . x1 > (n1 + 1) + n2 & (n1 + 1) + n2 >= n1 + 1 ) by A23, A21, XREAL_1:6, XREAL_1:31; hence (Special_Function4 (n1,n2)) . x2 <> (Special_Function4 (n1,n2)) . x1 by A21, A23, XXREAL_0:2; ::_thesis: verum end; hence x1 = x2 by A18; ::_thesis: verum end; supposeA24: ( x1 <= n1 + 1 & x2 > n1 + 1 ) ; ::_thesis: x1 = x2 A25: ( (Special_Function4 (n1,n2)) . x2 = IFGT (x2,(n1 + 1),(x2 + n2),x2) & (Special_Function4 (n1,n2)) . x1 = IFGT (x1,(n1 + 1),(x1 + n2),x1) & IFGT (x2,(n1 + 1),(x2 + n2),x2) = x2 + n2 & IFGT (x1,(n1 + 1),(x1 + n2),x1) = x1 ) by Def5, A24, XXREAL_0:def_11; ( x1 <> x2 implies (Special_Function4 (n1,n2)) . x2 <> (Special_Function4 (n1,n2)) . x1 ) proof assume x1 <> x2 ; ::_thesis: (Special_Function4 (n1,n2)) . x2 <> (Special_Function4 (n1,n2)) . x1 ( (Special_Function4 (n1,n2)) . x2 > (n1 + 1) + n2 & (n1 + 1) + n2 >= n1 + 1 ) by A25, A24, XREAL_1:6, XREAL_1:31; hence (Special_Function4 (n1,n2)) . x2 <> (Special_Function4 (n1,n2)) . x1 by A24, A25, XXREAL_0:2; ::_thesis: verum end; hence x1 = x2 by A18; ::_thesis: verum end; supposeA26: ( x1 > n1 + 1 & x2 > n1 + 1 ) ; ::_thesis: x1 = x2 A27: ( (Special_Function4 (n1,n2)) . x2 = IFGT (x2,(n1 + 1),(x2 + n2),x2) & (Special_Function4 (n1,n2)) . x1 = IFGT (x1,(n1 + 1),(x1 + n2),x1) ) by Def5; ( IFGT (x2,(n1 + 1),(x2 + n2),x2) = x2 + n2 & IFGT (x1,(n1 + 1),(x1 + n2),x1) = x1 + n2 ) by A26, XXREAL_0:def_11; then x1 + n2 = x2 + n2 by A27, A18; hence x1 = x2 ; ::_thesis: verum end; end; end; end; registration let n be Element of NAT ; cluster Special_Function2 n -> one-to-one ; coherence Special_Function2 n is one-to-one proof let x1, x2 be set ; :: according to FUNCT_1:def_4 ::_thesis: ( not x1 in dom (Special_Function2 n) or not x2 in dom (Special_Function2 n) or not (Special_Function2 n) . x1 = (Special_Function2 n) . x2 or x1 = x2 ) assume that A1: x1 in dom (Special_Function2 n) and A2: x2 in dom (Special_Function2 n) ; ::_thesis: ( not (Special_Function2 n) . x1 = (Special_Function2 n) . x2 or x1 = x2 ) assume A3: (Special_Function2 n) . x1 = (Special_Function2 n) . x2 ; ::_thesis: x1 = x2 reconsider x1 = x1 as Element of NAT by A1; reconsider x2 = x2 as Element of NAT by A2; (Special_Function2 n) . x2 = x2 + n by Def3; then x1 + n = x2 + n by A3, Def3; hence x1 = x2 ; ::_thesis: verum end; end; registration let Omega be non empty set ; let Sigma be SigmaField of Omega; let s be Element of NAT ; let A be SetSequence of Sigma; clusterA ^\ s -> Sigma -valued ; coherence A ^\ s is Sigma -valued ; end; theorem Th5: :: BOR_CANT:5 for Omega being non empty set for Sigma being SigmaField of Omega for Prob being Probability of Sigma for n, n1, n2 being Element of NAT holds ( ( for A, B being SetSequence of Sigma st n > n1 & B = A * (Special_Function (n1,n2)) holds (Partial_Product (Prob * B)) . n = ((Partial_Product (Prob * A)) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . ((n - n1) - 1)) ) & ( for A, B, C being SetSequence of Sigma for e being sequence of NAT st n > n1 & C = A * e & B = C * (Special_Function (n1,n2)) holds (Partial_Intersection B) . n = ((Partial_Intersection C) . n1) /\ ((Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1)) ) ) proof let Omega be non empty set ; ::_thesis: for Sigma being SigmaField of Omega for Prob being Probability of Sigma for n, n1, n2 being Element of NAT holds ( ( for A, B being SetSequence of Sigma st n > n1 & B = A * (Special_Function (n1,n2)) holds (Partial_Product (Prob * B)) . n = ((Partial_Product (Prob * A)) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . ((n - n1) - 1)) ) & ( for A, B, C being SetSequence of Sigma for e being sequence of NAT st n > n1 & C = A * e & B = C * (Special_Function (n1,n2)) holds (Partial_Intersection B) . n = ((Partial_Intersection C) . n1) /\ ((Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1)) ) ) let Sigma be SigmaField of Omega; ::_thesis: for Prob being Probability of Sigma for n, n1, n2 being Element of NAT holds ( ( for A, B being SetSequence of Sigma st n > n1 & B = A * (Special_Function (n1,n2)) holds (Partial_Product (Prob * B)) . n = ((Partial_Product (Prob * A)) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . ((n - n1) - 1)) ) & ( for A, B, C being SetSequence of Sigma for e being sequence of NAT st n > n1 & C = A * e & B = C * (Special_Function (n1,n2)) holds (Partial_Intersection B) . n = ((Partial_Intersection C) . n1) /\ ((Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1)) ) ) let Prob be Probability of Sigma; ::_thesis: for n, n1, n2 being Element of NAT holds ( ( for A, B being SetSequence of Sigma st n > n1 & B = A * (Special_Function (n1,n2)) holds (Partial_Product (Prob * B)) . n = ((Partial_Product (Prob * A)) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . ((n - n1) - 1)) ) & ( for A, B, C being SetSequence of Sigma for e being sequence of NAT st n > n1 & C = A * e & B = C * (Special_Function (n1,n2)) holds (Partial_Intersection B) . n = ((Partial_Intersection C) . n1) /\ ((Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1)) ) ) let n, n1, n2 be Element of NAT ; ::_thesis: ( ( for A, B being SetSequence of Sigma st n > n1 & B = A * (Special_Function (n1,n2)) holds (Partial_Product (Prob * B)) . n = ((Partial_Product (Prob * A)) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . ((n - n1) - 1)) ) & ( for A, B, C being SetSequence of Sigma for e being sequence of NAT st n > n1 & C = A * e & B = C * (Special_Function (n1,n2)) holds (Partial_Intersection B) . n = ((Partial_Intersection C) . n1) /\ ((Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1)) ) ) A1: for A, B being SetSequence of Sigma st n > n1 & B = A * (Special_Function (n1,n2)) holds (Partial_Product (Prob * B)) . n = ((Partial_Product (Prob * A)) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . ((n - n1) - 1)) proof let A, B be SetSequence of Sigma; ::_thesis: ( n > n1 & B = A * (Special_Function (n1,n2)) implies (Partial_Product (Prob * B)) . n = ((Partial_Product (Prob * A)) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . ((n - n1) - 1)) ) assume that A2: n > n1 and A3: B = A * (Special_Function (n1,n2)) ; ::_thesis: (Partial_Product (Prob * B)) . n = ((Partial_Product (Prob * A)) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . ((n - n1) - 1)) A4: for q being Element of NAT st q <= n1 holds (Partial_Product (Prob * B)) . q = (Partial_Product (Prob * A)) . q proof let q be Element of NAT ; ::_thesis: ( q <= n1 implies (Partial_Product (Prob * B)) . q = (Partial_Product (Prob * A)) . q ) assume A5: q <= n1 ; ::_thesis: (Partial_Product (Prob * B)) . q = (Partial_Product (Prob * A)) . q defpred S1[ Nat] means (Partial_Product (Prob * B)) . (\$1 * ((Special_Function3 n1) . \$1)) = (Partial_Product (Prob * A)) . (\$1 * ((Special_Function3 n1) . \$1)); A6: S1[ 0 ] proof A7: (Partial_Product (Prob * B)) . 0 = (Prob * B) . 0 by SERIES_3:def_1; A8: (Partial_Product (Prob * A)) . 0 = (Prob * A) . 0 by SERIES_3:def_1; dom (Prob * B) = NAT by FUNCT_2:def_1; then A9: (Prob * B) . 0 = Prob . (B . 0) by FUNCT_1:12; A10: dom (A * (Special_Function (n1,n2))) = NAT by FUNCT_2:def_1; ( (Special_Function (n1,n2)) . 0 = IFGT (0,n1,(0 + n2),0) & IFGT (0,n1,(0 + n2),0) = 0 ) by Def2, XXREAL_0:def_11; then A11: (Prob * B) . 0 = Prob . (A . 0) by A10, A3, A9, FUNCT_1:12; dom (Prob * A) = NAT by FUNCT_2:def_1; hence S1[ 0 ] by A11, A7, A8, FUNCT_1:12; ::_thesis: verum end; A12: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A13: S1[k] ; ::_thesis: S1[k + 1] percases ( k < n1 or not k < n1 ) ; supposeA14: k < n1 ; ::_thesis: S1[k + 1] then A15: ( (Special_Function3 n1) . k = IFGT (k,n1,0,1) & IFGT (k,n1,0,1) = 1 ) by Def4, XXREAL_0:def_11; k + 1 <= n1 by A14, NAT_1:13; then A16: ( (Special_Function3 n1) . (k + 1) = IFGT ((k + 1),n1,0,1) & IFGT ((k + 1),n1,0,1) = 1 ) by Def4, XXREAL_0:def_11; A17: (Prob * B) . (k + 1) = (Prob * A) . (k + 1) proof dom (Prob * B) = NAT by FUNCT_2:def_1; then A18: (Prob * B) . (k + 1) = Prob . (B . (k + 1)) by FUNCT_1:12; A19: dom (A * (Special_Function (n1,n2))) = NAT by FUNCT_2:def_1; k + 1 <= n1 by A14, NAT_1:13; then ( (Special_Function (n1,n2)) . (k + 1) = IFGT ((k + 1),n1,((k + 1) + n2),(k + 1)) & IFGT ((k + 1),n1,((k + 1) + n2),(k + 1)) = k + 1 ) by Def2, XXREAL_0:def_11; then A20: (Prob * B) . (k + 1) = Prob . (A . (k + 1)) by A19, A3, A18, FUNCT_1:12; dom (Prob * A) = NAT by FUNCT_2:def_1; hence (Prob * B) . (k + 1) = (Prob * A) . (k + 1) by A20, FUNCT_1:12; ::_thesis: verum end; (Partial_Product (Prob * B)) . (k + 1) = ((Partial_Product (Prob * A)) . k) * ((Prob * A) . (k + 1)) by A15, A13, A17, SERIES_3:def_1; hence S1[k + 1] by A16, SERIES_3:def_1; ::_thesis: verum end; supposeA21: not k < n1 ; ::_thesis: S1[k + 1] n1 < k + 1 by A21, XREAL_1:145; then A22: ( (Special_Function3 n1) . (k + 1) = IFGT ((k + 1),n1,0,1) & IFGT ((k + 1),n1,0,1) = 0 ) by Def4, XXREAL_0:def_11; A23: (Prob * B) . 0 = (Prob * A) . 0 proof dom (Prob * B) = NAT by FUNCT_2:def_1; then A24: (Prob * B) . 0 = Prob . (B . 0) by FUNCT_1:12; A25: dom (A * (Special_Function (n1,n2))) = NAT by FUNCT_2:def_1; ( (Special_Function (n1,n2)) . 0 = IFGT (0,n1,(0 + n2),0) & IFGT (0,n1,(0 + n2),0) = 0 ) by Def2, XXREAL_0:def_11; then A26: (Prob * B) . 0 = Prob . (A . 0) by A25, A3, A24, FUNCT_1:12; dom (Prob * A) = NAT by FUNCT_2:def_1; hence (Prob * B) . 0 = (Prob * A) . 0 by A26, FUNCT_1:12; ::_thesis: verum end; ( (Partial_Product (Prob * B)) . ((k + 1) * ((Special_Function3 n1) . (k + 1))) = (Prob * B) . 0 & (Partial_Product (Prob * A)) . ((k + 1) * ((Special_Function3 n1) . (k + 1))) = (Prob * A) . 0 ) by A22, SERIES_3:def_1; hence S1[k + 1] by A23; ::_thesis: verum end; end; end; A27: for k being Element of NAT holds S1[k] from NAT_1:sch_1(A6, A12); (Partial_Product (Prob * B)) . q = (Partial_Product (Prob * A)) . q proof ( (Special_Function3 n1) . q = IFGT (q,n1,0,1) & IFGT (q,n1,0,1) = 1 ) by Def4, A5, XXREAL_0:def_11; then q * ((Special_Function3 n1) . q) = q ; hence (Partial_Product (Prob * B)) . q = (Partial_Product (Prob * A)) . q by A27; ::_thesis: verum end; hence (Partial_Product (Prob * B)) . q = (Partial_Product (Prob * A)) . q ; ::_thesis: verum end; defpred S1[ Element of NAT ] means (Partial_Product (Prob * B)) . ((\$1 + n1) + 1) = ((Partial_Product (Prob * A)) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . ((((\$1 + n1) + 1) - n1) - 1)); ( (Partial_Product (Prob * B)) . ((0 + n1) + 1) = ((Partial_Product (Prob * B)) . n1) * ((Prob * B) . (n1 + 1)) & n1 <= n1 ) by SERIES_3:def_1; then A28: (Partial_Product (Prob * B)) . ((0 + n1) + 1) = ((Partial_Product (Prob * A)) . n1) * ((Prob * B) . (n1 + 1)) by A4; A29: dom (A * (Special_Function (n1,n2))) = NAT by FUNCT_2:def_1; A30: n1 < n1 + 1 by NAT_1:13; ( (Special_Function (n1,n2)) . (n1 + 1) = IFGT ((n1 + 1),n1,((n1 + 1) + n2),(n1 + 1)) & IFGT ((n1 + 1),n1,((n1 + 1) + n2),(n1 + 1)) = (n1 + 1) + n2 ) by Def2, A30, XXREAL_0:def_11; then A31: Prob . (B . (n1 + 1)) = Prob . (A . ((n1 + 1) + n2)) by A29, A3, FUNCT_1:12; A32: A . (((n1 + n2) + 1) + 0) = (A ^\ ((n1 + n2) + 1)) . 0 by NAT_1:def_3; dom (Prob * (A ^\ ((n1 + n2) + 1))) = NAT by FUNCT_2:def_1; then A33: Prob . (B . (n1 + 1)) = (Prob * (A ^\ ((n1 + n2) + 1))) . 0 by A32, A31, FUNCT_1:12; dom (Prob * B) = NAT by FUNCT_2:def_1; then (Prob * B) . (n1 + 1) = (Prob * (A ^\ ((n1 + n2) + 1))) . 0 by A33, FUNCT_1:12; then A34: S1[ 0 ] by A28, SERIES_3:def_1; A35: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A36: S1[k] ; ::_thesis: S1[k + 1] A37: (Partial_Product (Prob * B)) . (((k + 1) + n1) + 1) = (((Partial_Product (Prob * A)) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . k)) * ((Prob * B) . (((k + 1) + n1) + 1)) by A36, SERIES_3:def_1; A38: (Prob * B) . (((k + 1) + n1) + 1) = (Prob * (A ^\ ((n1 + n2) + 1))) . (k + 1) proof set j = ((k + 1) + n1) + 1; A39: dom (A * (Special_Function (n1,n2))) = NAT by FUNCT_2:def_1; 1 <= (k + 1) + 1 by XREAL_1:31; then n1 < n1 + ((k + 1) + 1) by NAT_1:19; then ( (Special_Function (n1,n2)) . (((k + 1) + n1) + 1) = IFGT ((((k + 1) + n1) + 1),n1,((((k + 1) + n1) + 1) + n2),(((k + 1) + n1) + 1)) & IFGT ((((k + 1) + n1) + 1),n1,((((k + 1) + n1) + 1) + n2),(((k + 1) + n1) + 1)) = (((k + 1) + n1) + 1) + n2 ) by Def2, XXREAL_0:def_11; then B . (((k + 1) + n1) + 1) = A . (((n1 + n2) + 1) + (k + 1)) by A39, A3, FUNCT_1:12; then A40: Prob . (B . (((k + 1) + n1) + 1)) = Prob . ((A ^\ ((n1 + n2) + 1)) . (k + 1)) by NAT_1:def_3; dom (Prob * B) = NAT by FUNCT_2:def_1; then A41: (Prob * B) . (((k + 1) + n1) + 1) = Prob . ((A ^\ ((n1 + n2) + 1)) . (k + 1)) by A40, FUNCT_1:12; dom (Prob * (A ^\ ((n1 + n2) + 1))) = NAT by FUNCT_2:def_1; hence (Prob * B) . (((k + 1) + n1) + 1) = (Prob * (A ^\ ((n1 + n2) + 1))) . (k + 1) by A41, FUNCT_1:12; ::_thesis: verum end; ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . k) * ((Prob * (A ^\ ((n1 + n2) + 1))) . (k + 1)) = (Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . (k + 1) by SERIES_3:def_1; hence S1[k + 1] by A38, A37; ::_thesis: verum end; A42: for k being Element of NAT holds S1[k] from NAT_1:sch_1(A34, A35); (n - n1) - 1 is Element of NAT proof n1 + 1 <= n by A2, NAT_1:13; then (n1 + 1) - 1 <= n - 1 by XREAL_1:9; then ( n1 <= n - 1 & n - 1 is Element of NAT ) by A2, NAT_1:20; then (n - 1) - n1 is Element of NAT by NAT_1:21; hence (n - n1) - 1 is Element of NAT ; ::_thesis: verum end; then consider k being Element of NAT such that A43: k = (n - n1) - 1 ; (Partial_Product (Prob * B)) . n = ((Partial_Product (Prob * A)) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . ((((((n - n1) - 1) + n1) + 1) - n1) - 1)) by A42, A43; hence (Partial_Product (Prob * B)) . n = ((Partial_Product (Prob * A)) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . ((n - n1) - 1)) ; ::_thesis: verum end; for A, B, C being SetSequence of Sigma for n1, n2, n being Element of NAT for e being sequence of NAT st n > n1 & C = A * e & B = C * (Special_Function (n1,n2)) holds (Partial_Intersection B) . n = ((Partial_Intersection C) . n1) /\ ((Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1)) proof let A, B, C be SetSequence of Sigma; ::_thesis: for n1, n2, n being Element of NAT for e being sequence of NAT st n > n1 & C = A * e & B = C * (Special_Function (n1,n2)) holds (Partial_Intersection B) . n = ((Partial_Intersection C) . n1) /\ ((Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1)) let n1, n2, n be Element of NAT ; ::_thesis: for e being sequence of NAT st n > n1 & C = A * e & B = C * (Special_Function (n1,n2)) holds (Partial_Intersection B) . n = ((Partial_Intersection C) . n1) /\ ((Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1)) let e be sequence of NAT; ::_thesis: ( n > n1 & C = A * e & B = C * (Special_Function (n1,n2)) implies (Partial_Intersection B) . n = ((Partial_Intersection C) . n1) /\ ((Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1)) ) assume A44: n > n1 ; ::_thesis: ( not C = A * e or not B = C * (Special_Function (n1,n2)) or (Partial_Intersection B) . n = ((Partial_Intersection C) . n1) /\ ((Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1)) ) assume C = A * e ; ::_thesis: ( not B = C * (Special_Function (n1,n2)) or (Partial_Intersection B) . n = ((Partial_Intersection C) . n1) /\ ((Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1)) ) assume A45: B = C * (Special_Function (n1,n2)) ; ::_thesis: (Partial_Intersection B) . n = ((Partial_Intersection C) . n1) /\ ((Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1)) reconsider B = B as SetSequence of Sigma ; A46: (Partial_Intersection B) . n1 = (Partial_Intersection C) . n1 proof for x being set holds ( x in (Partial_Intersection B) . n1 iff x in (Partial_Intersection C) . n1 ) proof let x be set ; ::_thesis: ( x in (Partial_Intersection B) . n1 iff x in (Partial_Intersection C) . n1 ) hereby ::_thesis: ( x in (Partial_Intersection C) . n1 implies x in (Partial_Intersection B) . n1 ) assume A47: x in (Partial_Intersection B) . n1 ; ::_thesis: x in (Partial_Intersection C) . n1 x in (Partial_Intersection C) . n1 proof A48: for knat being Nat st knat <= n1 holds x in C . knat proof let knat be Nat; ::_thesis: ( knat <= n1 implies x in C . knat ) assume A49: knat <= n1 ; ::_thesis: x in C . knat reconsider knat = knat as Element of NAT by ORDINAL1:def_12; A50: x in B . knat by A49, A47, PROB_3:25; A51: dom (C * (Special_Function (n1,n2))) = NAT by FUNCT_2:def_1; ( (Special_Function (n1,n2)) . knat = IFGT (knat,n1,(knat + n2),knat) & IFGT (knat,n1,(knat + n2),knat) = knat ) by Def2, A49, XXREAL_0:def_11; hence x in C . knat by A51, A45, A50, FUNCT_1:12; ::_thesis: verum end; reconsider n1 = n1 as Nat ; thus x in (Partial_Intersection C) . n1 by A48, PROB_3:25; ::_thesis: verum end; hence x in (Partial_Intersection C) . n1 ; ::_thesis: verum end; assume A52: x in (Partial_Intersection C) . n1 ; ::_thesis: x in (Partial_Intersection B) . n1 x in (Partial_Intersection B) . n1 proof for knat being Nat st knat <= n1 holds x in B . knat proof let knat be Nat; ::_thesis: ( knat <= n1 implies x in B . knat ) assume A53: knat <= n1 ; ::_thesis: x in B . knat reconsider knat = knat as Element of NAT by ORDINAL1:def_12; A54: x in C . knat by A53, A52, PROB_3:25; A55: dom (C * (Special_Function (n1,n2))) = NAT by FUNCT_2:def_1; ( (Special_Function (n1,n2)) . knat = IFGT (knat,n1,(knat + n2),knat) & IFGT (knat,n1,(knat + n2),knat) = knat ) by Def2, A53, XXREAL_0:def_11; hence x in B . knat by A55, A45, A54, FUNCT_1:12; ::_thesis: verum end; hence x in (Partial_Intersection B) . n1 by PROB_3:25; ::_thesis: verum end; hence x in (Partial_Intersection B) . n1 ; ::_thesis: verum end; hence (Partial_Intersection B) . n1 = (Partial_Intersection C) . n1 by TARSKI:1; ::_thesis: verum end; A56: for x being set st ( for knat being Nat st knat <= n holds x in B . knat ) holds ( ( for knat being Nat st knat <= n1 holds x in B . knat ) & ( for knat being Nat st n1 < knat & knat <= n holds x in B . knat ) ) by A44, XXREAL_0:2; A57: for x being set st x in (Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1) holds for qnat being Nat st n1 < qnat & qnat <= n holds x in B . qnat proof let x be set ; ::_thesis: ( x in (Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1) implies for qnat being Nat st n1 < qnat & qnat <= n holds x in B . qnat ) assume A58: x in (Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1) ; ::_thesis: for qnat being Nat st n1 < qnat & qnat <= n holds x in B . qnat A59: ( (n - n1) - 1 >= 0 & (n - n1) - 1 is Element of NAT ) proof n - n1 is Element of NAT by A44, NAT_1:21; hence ( (n - n1) - 1 >= 0 & (n - n1) - 1 is Element of NAT ) by A44, NAT_1:20, XREAL_1:50; ::_thesis: verum end; A60: for knat being Nat st knat <= (n - n1) - 1 holds x in C . (knat + ((n1 + n2) + 1)) proof let knat be Nat; ::_thesis: ( knat <= (n - n1) - 1 implies x in C . (knat + ((n1 + n2) + 1)) ) assume knat <= (n - n1) - 1 ; ::_thesis: x in C . (knat + ((n1 + n2) + 1)) then x in (C ^\ ((n1 + n2) + 1)) . knat by A59, A58, PROB_3:25; hence x in C . (knat + ((n1 + n2) + 1)) by NAT_1:def_3; ::_thesis: verum end; for qnat being Nat st n1 < qnat & qnat <= n holds x in B . qnat proof let qnat be Nat; ::_thesis: ( n1 < qnat & qnat <= n implies x in B . qnat ) assume that A61: n1 < qnat and A62: qnat <= n ; ::_thesis: x in B . qnat A63: ( n1 + 1 <= qnat & qnat <= n ) by A61, A62, NAT_1:13; A64: qnat - (n1 + 1) is Element of NAT by A63, NAT_1:21; consider knat being Nat such that A65: knat = (qnat - n1) - 1 by A64; A66: (qnat - n1) - 1 <= (n - n1) - 1 proof qnat - (n1 + 1) <= n - (n1 + 1) by A62, XREAL_1:9; hence (qnat - n1) - 1 <= (n - n1) - 1 ; ::_thesis: verum end; A67: x in C . (knat + ((n1 + n2) + 1)) by A65, A66, A60; x in B . qnat proof reconsider qnat = qnat as Element of NAT by ORDINAL1:def_12; A68: dom (C * (Special_Function (n1,n2))) = NAT by FUNCT_2:def_1; ( (Special_Function (n1,n2)) . qnat = IFGT (qnat,n1,(qnat + n2),qnat) & IFGT (qnat,n1,(qnat + n2),qnat) = qnat + n2 ) by Def2, A61, XXREAL_0:def_11; hence x in B . qnat by A68, A45, A65, A67, FUNCT_1:12; ::_thesis: verum end; hence x in B . qnat ; ::_thesis: verum end; hence for qnat being Nat st n1 < qnat & qnat <= n holds x in B . qnat ; ::_thesis: verum end; A69: for x being set st ( for qnat being Nat st n1 < qnat & qnat <= n holds x in B . qnat ) holds x in (Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1) proof let x be set ; ::_thesis: ( ( for qnat being Nat st n1 < qnat & qnat <= n holds x in B . qnat ) implies x in (Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1) ) assume A70: for qnat being Nat st n1 < qnat & qnat <= n holds x in B . qnat ; ::_thesis: x in (Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1) A71: ( (n - n1) - 1 >= 0 & (n - n1) - 1 is Element of NAT ) proof n - n1 is Element of NAT by A44, NAT_1:21; hence ( (n - n1) - 1 >= 0 & (n - n1) - 1 is Element of NAT ) by A44, NAT_1:20, XREAL_1:50; ::_thesis: verum end; x in (Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1) proof A72: for qnat being Nat st 0 <= (qnat - n1) - 1 & (qnat - n1) - 1 <= (n - n1) - 1 holds x in B . qnat proof let qnat be Nat; ::_thesis: ( 0 <= (qnat - n1) - 1 & (qnat - n1) - 1 <= (n - n1) - 1 implies x in B . qnat ) assume that A73: 0 <= (qnat - n1) - 1 and A74: (qnat - n1) - 1 <= (n - n1) - 1 ; ::_thesis: x in B . qnat 0 + (n1 + 1) <= (qnat - (n1 + 1)) + (n1 + 1) by A73, XREAL_1:6; then (n1 + 1) - 1 <= qnat - 1 by XREAL_1:9; then ( n1 <= qnat - 1 & qnat < qnat + 1 ) by NAT_1:13; then ( n1 <= qnat - 1 & qnat - 1 < (qnat + 1) - 1 ) by XREAL_1:9; then A75: n1 < qnat by XXREAL_0:2; (qnat - (n1 + 1)) + (n1 + 1) <= (n - (n1 + 1)) + (n1 + 1) by A74, XREAL_1:6; hence x in B . qnat by A75, A70; ::_thesis: verum end; for knat being Nat st 0 <= knat & knat <= (n - n1) - 1 holds x in (C ^\ ((n1 + n2) + 1)) . knat proof let knat be Nat; ::_thesis: ( 0 <= knat & knat <= (n - n1) - 1 implies x in (C ^\ ((n1 + n2) + 1)) . knat ) assume that 0 <= knat and A76: knat <= (n - n1) - 1 ; ::_thesis: x in (C ^\ ((n1 + n2) + 1)) . knat set qnat = (knat + n1) + 1; A77: (((knat + n1) + 1) - n1) - 1 <= (n - n1) - 1 by A76; A78: x in B . ((knat + n1) + 1) by A77, A72; A79: dom (C * (Special_Function (n1,n2))) = NAT by FUNCT_2:def_1; A80: n1 < (knat + n1) + 1 proof n1 + 1 <= (n1 + 1) + knat by XREAL_1:31; hence n1 < (knat + n1) + 1 by NAT_1:13; ::_thesis: verum end; ( (Special_Function (n1,n2)) . ((knat + n1) + 1) = IFGT (((knat + n1) + 1),n1,(((knat + n1) + 1) + n2),((knat + n1) + 1)) & IFGT (((knat + n1) + 1),n1,(((knat + n1) + 1) + n2),((knat + n1) + 1)) = ((knat + n1) + 1) + n2 ) by Def2, A80, XXREAL_0:def_11; then B . ((knat + n1) + 1) = C . (((n1 + n2) + 1) + knat) by A45, A79, FUNCT_1:12; hence x in (C ^\ ((n1 + n2) + 1)) . knat by A78, NAT_1:def_3; ::_thesis: verum end; then for knat being Nat st knat <= (n - n1) - 1 holds x in (C ^\ ((n1 + n2) + 1)) . knat ; hence x in (Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1) by A71, PROB_3:25; ::_thesis: verum end; hence x in (Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1) ; ::_thesis: verum end; A81: for x being set holds ( x in (Partial_Intersection B) . n iff ( ( for knat being Nat st knat <= n1 holds x in B . knat ) & ( for knat being Nat st n1 < knat & knat <= n holds x in B . knat ) ) ) proof let x be set ; ::_thesis: ( x in (Partial_Intersection B) . n iff ( ( for knat being Nat st knat <= n1 holds x in B . knat ) & ( for knat being Nat st n1 < knat & knat <= n holds x in B . knat ) ) ) ( x in (Partial_Intersection B) . n iff for knat being Nat st knat <= n holds x in B . knat ) by PROB_3:25; hence ( x in (Partial_Intersection B) . n iff ( ( for knat being Nat st knat <= n1 holds x in B . knat ) & ( for knat being Nat st n1 < knat & knat <= n holds x in B . knat ) ) ) by A56; ::_thesis: verum end; A82: for x being set holds ( x in (Partial_Intersection B) . n iff ( x in (Partial_Intersection B) . n1 & ( for knat being Nat st n1 < knat & knat <= n holds x in B . knat ) ) ) proof let x be set ; ::_thesis: ( x in (Partial_Intersection B) . n iff ( x in (Partial_Intersection B) . n1 & ( for knat being Nat st n1 < knat & knat <= n holds x in B . knat ) ) ) ( x in (Partial_Intersection B) . n1 iff for knat being Nat st knat <= n1 holds x in B . knat ) by PROB_3:25; hence ( x in (Partial_Intersection B) . n iff ( x in (Partial_Intersection B) . n1 & ( for knat being Nat st n1 < knat & knat <= n holds x in B . knat ) ) ) by A81; ::_thesis: verum end; for x being set holds ( x in (Partial_Intersection B) . n iff ( x in (Partial_Intersection B) . n1 & x in (Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1) ) ) proof let x be set ; ::_thesis: ( x in (Partial_Intersection B) . n iff ( x in (Partial_Intersection B) . n1 & x in (Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1) ) ) ( x in (Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1) iff for knat being Nat st n1 < knat & knat <= n holds x in B . knat ) by A57, A69; hence ( x in (Partial_Intersection B) . n iff ( x in (Partial_Intersection B) . n1 & x in (Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1) ) ) by A82; ::_thesis: verum end; hence (Partial_Intersection B) . n = ((Partial_Intersection C) . n1) /\ ((Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1)) by A46, XBOOLE_0:def_4; ::_thesis: verum end; hence ( ( for A, B being SetSequence of Sigma st n > n1 & B = A * (Special_Function (n1,n2)) holds (Partial_Product (Prob * B)) . n = ((Partial_Product (Prob * A)) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . ((n - n1) - 1)) ) & ( for A, B, C being SetSequence of Sigma for e being sequence of NAT st n > n1 & C = A * e & B = C * (Special_Function (n1,n2)) holds (Partial_Intersection B) . n = ((Partial_Intersection C) . n1) /\ ((Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1)) ) ) by A1; ::_thesis: verum end; definition let Omega be non empty set ; let Sigma be SigmaField of Omega; let Prob be Probability of Sigma; let A be SetSequence of Sigma; predA is_all_independent_wrt Prob means :Def6: :: BOR_CANT:def 6 for B being SetSequence of Sigma st ex e being sequence of NAT st ( e is one-to-one & ( for n being Element of NAT holds A . (e . n) = B . n ) ) holds for n being Element of NAT holds (Partial_Product (Prob * B)) . n = Prob . ((Partial_Intersection B) . n); end; :: deftheorem Def6 defines is_all_independent_wrt BOR_CANT:def_6_:_ for Omega being non empty set for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma holds ( A is_all_independent_wrt Prob iff for B being SetSequence of Sigma st ex e being sequence of NAT st ( e is one-to-one & ( for n being Element of NAT holds A . (e . n) = B . n ) ) holds for n being Element of NAT holds (Partial_Product (Prob * B)) . n = Prob . ((Partial_Intersection B) . n) ); theorem Th6: :: BOR_CANT:6 for Omega being non empty set for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma for n, n1, n2 being Element of NAT st n > n1 & A is_all_independent_wrt Prob holds Prob . (((Partial_Intersection (Complement A)) . n1) /\ ((Partial_Intersection (A ^\ ((n1 + n2) + 1))) . ((n - n1) - 1))) = ((Partial_Product (Prob * (Complement A))) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . ((n - n1) - 1)) proof let Omega be non empty set ; ::_thesis: for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma for n, n1, n2 being Element of NAT st n > n1 & A is_all_independent_wrt Prob holds Prob . (((Partial_Intersection (Complement A)) . n1) /\ ((Partial_Intersection (A ^\ ((n1 + n2) + 1))) . ((n - n1) - 1))) = ((Partial_Product (Prob * (Complement A))) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . ((n - n1) - 1)) let Sigma be SigmaField of Omega; ::_thesis: for Prob being Probability of Sigma for A being SetSequence of Sigma for n, n1, n2 being Element of NAT st n > n1 & A is_all_independent_wrt Prob holds Prob . (((Partial_Intersection (Complement A)) . n1) /\ ((Partial_Intersection (A ^\ ((n1 + n2) + 1))) . ((n - n1) - 1))) = ((Partial_Product (Prob * (Complement A))) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . ((n - n1) - 1)) let Prob be Probability of Sigma; ::_thesis: for A being SetSequence of Sigma for n, n1, n2 being Element of NAT st n > n1 & A is_all_independent_wrt Prob holds Prob . (((Partial_Intersection (Complement A)) . n1) /\ ((Partial_Intersection (A ^\ ((n1 + n2) + 1))) . ((n - n1) - 1))) = ((Partial_Product (Prob * (Complement A))) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . ((n - n1) - 1)) let A be SetSequence of Sigma; ::_thesis: for n, n1, n2 being Element of NAT st n > n1 & A is_all_independent_wrt Prob holds Prob . (((Partial_Intersection (Complement A)) . n1) /\ ((Partial_Intersection (A ^\ ((n1 + n2) + 1))) . ((n - n1) - 1))) = ((Partial_Product (Prob * (Complement A))) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . ((n - n1) - 1)) let n, n1, n2 be Element of NAT ; ::_thesis: ( n > n1 & A is_all_independent_wrt Prob implies Prob . (((Partial_Intersection (Complement A)) . n1) /\ ((Partial_Intersection (A ^\ ((n1 + n2) + 1))) . ((n - n1) - 1))) = ((Partial_Product (Prob * (Complement A))) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . ((n - n1) - 1)) ) assume that A1: n > n1 and A2: A is_all_independent_wrt Prob ; ::_thesis: Prob . (((Partial_Intersection (Complement A)) . n1) /\ ((Partial_Intersection (A ^\ ((n1 + n2) + 1))) . ((n - n1) - 1))) = ((Partial_Product (Prob * (Complement A))) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . ((n - n1) - 1)) A3: for A, B being SetSequence of Sigma for k, n being Element of NAT st B = A * (Special_Function2 k) holds (Partial_Product (Prob * (A ^\ k))) . n = (Partial_Product (Prob * B)) . n proof let A, B be SetSequence of Sigma; ::_thesis: for k, n being Element of NAT st B = A * (Special_Function2 k) holds (Partial_Product (Prob * (A ^\ k))) . n = (Partial_Product (Prob * B)) . n let k, n be Element of NAT ; ::_thesis: ( B = A * (Special_Function2 k) implies (Partial_Product (Prob * (A ^\ k))) . n = (Partial_Product (Prob * B)) . n ) assume A4: B = A * (Special_Function2 k) ; ::_thesis: (Partial_Product (Prob * (A ^\ k))) . n = (Partial_Product (Prob * B)) . n defpred S1[ Element of NAT ] means (Partial_Product (Prob * (A ^\ k))) . \$1 = (Partial_Product (Prob * B)) . \$1; dom (Prob * (A ^\ k)) = NAT by FUNCT_2:def_1; then A5: (Prob * (A ^\ k)) . 0 = Prob . ((A ^\ k) . 0) by FUNCT_1:12; (Prob * (A ^\ k)) . 0 = Prob . (A . (0 + k)) by A5, NAT_1:def_3; then A6: (Partial_Product (Prob * (A ^\ k))) . 0 = Prob . (A . k) by SERIES_3:def_1; A7: (Partial_Product (Prob * B)) . 0 = (Prob * B) . 0 by SERIES_3:def_1; A8: (Special_Function2 k) . 0 = 0 + k by Def3; dom (A * (Special_Function2 k)) = NAT by FUNCT_2:def_1; then A9: Prob . (B . 0) = Prob . (A . k) by A8, A4, FUNCT_1:12; dom (Prob * B) = NAT by FUNCT_2:def_1; then A10: S1[ 0 ] by A9, A7, A6, FUNCT_1:12; A11: for q being Element of NAT st S1[q] holds S1[q + 1] proof let q be Element of NAT ; ::_thesis: ( S1[q] implies S1[q + 1] ) assume A12: S1[q] ; ::_thesis: S1[q + 1] A13: (Partial_Product (Prob * (A ^\ k))) . (q + 1) = ((Partial_Product (Prob * B)) . q) * ((Prob * (A ^\ k)) . (q + 1)) by A12, SERIES_3:def_1; (Prob * (A ^\ k)) . (q + 1) = (Prob * B) . (q + 1) proof dom (Prob * (A ^\ k)) = NAT by FUNCT_2:def_1; then A14: (Prob * (A ^\ k)) . (q + 1) = Prob . ((A ^\ k) . (q + 1)) by FUNCT_1:12; dom (Prob * B) = NAT by FUNCT_2:def_1; then A15: (Prob * B) . (q + 1) = Prob . (B . (q + 1)) by FUNCT_1:12; dom (A * (Special_Function2 k)) = NAT by FUNCT_2:def_1; then A16: B . (q + 1) = A . ((Special_Function2 k) . (q + 1)) by A4, FUNCT_1:12; ( (Special_Function2 k) . (q + 1) = (q + 1) + k & (q + 1) + k = (q + 1) + k ) by Def3; hence (Prob * (A ^\ k)) . (q + 1) = (Prob * B) . (q + 1) by A16, A15, A14, NAT_1:def_3; ::_thesis: verum end; hence S1[q + 1] by A13, SERIES_3:def_1; ::_thesis: verum end; for k being Element of NAT holds S1[k] from NAT_1:sch_1(A10, A11); hence (Partial_Product (Prob * (A ^\ k))) . n = (Partial_Product (Prob * B)) . n ; ::_thesis: verum end; A17: for m, m1, m2 being Element of NAT for e being sequence of NAT for C, B being SetSequence of Sigma st m1 < m & e is one-to-one & C = A * e & B = C * (Special_Function (m1,m2)) holds Prob . ((Partial_Intersection B) . m) = ((Partial_Product (Prob * C)) . m1) * ((Partial_Product (Prob * (C ^\ ((m1 + m2) + 1)))) . ((m - m1) - 1)) proof let m, m1, m2 be Element of NAT ; ::_thesis: for e being sequence of NAT for C, B being SetSequence of Sigma st m1 < m & e is one-to-one & C = A * e & B = C * (Special_Function (m1,m2)) holds Prob . ((Partial_Intersection B) . m) = ((Partial_Product (Prob * C)) . m1) * ((Partial_Product (Prob * (C ^\ ((m1 + m2) + 1)))) . ((m - m1) - 1)) let e be sequence of NAT; ::_thesis: for C, B being SetSequence of Sigma st m1 < m & e is one-to-one & C = A * e & B = C * (Special_Function (m1,m2)) holds Prob . ((Partial_Intersection B) . m) = ((Partial_Product (Prob * C)) . m1) * ((Partial_Product (Prob * (C ^\ ((m1 + m2) + 1)))) . ((m - m1) - 1)) let C, B be SetSequence of Sigma; ::_thesis: ( m1 < m & e is one-to-one & C = A * e & B = C * (Special_Function (m1,m2)) implies Prob . ((Partial_Intersection B) . m) = ((Partial_Product (Prob * C)) . m1) * ((Partial_Product (Prob * (C ^\ ((m1 + m2) + 1)))) . ((m - m1) - 1)) ) assume that A18: m1 < m and A19: e is one-to-one and A20: C = A * e and A21: B = C * (Special_Function (m1,m2)) ; ::_thesis: Prob . ((Partial_Intersection B) . m) = ((Partial_Product (Prob * C)) . m1) * ((Partial_Product (Prob * (C ^\ ((m1 + m2) + 1)))) . ((m - m1) - 1)) ( B is SetSequence of Sigma & e * (Special_Function (m1,m2)) is sequence of NAT & e * (Special_Function (m1,m2)) is one-to-one & ( for n being Element of NAT holds A . ((e * (Special_Function (m1,m2))) . n) = B . n ) ) proof for n being Element of NAT holds A . ((e * (Special_Function (m1,m2))) . n) = B . n proof let n be Element of NAT ; ::_thesis: A . ((e * (Special_Function (m1,m2))) . n) = B . n A22: dom ((A * e) * (Special_Function (m1,m2))) = NAT by FUNCT_2:def_1; A23: B . n = (A * e) . ((Special_Function (m1,m2)) . n) by A21, A20, A22, FUNCT_1:12; dom (A * e) = NAT by FUNCT_2:def_1; then A24: B . n = A . (e . ((Special_Function (m1,m2)) . n)) by A23, FUNCT_1:12; dom (e * (Special_Function (m1,m2))) = NAT by FUNCT_2:def_1; hence A . ((e * (Special_Function (m1,m2))) . n) = B . n by A24, FUNCT_1:12; ::_thesis: verum end; hence ( B is SetSequence of Sigma & e * (Special_Function (m1,m2)) is sequence of NAT & e * (Special_Function (m1,m2)) is one-to-one & ( for n being Element of NAT holds A . ((e * (Special_Function (m1,m2))) . n) = B . n ) ) by A19, FUNCT_1:24; ::_thesis: verum end; then Prob . ((Partial_Intersection B) . m) = (Partial_Product (Prob * B)) . m by A2, Def6; hence Prob . ((Partial_Intersection B) . m) = ((Partial_Product (Prob * C)) . m1) * ((Partial_Product (Prob * (C ^\ ((m1 + m2) + 1)))) . ((m - m1) - 1)) by A18, A21, Th5; ::_thesis: verum end; A25: for m, m1 being Element of NAT for e being sequence of NAT for C, B being SetSequence of Sigma st C = A * e & e is one-to-one & B = C * (Special_Function2 m1) holds Prob . ((Partial_Intersection B) . m) = (Partial_Product (Prob * (C ^\ m1))) . m proof let m, m1 be Element of NAT ; ::_thesis: for e being sequence of NAT for C, B being SetSequence of Sigma st C = A * e & e is one-to-one & B = C * (Special_Function2 m1) holds Prob . ((Partial_Intersection B) . m) = (Partial_Product (Prob * (C ^\ m1))) . m let e be sequence of NAT; ::_thesis: for C, B being SetSequence of Sigma st C = A * e & e is one-to-one & B = C * (Special_Function2 m1) holds Prob . ((Partial_Intersection B) . m) = (Partial_Product (Prob * (C ^\ m1))) . m let C, B be SetSequence of Sigma; ::_thesis: ( C = A * e & e is one-to-one & B = C * (Special_Function2 m1) implies Prob . ((Partial_Intersection B) . m) = (Partial_Product (Prob * (C ^\ m1))) . m ) assume that A26: C = A * e and A27: e is one-to-one and A28: B = C * (Special_Function2 m1) ; ::_thesis: Prob . ((Partial_Intersection B) . m) = (Partial_Product (Prob * (C ^\ m1))) . m A29: ( B is SetSequence of Sigma & Special_Function2 m1 is sequence of NAT & dom (e * (Special_Function2 m1)) <> {} & e * (Special_Function2 m1) is one-to-one & ( for n being Element of NAT holds A . ((e * (Special_Function2 m1)) . n) = B . n ) ) proof A30: for n being Element of NAT holds A . ((e * (Special_Function2 m1)) . n) = B . n proof let n be Element of NAT ; ::_thesis: A . ((e * (Special_Function2 m1)) . n) = B . n dom (A * (e * (Special_Function2 m1))) = NAT by FUNCT_2:def_1; then A31: (A * (e * (Special_Function2 m1))) . n = A . ((e * (Special_Function2 m1)) . n) by FUNCT_1:12; dom (A * e) = NAT by FUNCT_2:def_1; then A32: (A * e) . ((Special_Function2 m1) . n) = A . (e . ((Special_Function2 m1) . n)) by FUNCT_1:12; dom (e * (Special_Function2 m1)) = NAT by FUNCT_2:def_1; then A33: (A * e) . ((Special_Function2 m1) . n) = (A * (e * (Special_Function2 m1))) . n by A32, A31, FUNCT_1:12; dom ((A * e) * (Special_Function2 m1)) = NAT by FUNCT_2:def_1; then A34: B . n = (A * (e * (Special_Function2 m1))) . n by A33, A28, A26, FUNCT_1:12; dom (A * (e * (Special_Function2 m1))) = NAT by FUNCT_2:def_1; hence A . ((e * (Special_Function2 m1)) . n) = B . n by A34, FUNCT_1:12; ::_thesis: verum end; thus ( B is SetSequence of Sigma & Special_Function2 m1 is sequence of NAT & dom (e * (Special_Function2 m1)) <> {} & e * (Special_Function2 m1) is one-to-one & ( for n being Element of NAT holds A . ((e * (Special_Function2 m1)) . n) = B . n ) ) by A27, A30, FUNCT_1:24; ::_thesis: verum end; Prob . ((Partial_Intersection B) . m) = (Partial_Product (Prob * B)) . m by A2, A29, Def6; hence Prob . ((Partial_Intersection B) . m) = (Partial_Product (Prob * (C ^\ m1))) . m by A28, A3; ::_thesis: verum end; A35: for q being Element of NAT holds Prob . (((Partial_Intersection (Complement A)) . n1) /\ ((Partial_Intersection (A ^\ ((n1 + n2) + 1))) . q)) = ((Partial_Product (Prob * (Complement A))) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . q) proof let q be Element of NAT ; ::_thesis: Prob . (((Partial_Intersection (Complement A)) . n1) /\ ((Partial_Intersection (A ^\ ((n1 + n2) + 1))) . q)) = ((Partial_Product (Prob * (Complement A))) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . q) defpred S1[ Element of NAT ] means for e being sequence of NAT for q, n2 being Element of NAT for C being SetSequence of Sigma st e is one-to-one & C = A * e holds Prob . (((Partial_Intersection (Complement C)) . \$1) /\ ((Partial_Intersection (C ^\ ((\$1 + n2) + 1))) . q)) = ((Partial_Product (Prob * (Complement C))) . \$1) * ((Partial_Product (Prob * (C ^\ ((\$1 + n2) + 1)))) . q); A36: S1[ 0 ] proof let e be sequence of NAT; ::_thesis: for q, n2 being Element of NAT for C being SetSequence of Sigma st e is one-to-one & C = A * e holds Prob . (((Partial_Intersection (Complement C)) . 0) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) = ((Partial_Product (Prob * (Complement C))) . 0) * ((Partial_Product (Prob * (C ^\ ((0 + n2) + 1)))) . q) let q, n2 be Element of NAT ; ::_thesis: for C being SetSequence of Sigma st e is one-to-one & C = A * e holds Prob . (((Partial_Intersection (Complement C)) . 0) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) = ((Partial_Product (Prob * (Complement C))) . 0) * ((Partial_Product (Prob * (C ^\ ((0 + n2) + 1)))) . q) let C be SetSequence of Sigma; ::_thesis: ( e is one-to-one & C = A * e implies Prob . (((Partial_Intersection (Complement C)) . 0) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) = ((Partial_Product (Prob * (Complement C))) . 0) * ((Partial_Product (Prob * (C ^\ ((0 + n2) + 1)))) . q) ) assume A37: e is one-to-one ; ::_thesis: ( not C = A * e or Prob . (((Partial_Intersection (Complement C)) . 0) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) = ((Partial_Product (Prob * (Complement C))) . 0) * ((Partial_Product (Prob * (C ^\ ((0 + n2) + 1)))) . q) ) assume A38: C = A * e ; ::_thesis: Prob . (((Partial_Intersection (Complement C)) . 0) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) = ((Partial_Product (Prob * (Complement C))) . 0) * ((Partial_Product (Prob * (C ^\ ((0 + n2) + 1)))) . q) Prob . (((Partial_Intersection (Complement C)) . 0) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) = Prob . (((Complement C) . 0) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) by PROB_3:21; then Prob . (((Partial_Intersection (Complement C)) . 0) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) = Prob . (((C . 0) `) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) by PROB_1:def_2; then Prob . (((Partial_Intersection (Complement C)) . 0) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) = Prob . ((Omega \ (C . 0)) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) by SUBSET_1:def_4; then A39: Prob . (((Partial_Intersection (Complement C)) . 0) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) = Prob . ((Omega /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) \ ((C . 0) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q))) by XBOOLE_1:111; A40: Prob . (((Partial_Intersection (Complement C)) . 0) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) = Prob . (((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q) \ ((C . 0) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q))) by A39, XBOOLE_1:28; A41: Prob . (((Partial_Intersection (Complement C)) . 0) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) = (Prob . ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) - (Prob . ((C . 0) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q))) by A40, PROB_1:33, XBOOLE_1:17; consider m1 being Element of NAT such that A42: m1 = 0 ; consider m being Element of NAT such that A43: m = (m1 + 1) + q ; consider m2 being Element of NAT such that A44: m2 = n2 ; consider B being SetSequence of Omega such that A45: B = C * (Special_Function (m1,m2)) ; reconsider B = B as SetSequence of Sigma by A45; A46: ( m1 < m & C = A * e & B = C * (Special_Function (m1,m2)) ) proof ( m1 < m1 + 1 & m1 + 1 <= (m1 + 1) + q ) by NAT_1:13, XREAL_1:31; hence ( m1 < m & C = A * e & B = C * (Special_Function (m1,m2)) ) by A43, A38, A45, XXREAL_0:2; ::_thesis: verum end; then Prob . ((Partial_Intersection B) . m) = Prob . (((Partial_Intersection C) . m1) /\ ((Partial_Intersection (C ^\ ((m1 + m2) + 1))) . ((m - m1) - 1))) by Th5; then ((Partial_Product (Prob * C)) . 0) * ((Partial_Product (Prob * (C ^\ ((0 + n2) + 1)))) . q) = Prob . (((Partial_Intersection C) . 0) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) by A46, A37, A17, A42, A44, A43; then (Prob . ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) - (((Partial_Product (Prob * C)) . 0) * ((Partial_Product (Prob * (C ^\ ((0 + n2) + 1)))) . q)) = (Prob . ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) - (Prob . ((C . 0) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q))) by PROB_3:21; then A47: Prob . (((Partial_Intersection (Complement C)) . 0) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) = (Prob . ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) - (((Prob * C) . 0) * ((Partial_Product (Prob * (C ^\ ((0 + n2) + 1)))) . q)) by A41, SERIES_3:def_1; (Prob * C) . 0 = 1 - ((Prob * (Complement C)) . 0) proof ( C . 0 = ((C . 0) `) ` & ((C . 0) `) ` = Omega \ ((C . 0) `) ) by SUBSET_1:def_4; then ( Prob . (C . 0) = Prob . (([#] Sigma) \ ((C . 0) `)) & (C . 0) ` is Event of Sigma ) by PROB_1:20; then A48: Prob . (C . 0) = 1 - (Prob . ((C . 0) `)) by PROB_1:32; dom (Prob * C) = NAT by FUNCT_2:def_1; then A49: (Prob * C) . 0 = 1 - (Prob . ((C . 0) `)) by A48, FUNCT_1:12; dom (Prob * (Complement C)) = NAT by FUNCT_2:def_1; then (Prob * (Complement C)) . 0 = Prob . ((Complement C) . 0) by FUNCT_1:12; hence (Prob * C) . 0 = 1 - ((Prob * (Complement C)) . 0) by A49, PROB_1:def_2; ::_thesis: verum end; then A50: Prob . (((Partial_Intersection (Complement C)) . 0) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) = (Prob . ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) - (((Partial_Product (Prob * (C ^\ ((0 + n2) + 1)))) . q) - (((Prob * (Complement C)) . 0) * ((Partial_Product (Prob * (C ^\ ((0 + n2) + 1)))) . q))) by A47; set m1 = (0 + n2) + 1; set m = q; set B = C * (Special_Function2 ((0 + n2) + 1)); reconsider B = C * (Special_Function2 ((0 + n2) + 1)) as SetSequence of Sigma ; A51: for A, B, C being SetSequence of Sigma for k, n being Element of NAT for e being sequence of NAT st C = A * e & B = C * (Special_Function2 k) holds (Partial_Intersection (C ^\ k)) . n = (Partial_Intersection B) . n proof let A, B, C be SetSequence of Sigma; ::_thesis: for k, n being Element of NAT for e being sequence of NAT st C = A * e & B = C * (Special_Function2 k) holds (Partial_Intersection (C ^\ k)) . n = (Partial_Intersection B) . n let k, n be Element of NAT ; ::_thesis: for e being sequence of NAT st C = A * e & B = C * (Special_Function2 k) holds (Partial_Intersection (C ^\ k)) . n = (Partial_Intersection B) . n let e be sequence of NAT; ::_thesis: ( C = A * e & B = C * (Special_Function2 k) implies (Partial_Intersection (C ^\ k)) . n = (Partial_Intersection B) . n ) assume C = A * e ; ::_thesis: ( not B = C * (Special_Function2 k) or (Partial_Intersection (C ^\ k)) . n = (Partial_Intersection B) . n ) assume A52: B = C * (Special_Function2 k) ; ::_thesis: (Partial_Intersection (C ^\ k)) . n = (Partial_Intersection B) . n A53: for x being set holds ( ( for knat being Nat st knat <= n holds x in (C ^\ k) . knat ) iff for knat being Nat st knat <= n holds x in B . knat ) proof let x be set ; ::_thesis: ( ( for knat being Nat st knat <= n holds x in (C ^\ k) . knat ) iff for knat being Nat st knat <= n holds x in B . knat ) hereby ::_thesis: ( ( for knat being Nat st knat <= n holds x in B . knat ) implies for knat being Nat st knat <= n holds x in (C ^\ k) . knat ) assume A54: for knat being Nat st knat <= n holds x in (C ^\ k) . knat ; ::_thesis: for knat being Nat st knat <= n holds x in B . knat thus for knat being Nat st knat <= n holds x in B . knat ::_thesis: verum proof let knat be Nat; ::_thesis: ( knat <= n implies x in B . knat ) assume A55: knat <= n ; ::_thesis: x in B . knat reconsider knat = knat as Element of NAT by ORDINAL1:def_12; dom (C * (Special_Function2 k)) = NAT by FUNCT_2:def_1; then A56: (C * (Special_Function2 k)) . knat = C . ((Special_Function2 k) . knat) by FUNCT_1:12; ( (Special_Function2 k) . knat = knat + k & knat + k = knat + k ) by Def3; then ( x in B . knat iff x in (C ^\ k) . knat ) by A52, A56, NAT_1:def_3; hence x in B . knat by A55, A54; ::_thesis: verum end; end; assume A57: for knat being Nat st knat <= n holds x in B . knat ; ::_thesis: for knat being Nat st knat <= n holds x in (C ^\ k) . knat thus for knat being Nat st knat <= n holds x in (C ^\ k) . knat ::_thesis: verum proof let knat be Nat; ::_thesis: ( knat <= n implies x in (C ^\ k) . knat ) assume A58: knat <= n ; ::_thesis: x in (C ^\ k) . knat reconsider knat = knat as Element of NAT by ORDINAL1:def_12; dom (C * (Special_Function2 k)) = NAT by FUNCT_2:def_1; then A59: (C * (Special_Function2 k)) . knat = C . ((Special_Function2 k) . knat) by FUNCT_1:12; ( (Special_Function2 k) . knat = knat + k & knat + k = knat + k ) by Def3; then ( x in B . knat iff x in (C ^\ k) . knat ) by A52, A59, NAT_1:def_3; hence x in (C ^\ k) . knat by A57, A58; ::_thesis: verum end; end; for x being set holds ( x in (Partial_Intersection (C ^\ k)) . n iff x in (Partial_Intersection B) . n ) proof let x be set ; ::_thesis: ( x in (Partial_Intersection (C ^\ k)) . n iff x in (Partial_Intersection B) . n ) ( ( x in (Partial_Intersection (C ^\ k)) . n implies for knat being Nat st knat <= n holds x in (C ^\ k) . knat ) & ( ( for knat being Nat st knat <= n holds x in (C ^\ k) . knat ) implies x in (Partial_Intersection (C ^\ k)) . n ) & ( x in (Partial_Intersection B) . n implies for knat being Nat st knat <= n holds x in B . knat ) & ( ( for knat being Nat st knat <= n holds x in B . knat ) implies x in (Partial_Intersection B) . n ) ) by PROB_3:25; hence ( x in (Partial_Intersection (C ^\ k)) . n iff x in (Partial_Intersection B) . n ) by A53; ::_thesis: verum end; hence (Partial_Intersection (C ^\ k)) . n = (Partial_Intersection B) . n by TARSKI:1; ::_thesis: verum end; A60: Prob . ((Partial_Intersection B) . q) = (Partial_Product (Prob * (C ^\ ((0 + n2) + 1)))) . q by A38, A37, A25; Prob . ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q) = (Partial_Product (Prob * (C ^\ ((0 + n2) + 1)))) . q by A38, A51, A60; hence Prob . (((Partial_Intersection (Complement C)) . 0) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) = ((Partial_Product (Prob * (Complement C))) . 0) * ((Partial_Product (Prob * (C ^\ ((0 + n2) + 1)))) . q) by A50, SERIES_3:def_1; ::_thesis: verum end; A61: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A62: S1[k] ; ::_thesis: S1[k + 1] let e be sequence of NAT; ::_thesis: for q, n2 being Element of NAT for C being SetSequence of Sigma st e is one-to-one & C = A * e holds Prob . (((Partial_Intersection (Complement C)) . (k + 1)) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) = ((Partial_Product (Prob * (Complement C))) . (k + 1)) * ((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . q) let q, n2 be Element of NAT ; ::_thesis: for C being SetSequence of Sigma st e is one-to-one & C = A * e holds Prob . (((Partial_Intersection (Complement C)) . (k + 1)) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) = ((Partial_Product (Prob * (Complement C))) . (k + 1)) * ((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . q) let C be SetSequence of Sigma; ::_thesis: ( e is one-to-one & C = A * e implies Prob . (((Partial_Intersection (Complement C)) . (k + 1)) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) = ((Partial_Product (Prob * (Complement C))) . (k + 1)) * ((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . q) ) assume A63: e is one-to-one ; ::_thesis: ( not C = A * e or Prob . (((Partial_Intersection (Complement C)) . (k + 1)) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) = ((Partial_Product (Prob * (Complement C))) . (k + 1)) * ((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . q) ) assume A64: C = A * e ; ::_thesis: Prob . (((Partial_Intersection (Complement C)) . (k + 1)) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) = ((Partial_Product (Prob * (Complement C))) . (k + 1)) * ((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . q) Prob . (((Partial_Intersection (Complement C)) . (k + 1)) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) = Prob . ((((Partial_Intersection (Complement C)) . k) /\ ((Complement C) . (k + 1))) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) by PROB_3:21; then Prob . (((Partial_Intersection (Complement C)) . (k + 1)) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) = Prob . ((((C . (k + 1)) `) /\ ((Partial_Intersection (Complement C)) . k)) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) by PROB_1:def_2; then Prob . (((Partial_Intersection (Complement C)) . (k + 1)) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) = Prob . (((C . (k + 1)) `) /\ (((Partial_Intersection (Complement C)) . k) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q))) by XBOOLE_1:16; then Prob . (((Partial_Intersection (Complement C)) . (k + 1)) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) = Prob . ((Omega \ (C . (k + 1))) /\ (((Partial_Intersection (Complement C)) . k) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q))) by SUBSET_1:def_4; then A65: Prob . (((Partial_Intersection (Complement C)) . (k + 1)) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) = Prob . ((Omega /\ (((Partial_Intersection (Complement C)) . k) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q))) \ ((C . (k + 1)) /\ (((Partial_Intersection (Complement C)) . k) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)))) by XBOOLE_1:50; A66: Prob . (((Partial_Intersection (Complement C)) . (k + 1)) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) = Prob . ((((Partial_Intersection (Complement C)) . k) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) \ ((C . (k + 1)) /\ (((Partial_Intersection (Complement C)) . k) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)))) by A65, XBOOLE_1:28; A67: Prob . (((Partial_Intersection (Complement C)) . k) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) = ((Partial_Product (Prob * (Complement C))) . k) * ((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . q) proof Prob . (((Partial_Intersection (Complement C)) . k) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) = ((Partial_Product (Prob * (Complement C))) . k) * ((Partial_Product (Prob * (C ^\ ((k + (1 + n2)) + 1)))) . q) by A63, A64, A62; hence Prob . (((Partial_Intersection (Complement C)) . k) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) = ((Partial_Product (Prob * (Complement C))) . k) * ((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . q) ; ::_thesis: verum end; A68: Prob . (((Partial_Intersection (Complement C)) . (k + 1)) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) = (((Partial_Product (Prob * (Complement C))) . k) * ((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . q)) - ((((Prob * C) . (k + 1)) * ((Partial_Product (Prob * (Complement C))) . k)) * ((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . q)) proof (((Prob * C) . (k + 1)) * ((Partial_Product (Prob * (Complement C))) . k)) * ((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . q) = Prob . ((C . (k + 1)) /\ (((Partial_Intersection (Complement C)) . k) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q))) proof consider F being SetSequence of Omega such that A69: F = C * (Special_Function4 (k,n2)) ; F is SetSequence of Sigma proof for n being Element of NAT holds F . n is Event of Sigma proof let n be Element of NAT ; ::_thesis: F . n is Event of Sigma A70: dom (C * (Special_Function4 (k,n2))) = NAT by FUNCT_2:def_1; F . n = C . ((Special_Function4 (k,n2)) . n) by A69, A70, FUNCT_1:12; hence F . n is Event of Sigma ; ::_thesis: verum end; hence F is SetSequence of Sigma by PROB_1:25; ::_thesis: verum end; then reconsider F = F as SetSequence of Sigma ; ( e * (Special_Function4 (k,n2)) is one-to-one & dom (e * (Special_Function4 (k,n2))) <> {} ) by A63, FUNCT_1:24; then consider f being sequence of NAT such that A71: ( f = e * (Special_Function4 (k,n2)) & f is one-to-one & dom f <> {} ) ; A72: for q being set st q in NAT holds F . q = (A * f) . q proof let q be set ; ::_thesis: ( q in NAT implies F . q = (A * f) . q ) assume q in NAT ; ::_thesis: F . q = (A * f) . q then reconsider q = q as Element of NAT ; dom (A * e) = NAT by FUNCT_2:def_1; then A73: (A * e) . ((Special_Function4 (k,n2)) . q) = A . (e . ((Special_Function4 (k,n2)) . q)) by FUNCT_1:12; dom ((A * e) * (Special_Function4 (k,n2))) = NAT by FUNCT_2:def_1; then A74: ((A * e) * (Special_Function4 (k,n2))) . q = A . (e . ((Special_Function4 (k,n2)) . q)) by A73, FUNCT_1:12; dom (e * (Special_Function4 (k,n2))) = NAT by FUNCT_2:def_1; then A75: ((A * e) * (Special_Function4 (k,n2))) . q = A . ((e * (Special_Function4 (k,n2))) . q) by A74, FUNCT_1:12; dom (A * f) = NAT by FUNCT_2:def_1; hence F . q = (A * f) . q by A71, A75, A64, A69, FUNCT_1:12; ::_thesis: verum end; A76: Prob . (((Partial_Intersection (Complement F)) . k) /\ ((Partial_Intersection (F ^\ ((k + 0) + 1))) . (q + 1))) = ((Partial_Product (Prob * (Complement F))) . k) * ((Partial_Product (Prob * (F ^\ ((k + 0) + 1)))) . (q + 1)) by A71, A72, A62, FUNCT_2:12; A77: (Partial_Intersection (Complement C)) . k = (Partial_Intersection (Complement F)) . k proof A78: for x being set for knat being Nat st knat <= k holds ( x in (Complement C) . knat iff x in (Complement F) . knat ) proof let x be set ; ::_thesis: for knat being Nat st knat <= k holds ( x in (Complement C) . knat iff x in (Complement F) . knat ) let knat be Nat; ::_thesis: ( knat <= k implies ( x in (Complement C) . knat iff x in (Complement F) . knat ) ) assume knat <= k ; ::_thesis: ( x in (Complement C) . knat iff x in (Complement F) . knat ) then A79: knat <= k + 1 by NAT_1:13; reconsider knat = knat as Element of NAT by ORDINAL1:def_12; dom (C * (Special_Function4 (k,n2))) = NAT by FUNCT_2:def_1; then A80: (C * (Special_Function4 (k,n2))) . knat = C . ((Special_Function4 (k,n2)) . knat) by FUNCT_1:12; ( (Special_Function4 (k,n2)) . knat = IFGT (knat,(k + 1),(knat + n2),knat) & IFGT (knat,(k + 1),(knat + n2),knat) = knat ) by Def5, A79, XXREAL_0:def_11; then (Complement F) . knat = (C . knat) ` by A69, A80, PROB_1:def_2; hence ( x in (Complement C) . knat iff x in (Complement F) . knat ) by PROB_1:def_2; ::_thesis: verum end; A81: for x being set holds ( ( for knat being Nat st knat <= k holds x in (Complement C) . knat ) iff for knat being Nat st knat <= k holds x in (Complement F) . knat ) proof let x be set ; ::_thesis: ( ( for knat being Nat st knat <= k holds x in (Complement C) . knat ) iff for knat being Nat st knat <= k holds x in (Complement F) . knat ) hereby ::_thesis: ( ( for knat being Nat st knat <= k holds x in (Complement F) . knat ) implies for knat being Nat st knat <= k holds x in (Complement C) . knat ) assume A82: for knat being Nat st knat <= k holds x in (Complement C) . knat ; ::_thesis: for knat being Nat st knat <= k holds x in (Complement F) . knat thus for knat being Nat st knat <= k holds x in (Complement F) . knat ::_thesis: verum proof let knat be Nat; ::_thesis: ( knat <= k implies x in (Complement F) . knat ) assume A83: knat <= k ; ::_thesis: x in (Complement F) . knat then ( x in (Complement C) . knat iff x in (Complement F) . knat ) by A78; hence x in (Complement F) . knat by A83, A82; ::_thesis: verum end; end; assume A84: for knat being Nat st knat <= k holds x in (Complement F) . knat ; ::_thesis: for knat being Nat st knat <= k holds x in (Complement C) . knat thus for knat being Nat st knat <= k holds x in (Complement C) . knat ::_thesis: verum proof let knat be Nat; ::_thesis: ( knat <= k implies x in (Complement C) . knat ) assume A85: knat <= k ; ::_thesis: x in (Complement C) . knat then ( x in (Complement C) . knat iff x in (Complement F) . knat ) by A78; hence x in (Complement C) . knat by A85, A84; ::_thesis: verum end; end; for x being set holds ( x in (Partial_Intersection (Complement C)) . k iff x in (Partial_Intersection (Complement F)) . k ) proof let x be set ; ::_thesis: ( x in (Partial_Intersection (Complement C)) . k iff x in (Partial_Intersection (Complement F)) . k ) ( x in (Partial_Intersection (Complement C)) . k iff for knat being Nat st knat <= k holds x in (Complement C) . knat ) by PROB_3:25; then ( x in (Partial_Intersection (Complement C)) . k iff for knat being Nat st knat <= k holds x in (Complement F) . knat ) by A81; hence ( x in (Partial_Intersection (Complement C)) . k iff x in (Partial_Intersection (Complement F)) . k ) by PROB_3:25; ::_thesis: verum end; hence (Partial_Intersection (Complement C)) . k = (Partial_Intersection (Complement F)) . k by TARSKI:1; ::_thesis: verum end; A86: (Partial_Intersection (F ^\ (k + 1))) . (q + 1) = (C . (k + 1)) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q) proof A87: for x being set for knat being Nat st knat <= q holds ( x in (F ^\ ((k + 1) + 1)) . knat iff x in (C ^\ (((k + 1) + n2) + 1)) . knat ) proof let x be set ; ::_thesis: for knat being Nat st knat <= q holds ( x in (F ^\ ((k + 1) + 1)) . knat iff x in (C ^\ (((k + 1) + n2) + 1)) . knat ) let knat be Nat; ::_thesis: ( knat <= q implies ( x in (F ^\ ((k + 1) + 1)) . knat iff x in (C ^\ (((k + 1) + n2) + 1)) . knat ) ) assume knat <= q ; ::_thesis: ( x in (F ^\ ((k + 1) + 1)) . knat iff x in (C ^\ (((k + 1) + n2) + 1)) . knat ) reconsider knat = knat as Element of NAT by ORDINAL1:def_12; A88: dom (C * (Special_Function4 (k,n2))) = NAT by FUNCT_2:def_1; set j = ((knat + k) + 1) + 1; ((knat + k) + 1) + 1 > k + 1 proof ( k + 1 < (k + 1) + 1 & k + 2 <= (k + 2) + knat ) by NAT_1:12, NAT_1:13; hence ((knat + k) + 1) + 1 > k + 1 by XXREAL_0:2; ::_thesis: verum end; then ( (Special_Function4 (k,n2)) . (((knat + k) + 1) + 1) = IFGT ((((knat + k) + 1) + 1),(k + 1),((((knat + k) + 1) + 1) + n2),(((knat + k) + 1) + 1)) & IFGT ((((knat + k) + 1) + 1),(k + 1),((((knat + k) + 1) + 1) + n2),(((knat + k) + 1) + 1)) = (((knat + k) + 1) + 1) + n2 ) by Def5, XXREAL_0:def_11; then F . (knat + ((k + 1) + 1)) = C . (knat + (((k + 1) + n2) + 1)) by A69, A88, FUNCT_1:12; then (F ^\ ((k + 1) + 1)) . knat = C . (knat + (((k + 1) + n2) + 1)) by NAT_1:def_3; hence ( x in (F ^\ ((k + 1) + 1)) . knat iff x in (C ^\ (((k + 1) + n2) + 1)) . knat ) by NAT_1:def_3; ::_thesis: verum end; A89: for x being set holds ( ( for knat being Nat st knat <= q holds x in (C ^\ (((k + 1) + n2) + 1)) . knat ) iff for knat being Nat st knat <= q holds x in (F ^\ ((k + 1) + 1)) . knat ) proof let x be set ; ::_thesis: ( ( for knat being Nat st knat <= q holds x in (C ^\ (((k + 1) + n2) + 1)) . knat ) iff for knat being Nat st knat <= q holds x in (F ^\ ((k + 1) + 1)) . knat ) hereby ::_thesis: ( ( for knat being Nat st knat <= q holds x in (F ^\ ((k + 1) + 1)) . knat ) implies for knat being Nat st knat <= q holds x in (C ^\ (((k + 1) + n2) + 1)) . knat ) assume A90: for knat being Nat st knat <= q holds x in (C ^\ (((k + 1) + n2) + 1)) . knat ; ::_thesis: for knat being Nat st knat <= q holds x in (F ^\ ((k + 1) + 1)) . knat thus for knat being Nat st knat <= q holds x in (F ^\ ((k + 1) + 1)) . knat ::_thesis: verum proof let knat be Nat; ::_thesis: ( knat <= q implies x in (F ^\ ((k + 1) + 1)) . knat ) assume A91: knat <= q ; ::_thesis: x in (F ^\ ((k + 1) + 1)) . knat then ( x in (C ^\ (((k + 1) + n2) + 1)) . knat iff x in (F ^\ ((k + 1) + 1)) . knat ) by A87; hence x in (F ^\ ((k + 1) + 1)) . knat by A91, A90; ::_thesis: verum end; end; assume A92: for knat being Nat st knat <= q holds x in (F ^\ ((k + 1) + 1)) . knat ; ::_thesis: for knat being Nat st knat <= q holds x in (C ^\ (((k + 1) + n2) + 1)) . knat thus for knat being Nat st knat <= q holds x in (C ^\ (((k + 1) + n2) + 1)) . knat ::_thesis: verum proof let knat be Nat; ::_thesis: ( knat <= q implies x in (C ^\ (((k + 1) + n2) + 1)) . knat ) assume A93: knat <= q ; ::_thesis: x in (C ^\ (((k + 1) + n2) + 1)) . knat then ( x in (C ^\ (((k + 1) + n2) + 1)) . knat iff x in (F ^\ ((k + 1) + 1)) . knat ) by A87; hence x in (C ^\ (((k + 1) + n2) + 1)) . knat by A93, A92; ::_thesis: verum end; end; A94: for x being set holds ( x in (Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q iff x in (Partial_Intersection (F ^\ ((k + 1) + 1))) . q ) proof let x be set ; ::_thesis: ( x in (Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q iff x in (Partial_Intersection (F ^\ ((k + 1) + 1))) . q ) ( x in (Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q iff for knat being Nat st knat <= q holds x in (C ^\ (((k + 1) + n2) + 1)) . knat ) by PROB_3:25; then ( x in (Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q iff for knat being Nat st knat <= q holds x in (F ^\ ((k + 1) + 1)) . knat ) by A89; hence ( x in (Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q iff x in (Partial_Intersection (F ^\ ((k + 1) + 1))) . q ) by PROB_3:25; ::_thesis: verum end; ((Partial_Intersection (F ^\ ((k + 1) + 1))) . q) /\ (C . (k + 1)) = (Partial_Intersection (F ^\ (k + 1))) . (q + 1) proof defpred S2[ Element of NAT ] means ((Partial_Intersection (F ^\ ((k + 1) + 1))) . \$1) /\ (C . (k + 1)) = (Partial_Intersection (F ^\ (k + 1))) . (\$1 + 1); A95: S2[ 0 ] proof ((Partial_Intersection (F ^\ ((k + 1) + 1))) . 0) /\ (C . (k + 1)) = ((F ^\ ((k + 1) + 1)) . 0) /\ (C . (k + 1)) by PROB_3:21; then ((Partial_Intersection (F ^\ ((k + 1) + 1))) . 0) /\ (C . (k + 1)) = (F . (0 + ((k + 1) + 1))) /\ (C . (k + 1)) by NAT_1:def_3; then A96: ((Partial_Intersection (F ^\ ((k + 1) + 1))) . 0) /\ (C . (k + 1)) = ((F ^\ (k + 1)) . (0 + 1)) /\ (C . (k + 1)) by NAT_1:def_3; A97: dom (C * (Special_Function4 (k,n2))) = NAT by FUNCT_2:def_1; ( (Special_Function4 (k,n2)) . (k + 1) = IFGT ((k + 1),(k + 1),((k + 1) + n2),(k + 1)) & IFGT ((k + 1),(k + 1),((k + 1) + n2),(k + 1)) = k + 1 ) by Def5, XXREAL_0:def_11; then ((Partial_Intersection (F ^\ ((k + 1) + 1))) . 0) /\ (C . (k + 1)) = ((F ^\ (k + 1)) . (0 + 1)) /\ (F . (0 + (k + 1))) by A69, A97, A96, FUNCT_1:12; then ((Partial_Intersection (F ^\ ((k + 1) + 1))) . 0) /\ (C . (k + 1)) = ((F ^\ (k + 1)) . (0 + 1)) /\ ((F ^\ (k + 1)) . 0) by NAT_1:def_3; then ((Partial_Intersection (F ^\ ((k + 1) + 1))) . 0) /\ (C . (k + 1)) = ((Partial_Intersection (F ^\ (k + 1))) . 0) /\ ((F ^\ (k + 1)) . (0 + 1)) by PROB_3:21; hence S2[ 0 ] by PROB_3:21; ::_thesis: verum end; A98: for q being Element of NAT st S2[q] holds S2[q + 1] proof let q be Element of NAT ; ::_thesis: ( S2[q] implies S2[q + 1] ) assume A99: S2[q] ; ::_thesis: S2[q + 1] ((Partial_Intersection (F ^\ ((k + 1) + 1))) . (q + 1)) /\ (C . (k + 1)) = (((Partial_Intersection (F ^\ ((k + 1) + 1))) . q) /\ ((F ^\ ((k + 1) + 1)) . (q + 1))) /\ (C . (k + 1)) by PROB_3:21; then A100: ((Partial_Intersection (F ^\ ((k + 1) + 1))) . (q + 1)) /\ (C . (k + 1)) = ((Partial_Intersection (F ^\ (k + 1))) . (q + 1)) /\ ((F ^\ ((k + 1) + 1)) . (q + 1)) by A99, XBOOLE_1:16; (F ^\ ((k + 1) + 1)) . (q + 1) = (F ^\ (k + 1)) . ((q + 1) + 1) proof (F ^\ ((k + 1) + 1)) . (q + 1) = F . ((q + 1) + ((k + 1) + 1)) by NAT_1:def_3; then (F ^\ ((k + 1) + 1)) . (q + 1) = F . (((q + 1) + 1) + (k + 1)) ; hence (F ^\ ((k + 1) + 1)) . (q + 1) = (F ^\ (k + 1)) . ((q + 1) + 1) by NAT_1:def_3; ::_thesis: verum end; hence S2[q + 1] by A100, PROB_3:21; ::_thesis: verum end; for k being Element of NAT holds S2[k] from NAT_1:sch_1(A95, A98); hence ((Partial_Intersection (F ^\ ((k + 1) + 1))) . q) /\ (C . (k + 1)) = (Partial_Intersection (F ^\ (k + 1))) . (q + 1) ; ::_thesis: verum end; hence (Partial_Intersection (F ^\ (k + 1))) . (q + 1) = (C . (k + 1)) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q) by A94, TARSKI:1; ::_thesis: verum end; A101: (Partial_Product (Prob * (F ^\ (k + 1)))) . (q + 1) = ((Prob * C) . (k + 1)) * ((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . q) proof defpred S2[ Element of NAT ] means (Partial_Product (Prob * (F ^\ (k + 1)))) . (\$1 + 1) = ((Prob * C) . (k + 1)) * ((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . \$1); A102: S2[ 0 ] proof A103: (F ^\ (k + 1)) . (0 + 1) = (C * (Special_Function4 (k,n2))) . ((k + 1) + 1) by A69, NAT_1:def_3; A104: dom (C * (Special_Function4 (k,n2))) = NAT by FUNCT_2:def_1; set j = (k + 1) + 1; (k + 1) + 1 > k + 1 by NAT_1:13; then ( (Special_Function4 (k,n2)) . ((k + 1) + 1) = IFGT (((k + 1) + 1),(k + 1),(((k + 1) + 1) + n2),((k + 1) + 1)) & IFGT (((k + 1) + 1),(k + 1),(((k + 1) + 1) + n2),((k + 1) + 1)) = ((k + 1) + 1) + n2 ) by Def5, XXREAL_0:def_11; then (F ^\ (k + 1)) . (0 + 1) = C . (0 + (((k + 1) + n2) + 1)) by A104, A103, FUNCT_1:12; then A105: Prob . ((F ^\ (k + 1)) . (0 + 1)) = Prob . ((C ^\ (((k + 1) + n2) + 1)) . 0) by NAT_1:def_3; ( dom (Prob * (F ^\ (k + 1))) = NAT & dom (Prob * (C ^\ (((k + 1) + n2) + 1))) = NAT ) by FUNCT_2:def_1; then ( (Prob * (F ^\ (k + 1))) . (0 + 1) = Prob . ((C ^\ (((k + 1) + n2) + 1)) . 0) & Prob . ((F ^\ (k + 1)) . (0 + 1)) = (Prob * (C ^\ (((k + 1) + n2) + 1))) . 0 & Prob . ((F ^\ (k + 1)) . (0 + 1)) = Prob . ((C ^\ (((k + 1) + n2) + 1)) . 0) ) by A105, FUNCT_1:12; then A106: ((Partial_Product (Prob * (F ^\ (k + 1)))) . 0) * ((Prob * (F ^\ (k + 1))) . (0 + 1)) = ((Prob * (F ^\ (k + 1))) . 0) * ((Prob * (C ^\ (((k + 1) + n2) + 1))) . 0) by SERIES_3:def_1; (Prob * (F ^\ (k + 1))) . 0 = (Prob * C) . (k + 1) proof A107: (F ^\ (k + 1)) . 0 = F . (0 + (k + 1)) by NAT_1:def_3; A108: dom (C * (Special_Function4 (k,n2))) = NAT by FUNCT_2:def_1; A109: F . (k + 1) = C . ((Special_Function4 (k,n2)) . (k + 1)) by A69, A108, FUNCT_1:12; A110: ( (Special_Function4 (k,n2)) . (k + 1) = IFGT ((k + 1),(k + 1),((k + 1) + n2),(k + 1)) & IFGT ((k + 1),(k + 1),((k + 1) + n2),(k + 1)) = k + 1 ) by Def5, XXREAL_0:def_11; dom (Prob * C) = NAT by FUNCT_2:def_1; then A111: Prob . ((F ^\ (k + 1)) . 0) = (Prob * C) . (k + 1) by A110, A109, A107, FUNCT_1:12; dom (Prob * (F ^\ (k + 1))) = NAT by FUNCT_2:def_1; hence (Prob * (F ^\ (k + 1))) . 0 = (Prob * C) . (k + 1) by A111, FUNCT_1:12; ::_thesis: verum end; then (Partial_Product (Prob * (F ^\ (k + 1)))) . (0 + 1) = ((Prob * C) . (k + 1)) * ((Prob * (C ^\ (((k + 1) + n2) + 1))) . 0) by A106, SERIES_3:def_1; hence S2[ 0 ] by SERIES_3:def_1; ::_thesis: verum end; A112: for q being Element of NAT st S2[q] holds S2[q + 1] proof let q be Element of NAT ; ::_thesis: ( S2[q] implies S2[q + 1] ) assume A113: S2[q] ; ::_thesis: S2[q + 1] A114: (Prob * (F ^\ (k + 1))) . ((q + 1) + 1) = (Prob * (C ^\ (((k + 1) + n2) + 1))) . (q + 1) proof A115: (F ^\ (k + 1)) . ((q + 1) + 1) = (C * (Special_Function4 (k,n2))) . (((q + 1) + 1) + (k + 1)) by A69, NAT_1:def_3; A116: dom (C * (Special_Function4 (k,n2))) = NAT by FUNCT_2:def_1; set j = ((q + 1) + 1) + (k + 1); ((q + 1) + 1) + (k + 1) > k + 1 proof ( k + 1 < (k + 1) + 1 & (k + 1) + 1 <= ((k + 1) + 1) + (q + 1) ) by NAT_1:13, XREAL_1:31; hence ((q + 1) + 1) + (k + 1) > k + 1 by XXREAL_0:2; ::_thesis: verum end; then ( (Special_Function4 (k,n2)) . (((q + 1) + 1) + (k + 1)) = IFGT ((((q + 1) + 1) + (k + 1)),(k + 1),((((q + 1) + 1) + (k + 1)) + n2),(((q + 1) + 1) + (k + 1))) & IFGT ((((q + 1) + 1) + (k + 1)),(k + 1),((((q + 1) + 1) + (k + 1)) + n2),(((q + 1) + 1) + (k + 1))) = (((q + 1) + 1) + (k + 1)) + n2 ) by Def5, XXREAL_0:def_11; then (F ^\ (k + 1)) . ((q + 1) + 1) = C . ((q + 1) + (((k + 1) + n2) + 1)) by A116, A115, FUNCT_1:12; then A117: Prob . ((F ^\ (k + 1)) . ((q + 1) + 1)) = Prob . ((C ^\ (((k + 1) + n2) + 1)) . (q + 1)) by NAT_1:def_3; ( dom (Prob * (F ^\ (k + 1))) = NAT & dom (Prob * (C ^\ (((k + 1) + n2) + 1))) = NAT ) by FUNCT_2:def_1; then ( (Prob * (F ^\ (k + 1))) . ((q + 1) + 1) = Prob . ((C ^\ (((k + 1) + n2) + 1)) . (q + 1)) & Prob . ((F ^\ (k + 1)) . ((q + 1) + 1)) = (Prob * (C ^\ (((k + 1) + n2) + 1))) . (q + 1) & Prob . ((F ^\ (k + 1)) . ((q + 1) + 1)) = Prob . ((C ^\ (((k + 1) + n2) + 1)) . (q + 1)) ) by A117, FUNCT_1:12; hence (Prob * (F ^\ (k + 1))) . ((q + 1) + 1) = (Prob * (C ^\ (((k + 1) + n2) + 1))) . (q + 1) ; ::_thesis: verum end; (Partial_Product (Prob * (F ^\ (k + 1)))) . ((q + 1) + 1) = (((Prob * C) . (k + 1)) * ((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . q)) * ((Prob * (C ^\ (((k + 1) + n2) + 1))) . (q + 1)) by A113, A114, SERIES_3:def_1; then (Partial_Product (Prob * (F ^\ (k + 1)))) . ((q + 1) + 1) = ((Prob * C) . (k + 1)) * (((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . q) * ((Prob * (C ^\ (((k + 1) + n2) + 1))) . (q + 1))) ; hence S2[q + 1] by SERIES_3:def_1; ::_thesis: verum end; for k being Element of NAT holds S2[k] from NAT_1:sch_1(A102, A112); hence (Partial_Product (Prob * (F ^\ (k + 1)))) . (q + 1) = ((Prob * C) . (k + 1)) * ((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . q) ; ::_thesis: verum end; defpred S2[ Element of NAT ] means ( ( for k being Element of NAT st k <= \$1 holds C . k = F . k ) implies (Partial_Product (Prob * (Complement F))) . \$1 = (Partial_Product (Prob * (Complement C))) . \$1 ); dom (C * (Special_Function4 (k,n2))) = NAT by FUNCT_2:def_1; then A118: (C * (Special_Function4 (k,n2))) . 0 = C . ((Special_Function4 (k,n2)) . 0) by FUNCT_1:12; A119: IFGT (0,(k + 1),(0 + n2),0) = 0 by XXREAL_0:def_11; then (F . 0) ` = (C . 0) ` by Def5, A118, A69; then (Complement F) . 0 = (C . 0) ` by PROB_1:def_2; then ( Prob . ((Complement F) . 0) = Prob . ((Complement C) . 0) & dom (Prob * (Complement F)) = NAT & dom (Prob * (Complement C)) = NAT ) by FUNCT_2:def_1, PROB_1:def_2; then ( Prob . ((Complement F) . 0) = Prob . ((Complement C) . 0) & (Prob * (Complement F)) . 0 = Prob . ((Complement F) . 0) & (Prob * (Complement C)) . 0 = Prob . ((Complement C) . 0) ) by FUNCT_1:12; then A120: ( (Partial_Product (Prob * (Complement F))) . 0 = (Prob * (Complement C)) . 0 & F . 0 = C . 0 ) by A119, Def5, A118, A69, SERIES_3:def_1; A121: S2[ 0 ] by A120, SERIES_3:def_1; A122: for q being Element of NAT st S2[q] holds S2[q + 1] proof let q be Element of NAT ; ::_thesis: ( S2[q] implies S2[q + 1] ) assume A123: S2[q] ; ::_thesis: S2[q + 1] A124: ( ( for k being Element of NAT st k <= q + 1 holds C . k = F . k ) implies for k being Element of NAT st k <= q holds C . k = F . k ) proof assume A125: for k being Element of NAT st k <= q + 1 holds C . k = F . k ; ::_thesis: for k being Element of NAT st k <= q holds C . k = F . k let k be Element of NAT ; ::_thesis: ( k <= q implies C . k = F . k ) assume k <= q ; ::_thesis: C . k = F . k then k <= q + 1 by NAT_1:13; hence C . k = F . k by A125; ::_thesis: verum end; ( ( for k being Element of NAT st k <= q + 1 holds C . k = F . k ) implies (Partial_Product (Prob * (Complement F))) . (q + 1) = (Partial_Product (Prob * (Complement C))) . (q + 1) ) proof assume A126: for k being Element of NAT st k <= q + 1 holds C . k = F . k ; ::_thesis: (Partial_Product (Prob * (Complement F))) . (q + 1) = (Partial_Product (Prob * (Complement C))) . (q + 1) then ( q + 1 <= q + 1 implies (C . (q + 1)) ` = (F . (q + 1)) ` ) ; then ( q + 1 <= q + 1 implies (Complement C) . (q + 1) = (F . (q + 1)) ` ) by PROB_1:def_2; then A127: ((Partial_Product (Prob * (Complement F))) . q) * (Prob . ((Complement F) . (q + 1))) = ((Partial_Product (Prob * (Complement C))) . q) * (Prob . ((Complement C) . (q + 1))) by A126, A124, A123, PROB_1:def_2; ( dom (Prob * (Complement C)) = NAT & dom (Prob * (Complement F)) = NAT ) by FUNCT_2:def_1; then ( (Prob * (Complement C)) . (q + 1) = Prob . ((Complement C) . (q + 1)) & (Prob * (Complement F)) . (q + 1) = Prob . ((Complement F) . (q + 1)) ) by FUNCT_1:12; then (Partial_Product (Prob * (Complement F))) . (q + 1) = ((Partial_Product (Prob * (Complement C))) . q) * ((Prob * (Complement C)) . (q + 1)) by A127, SERIES_3:def_1; hence (Partial_Product (Prob * (Complement F))) . (q + 1) = (Partial_Product (Prob * (Complement C))) . (q + 1) by SERIES_3:def_1; ::_thesis: verum end; hence S2[q + 1] ; ::_thesis: verum end; A128: for k being Element of NAT holds S2[k] from NAT_1:sch_1(A121, A122); for q being Element of NAT st q <= k holds C . q = F . q proof let q be Element of NAT ; ::_thesis: ( q <= k implies C . q = F . q ) assume q <= k ; ::_thesis: C . q = F . q then A129: q <= k + 1 by NAT_1:13; A130: dom (C * (Special_Function4 (k,n2))) = NAT by FUNCT_2:def_1; ( (Special_Function4 (k,n2)) . q = IFGT (q,(k + 1),(q + n2),q) & IFGT (q,(k + 1),(q + n2),q) = q ) by Def5, A129, XXREAL_0:def_11; hence C . q = F . q by A130, A69, FUNCT_1:12; ::_thesis: verum end; then Prob . (((Partial_Intersection (Complement C)) . k) /\ ((C . (k + 1)) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q))) = ((Partial_Product (Prob * (Complement C))) . k) * (((Prob * C) . (k + 1)) * ((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . q)) by A128, A101, A86, A77, A76; hence (((Prob * C) . (k + 1)) * ((Partial_Product (Prob * (Complement C))) . k)) * ((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . q) = Prob . ((C . (k + 1)) /\ (((Partial_Intersection (Complement C)) . k) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q))) by XBOOLE_1:16; ::_thesis: verum end; hence Prob . (((Partial_Intersection (Complement C)) . (k + 1)) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) = (((Partial_Product (Prob * (Complement C))) . k) * ((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . q)) - ((((Prob * C) . (k + 1)) * ((Partial_Product (Prob * (Complement C))) . k)) * ((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . q)) by A67, A66, PROB_1:33, XBOOLE_1:17; ::_thesis: verum end; (Prob * C) . (k + 1) = 1 - ((Prob * (Complement C)) . (k + 1)) proof ( C . (k + 1) = ((C . (k + 1)) `) ` & ((C . (k + 1)) `) ` = Omega \ ((C . (k + 1)) `) ) by SUBSET_1:def_4; then ( Prob . (C . (k + 1)) = Prob . (([#] Sigma) \ ((C . (k + 1)) `)) & (C . (k + 1)) ` is Event of Sigma ) by PROB_1:20; then A131: Prob . (C . (k + 1)) = 1 - (Prob . ((C . (k + 1)) `)) by PROB_1:32; dom (Prob * C) = NAT by FUNCT_2:def_1; then A132: (Prob * C) . (k + 1) = 1 - (Prob . ((C . (k + 1)) `)) by A131, FUNCT_1:12; dom (Prob * (Complement C)) = NAT by FUNCT_2:def_1; then (Prob * (Complement C)) . (k + 1) = Prob . ((Complement C) . (k + 1)) by FUNCT_1:12; hence (Prob * C) . (k + 1) = 1 - ((Prob * (Complement C)) . (k + 1)) by A132, PROB_1:def_2; ::_thesis: verum end; then Prob . (((Partial_Intersection (Complement C)) . (k + 1)) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) = (((Prob * (Complement C)) . (k + 1)) * ((Partial_Product (Prob * (Complement C))) . k)) * ((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . q) by A68; hence Prob . (((Partial_Intersection (Complement C)) . (k + 1)) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) = ((Partial_Product (Prob * (Complement C))) . (k + 1)) * ((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . q) by SERIES_3:def_1; ::_thesis: verum end; A133: for k being Element of NAT holds S1[k] from NAT_1:sch_1(A36, A61); ex e being sequence of NAT st ( A * e = A & e is one-to-one & dom e <> {} ) proof set e = Special_Function2 0; A134: dom (Special_Function2 0) <> {} ; ( A is sequence of (bool Omega) & A * (Special_Function2 0) is sequence of (bool Omega) & ( for n being set st n in NAT holds (A * (Special_Function2 0)) . n = A . n ) ) proof A135: for n being set st n in NAT holds ( (A * (Special_Function2 0)) . n = A . n & A . ((Special_Function2 0) . n) = A . n ) proof let n be set ; ::_thesis: ( n in NAT implies ( (A * (Special_Function2 0)) . n = A . n & A . ((Special_Function2 0) . n) = A . n ) ) assume n in NAT ; ::_thesis: ( (A * (Special_Function2 0)) . n = A . n & A . ((Special_Function2 0) . n) = A . n ) then reconsider n = n as Element of NAT ; A136: (Special_Function2 0) . n = n + 0 by Def3; dom (A * (Special_Function2 0)) = NAT by FUNCT_2:def_1; hence ( (A * (Special_Function2 0)) . n = A . n & A . ((Special_Function2 0) . n) = A . n ) by A136, FUNCT_1:12; ::_thesis: verum end; thus ( A is sequence of (bool Omega) & A * (Special_Function2 0) is sequence of (bool Omega) & ( for n being set st n in NAT holds (A * (Special_Function2 0)) . n = A . n ) ) by A135; ::_thesis: verum end; hence ex e being sequence of NAT st ( A * e = A & e is one-to-one & dom e <> {} ) by A134, FUNCT_2:12; ::_thesis: verum end; hence Prob . (((Partial_Intersection (Complement A)) . n1) /\ ((Partial_Intersection (A ^\ ((n1 + n2) + 1))) . q)) = ((Partial_Product (Prob * (Complement A))) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . q) by A133; ::_thesis: verum end; (n - n1) - 1 is Element of NAT proof n1 + 1 <= n by A1, NAT_1:13; then (n1 + 1) - 1 <= n - 1 by XREAL_1:9; then ( n1 <= n - 1 & n - 1 is Element of NAT ) by A1, NAT_1:20; then (n - 1) - n1 is Element of NAT by NAT_1:21; hence (n - n1) - 1 is Element of NAT ; ::_thesis: verum end; hence Prob . (((Partial_Intersection (Complement A)) . n1) /\ ((Partial_Intersection (A ^\ ((n1 + n2) + 1))) . ((n - n1) - 1))) = ((Partial_Product (Prob * (Complement A))) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . ((n - n1) - 1)) by A35; ::_thesis: verum end; theorem Th7: :: BOR_CANT:7 for Omega being non empty set for Sigma being SigmaField of Omega for A being SetSequence of Sigma for n being Element of NAT holds (Partial_Intersection (Complement A)) . n = ((Partial_Union A) . n) ` proof let Omega be non empty set ; ::_thesis: for Sigma being SigmaField of Omega for A being SetSequence of Sigma for n being Element of NAT holds (Partial_Intersection (Complement A)) . n = ((Partial_Union A) . n) ` let Sigma be SigmaField of Omega; ::_thesis: for A being SetSequence of Sigma for n being Element of NAT holds (Partial_Intersection (Complement A)) . n = ((Partial_Union A) . n) ` let A be SetSequence of Sigma; ::_thesis: for n being Element of NAT holds (Partial_Intersection (Complement A)) . n = ((Partial_Union A) . n) ` let n be Element of NAT ; ::_thesis: (Partial_Intersection (Complement A)) . n = ((Partial_Union A) . n) ` for x being set holds ( x in (Partial_Intersection (Complement A)) . n iff x in ((Partial_Union A) . n) ` ) proof let x be set ; ::_thesis: ( x in (Partial_Intersection (Complement A)) . n iff x in ((Partial_Union A) . n) ` ) hereby ::_thesis: ( x in ((Partial_Union A) . n) ` implies x in (Partial_Intersection (Complement A)) . n ) assume A1: x in (Partial_Intersection (Complement A)) . n ; ::_thesis: x in ((Partial_Union A) . n) ` for knat being Nat st knat <= n holds not x in A . knat proof let knat be Nat; ::_thesis: ( knat <= n implies not x in A . knat ) assume knat <= n ; ::_thesis: not x in A . knat then A2: x in (Complement A) . knat by A1, PROB_3:25; reconsider knat = knat as Element of NAT by ORDINAL1:def_12; (Complement A) . knat = (A . knat) ` by PROB_1:def_2; then (Complement A) . knat = Omega \ (A . knat) by SUBSET_1:def_4; hence not x in A . knat by A2, XBOOLE_0:def_5; ::_thesis: verum end; then A3: not x in (Partial_Union A) . n by PROB_3:26; x in Omega \ ((Partial_Union A) . n) by A1, A3, XBOOLE_0:def_5; hence x in ((Partial_Union A) . n) ` by SUBSET_1:def_4; ::_thesis: verum end; assume A4: x in ((Partial_Union A) . n) ` ; ::_thesis: x in (Partial_Intersection (Complement A)) . n x in Omega \ ((Partial_Union A) . n) by A4, SUBSET_1:def_4; then A5: ( x in Omega & not x in (Partial_Union A) . n ) by XBOOLE_0:def_5; for knat being Nat st knat <= n holds x in (Complement A) . knat proof let knat be Nat; ::_thesis: ( knat <= n implies x in (Complement A) . knat ) assume knat <= n ; ::_thesis: x in (Complement A) . knat then ( x in Omega & not x in A . knat ) by A5, PROB_3:26; then A6: x in Omega \ (A . knat) by XBOOLE_0:def_5; reconsider knat = knat as Element of NAT by ORDINAL1:def_12; x in (A . knat) ` by A6, SUBSET_1:def_4; hence x in (Complement A) . knat by PROB_1:def_2; ::_thesis: verum end; hence x in (Partial_Intersection (Complement A)) . n by PROB_3:25; ::_thesis: verum end; hence (Partial_Intersection (Complement A)) . n = ((Partial_Union A) . n) ` by TARSKI:1; ::_thesis: verum end; theorem Th8: :: BOR_CANT:8 for Omega being non empty set for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma for n being Element of NAT holds Prob . ((Partial_Intersection (Complement A)) . n) = 1 - (Prob . ((Partial_Union A) . n)) proof let Omega be non empty set ; ::_thesis: for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma for n being Element of NAT holds Prob . ((Partial_Intersection (Complement A)) . n) = 1 - (Prob . ((Partial_Union A) . n)) let Sigma be SigmaField of Omega; ::_thesis: for Prob being Probability of Sigma for A being SetSequence of Sigma for n being Element of NAT holds Prob . ((Partial_Intersection (Complement A)) . n) = 1 - (Prob . ((Partial_Union A) . n)) let Prob be Probability of Sigma; ::_thesis: for A being SetSequence of Sigma for n being Element of NAT holds Prob . ((Partial_Intersection (Complement A)) . n) = 1 - (Prob . ((Partial_Union A) . n)) let A be SetSequence of Sigma; ::_thesis: for n being Element of NAT holds Prob . ((Partial_Intersection (Complement A)) . n) = 1 - (Prob . ((Partial_Union A) . n)) let n be Element of NAT ; ::_thesis: Prob . ((Partial_Intersection (Complement A)) . n) = 1 - (Prob . ((Partial_Union A) . n)) A1: Prob . ((Partial_Intersection (Complement A)) . n) = Prob . (((Partial_Union A) . n) `) by Th7; Prob . (((Partial_Union A) . n) `) = Prob . (([#] Sigma) \ ((Partial_Union A) . n)) by SUBSET_1:def_4; hence Prob . ((Partial_Intersection (Complement A)) . n) = 1 - (Prob . ((Partial_Union A) . n)) by A1, PROB_1:32; ::_thesis: verum end; definition let X be set ; let A be SetSequence of X; func Union_Shift_Seq A -> SetSequence of X means :Def7: :: BOR_CANT:def 7 for n being Element of NAT holds it . n = Union (A ^\ n); existence ex b1 being SetSequence of X st for n being Element of NAT holds b1 . n = Union (A ^\ n) proof for X being set for A being SetSequence of X ex S being SetSequence of X st for n being Element of NAT holds S . n = Union (A ^\ n) proof let X be set ; ::_thesis: for A being SetSequence of X ex S being SetSequence of X st for n being Element of NAT holds S . n = Union (A ^\ n) let A be SetSequence of X; ::_thesis: ex S being SetSequence of X st for n being Element of NAT holds S . n = Union (A ^\ n) ex J being SetSequence of X st ( J . 0 = Union (A ^\ 0) & ( for n being Element of NAT holds J . (n + 1) = Union (A ^\ (n + 1)) ) ) proof defpred S1[ set , set , set ] means for x, y being Subset of X for k being Element of NAT st k = \$1 & x = \$2 & y = \$3 holds y = Union (A ^\ (k + 1)); A1: for n being Element of NAT for x being Subset of X ex y being Subset of X st S1[n,x,y] proof let n be Element of NAT ; ::_thesis: for x being Subset of X ex y being Subset of X st S1[n,x,y] let x be Subset of X; ::_thesis: ex y being Subset of X st S1[n,x,y] take y = Union (A ^\ (n + 1)); ::_thesis: S1[n,x,y] thus S1[n,x,y] ; ::_thesis: verum end; consider J being SetSequence of X such that A2: J . 0 = Union (A ^\ 0) and A3: for n being Element of NAT holds S1[n,J . n,J . (n + 1)] from RECDEF_1:sch_2(A1); take J ; ::_thesis: ( J . 0 = Union (A ^\ 0) & ( for n being Element of NAT holds J . (n + 1) = Union (A ^\ (n + 1)) ) ) thus J . 0 = Union (A ^\ 0) by A2; ::_thesis: for n being Element of NAT holds J . (n + 1) = Union (A ^\ (n + 1)) let n be Element of NAT ; ::_thesis: J . (n + 1) = Union (A ^\ (n + 1)) S1[n,J . n,J . (n + 1)] by A3; hence J . (n + 1) = Union (A ^\ (n + 1)) ; ::_thesis: verum end; then consider J being SetSequence of X such that A4: ( J . 0 = Union (A ^\ 0) & ( for n being Element of NAT holds J . (n + 1) = Union (A ^\ (n + 1)) ) ) ; A5: ( J . 0 = Union (A ^\ 0) & ( for n being Element of NAT holds J . (n + 1) = Union (A ^\ (n + 1)) ) implies for n being Element of NAT holds J . n = Union (A ^\ n) ) proof assume A6: ( J . 0 = Union (A ^\ 0) & ( for n being Element of NAT holds J . (n + 1) = Union (A ^\ (n + 1)) ) ) ; ::_thesis: for n being Element of NAT holds J . n = Union (A ^\ n) let n be Nat; ::_thesis: ( n is Element of REAL & n is Element of NAT implies J . n = Union (A ^\ n) ) percases ( n = 0 or ex q being Nat st n = q + 1 ) by NAT_1:6; suppose n = 0 ; ::_thesis: ( n is Element of REAL & n is Element of NAT implies J . n = Union (A ^\ n) ) hence ( n is Element of REAL & n is Element of NAT implies J . n = Union (A ^\ n) ) by A6; ::_thesis: verum end; suppose ex q being Nat st n = q + 1 ; ::_thesis: ( n is Element of REAL & n is Element of NAT implies J . n = Union (A ^\ n) ) then consider q being Nat such that A7: n = q + 1 ; reconsider q = q as Element of NAT by ORDINAL1:def_12; J . (q + 1) = Union (A ^\ (q + 1)) by A6; hence ( n is Element of REAL & n is Element of NAT implies J . n = Union (A ^\ n) ) by A7; ::_thesis: verum end; end; end; take J ; ::_thesis: for n being Element of NAT holds J . n = Union (A ^\ n) thus for n being Element of NAT holds J . n = Union (A ^\ n) by A4, A5; ::_thesis: verum end; hence ex b1 being SetSequence of X st for n being Element of NAT holds b1 . n = Union (A ^\ n) ; ::_thesis: verum end; uniqueness for b1, b2 being SetSequence of X st ( for n being Element of NAT holds b1 . n = Union (A ^\ n) ) & ( for n being Element of NAT holds b2 . n = Union (A ^\ n) ) holds b1 = b2 proof let J1, J2 be SetSequence of X; ::_thesis: ( ( for n being Element of NAT holds J1 . n = Union (A ^\ n) ) & ( for n being Element of NAT holds J2 . n = Union (A ^\ n) ) implies J1 = J2 ) assume that A8: for n being Element of NAT holds J1 . n = Union (A ^\ n) and A9: for n being Element of NAT holds J2 . n = Union (A ^\ n) ; ::_thesis: J1 = J2 for n being Element of NAT holds J1 . n = J2 . n proof let n be Element of NAT ; ::_thesis: J1 . n = J2 . n J1 . n = Union (A ^\ n) by A8; hence J1 . n = J2 . n by A9; ::_thesis: verum end; then for n being set st n in NAT holds J1 . n = J2 . n ; hence J1 = J2 by FUNCT_2:12; ::_thesis: verum end; end; :: deftheorem Def7 defines Union_Shift_Seq BOR_CANT:def_7_:_ for X being set for A, b3 being SetSequence of X holds ( b3 = Union_Shift_Seq A iff for n being Element of NAT holds b3 . n = Union (A ^\ n) ); registration let Omega be non empty set ; let Sigma be SigmaField of Omega; let A be SetSequence of Sigma; cluster Union_Shift_Seq A -> Sigma -valued ; coherence Union_Shift_Seq A is Sigma -valued proof defpred S1[ set ] means (Union_Shift_Seq A) . Omega is Event of Sigma; (Union_Shift_Seq A) . 0 = Union (A ^\ 0) by Def7; then A1: S1[ 0 ] by PROB_1:17; A2: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume (Union_Shift_Seq A) . k is Event of Sigma ; ::_thesis: S1[k + 1] Union (A ^\ (k + 1)) in Sigma by PROB_1:17; hence S1[k + 1] by Def7; ::_thesis: verum end; for k being Element of NAT holds S1[k] from NAT_1:sch_1(A1, A2); hence Union_Shift_Seq A is Sigma -valued by PROB_1:25; ::_thesis: verum end; end; definition let Omega be non empty set ; let Sigma be SigmaField of Omega; let A be SetSequence of Sigma; func @lim_sup A -> Event of Sigma equals :: BOR_CANT:def 8 @Intersection (Union_Shift_Seq A); correctness coherence @Intersection (Union_Shift_Seq A) is Event of Sigma; ; end; :: deftheorem defines @lim_sup BOR_CANT:def_8_:_ for Omega being non empty set for Sigma being SigmaField of Omega for A being SetSequence of Sigma holds @lim_sup A = @Intersection (Union_Shift_Seq A); definition let X be set ; let A be SetSequence of X; func Intersect_Shift_Seq A -> SetSequence of X means :Def9: :: BOR_CANT:def 9 for n being Element of NAT holds it . n = Intersection (A ^\ n); existence ex b1 being SetSequence of X st for n being Element of NAT holds b1 . n = Intersection (A ^\ n) proof for X being set for A being SetSequence of X ex S being SetSequence of X st for n being Element of NAT holds S . n = Intersection (A ^\ n) proof let X be set ; ::_thesis: for A being SetSequence of X ex S being SetSequence of X st for n being Element of NAT holds S . n = Intersection (A ^\ n) let A be SetSequence of X; ::_thesis: ex S being SetSequence of X st for n being Element of NAT holds S . n = Intersection (A ^\ n) A1: ex J being SetSequence of X st ( J . 0 = Intersection (A ^\ 0) & ( for n being Element of NAT holds J . (n + 1) = Intersection (A ^\ (n + 1)) ) ) proof defpred S1[ set , set , set ] means for x, y being Subset of X for k being Element of NAT st k = \$1 & x = \$2 & y = \$3 holds y = Intersection (A ^\ (k + 1)); A2: for n being Element of NAT for x being Subset of X ex y being Subset of X st S1[n,x,y] proof let n be Element of NAT ; ::_thesis: for x being Subset of X ex y being Subset of X st S1[n,x,y] let x be Subset of X; ::_thesis: ex y being Subset of X st S1[n,x,y] take y = Intersection (A ^\ (n + 1)); ::_thesis: S1[n,x,y] thus S1[n,x,y] ; ::_thesis: verum end; consider J being SetSequence of X such that A3: J . 0 = Intersection (A ^\ 0) and A4: for n being Element of NAT holds S1[n,J . n,J . (n + 1)] from RECDEF_1:sch_2(A2); take J ; ::_thesis: ( J . 0 = Intersection (A ^\ 0) & ( for n being Element of NAT holds J . (n + 1) = Intersection (A ^\ (n + 1)) ) ) thus J . 0 = Intersection (A ^\ 0) by A3; ::_thesis: for n being Element of NAT holds J . (n + 1) = Intersection (A ^\ (n + 1)) let n be Element of NAT ; ::_thesis: J . (n + 1) = Intersection (A ^\ (n + 1)) S1[n,J . n,J . (n + 1)] by A4; hence J . (n + 1) = Intersection (A ^\ (n + 1)) ; ::_thesis: verum end; consider J being SetSequence of X such that A5: ( J . 0 = Intersection (A ^\ 0) & ( for n being Element of NAT holds J . (n + 1) = Intersection (A ^\ (n + 1)) ) ) by A1; A6: ( J . 0 = Intersection (A ^\ 0) & ( for n being Element of NAT holds J . (n + 1) = Intersection (A ^\ (n + 1)) ) implies for n being Element of NAT holds J . n = Intersection (A ^\ n) ) proof assume A7: ( J . 0 = Intersection (A ^\ 0) & ( for n being Element of NAT holds J . (n + 1) = Intersection (A ^\ (n + 1)) ) ) ; ::_thesis: for n being Element of NAT holds J . n = Intersection (A ^\ n) let n be Nat; ::_thesis: ( n is Element of REAL & n is Element of NAT implies J . n = Intersection (A ^\ n) ) percases ( n = 0 or ex q being Nat st n = q + 1 ) by NAT_1:6; suppose n = 0 ; ::_thesis: ( n is Element of REAL & n is Element of NAT implies J . n = Intersection (A ^\ n) ) hence ( n is Element of REAL & n is Element of NAT implies J . n = Intersection (A ^\ n) ) by A7; ::_thesis: verum end; suppose ex q being Nat st n = q + 1 ; ::_thesis: ( n is Element of REAL & n is Element of NAT implies J . n = Intersection (A ^\ n) ) then consider q being Nat such that A8: n = q + 1 ; reconsider q = q as Element of NAT by ORDINAL1:def_12; J . (q + 1) = Intersection (A ^\ (q + 1)) by A7; hence ( n is Element of REAL & n is Element of NAT implies J . n = Intersection (A ^\ n) ) by A8; ::_thesis: verum end; end; end; take J ; ::_thesis: for n being Element of NAT holds J . n = Intersection (A ^\ n) thus for n being Element of NAT holds J . n = Intersection (A ^\ n) by A5, A6; ::_thesis: verum end; hence ex b1 being SetSequence of X st for n being Element of NAT holds b1 . n = Intersection (A ^\ n) ; ::_thesis: verum end; uniqueness for b1, b2 being SetSequence of X st ( for n being Element of NAT holds b1 . n = Intersection (A ^\ n) ) & ( for n being Element of NAT holds b2 . n = Intersection (A ^\ n) ) holds b1 = b2 proof let J1, J2 be SetSequence of X; ::_thesis: ( ( for n being Element of NAT holds J1 . n = Intersection (A ^\ n) ) & ( for n being Element of NAT holds J2 . n = Intersection (A ^\ n) ) implies J1 = J2 ) assume that A9: for n being Element of NAT holds J1 . n = Intersection (A ^\ n) and A10: for n being Element of NAT holds J2 . n = Intersection (A ^\ n) ; ::_thesis: J1 = J2 for n being Element of NAT holds J1 . n = J2 . n proof let n be Element of NAT ; ::_thesis: J1 . n = J2 . n J1 . n = Intersection (A ^\ n) by A9; hence J1 . n = J2 . n by A10; ::_thesis: verum end; then for n being set st n in NAT holds J1 . n = J2 . n ; hence J1 = J2 by FUNCT_2:12; ::_thesis: verum end; end; :: deftheorem Def9 defines Intersect_Shift_Seq BOR_CANT:def_9_:_ for X being set for A, b3 being SetSequence of X holds ( b3 = Intersect_Shift_Seq A iff for n being Element of NAT holds b3 . n = Intersection (A ^\ n) ); registration let Omega be non empty set ; let Sigma be SigmaField of Omega; let A be SetSequence of Sigma; cluster Intersect_Shift_Seq A -> Sigma -valued ; coherence Intersect_Shift_Seq A is Sigma -valued proof defpred S1[ set ] means (Intersect_Shift_Seq A) . Omega is Event of Sigma; A1: Union (Complement (A ^\ 0)) is Event of Sigma by PROB_1:26; (Intersect_Shift_Seq A) . 0 = Intersection (A ^\ 0) by Def9; then A2: S1[ 0 ] by A1, PROB_1:20; A3: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume (Intersect_Shift_Seq A) . k is Event of Sigma ; ::_thesis: S1[k + 1] A4: Union (Complement (A ^\ (k + 1))) is Event of Sigma by PROB_1:26; (Intersect_Shift_Seq A) . (k + 1) = Intersection (A ^\ (k + 1)) by Def9; hence S1[k + 1] by A4, PROB_1:20; ::_thesis: verum end; for k being Element of NAT holds S1[k] from NAT_1:sch_1(A2, A3); hence Intersect_Shift_Seq A is Sigma -valued by PROB_1:25; ::_thesis: verum end; end; definition let Omega be non empty set ; let Sigma be SigmaField of Omega; let A be SetSequence of Sigma; func @lim_inf A -> Event of Sigma equals :: BOR_CANT:def 10 Union (Intersect_Shift_Seq A); correctness coherence Union (Intersect_Shift_Seq A) is Event of Sigma; by PROB_1:26; end; :: deftheorem defines @lim_inf BOR_CANT:def_10_:_ for Omega being non empty set for Sigma being SigmaField of Omega for A being SetSequence of Sigma holds @lim_inf A = Union (Intersect_Shift_Seq A); theorem Th9: :: BOR_CANT:9 for Omega being non empty set for Sigma being SigmaField of Omega for A being SetSequence of Sigma for n being Element of NAT holds (Intersect_Shift_Seq (Complement A)) . n = ((Union_Shift_Seq A) . n) ` proof let Omega be non empty set ; ::_thesis: for Sigma being SigmaField of Omega for A being SetSequence of Sigma for n being Element of NAT holds (Intersect_Shift_Seq (Complement A)) . n = ((Union_Shift_Seq A) . n) ` let Sigma be SigmaField of Omega; ::_thesis: for A being SetSequence of Sigma for n being Element of NAT holds (Intersect_Shift_Seq (Complement A)) . n = ((Union_Shift_Seq A) . n) ` let A be SetSequence of Sigma; ::_thesis: for n being Element of NAT holds (Intersect_Shift_Seq (Complement A)) . n = ((Union_Shift_Seq A) . n) ` let n be Element of NAT ; ::_thesis: (Intersect_Shift_Seq (Complement A)) . n = ((Union_Shift_Seq A) . n) ` for x being set holds ( x in (Intersect_Shift_Seq (Complement A)) . n iff x in ((Union_Shift_Seq A) . n) ` ) proof let x be set ; ::_thesis: ( x in (Intersect_Shift_Seq (Complement A)) . n iff x in ((Union_Shift_Seq A) . n) ` ) hereby ::_thesis: ( x in ((Union_Shift_Seq A) . n) ` implies x in (Intersect_Shift_Seq (Complement A)) . n ) assume A1: x in (Intersect_Shift_Seq (Complement A)) . n ; ::_thesis: x in ((Union_Shift_Seq A) . n) ` then A2: x in Intersection ((Complement A) ^\ n) by Def9; A3: for k being Element of NAT holds not x in (A ^\ n) . k proof let k be Element of NAT ; ::_thesis: not x in (A ^\ n) . k x in ((Complement A) ^\ n) . k by A2, PROB_1:13; then x in (Complement A) . (n + k) by NAT_1:def_3; then A4: x in (A . (n + k)) ` by PROB_1:def_2; x in Omega \ (A . (n + k)) by A4, SUBSET_1:def_4; then ( x in Omega & not x in A . (n + k) ) by XBOOLE_0:def_5; hence not x in (A ^\ n) . k by NAT_1:def_3; ::_thesis: verum end; A5: not x in Union (A ^\ n) by A3, PROB_1:12; A6: not x in (Union_Shift_Seq A) . n by Def7, A5; A7: x in Omega \ ((Union_Shift_Seq A) . n) by A1, A6, XBOOLE_0:def_5; thus x in ((Union_Shift_Seq A) . n) ` by A7, SUBSET_1:def_4; ::_thesis: verum end; assume A8: x in ((Union_Shift_Seq A) . n) ` ; ::_thesis: x in (Intersect_Shift_Seq (Complement A)) . n A9: ( x in ((Union_Shift_Seq A) . n) ` iff x in Omega \ ((Union_Shift_Seq A) . n) ) by SUBSET_1:def_4; A10: ( x in (Union_Shift_Seq A) . n iff x in Union (A ^\ n) ) by Def7; A11: for k being Element of NAT holds x in ((Complement A) ^\ n) . k proof let k be Element of NAT ; ::_thesis: x in ((Complement A) ^\ n) . k A12: not x in (A ^\ n) . k by A10, A8, A9, PROB_1:12, XBOOLE_0:def_5; A13: not x in A . (n + k) by A12, NAT_1:def_3; A14: x in Omega \ (A . (n + k)) by A8, A13, XBOOLE_0:def_5; ( x in (A . (n + k)) ` iff x in (Complement A) . (n + k) ) by PROB_1:def_2; hence x in ((Complement A) ^\ n) . k by A14, NAT_1:def_3, SUBSET_1:def_4; ::_thesis: verum end; x in Intersection ((Complement A) ^\ n) by A11, PROB_1:13; hence x in (Intersect_Shift_Seq (Complement A)) . n by Def9; ::_thesis: verum end; hence (Intersect_Shift_Seq (Complement A)) . n = ((Union_Shift_Seq A) . n) ` by TARSKI:1; ::_thesis: verum end; theorem Th10: :: BOR_CANT:10 for Omega being non empty set for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma for n being Element of NAT st A is_all_independent_wrt Prob holds Prob . ((Partial_Intersection (Complement A)) . n) = (Partial_Product (Prob * (Complement A))) . n proof let Omega be non empty set ; ::_thesis: for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma for n being Element of NAT st A is_all_independent_wrt Prob holds Prob . ((Partial_Intersection (Complement A)) . n) = (Partial_Product (Prob * (Complement A))) . n let Sigma be SigmaField of Omega; ::_thesis: for Prob being Probability of Sigma for A being SetSequence of Sigma for n being Element of NAT st A is_all_independent_wrt Prob holds Prob . ((Partial_Intersection (Complement A)) . n) = (Partial_Product (Prob * (Complement A))) . n let Prob be Probability of Sigma; ::_thesis: for A being SetSequence of Sigma for n being Element of NAT st A is_all_independent_wrt Prob holds Prob . ((Partial_Intersection (Complement A)) . n) = (Partial_Product (Prob * (Complement A))) . n let A be SetSequence of Sigma; ::_thesis: for n being Element of NAT st A is_all_independent_wrt Prob holds Prob . ((Partial_Intersection (Complement A)) . n) = (Partial_Product (Prob * (Complement A))) . n let n be Element of NAT ; ::_thesis: ( A is_all_independent_wrt Prob implies Prob . ((Partial_Intersection (Complement A)) . n) = (Partial_Product (Prob * (Complement A))) . n ) assume A1: A is_all_independent_wrt Prob ; ::_thesis: Prob . ((Partial_Intersection (Complement A)) . n) = (Partial_Product (Prob * (Complement A))) . n defpred S1[ Element of NAT ] means Prob . ((Partial_Intersection (Complement A)) . \$1) = (Partial_Product (Prob * (Complement A))) . \$1; dom (Prob * (Complement A)) = NAT by FUNCT_2:def_1; then A2: (Prob * (Complement A)) . 0 = Prob . ((Complement A) . 0) by FUNCT_1:12; A3: (Partial_Product (Prob * (Complement A))) . 0 = (Prob * (Complement A)) . 0 by SERIES_3:def_1; A4: S1[ 0 ] by A2, A3, PROB_3:21; A5: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A6: S1[k] ; ::_thesis: S1[k + 1] (((Partial_Intersection (Complement A)) . k) /\ ((Partial_Intersection (Complement A)) . k)) /\ ((Complement A) . (k + 1)) = ((Partial_Intersection (Complement A)) . k) /\ ((A . (k + 1)) `) by PROB_1:def_2; then (((Partial_Intersection (Complement A)) . k) /\ ((Partial_Intersection (Complement A)) . k)) /\ ((Complement A) . (k + 1)) = ((Partial_Intersection (Complement A)) . k) /\ (Omega \ (A . (k + 1))) by SUBSET_1:def_4; then A7: (((Partial_Intersection (Complement A)) . k) /\ ((Partial_Intersection (Complement A)) . k)) /\ ((Complement A) . (k + 1)) = (((Partial_Intersection (Complement A)) . k) /\ Omega) \ (((Partial_Intersection (Complement A)) . k) /\ (A . (k + 1))) by XBOOLE_1:50; A8: ((Partial_Intersection (Complement A)) . k) /\ Omega = (Partial_Intersection (Complement A)) . k by XBOOLE_1:28; A9: Prob . (((Partial_Intersection (Complement A)) . k) \ (((Partial_Intersection (Complement A)) . k) /\ (A . (k + 1)))) = (Prob . ((Partial_Intersection (Complement A)) . k)) - (Prob . (((Partial_Intersection (Complement A)) . k) /\ (A . (k + 1)))) by PROB_1:33, XBOOLE_1:17; A10: Prob . ((Partial_Intersection (Complement A)) . (k + 1)) = (Prob . ((Partial_Intersection (Complement A)) . k)) - (Prob . (((Partial_Intersection (Complement A)) . k) /\ (A . (k + 1)))) by A7, A8, A9, PROB_3:21; for A being SetSequence of Sigma for k being Element of NAT st A is_all_independent_wrt Prob holds Prob . (((Partial_Intersection (Complement A)) . k) /\ (A . (k + 1))) = ((Partial_Product (Prob * (Complement A))) . k) * ((Prob * A) . (k + 1)) proof let A be SetSequence of Sigma; ::_thesis: for k being Element of NAT st A is_all_independent_wrt Prob holds Prob . (((Partial_Intersection (Complement A)) . k) /\ (A . (k + 1))) = ((Partial_Product (Prob * (Complement A))) . k) * ((Prob * A) . (k + 1)) let k be Element of NAT ; ::_thesis: ( A is_all_independent_wrt Prob implies Prob . (((Partial_Intersection (Complement A)) . k) /\ (A . (k + 1))) = ((Partial_Product (Prob * (Complement A))) . k) * ((Prob * A) . (k + 1)) ) assume A11: A is_all_independent_wrt Prob ; ::_thesis: Prob . (((Partial_Intersection (Complement A)) . k) /\ (A . (k + 1))) = ((Partial_Product (Prob * (Complement A))) . k) * ((Prob * A) . (k + 1)) consider n being Element of NAT such that A12: n = k + 1 ; consider n1 being Element of NAT such that A13: n1 = k ; n1 < k + 1 by A13, NAT_1:13; then Prob . (((Partial_Intersection (Complement A)) . k) /\ ((Partial_Intersection (A ^\ ((k + 0) + 1))) . ((n - k) - 1))) = ((Partial_Product (Prob * (Complement A))) . k) * ((Partial_Product (Prob * (A ^\ ((k + 0) + 1)))) . ((n - k) - 1)) by A12, A11, Th6, A13; then A14: Prob . (((Partial_Intersection (Complement A)) . k) /\ ((A ^\ (k + 1)) . 0)) = ((Partial_Product (Prob * (Complement A))) . k) * ((Partial_Product (Prob * (A ^\ (k + 1)))) . 0) by A12, PROB_3:21; A15: (A ^\ (k + 1)) . 0 = A . (0 + (k + 1)) by NAT_1:def_3; then A16: Prob . (((Partial_Intersection (Complement A)) . k) /\ (A . (k + 1))) = ((Partial_Product (Prob * (Complement A))) . k) * ((Prob * (A ^\ (k + 1))) . 0) by A14, SERIES_3:def_1; dom (Prob * (A ^\ (k + 1))) = NAT by FUNCT_2:def_1; then A17: Prob . (((Partial_Intersection (Complement A)) . k) /\ (A . (k + 1))) = ((Partial_Product (Prob * (Complement A))) . k) * (Prob . (A . (k + 1))) by A15, A16, FUNCT_1:12; dom (Prob * A) = NAT by FUNCT_2:def_1; hence Prob . (((Partial_Intersection (Complement A)) . k) /\ (A . (k + 1))) = ((Partial_Product (Prob * (Complement A))) . k) * ((Prob * A) . (k + 1)) by A17, FUNCT_1:12; ::_thesis: verum end; then A18: Prob . ((Partial_Intersection (Complement A)) . (k + 1)) = ((Partial_Product (Prob * (Complement A))) . k) - (((Partial_Product (Prob * (Complement A))) . k) * ((Prob * A) . (k + 1))) by A6, A10, A1; ( A . (k + 1) = ((A . (k + 1)) `) ` & ((A . (k + 1)) `) ` = Omega \ ((A . (k + 1)) `) ) by SUBSET_1:def_4; then ( Prob . (A . (k + 1)) = Prob . (([#] Sigma) \ ((A . (k + 1)) `)) & (A . (k + 1)) ` is Event of Sigma ) by PROB_1:20; then A19: Prob . (A . (k + 1)) = 1 - (Prob . ((A . (k + 1)) `)) by PROB_1:32; dom (Prob * A) = NAT by FUNCT_2:def_1; then A20: (Prob * A) . (k + 1) = 1 - (Prob . ((A . (k + 1)) `)) by A19, FUNCT_1:12; dom (Prob * (Complement A)) = NAT by FUNCT_2:def_1; then A21: (Prob * (Complement A)) . (k + 1) = Prob . ((Complement A) . (k + 1)) by FUNCT_1:12; (Prob * A) . (k + 1) = 1 - ((Prob * (Complement A)) . (k + 1)) by A21, A20, PROB_1:def_2; then Prob . ((Partial_Intersection (Complement A)) . (k + 1)) = (((Partial_Product (Prob * (Complement A))) . k) - ((Partial_Product (Prob * (Complement A))) . k)) + (((Partial_Product (Prob * (Complement A))) . k) * ((Prob * (Complement A)) . (k + 1))) by A18; hence S1[k + 1] by SERIES_3:def_1; ::_thesis: verum end; for k being Element of NAT holds S1[k] from NAT_1:sch_1(A4, A5); hence Prob . ((Partial_Intersection (Complement A)) . n) = (Partial_Product (Prob * (Complement A))) . n ; ::_thesis: verum end; theorem Th11: :: BOR_CANT:11 for X being set for A being SetSequence of X holds ( superior_setsequence A = Union_Shift_Seq A & inferior_setsequence A = Intersect_Shift_Seq A ) proof let X be set ; ::_thesis: for A being SetSequence of X holds ( superior_setsequence A = Union_Shift_Seq A & inferior_setsequence A = Intersect_Shift_Seq A ) let A be SetSequence of X; ::_thesis: ( superior_setsequence A = Union_Shift_Seq A & inferior_setsequence A = Intersect_Shift_Seq A ) thus superior_setsequence A = Union_Shift_Seq A ::_thesis: inferior_setsequence A = Intersect_Shift_Seq A proof let n be Element of NAT ; :: according to FUNCT_2:def_8 ::_thesis: (superior_setsequence A) . n = (Union_Shift_Seq A) . n for x being set holds ( x in (superior_setsequence A) . n iff x in (Union_Shift_Seq A) . n ) proof let x be set ; ::_thesis: ( x in (superior_setsequence A) . n iff x in (Union_Shift_Seq A) . n ) hereby ::_thesis: ( x in (Union_Shift_Seq A) . n implies x in (superior_setsequence A) . n ) assume x in (superior_setsequence A) . n ; ::_thesis: x in (Union_Shift_Seq A) . n then consider k being Element of NAT such that A1: x in A . (n + k) by SETLIM_1:20; x in (A ^\ n) . k by A1, NAT_1:def_3; then x in Union (A ^\ n) by PROB_1:12; hence x in (Union_Shift_Seq A) . n by Def7; ::_thesis: verum end; assume x in (Union_Shift_Seq A) . n ; ::_thesis: x in (superior_setsequence A) . n then x in Union (A ^\ n) by Def7; then consider k being Element of NAT such that A2: x in (A ^\ n) . k by PROB_1:12; x in A . (n + k) by A2, NAT_1:def_3; hence x in (superior_setsequence A) . n by SETLIM_1:20; ::_thesis: verum end; hence (superior_setsequence A) . n = (Union_Shift_Seq A) . n by TARSKI:1; ::_thesis: verum end; let n be Element of NAT ; :: according to FUNCT_2:def_8 ::_thesis: (inferior_setsequence A) . n = (Intersect_Shift_Seq A) . n for x being set holds ( x in (inferior_setsequence A) . n iff x in (Intersect_Shift_Seq A) . n ) proof let x be set ; ::_thesis: ( x in (inferior_setsequence A) . n iff x in (Intersect_Shift_Seq A) . n ) hereby ::_thesis: ( x in (Intersect_Shift_Seq A) . n implies x in (inferior_setsequence A) . n ) assume A3: x in (inferior_setsequence A) . n ; ::_thesis: x in (Intersect_Shift_Seq A) . n A4: for k being Element of NAT holds x in (A ^\ n) . k proof let k be Element of NAT ; ::_thesis: x in (A ^\ n) . k x in A . (k + n) by A3, SETLIM_1:19; hence x in (A ^\ n) . k by NAT_1:def_3; ::_thesis: verum end; x in Intersection (A ^\ n) by A4, PROB_1:13; hence x in (Intersect_Shift_Seq A) . n by Def9; ::_thesis: verum end; assume x in (Intersect_Shift_Seq A) . n ; ::_thesis: x in (inferior_setsequence A) . n then A5: x in Intersection (A ^\ n) by Def9; for k being Element of NAT holds x in A . (n + k) proof let k be Element of NAT ; ::_thesis: x in A . (n + k) x in (A ^\ n) . k by A5, PROB_1:13; hence x in A . (n + k) by NAT_1:def_3; ::_thesis: verum end; hence x in (inferior_setsequence A) . n by SETLIM_1:19; ::_thesis: verum end; hence (inferior_setsequence A) . n = (Intersect_Shift_Seq A) . n by TARSKI:1; ::_thesis: verum end; theorem :: BOR_CANT:12 for Omega being non empty set for Sigma being SigmaField of Omega for A being SetSequence of Sigma holds ( superior_setsequence A = Union_Shift_Seq A & inferior_setsequence A = Intersect_Shift_Seq A ) by Th11; definition let Omega be non empty set ; let Sigma be SigmaField of Omega; let Prob be Probability of Sigma; let A be SetSequence of Sigma; func Sum_Shift_Seq (Prob,A) -> Real_Sequence means :Def11: :: BOR_CANT:def 11 for n being Element of NAT holds it . n = Sum (Prob * (A ^\ n)); existence ex b1 being Real_Sequence st for n being Element of NAT holds b1 . n = Sum (Prob * (A ^\ n)) proof deffunc H1( Element of NAT ) -> Element of REAL = Sum (Prob * (A ^\ \$1)); consider f being Real_Sequence such that A1: for k being Element of NAT holds f . k = H1(k) from FUNCT_2:sch_4(); take f ; ::_thesis: for n being Element of NAT holds f . n = Sum (Prob * (A ^\ n)) let knat be Nat; ::_thesis: ( knat is Element of REAL & knat is Element of NAT implies f . knat = Sum (Prob * (A ^\ knat)) ) thus ( knat is Element of REAL & knat is Element of NAT implies f . knat = Sum (Prob * (A ^\ knat)) ) by A1; ::_thesis: verum end; uniqueness for b1, b2 being Real_Sequence st ( for n being Element of NAT holds b1 . n = Sum (Prob * (A ^\ n)) ) & ( for n being Element of NAT holds b2 . n = Sum (Prob * (A ^\ n)) ) holds b1 = b2 proof let J1, J2 be Real_Sequence; ::_thesis: ( ( for n being Element of NAT holds J1 . n = Sum (Prob * (A ^\ n)) ) & ( for n being Element of NAT holds J2 . n = Sum (Prob * (A ^\ n)) ) implies J1 = J2 ) assume A2: for n being Element of NAT holds J1 . n = Sum (Prob * (A ^\ n)) ; ::_thesis: ( ex n being Element of NAT st not J2 . n = Sum (Prob * (A ^\ n)) or J1 = J2 ) assume A3: for n being Element of NAT holds J2 . n = Sum (Prob * (A ^\ n)) ; ::_thesis: J1 = J2 let n be Element of NAT ; :: according to FUNCT_2:def_8 ::_thesis: J1 . n = J2 . n J1 . n = Sum (Prob * (A ^\ n)) by A2; hence J1 . n = J2 . n by A3; ::_thesis: verum end; end; :: deftheorem Def11 defines Sum_Shift_Seq BOR_CANT:def_11_:_ for Omega being non empty set for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma for b5 being Real_Sequence holds ( b5 = Sum_Shift_Seq (Prob,A) iff for n being Element of NAT holds b5 . n = Sum (Prob * (A ^\ n)) ); theorem Th13: :: BOR_CANT:13 for Omega being non empty set for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma st Partial_Sums (Prob * A) is convergent holds ( Prob . (@lim_sup A) = 0 & lim (Sum_Shift_Seq (Prob,A)) = 0 & Sum_Shift_Seq (Prob,A) is convergent ) proof let Omega be non empty set ; ::_thesis: for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma st Partial_Sums (Prob * A) is convergent holds ( Prob . (@lim_sup A) = 0 & lim (Sum_Shift_Seq (Prob,A)) = 0 & Sum_Shift_Seq (Prob,A) is convergent ) let Sigma be SigmaField of Omega; ::_thesis: for Prob being Probability of Sigma for A being SetSequence of Sigma st Partial_Sums (Prob * A) is convergent holds ( Prob . (@lim_sup A) = 0 & lim (Sum_Shift_Seq (Prob,A)) = 0 & Sum_Shift_Seq (Prob,A) is convergent ) let Prob be Probability of Sigma; ::_thesis: for A being SetSequence of Sigma st Partial_Sums (Prob * A) is convergent holds ( Prob . (@lim_sup A) = 0 & lim (Sum_Shift_Seq (Prob,A)) = 0 & Sum_Shift_Seq (Prob,A) is convergent ) let A be SetSequence of Sigma; ::_thesis: ( Partial_Sums (Prob * A) is convergent implies ( Prob . (@lim_sup A) = 0 & lim (Sum_Shift_Seq (Prob,A)) = 0 & Sum_Shift_Seq (Prob,A) is convergent ) ) assume A1: Partial_Sums (Prob * A) is convergent ; ::_thesis: ( Prob . (@lim_sup A) = 0 & lim (Sum_Shift_Seq (Prob,A)) = 0 & Sum_Shift_Seq (Prob,A) is convergent ) A2: Prob * A is summable by A1, SERIES_1:def_2; A3: for n being Element of NAT holds 0 <= (Prob * (Partial_Intersection (Union_Shift_Seq A))) . n proof let n be Element of NAT ; ::_thesis: 0 <= (Prob * (Partial_Intersection (Union_Shift_Seq A))) . n A4: dom (Prob * (Partial_Intersection (Union_Shift_Seq A))) = NAT by FUNCT_2:def_1; (Prob * (Partial_Intersection (Union_Shift_Seq A))) . n = Prob . ((Partial_Intersection (Union_Shift_Seq A)) . n) by A4, FUNCT_1:12; hence 0 <= (Prob * (Partial_Intersection (Union_Shift_Seq A))) . n by PROB_1:def_8; ::_thesis: verum end; A5: Intersection (Partial_Intersection (Union_Shift_Seq A)) = Intersection (Union_Shift_Seq A) by PROB_3:29; A6: Partial_Intersection (Union_Shift_Seq A) is non-ascending by PROB_3:27; A7: ( lim (Prob * (Partial_Intersection (Union_Shift_Seq A))) = Prob . (Intersection (Partial_Intersection (Union_Shift_Seq A))) & Prob * (Partial_Intersection (Union_Shift_Seq A)) is convergent ) by A6, PROB_1:def_8; A8: for A being SetSequence of Sigma for n, s being Element of NAT holds (Prob * (Partial_Union (A ^\ s))) . n <= (Partial_Sums (Prob * (A ^\ s))) . n proof let A be SetSequence of Sigma; ::_thesis: for n, s being Element of NAT holds (Prob * (Partial_Union (A ^\ s))) . n <= (Partial_Sums (Prob * (A ^\ s))) . n let n, s be Element of NAT ; ::_thesis: (Prob * (Partial_Union (A ^\ s))) . n <= (Partial_Sums (Prob * (A ^\ s))) . n defpred S1[ set ] means (Prob * (Partial_Union (A ^\ s))) . \$1 <= (Partial_Sums (Prob * (A ^\ s))) . \$1; A9: (Partial_Sums (Prob * (A ^\ s))) . 0 = (Prob * (A ^\ s)) . 0 by SERIES_1:def_1; A10: dom (Prob * (A ^\ s)) = NAT by FUNCT_2:def_1; A11: (Prob * (A ^\ s)) . 0 = Prob . ((A ^\ s) . 0) by A10, FUNCT_1:12; A12: Prob . ((Partial_Union (A ^\ s)) . 0) = Prob . ((A ^\ s) . 0) by PROB_3:def_2; A13: dom (Prob * (Partial_Union (A ^\ s))) = NAT by FUNCT_2:def_1; A14: S1[ 0 ] by A13, A12, A11, A9, FUNCT_1:12; A15: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A16: (Prob * (Partial_Union (A ^\ s))) . k <= (Partial_Sums (Prob * (A ^\ s))) . k ; ::_thesis: S1[k + 1] A17: dom (Prob * (Partial_Union (A ^\ s))) = NAT by FUNCT_2:def_1; A18: Prob . (((Partial_Union (A ^\ s)) . k) \/ ((A ^\ s) . (k + 1))) <= (Prob . ((Partial_Union (A ^\ s)) . k)) + (Prob . ((A ^\ s) . (k + 1))) by PROB_1:39; dom (Prob * (A ^\ s)) = NAT by FUNCT_2:def_1; then A19: (Prob * (A ^\ s)) . (k + 1) = Prob . ((A ^\ s) . (k + 1)) by FUNCT_1:12; A20: ( Prob . ((Partial_Union (A ^\ s)) . (k + 1)) <= (Prob . ((Partial_Union (A ^\ s)) . k)) + ((Prob * (A ^\ s)) . (k + 1)) implies (Prob . ((Partial_Union (A ^\ s)) . (k + 1))) - (Prob . ((Partial_Union (A ^\ s)) . k)) <= (Prob * (A ^\ s)) . (k + 1) ) by XREAL_1:20; A21: ( (Prob . ((Partial_Union (A ^\ s)) . (k + 1))) - ((Prob * (A ^\ s)) . (k + 1)) <= Prob . ((Partial_Union (A ^\ s)) . k) & Prob . ((Partial_Union (A ^\ s)) . k) <= (Partial_Sums (Prob * (A ^\ s))) . k implies (Prob . ((Partial_Union (A ^\ s)) . (k + 1))) - ((Prob * (A ^\ s)) . (k + 1)) <= (Partial_Sums (Prob * (A ^\ s))) . k ) by XXREAL_0:2; A22: ( (Prob . ((Partial_Union (A ^\ s)) . (k + 1))) - ((Prob * (A ^\ s)) . (k + 1)) <= (Partial_Sums (Prob * (A ^\ s))) . k implies Prob . ((Partial_Union (A ^\ s)) . (k + 1)) <= ((Partial_Sums (Prob * (A ^\ s))) . k) + ((Prob * (A ^\ s)) . (k + 1)) ) by XREAL_1:20; A23: Prob . ((Partial_Union (A ^\ s)) . (k + 1)) <= (Partial_Sums (Prob * (A ^\ s))) . (k + 1) by A18, A19, A20, A17, A16, A21, A22, FUNCT_1:12, PROB_3:def_2, SERIES_1:def_1, XREAL_1:12; dom (Prob * (Partial_Union (A ^\ s))) = NAT by FUNCT_2:def_1; hence S1[k + 1] by A23, FUNCT_1:12; ::_thesis: verum end; for k being Element of NAT holds S1[k] from NAT_1:sch_1(A14, A15); hence (Prob * (Partial_Union (A ^\ s))) . n <= (Partial_Sums (Prob * (A ^\ s))) . n ; ::_thesis: verum end; A24: for k being Element of NAT holds Partial_Sums ((Prob * A) ^\ k) is convergent proof let k be Element of NAT ; ::_thesis: Partial_Sums ((Prob * A) ^\ k) is convergent (Prob * A) ^\ k is summable by A2, SERIES_1:12; hence Partial_Sums ((Prob * A) ^\ k) is convergent by SERIES_1:def_2; ::_thesis: verum end; A25: for A being SetSequence of Sigma for n being Element of NAT holds Prob * (A ^\ n) = (Prob * A) ^\ n proof let A be SetSequence of Sigma; ::_thesis: for n being Element of NAT holds Prob * (A ^\ n) = (Prob * A) ^\ n let n be Element of NAT ; ::_thesis: Prob * (A ^\ n) = (Prob * A) ^\ n for k being Element of NAT holds (Prob * (A ^\ n)) . k = ((Prob * A) ^\ n) . k proof let k be Element of NAT ; ::_thesis: (Prob * (A ^\ n)) . k = ((Prob * A) ^\ n) . k dom (Prob * (A ^\ n)) = NAT by FUNCT_2:def_1; then A26: (Prob * (A ^\ n)) . k = Prob . ((A ^\ n) . k) by FUNCT_1:12; dom (Prob * A) = NAT by FUNCT_2:def_1; then A27: Prob . (A . (n + k)) = (Prob * A) . (n + k) by FUNCT_1:12; (Prob * A) . (k + n) = ((Prob * A) ^\ n) . k by NAT_1:def_3; hence (Prob * (A ^\ n)) . k = ((Prob * A) ^\ n) . k by A26, A27, NAT_1:def_3; ::_thesis: verum end; hence Prob * (A ^\ n) = (Prob * A) ^\ n by FUNCT_2:63; ::_thesis: verum end; A28: for n being Element of NAT holds Partial_Sums (Prob * (A ^\ n)) is convergent proof let n be Element of NAT ; ::_thesis: Partial_Sums (Prob * (A ^\ n)) is convergent Partial_Sums (Prob * (A ^\ n)) = Partial_Sums ((Prob * A) ^\ n) by A25; hence Partial_Sums (Prob * (A ^\ n)) is convergent by A24; ::_thesis: verum end; A29: for n being Element of NAT holds lim (Prob * (Partial_Union (A ^\ n))) <= lim (Partial_Sums (Prob * (A ^\ n))) proof let n be Element of NAT ; ::_thesis: lim (Prob * (Partial_Union (A ^\ n))) <= lim (Partial_Sums (Prob * (A ^\ n))) A30: for k being Element of NAT holds (Prob * (Partial_Union (A ^\ n))) . k <= (Partial_Sums (Prob * (A ^\ n))) . k by A8; A31: Prob * (Partial_Union (A ^\ n)) is convergent by PROB_3:41; Partial_Sums (Prob * (A ^\ n)) is convergent by A28; hence lim (Prob * (Partial_Union (A ^\ n))) <= lim (Partial_Sums (Prob * (A ^\ n))) by A31, A30, SEQ_2:18; ::_thesis: verum end; A32: for n being Element of NAT holds Prob . (Union (A ^\ n)) <= lim (Partial_Sums (Prob * (A ^\ n))) proof let n be Element of NAT ; ::_thesis: Prob . (Union (A ^\ n)) <= lim (Partial_Sums (Prob * (A ^\ n))) lim (Prob * (Partial_Union (A ^\ n))) <= lim (Partial_Sums (Prob * (A ^\ n))) by A29; hence Prob . (Union (A ^\ n)) <= lim (Partial_Sums (Prob * (A ^\ n))) by PROB_3:41; ::_thesis: verum end; A33: for n being Element of NAT holds Prob . (Union (A ^\ n)) <= Sum (Prob * (A ^\ n)) proof let n be Element of NAT ; ::_thesis: Prob . (Union (A ^\ n)) <= Sum (Prob * (A ^\ n)) lim (Partial_Sums (Prob * (A ^\ n))) = Sum (Prob * (A ^\ n)) by SERIES_1:def_3; hence Prob . (Union (A ^\ n)) <= Sum (Prob * (A ^\ n)) by A32; ::_thesis: verum end; A34: for n being Element of NAT holds (Prob * (Union_Shift_Seq A)) . n <= (Sum_Shift_Seq (Prob,A)) . n proof let n be Element of NAT ; ::_thesis: (Prob * (Union_Shift_Seq A)) . n <= (Sum_Shift_Seq (Prob,A)) . n A35: dom (Prob * (Union_Shift_Seq A)) = NAT by FUNCT_2:def_1; A36: (Prob * (Union_Shift_Seq A)) . n = Prob . ((Union_Shift_Seq A) . n) by A35, FUNCT_1:12; A37: Prob . (Union (A ^\ n)) <= Sum (Prob * (A ^\ n)) by A33; Sum (Prob * (A ^\ n)) = (Sum_Shift_Seq (Prob,A)) . n by Def11; hence (Prob * (Union_Shift_Seq A)) . n <= (Sum_Shift_Seq (Prob,A)) . n by Def7, A36, A37; ::_thesis: verum end; A38: 0 <= lim (Prob * (Partial_Intersection (Union_Shift_Seq A))) by A7, A3, SEQ_2:17; A39: ( Sum_Shift_Seq (Prob,A) is convergent implies lim (Prob * (Partial_Intersection (Union_Shift_Seq A))) <= lim (Sum_Shift_Seq (Prob,A)) ) proof assume A40: Sum_Shift_Seq (Prob,A) is convergent ; ::_thesis: lim (Prob * (Partial_Intersection (Union_Shift_Seq A))) <= lim (Sum_Shift_Seq (Prob,A)) A41: for n being Element of NAT holds (Prob * (Partial_Intersection (Union_Shift_Seq A))) . n <= (Prob * (Union_Shift_Seq A)) . n proof let n be Element of NAT ; ::_thesis: (Prob * (Partial_Intersection (Union_Shift_Seq A))) . n <= (Prob * (Union_Shift_Seq A)) . n A42: Prob . ((Partial_Intersection (Union_Shift_Seq A)) . n) <= Prob . ((Union_Shift_Seq A) . n) by PROB_1:34, PROB_3:23; A43: dom (Prob * (Partial_Intersection (Union_Shift_Seq A))) = NAT by FUNCT_2:def_1; A44: dom (Prob * (Union_Shift_Seq A)) = NAT by FUNCT_2:def_1; (Prob * (Union_Shift_Seq A)) . n = Prob . ((Union_Shift_Seq A) . n) by A44, FUNCT_1:12; hence (Prob * (Partial_Intersection (Union_Shift_Seq A))) . n <= (Prob * (Union_Shift_Seq A)) . n by A43, A42, FUNCT_1:12; ::_thesis: verum end; lim (Prob * (Partial_Intersection (Union_Shift_Seq A))) <= lim (Sum_Shift_Seq (Prob,A)) proof A45: for n being Element of NAT holds (Prob * (Partial_Intersection (Union_Shift_Seq A))) . n <= (Sum_Shift_Seq (Prob,A)) . n proof let n be Element of NAT ; ::_thesis: (Prob * (Partial_Intersection (Union_Shift_Seq A))) . n <= (Sum_Shift_Seq (Prob,A)) . n A46: (Prob * (Partial_Intersection (Union_Shift_Seq A))) . n <= (Prob * (Union_Shift_Seq A)) . n by A41; A47: (Prob * (Union_Shift_Seq A)) . n <= (Sum_Shift_Seq (Prob,A)) . n by A34; thus (Prob * (Partial_Intersection (Union_Shift_Seq A))) . n <= (Sum_Shift_Seq (Prob,A)) . n by A46, A47, XXREAL_0:2; ::_thesis: verum end; thus lim (Prob * (Partial_Intersection (Union_Shift_Seq A))) <= lim (Sum_Shift_Seq (Prob,A)) by A7, A40, A45, SEQ_2:18; ::_thesis: verum end; hence lim (Prob * (Partial_Intersection (Union_Shift_Seq A))) <= lim (Sum_Shift_Seq (Prob,A)) ; ::_thesis: verum end; A48: for A being SetSequence of Sigma st Partial_Sums (Prob * A) is convergent holds ( 0 = lim (Sum_Shift_Seq (Prob,A)) & Sum_Shift_Seq (Prob,A) is convergent ) proof let A be SetSequence of Sigma; ::_thesis: ( Partial_Sums (Prob * A) is convergent implies ( 0 = lim (Sum_Shift_Seq (Prob,A)) & Sum_Shift_Seq (Prob,A) is convergent ) ) assume A49: Partial_Sums (Prob * A) is convergent ; ::_thesis: ( 0 = lim (Sum_Shift_Seq (Prob,A)) & Sum_Shift_Seq (Prob,A) is convergent ) then A50: Prob * A is summable by SERIES_1:def_2; A51: for n being Element of NAT holds (Sum (Prob * A)) - (Sum ((Prob * A) ^\ (n + 1))) = (Partial_Sums (Prob * A)) . n proof let n be Element of NAT ; ::_thesis: (Sum (Prob * A)) - (Sum ((Prob * A) ^\ (n + 1))) = (Partial_Sums (Prob * A)) . n (Sum (Prob * A)) - (Sum ((Prob * A) ^\ (n + 1))) = (((Partial_Sums (Prob * A)) . n) + (Sum ((Prob * A) ^\ (n + 1)))) - (Sum ((Prob * A) ^\ (n + 1))) by A50, SERIES_1:15; hence (Sum (Prob * A)) - (Sum ((Prob * A) ^\ (n + 1))) = (Partial_Sums (Prob * A)) . n ; ::_thesis: verum end; A52: for n, m being Element of NAT st n <= m holds abs (((Partial_Sums (Prob * A)) . m) - ((Partial_Sums (Prob * A)) . n)) = abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n)) proof let n, m be Element of NAT ; ::_thesis: ( n <= m implies abs (((Partial_Sums (Prob * A)) . m) - ((Partial_Sums (Prob * A)) . n)) = abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n)) ) assume n <= m ; ::_thesis: abs (((Partial_Sums (Prob * A)) . m) - ((Partial_Sums (Prob * A)) . n)) = abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n)) A53: ((Partial_Sums (Prob * A)) . m) - ((Partial_Sums (Prob * A)) . n) = ((Partial_Sums (Prob * A)) . m) - ((Sum (Prob * A)) - (Sum ((Prob * A) ^\ (n + 1)))) by A51; A54: ((Partial_Sums (Prob * A)) . m) - ((Partial_Sums (Prob * A)) . n) = ((Sum (Prob * A)) - (Sum ((Prob * A) ^\ (m + 1)))) - ((Sum (Prob * A)) - (Sum ((Prob * A) ^\ (n + 1)))) by A51, A53; A55: ((Partial_Sums (Prob * A)) . m) - ((Partial_Sums (Prob * A)) . n) = (Sum ((Prob * A) ^\ (n + 1))) - (Sum ((Prob * A) ^\ (m + 1))) by A54; A56: for A being SetSequence of Sigma for n being Element of NAT holds Prob * (A ^\ n) = (Prob * A) ^\ n proof let A be SetSequence of Sigma; ::_thesis: for n being Element of NAT holds Prob * (A ^\ n) = (Prob * A) ^\ n let n be Element of NAT ; ::_thesis: Prob * (A ^\ n) = (Prob * A) ^\ n for k being Element of NAT holds (Prob * (A ^\ n)) . k = ((Prob * A) ^\ n) . k proof let k be Element of NAT ; ::_thesis: (Prob * (A ^\ n)) . k = ((Prob * A) ^\ n) . k dom (Prob * (A ^\ n)) = NAT by FUNCT_2:def_1; then A57: (Prob * (A ^\ n)) . k = Prob . ((A ^\ n) . k) by FUNCT_1:12; dom (Prob * A) = NAT by FUNCT_2:def_1; then A58: Prob . (A . (n + k)) = (Prob * A) . (n + k) by FUNCT_1:12; (Prob * A) . (k + n) = ((Prob * A) ^\ n) . k by NAT_1:def_3; hence (Prob * (A ^\ n)) . k = ((Prob * A) ^\ n) . k by A57, A58, NAT_1:def_3; ::_thesis: verum end; hence Prob * (A ^\ n) = (Prob * A) ^\ n by FUNCT_2:63; ::_thesis: verum end; A59: for n being Element of NAT holds ((Sum_Shift_Seq (Prob,A)) ^\ 1) . n = Sum ((Prob * A) ^\ (n + 1)) proof let n be Element of NAT ; ::_thesis: ((Sum_Shift_Seq (Prob,A)) ^\ 1) . n = Sum ((Prob * A) ^\ (n + 1)) A60: ((Sum_Shift_Seq (Prob,A)) ^\ 1) . n = (Sum_Shift_Seq (Prob,A)) . (n + 1) by NAT_1:def_3; (Sum_Shift_Seq (Prob,A)) . (n + 1) = Sum (Prob * (A ^\ (n + 1))) by Def11; hence ((Sum_Shift_Seq (Prob,A)) ^\ 1) . n = Sum ((Prob * A) ^\ (n + 1)) by A56, A60; ::_thesis: verum end; A61: abs (((Partial_Sums (Prob * A)) . m) - ((Partial_Sums (Prob * A)) . n)) = abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . n) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . m)) proof ((Partial_Sums (Prob * A)) . m) - ((Partial_Sums (Prob * A)) . n) = (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n) - (Sum ((Prob * A) ^\ (m + 1))) by A55, A59; hence abs (((Partial_Sums (Prob * A)) . m) - ((Partial_Sums (Prob * A)) . n)) = abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . n) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . m)) by A59; ::_thesis: verum end; abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . n) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . m)) = abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n)) proof percases ( (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) = 0 or 0 < (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) or (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) < 0 ) ; suppose (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) = 0 ; ::_thesis: abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . n) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . m)) = abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n)) hence abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . n) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . m)) = abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n)) ; ::_thesis: verum end; suppose 0 < (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) ; ::_thesis: abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . n) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . m)) = abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n)) then A62: - 0 > - ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . n) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . m)) ; abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n)) = - ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n)) by A62, ABSVALUE:def_1; hence abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . n) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . m)) = abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n)) ; ::_thesis: verum end; supposeA63: (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) < 0 ; ::_thesis: abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . n) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . m)) = abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n)) abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . n) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . m)) = - ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . n) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . m)) by A63, ABSVALUE:def_1; hence abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . n) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . m)) = abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n)) ; ::_thesis: verum end; end; end; hence abs (((Partial_Sums (Prob * A)) . m) - ((Partial_Sums (Prob * A)) . n)) = abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n)) by A61; ::_thesis: verum end; A64: ( ( for sr being real number st 0 < sr holds ex n being Element of NAT st for m being Element of NAT st n <= m holds abs (((Partial_Sums (Prob * A)) . m) - ((Partial_Sums (Prob * A)) . n)) < sr ) implies for sr being real number st 0 < sr holds ex n being Element of NAT st for m being Element of NAT st n <= m holds abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n)) < sr ) proof assume A65: for sr being real number st 0 < sr holds ex n being Element of NAT st for m being Element of NAT st n <= m holds abs (((Partial_Sums (Prob * A)) . m) - ((Partial_Sums (Prob * A)) . n)) < sr ; ::_thesis: for sr being real number st 0 < sr holds ex n being Element of NAT st for m being Element of NAT st n <= m holds abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n)) < sr let sr be real number ; ::_thesis: ( 0 < sr implies ex n being Element of NAT st for m being Element of NAT st n <= m holds abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n)) < sr ) assume A66: 0 < sr ; ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n)) < sr consider n being Element of NAT such that A67: for m being Element of NAT st n <= m holds abs (((Partial_Sums (Prob * A)) . m) - ((Partial_Sums (Prob * A)) . n)) < sr by A65, A66; take n ; ::_thesis: for m being Element of NAT st n <= m holds abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n)) < sr let m be Element of NAT ; ::_thesis: ( n <= m implies abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n)) < sr ) assume A68: n <= m ; ::_thesis: abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n)) < sr abs (((Partial_Sums (Prob * A)) . m) - ((Partial_Sums (Prob * A)) . n)) = abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n)) by A68, A52; hence abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n)) < sr by A67, A68; ::_thesis: verum end; A69: ( Partial_Sums (Prob * A) is convergent & (Sum_Shift_Seq (Prob,A)) ^\ 1 is convergent ) by A49, A64, SEQ_4:41; A70: dom (((Sum_Shift_Seq (Prob,A)) ^\ 1) + (Partial_Sums (Prob * A))) = NAT by FUNCT_2:def_1; consider B being Real_Sequence such that A71: B = ((Sum_Shift_Seq (Prob,A)) ^\ 1) + (Partial_Sums (Prob * A)) ; set B1 = NAT --> (Sum (Prob * A)); A72: Prob * A is summable by A49, SERIES_1:def_2; A73: for n being Element of NAT holds (NAT --> (Sum (Prob * A))) . n = B . n proof let n be Element of NAT ; ::_thesis: (NAT --> (Sum (Prob * A))) . n = B . n A74: for n being Element of NAT holds ((Sum_Shift_Seq (Prob,A)) ^\ 1) . n = Sum ((Prob * A) ^\ (n + 1)) proof let n be Element of NAT ; ::_thesis: ((Sum_Shift_Seq (Prob,A)) ^\ 1) . n = Sum ((Prob * A) ^\ (n + 1)) A75: ((Sum_Shift_Seq (Prob,A)) ^\ 1) . n = (Sum_Shift_Seq (Prob,A)) . (n + 1) by NAT_1:def_3; A76: for A being SetSequence of Sigma for n being Element of NAT holds Prob * (A ^\ n) = (Prob * A) ^\ n proof let A be SetSequence of Sigma; ::_thesis: for n being Element of NAT holds Prob * (A ^\ n) = (Prob * A) ^\ n let n be Element of NAT ; ::_thesis: Prob * (A ^\ n) = (Prob * A) ^\ n for k being Element of NAT holds (Prob * (A ^\ n)) . k = ((Prob * A) ^\ n) . k proof let k be Element of NAT ; ::_thesis: (Prob * (A ^\ n)) . k = ((Prob * A) ^\ n) . k dom (Prob * (A ^\ n)) = NAT by FUNCT_2:def_1; then A77: (Prob * (A ^\ n)) . k = Prob . ((A ^\ n) . k) by FUNCT_1:12; dom (Prob * A) = NAT by FUNCT_2:def_1; then A78: Prob . (A . (n + k)) = (Prob * A) . (n + k) by FUNCT_1:12; (Prob * A) . (k + n) = ((Prob * A) ^\ n) . k by NAT_1:def_3; hence (Prob * (A ^\ n)) . k = ((Prob * A) ^\ n) . k by A77, A78, NAT_1:def_3; ::_thesis: verum end; hence Prob * (A ^\ n) = (Prob * A) ^\ n by FUNCT_2:63; ::_thesis: verum end; (Sum_Shift_Seq (Prob,A)) . (n + 1) = Sum (Prob * (A ^\ (n + 1))) by Def11; hence ((Sum_Shift_Seq (Prob,A)) ^\ 1) . n = Sum ((Prob * A) ^\ (n + 1)) by A75, A76; ::_thesis: verum end; A79: ((Sum_Shift_Seq (Prob,A)) ^\ 1) . n = Sum ((Prob * A) ^\ (n + 1)) by A74; Sum (Prob * A) = ((Partial_Sums (Prob * A)) . n) + (Sum ((Prob * A) ^\ (n + 1))) by A72, SERIES_1:15; then (NAT --> (Sum (Prob * A))) . n = ((Partial_Sums (Prob * A)) . n) + (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n) by A79, FUNCOP_1:7; hence (NAT --> (Sum (Prob * A))) . n = B . n by A70, A71, VALUED_1:def_1; ::_thesis: verum end; A80: lim (NAT --> (Sum (Prob * A))) = lim B proof ex k being Element of NAT st for n being Element of NAT st k <= n holds (NAT --> (Sum (Prob * A))) . n = B . n proof take 1 ; ::_thesis: for n being Element of NAT st 1 <= n holds (NAT --> (Sum (Prob * A))) . n = B . n thus for n being Element of NAT st 1 <= n holds (NAT --> (Sum (Prob * A))) . n = B . n by A73; ::_thesis: verum end; hence lim (NAT --> (Sum (Prob * A))) = lim B by SEQ_4:19; ::_thesis: verum end; A81: Sum (Prob * A) = (NAT --> (Sum (Prob * A))) . 1 by FUNCOP_1:7 .= lim B by A80, SEQ_4:26 ; A82: lim B = (lim ((Sum_Shift_Seq (Prob,A)) ^\ 1)) + (lim (Partial_Sums (Prob * A))) by A71, A69, SEQ_2:6; Sum (Prob * A) = (lim ((Sum_Shift_Seq (Prob,A)) ^\ 1)) + (Sum (Prob * A)) by A81, A82, SERIES_1:def_3; hence ( 0 = lim (Sum_Shift_Seq (Prob,A)) & Sum_Shift_Seq (Prob,A) is convergent ) by A69, SEQ_4:21, SEQ_4:22; ::_thesis: verum end; ( lim (Sum_Shift_Seq (Prob,A)) = 0 & Sum_Shift_Seq (Prob,A) is convergent ) by A1, A48; hence ( Prob . (@lim_sup A) = 0 & lim (Sum_Shift_Seq (Prob,A)) = 0 & Sum_Shift_Seq (Prob,A) is convergent ) by A5, A7, A38, A39, PROB_2:def_1; ::_thesis: verum end; theorem Th14: :: BOR_CANT:14 for Omega being non empty set for Sigma being SigmaField of Omega holds ( ( for X being set for A being SetSequence of X for n being Element of NAT for x being set holds ( ex k being Element of NAT st x in (A ^\ n) . k iff ex k being Element of NAT st ( k >= n & x in A . k ) ) ) & ( for X being set for A being SetSequence of X for x being set holds ( x in Intersection (Union_Shift_Seq A) iff for m being Element of NAT ex n being Element of NAT st ( n >= m & x in A . n ) ) ) & ( for A being SetSequence of Sigma for x being set holds ( x in @Intersection (Union_Shift_Seq A) iff for m being Element of NAT ex n being Element of NAT st ( n >= m & x in A . n ) ) ) & ( for X being set for A being SetSequence of X for x being set holds ( x in Union (Intersect_Shift_Seq A) iff ex n being Element of NAT st for k being Element of NAT st k >= n holds x in A . k ) ) & ( for A being SetSequence of Sigma for x being set holds ( x in Union (Intersect_Shift_Seq A) iff ex n being Element of NAT st for k being Element of NAT st k >= n holds x in A . k ) ) & ( for A being SetSequence of Sigma for x being Element of Omega holds ( x in Union (Intersect_Shift_Seq (Complement A)) iff ex n being Element of NAT st for k being Element of NAT st k >= n holds not x in A . k ) ) ) proof let Omega be non empty set ; ::_thesis: for Sigma being SigmaField of Omega holds ( ( for X being set for A being SetSequence of X for n being Element of NAT for x being set holds ( ex k being Element of NAT st x in (A ^\ n) . k iff ex k being Element of NAT st ( k >= n & x in A . k ) ) ) & ( for X being set for A being SetSequence of X for x being set holds ( x in Intersection (Union_Shift_Seq A) iff for m being Element of NAT ex n being Element of NAT st ( n >= m & x in A . n ) ) ) & ( for A being SetSequence of Sigma for x being set holds ( x in @Intersection (Union_Shift_Seq A) iff for m being Element of NAT ex n being Element of NAT st ( n >= m & x in A . n ) ) ) & ( for X being set for A being SetSequence of X for x being set holds ( x in Union (Intersect_Shift_Seq A) iff ex n being Element of NAT st for k being Element of NAT st k >= n holds x in A . k ) ) & ( for A being SetSequence of Sigma for x being set holds ( x in Union (Intersect_Shift_Seq A) iff ex n being Element of NAT st for k being Element of NAT st k >= n holds x in A . k ) ) & ( for A being SetSequence of Sigma for x being Element of Omega holds ( x in Union (Intersect_Shift_Seq (Complement A)) iff ex n being Element of NAT st for k being Element of NAT st k >= n holds not x in A . k ) ) ) let Sigma be SigmaField of Omega; ::_thesis: ( ( for X being set for A being SetSequence of X for n being Element of NAT for x being set holds ( ex k being Element of NAT st x in (A ^\ n) . k iff ex k being Element of NAT st ( k >= n & x in A . k ) ) ) & ( for X being set for A being SetSequence of X for x being set holds ( x in Intersection (Union_Shift_Seq A) iff for m being Element of NAT ex n being Element of NAT st ( n >= m & x in A . n ) ) ) & ( for A being SetSequence of Sigma for x being set holds ( x in @Intersection (Union_Shift_Seq A) iff for m being Element of NAT ex n being Element of NAT st ( n >= m & x in A . n ) ) ) & ( for X being set for A being SetSequence of X for x being set holds ( x in Union (Intersect_Shift_Seq A) iff ex n being Element of NAT st for k being Element of NAT st k >= n holds x in A . k ) ) & ( for A being SetSequence of Sigma for x being set holds ( x in Union (Intersect_Shift_Seq A) iff ex n being Element of NAT st for k being Element of NAT st k >= n holds x in A . k ) ) & ( for A being SetSequence of Sigma for x being Element of Omega holds ( x in Union (Intersect_Shift_Seq (Complement A)) iff ex n being Element of NAT st for k being Element of NAT st k >= n holds not x in A . k ) ) ) A1: for X being set for A being SetSequence of X for n being Element of NAT for x being set holds ( ex k being Element of NAT st x in (A ^\ n) . k iff ex k being Element of NAT st ( k >= n & x in A . k ) ) proof let X be set ; ::_thesis: for A being SetSequence of X for n being Element of NAT for x being set holds ( ex k being Element of NAT st x in (A ^\ n) . k iff ex k being Element of NAT st ( k >= n & x in A . k ) ) let A be SetSequence of X; ::_thesis: for n being Element of NAT for x being set holds ( ex k being Element of NAT st x in (A ^\ n) . k iff ex k being Element of NAT st ( k >= n & x in A . k ) ) let n be Element of NAT ; ::_thesis: for x being set holds ( ex k being Element of NAT st x in (A ^\ n) . k iff ex k being Element of NAT st ( k >= n & x in A . k ) ) let x be set ; ::_thesis: ( ex k being Element of NAT st x in (A ^\ n) . k iff ex k being Element of NAT st ( k >= n & x in A . k ) ) hereby ::_thesis: ( ex k being Element of NAT st ( k >= n & x in A . k ) implies ex k being Element of NAT st x in (A ^\ n) . k ) assume ex k being Element of NAT st x in (A ^\ n) . k ; ::_thesis: ex k being Element of NAT st ( k >= n & x in A . k ) then consider k being Element of NAT such that A2: x in (A ^\ n) . k ; A3: x in A . (k + n) by A2, NAT_1:def_3; consider k being Element of NAT such that A4: x in A . (k + n) by A3; consider k being Element of NAT such that A5: ( k >= n & x in A . k ) by A4, NAT_1:11; thus ex k being Element of NAT st ( k >= n & x in A . k ) by A5; ::_thesis: verum end; assume ex k being Element of NAT st ( k >= n & x in A . k ) ; ::_thesis: ex k being Element of NAT st x in (A ^\ n) . k then consider k being Element of NAT such that A6: ( k >= n & x in A . k ) ; consider knat being Nat such that A7: k = n + knat by A6, NAT_1:10; reconsider knat = knat as Element of NAT by ORDINAL1:def_12; ( x in A . k implies x in (A ^\ n) . knat ) by A7, NAT_1:def_3; hence ex k being Element of NAT st x in (A ^\ n) . k by A6; ::_thesis: verum end; A8: for X being set for A being SetSequence of X for x being set holds ( x in Intersection (Union_Shift_Seq A) iff for m being Element of NAT ex n being Element of NAT st ( n >= m & x in A . n ) ) proof let X be set ; ::_thesis: for A being SetSequence of X for x being set holds ( x in Intersection (Union_Shift_Seq A) iff for m being Element of NAT ex n being Element of NAT st ( n >= m & x in A . n ) ) let A be SetSequence of X; ::_thesis: for x being set holds ( x in Intersection (Union_Shift_Seq A) iff for m being Element of NAT ex n being Element of NAT st ( n >= m & x in A . n ) ) let x be set ; ::_thesis: ( x in Intersection (Union_Shift_Seq A) iff for m being Element of NAT ex n being Element of NAT st ( n >= m & x in A . n ) ) hereby ::_thesis: ( ( for m being Element of NAT ex n being Element of NAT st ( n >= m & x in A . n ) ) implies x in Intersection (Union_Shift_Seq A) ) assume A9: x in Intersection (Union_Shift_Seq A) ; ::_thesis: for m being Element of NAT ex n being Element of NAT st ( n >= m & x in A . n ) A10: for n being Element of NAT st x in (Union_Shift_Seq A) . n holds ex k being Element of NAT st ( k >= n & x in A . k ) proof let n be Element of NAT ; ::_thesis: ( x in (Union_Shift_Seq A) . n implies ex k being Element of NAT st ( k >= n & x in A . k ) ) assume A11: x in (Union_Shift_Seq A) . n ; ::_thesis: ex k being Element of NAT st ( k >= n & x in A . k ) A12: ( x in (Union_Shift_Seq A) . n implies x in Union (A ^\ n) ) by Def7; A13: ex k being Element of NAT st x in (A ^\ n) . k by A11, A12, PROB_1:12; consider k being Element of NAT such that A14: ( k >= n & x in A . k ) by A13, A1; take k ; ::_thesis: ( k >= n & x in A . k ) thus ( k >= n & x in A . k ) by A14; ::_thesis: verum end; A15: for m being Element of NAT ex n being Element of NAT st ( n >= m & x in A . n ) proof let m be Element of NAT ; ::_thesis: ex n being Element of NAT st ( n >= m & x in A . n ) x in (Union_Shift_Seq A) . m by A9, PROB_1:13; hence ex n being Element of NAT st ( n >= m & x in A . n ) by A10; ::_thesis: verum end; thus for m being Element of NAT ex n being Element of NAT st ( n >= m & x in A . n ) by A15; ::_thesis: verum end; assume A16: for m being Element of NAT ex n being Element of NAT st ( n >= m & x in A . n ) ; ::_thesis: x in Intersection (Union_Shift_Seq A) A17: for m being Element of NAT st ex n being Element of NAT st ( n >= m & x in A . n ) holds x in (Union_Shift_Seq A) . m proof let m be Element of NAT ; ::_thesis: ( ex n being Element of NAT st ( n >= m & x in A . n ) implies x in (Union_Shift_Seq A) . m ) assume ex n being Element of NAT st ( n >= m & x in A . n ) ; ::_thesis: x in (Union_Shift_Seq A) . m then consider n being Element of NAT such that A18: ( n >= m & x in A . n ) ; ex k being Element of NAT st x in (A ^\ m) . k by A18, A1; then x in Union (A ^\ m) by PROB_1:12; hence x in (Union_Shift_Seq A) . m by Def7; ::_thesis: verum end; for m being Element of NAT holds x in (Union_Shift_Seq A) . m proof let m be Element of NAT ; ::_thesis: x in (Union_Shift_Seq A) . m ex n being Element of NAT st ( n >= m & x in A . n ) by A16; hence x in (Union_Shift_Seq A) . m by A17; ::_thesis: verum end; hence x in Intersection (Union_Shift_Seq A) by PROB_1:13; ::_thesis: verum end; A19: for A being SetSequence of Sigma for x being set holds ( x in @Intersection (Union_Shift_Seq A) iff for m being Element of NAT ex n being Element of NAT st ( n >= m & x in A . n ) ) proof let A be SetSequence of Sigma; ::_thesis: for x being set holds ( x in @Intersection (Union_Shift_Seq A) iff for m being Element of NAT ex n being Element of NAT st ( n >= m & x in A . n ) ) let x be set ; ::_thesis: ( x in @Intersection (Union_Shift_Seq A) iff for m being Element of NAT ex n being Element of NAT st ( n >= m & x in A . n ) ) @Intersection (Union_Shift_Seq A) = Intersection (Union_Shift_Seq A) by PROB_2:def_1; hence ( x in @Intersection (Union_Shift_Seq A) iff for m being Element of NAT ex n being Element of NAT st ( n >= m & x in A . n ) ) by A8; ::_thesis: verum end; A20: for X being set for A being SetSequence of X for x being set holds ( x in Union (Intersect_Shift_Seq A) iff ex n being Element of NAT st for k being Element of NAT st k >= n holds x in A . k ) proof let X be set ; ::_thesis: for A being SetSequence of X for x being set holds ( x in Union (Intersect_Shift_Seq A) iff ex n being Element of NAT st for k being Element of NAT st k >= n holds x in A . k ) let A be SetSequence of X; ::_thesis: for x being set holds ( x in Union (Intersect_Shift_Seq A) iff ex n being Element of NAT st for k being Element of NAT st k >= n holds x in A . k ) let x be set ; ::_thesis: ( x in Union (Intersect_Shift_Seq A) iff ex n being Element of NAT st for k being Element of NAT st k >= n holds x in A . k ) hereby ::_thesis: ( ex n being Element of NAT st for k being Element of NAT st k >= n holds x in A . k implies x in Union (Intersect_Shift_Seq A) ) assume x in Union (Intersect_Shift_Seq A) ; ::_thesis: ex n being Element of NAT st for k being Element of NAT st k >= n holds x in A . k then consider n being Element of NAT such that A21: x in (Intersect_Shift_Seq A) . n by PROB_1:12; A22: (Intersect_Shift_Seq A) . n = Intersection (A ^\ n) by Def9; for k being Element of NAT st k >= n holds x in A . k proof let k be Element of NAT ; ::_thesis: ( k >= n implies x in A . k ) assume A23: n <= k ; ::_thesis: x in A . k consider knat being Nat such that A24: k = n + knat by A23, NAT_1:10; reconsider knat = knat as Element of NAT by ORDINAL1:def_12; A25: ( x in A . k iff x in (A ^\ n) . knat ) by A24, NAT_1:def_3; thus x in A . k by A22, A21, A25, PROB_1:13; ::_thesis: verum end; hence ex n being Element of NAT st for k being Element of NAT st k >= n holds x in A . k ; ::_thesis: verum end; assume ex n being Element of NAT st for k being Element of NAT st k >= n holds x in A . k ; ::_thesis: x in Union (Intersect_Shift_Seq A) then consider n being Element of NAT such that A26: for k being Element of NAT st k >= n holds x in A . k ; set knat = the Nat; for s being Element of NAT holds x in (A ^\ n) . s proof let s be Element of NAT ; ::_thesis: x in (A ^\ n) . s ( x in (A ^\ n) . s iff x in A . (n + s) ) by NAT_1:def_3; hence x in (A ^\ n) . s by A26, NAT_1:12; ::_thesis: verum end; then x in Intersection (A ^\ n) by PROB_1:13; then x in (Intersect_Shift_Seq A) . n by Def9; hence x in Union (Intersect_Shift_Seq A) by PROB_1:12; ::_thesis: verum end; for A being SetSequence of Sigma for x being Element of Omega holds ( x in Union (Intersect_Shift_Seq (Complement A)) iff ex n being Element of NAT st for k being Element of NAT st k >= n holds not x in A . k ) proof let A be SetSequence of Sigma; ::_thesis: for x being Element of Omega holds ( x in Union (Intersect_Shift_Seq (Complement A)) iff ex n being Element of NAT st for k being Element of NAT st k >= n holds not x in A . k ) let x be Element of Omega; ::_thesis: ( x in Union (Intersect_Shift_Seq (Complement A)) iff ex n being Element of NAT st for k being Element of NAT st k >= n holds not x in A . k ) hereby ::_thesis: ( ex n being Element of NAT st for k being Element of NAT st k >= n holds not x in A . k implies x in Union (Intersect_Shift_Seq (Complement A)) ) assume x in Union (Intersect_Shift_Seq (Complement A)) ; ::_thesis: ex n being Element of NAT st for k being Element of NAT st k >= n holds not x in A . k then consider n being Element of NAT such that A27: x in (Intersect_Shift_Seq (Complement A)) . n by PROB_1:12; A28: (Intersect_Shift_Seq (Complement A)) . n = Intersection ((Complement A) ^\ n) by Def9; set m = the Element of NAT ; for k being Element of NAT st k >= n holds not x in A . k proof let k be Element of NAT ; ::_thesis: ( k >= n implies not x in A . k ) assume A29: n <= k ; ::_thesis: not x in A . k consider knat being Nat such that A30: k = n + knat by A29, NAT_1:10; reconsider knat = knat as Element of NAT by ORDINAL1:def_12; A31: ( x in (Complement A) . k iff x in ((Complement A) ^\ n) . knat ) by A30, NAT_1:def_3; x in (A . k) ` by A28, A27, A31, PROB_1:13, PROB_1:def_2; then x in Omega \ (A . k) by SUBSET_1:def_4; hence not x in A . k by XBOOLE_0:def_5; ::_thesis: verum end; hence ex n being Element of NAT st for k being Element of NAT st k >= n holds not x in A . k ; ::_thesis: verum end; assume ex n being Element of NAT st for k being Element of NAT st k >= n holds not x in A . k ; ::_thesis: x in Union (Intersect_Shift_Seq (Complement A)) then consider n being Element of NAT such that A32: for k being Element of NAT st k >= n holds not x in A . k ; set k = the Element of NAT ; A33: for k being Element of NAT st n <= k holds x in (Complement A) . k proof let k be Element of NAT ; ::_thesis: ( n <= k implies x in (Complement A) . k ) assume A34: n <= k ; ::_thesis: x in (Complement A) . k A35: not x in A . k by A34, A32; x in Omega \ (A . k) by A35, XBOOLE_0:def_5; then x in (A . k) ` by SUBSET_1:def_4; hence x in (Complement A) . k by PROB_1:def_2; ::_thesis: verum end; for s being Element of NAT holds x in ((Complement A) ^\ n) . s proof let s be Element of NAT ; ::_thesis: x in ((Complement A) ^\ n) . s ( x in ((Complement A) ^\ n) . s iff x in (Complement A) . (n + s) ) by NAT_1:def_3; hence x in ((Complement A) ^\ n) . s by A33, NAT_1:12; ::_thesis: verum end; then x in Intersection ((Complement A) ^\ n) by PROB_1:13; then x in (Intersect_Shift_Seq (Complement A)) . n by Def9; hence x in Union (Intersect_Shift_Seq (Complement A)) by PROB_1:12; ::_thesis: verum end; hence ( ( for X being set for A being SetSequence of X for n being Element of NAT for x being set holds ( ex k being Element of NAT st x in (A ^\ n) . k iff ex k being Element of NAT st ( k >= n & x in A . k ) ) ) & ( for X being set for A being SetSequence of X for x being set holds ( x in Intersection (Union_Shift_Seq A) iff for m being Element of NAT ex n being Element of NAT st ( n >= m & x in A . n ) ) ) & ( for A being SetSequence of Sigma for x being set holds ( x in @Intersection (Union_Shift_Seq A) iff for m being Element of NAT ex n being Element of NAT st ( n >= m & x in A . n ) ) ) & ( for X being set for A being SetSequence of X for x being set holds ( x in Union (Intersect_Shift_Seq A) iff ex n being Element of NAT st for k being Element of NAT st k >= n holds x in A . k ) ) & ( for A being SetSequence of Sigma for x being set holds ( x in Union (Intersect_Shift_Seq A) iff ex n being Element of NAT st for k being Element of NAT st k >= n holds x in A . k ) ) & ( for A being SetSequence of Sigma for x being Element of Omega holds ( x in Union (Intersect_Shift_Seq (Complement A)) iff ex n being Element of NAT st for k being Element of NAT st k >= n holds not x in A . k ) ) ) by A1, A8, A19, A20; ::_thesis: verum end; theorem Th15: :: BOR_CANT:15 for Omega being non empty set for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma holds ( lim_sup A = @lim_sup A & lim_inf A = @lim_inf A & @lim_inf (Complement A) = (@lim_sup A) ` & (Prob . (@lim_inf (Complement A))) + (Prob . (@lim_sup A)) = 1 & (Prob . (lim_inf (Complement A))) + (Prob . (lim_sup A)) = 1 ) proof let Omega be non empty set ; ::_thesis: for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma holds ( lim_sup A = @lim_sup A & lim_inf A = @lim_inf A & @lim_inf (Complement A) = (@lim_sup A) ` & (Prob . (@lim_inf (Complement A))) + (Prob . (@lim_sup A)) = 1 & (Prob . (lim_inf (Complement A))) + (Prob . (lim_sup A)) = 1 ) let Sigma be SigmaField of Omega; ::_thesis: for Prob being Probability of Sigma for A being SetSequence of Sigma holds ( lim_sup A = @lim_sup A & lim_inf A = @lim_inf A & @lim_inf (Complement A) = (@lim_sup A) ` & (Prob . (@lim_inf (Complement A))) + (Prob . (@lim_sup A)) = 1 & (Prob . (lim_inf (Complement A))) + (Prob . (lim_sup A)) = 1 ) let Prob be Probability of Sigma; ::_thesis: for A being SetSequence of Sigma holds ( lim_sup A = @lim_sup A & lim_inf A = @lim_inf A & @lim_inf (Complement A) = (@lim_sup A) ` & (Prob . (@lim_inf (Complement A))) + (Prob . (@lim_sup A)) = 1 & (Prob . (lim_inf (Complement A))) + (Prob . (lim_sup A)) = 1 ) let A be SetSequence of Sigma; ::_thesis: ( lim_sup A = @lim_sup A & lim_inf A = @lim_inf A & @lim_inf (Complement A) = (@lim_sup A) ` & (Prob . (@lim_inf (Complement A))) + (Prob . (@lim_sup A)) = 1 & (Prob . (lim_inf (Complement A))) + (Prob . (lim_sup A)) = 1 ) thus A1: lim_sup A = @lim_sup A ::_thesis: ( lim_inf A = @lim_inf A & @lim_inf (Complement A) = (@lim_sup A) ` & (Prob . (@lim_inf (Complement A))) + (Prob . (@lim_sup A)) = 1 & (Prob . (lim_inf (Complement A))) + (Prob . (lim_sup A)) = 1 ) proof A2: for n being Element of NAT for x being set holds ( ex k being Element of NAT st x in (A ^\ n) . k iff ex k being Element of NAT st ( k >= n & x in A . k ) ) proof let n be Element of NAT ; ::_thesis: for x being set holds ( ex k being Element of NAT st x in (A ^\ n) . k iff ex k being Element of NAT st ( k >= n & x in A . k ) ) let x be set ; ::_thesis: ( ex k being Element of NAT st x in (A ^\ n) . k iff ex k being Element of NAT st ( k >= n & x in A . k ) ) hereby ::_thesis: ( ex k being Element of NAT st ( k >= n & x in A . k ) implies ex k being Element of NAT st x in (A ^\ n) . k ) assume ex k being Element of NAT st x in (A ^\ n) . k ; ::_thesis: ex k being Element of NAT st ( k >= n & x in A . k ) then consider k being Element of NAT such that A3: x in (A ^\ n) . k ; A4: x in A . (k + n) by A3, NAT_1:def_3; consider k being Element of NAT such that A5: x in A . (k + n) by A4; consider k being Element of NAT such that A6: ( k >= n & x in A . k ) by A5, NAT_1:11; thus ex k being Element of NAT st ( k >= n & x in A . k ) by A6; ::_thesis: verum end; assume ex k being Element of NAT st ( k >= n & x in A . k ) ; ::_thesis: ex k being Element of NAT st x in (A ^\ n) . k then consider k being Element of NAT such that A7: ( k >= n & x in A . k ) ; consider knat being Nat such that A8: k = n + knat by A7, NAT_1:10; reconsider knat = knat as Element of NAT by ORDINAL1:def_12; A9: ( x in A . k implies x in (A ^\ n) . knat ) by A8, NAT_1:def_3; thus ex k being Element of NAT st x in (A ^\ n) . k by A7, A9; ::_thesis: verum end; A10: for x being set holds ( ( for m being Element of NAT ex n being Element of NAT st ( n >= m & x in A . n ) ) iff for m being Element of NAT ex n being Element of NAT st x in A . (m + n) ) proof let x be set ; ::_thesis: ( ( for m being Element of NAT ex n being Element of NAT st ( n >= m & x in A . n ) ) iff for m being Element of NAT ex n being Element of NAT st x in A . (m + n) ) hereby ::_thesis: ( ( for m being Element of NAT ex n being Element of NAT st x in A . (m + n) ) implies for m being Element of NAT ex n being Element of NAT st ( n >= m & x in A . n ) ) assume A11: for m being Element of NAT ex n being Element of NAT st ( n >= m & x in A . n ) ; ::_thesis: for m being Element of NAT ex n being Element of NAT st x in A . (m + n) thus for m being Element of NAT ex n being Element of NAT st x in A . (m + n) ::_thesis: verum proof let m be Element of NAT ; ::_thesis: ex n being Element of NAT st x in A . (m + n) ex n being Element of NAT st ( n >= m & x in A . n ) by A11; then consider n being Element of NAT such that A12: x in (A ^\ m) . n by A2; x in A . (m + n) by A12, NAT_1:def_3; hence ex n being Element of NAT st x in A . (m + n) ; ::_thesis: verum end; end; assume A13: for m being Element of NAT ex n being Element of NAT st x in A . (m + n) ; ::_thesis: for m being Element of NAT ex n being Element of NAT st ( n >= m & x in A . n ) thus for m being Element of NAT ex n being Element of NAT st ( n >= m & x in A . n ) ::_thesis: verum proof let m be Element of NAT ; ::_thesis: ex n being Element of NAT st ( n >= m & x in A . n ) consider n being Element of NAT such that A14: x in A . (m + n) by A13; x in (A ^\ m) . n by A14, NAT_1:def_3; hence ex n being Element of NAT st ( n >= m & x in A . n ) by A2; ::_thesis: verum end; end; A15: for x being set holds ( x in @Intersection (Union_Shift_Seq A) iff for m being Element of NAT ex n being Element of NAT st x in A . (m + n) ) proof let x be set ; ::_thesis: ( x in @Intersection (Union_Shift_Seq A) iff for m being Element of NAT ex n being Element of NAT st x in A . (m + n) ) hereby ::_thesis: ( ( for m being Element of NAT ex n being Element of NAT st x in A . (m + n) ) implies x in @Intersection (Union_Shift_Seq A) ) assume x in @Intersection (Union_Shift_Seq A) ; ::_thesis: for m being Element of NAT ex n being Element of NAT st x in A . (m + n) then A16: for m being Element of NAT ex n being Element of NAT st ( n >= m & x in A . n ) by Th14; thus for m being Element of NAT ex n being Element of NAT st x in A . (m + n) by A16, A10; ::_thesis: verum end; assume for m being Element of NAT ex n being Element of NAT st x in A . (m + n) ; ::_thesis: x in @Intersection (Union_Shift_Seq A) then for m being Element of NAT ex n being Element of NAT st ( n >= m & x in A . n ) by A10; hence x in @Intersection (Union_Shift_Seq A) by Th14; ::_thesis: verum end; for x being set holds ( x in lim_sup A iff x in @Intersection (Union_Shift_Seq A) ) proof let x be set ; ::_thesis: ( x in lim_sup A iff x in @Intersection (Union_Shift_Seq A) ) hereby ::_thesis: ( x in @Intersection (Union_Shift_Seq A) implies x in lim_sup A ) assume x in lim_sup A ; ::_thesis: x in @Intersection (Union_Shift_Seq A) then A17: for m being Element of NAT ex n being Element of NAT st x in A . (m + n) by SETLIM_1:66; thus x in @Intersection (Union_Shift_Seq A) by A17, A15; ::_thesis: verum end; assume x in @Intersection (Union_Shift_Seq A) ; ::_thesis: x in lim_sup A then for m being Element of NAT ex n being Element of NAT st x in A . (m + n) by A15; hence x in lim_sup A by SETLIM_1:66; ::_thesis: verum end; hence lim_sup A = @lim_sup A by TARSKI:1; ::_thesis: verum end; A18: for A being SetSequence of Sigma holds lim_inf A = @lim_inf A proof let A be SetSequence of Sigma; ::_thesis: lim_inf A = @lim_inf A A19: for x being set holds ( ex n being Element of NAT st for k being Element of NAT st k >= n holds x in A . k iff ex n being Element of NAT st for k being Element of NAT holds x in A . (n + k) ) proof let x be set ; ::_thesis: ( ex n being Element of NAT st for k being Element of NAT st k >= n holds x in A . k iff ex n being Element of NAT st for k being Element of NAT holds x in A . (n + k) ) hereby ::_thesis: ( ex n being Element of NAT st for k being Element of NAT holds x in A . (n + k) implies ex n being Element of NAT st for k being Element of NAT st k >= n holds x in A . k ) assume ex n being Element of NAT st for k being Element of NAT st k >= n holds x in A . k ; ::_thesis: ex n being Element of NAT st for k being Element of NAT holds x in A . (n + k) then consider n being Element of NAT such that A20: for k being Element of NAT st k >= n holds x in A . k ; for k being Element of NAT holds x in A . (n + k) by A20, NAT_1:11; hence ex n being Element of NAT st for k being Element of NAT holds x in A . (n + k) ; ::_thesis: verum end; assume ex n being Element of NAT st for k being Element of NAT holds x in A . (n + k) ; ::_thesis: ex n being Element of NAT st for k being Element of NAT st k >= n holds x in A . k then consider n being Element of NAT such that A21: for k being Element of NAT holds x in A . (n + k) ; for k being Element of NAT st k >= n holds x in A . k proof let k be Element of NAT ; ::_thesis: ( k >= n implies x in A . k ) assume n <= k ; ::_thesis: x in A . k then consider knat being Nat such that A22: k = n + knat by NAT_1:10; reconsider knat = knat as Element of NAT by ORDINAL1:def_12; x in A . (n + knat) by A21; hence x in A . k by A22; ::_thesis: verum end; hence ex n being Element of NAT st for k being Element of NAT st k >= n holds x in A . k ; ::_thesis: verum end; for x being set holds ( x in @lim_inf A iff x in lim_inf A ) proof let x be set ; ::_thesis: ( x in @lim_inf A iff x in lim_inf A ) hereby ::_thesis: ( x in lim_inf A implies x in @lim_inf A ) assume x in @lim_inf A ; ::_thesis: x in lim_inf A then ex n being Element of NAT st for k being Element of NAT st k >= n holds x in A . k by Th14; then ex n being Element of NAT st for k being Element of NAT holds x in A . (n + k) by A19; hence x in lim_inf A by SETLIM_1:67; ::_thesis: verum end; assume x in lim_inf A ; ::_thesis: x in @lim_inf A then ex n being Element of NAT st for k being Element of NAT holds x in A . (n + k) by SETLIM_1:67; then ex n being Element of NAT st for k being Element of NAT st k >= n holds x in A . k by A19; hence x in @lim_inf A by Th14; ::_thesis: verum end; hence lim_inf A = @lim_inf A by TARSKI:1; ::_thesis: verum end; A23: @lim_inf (Complement A) = (@lim_sup A) ` proof reconsider CA = Complement A as SetSequence of Sigma ; for x being set holds ( x in @lim_inf (Complement A) iff x in (@lim_sup A) ` ) proof let x be set ; ::_thesis: ( x in @lim_inf (Complement A) iff x in (@lim_sup A) ` ) hereby ::_thesis: ( x in (@lim_sup A) ` implies x in @lim_inf (Complement A) ) assume x in @lim_inf (Complement A) ; ::_thesis: x in (@lim_sup A) ` then x in @lim_inf CA ; then ( x in Omega & ex n being Element of NAT st for k being Element of NAT st k >= n holds not x in A . k ) by Th14; then ( x in Omega & not x in @lim_sup A ) by Th14; then x in Omega \ (@lim_sup A) by XBOOLE_0:def_5; hence x in (@lim_sup A) ` by SUBSET_1:def_4; ::_thesis: verum end; assume A24: x in (@lim_sup A) ` ; ::_thesis: x in @lim_inf (Complement A) x in Omega \ (@lim_sup A) by A24, SUBSET_1:def_4; then not x in @Intersection (Union_Shift_Seq A) by XBOOLE_0:def_5; then ex m being Element of NAT st for n being Element of NAT st n >= m holds not x in A . n by Th14; then x in @lim_inf CA by A24, Th14; hence x in @lim_inf (Complement A) ; ::_thesis: verum end; hence @lim_inf (Complement A) = (@lim_sup A) ` by TARSKI:1; ::_thesis: verum end; (Prob . (@lim_inf (Complement A))) + (Prob . (@lim_sup A)) = 1 proof (Prob . (([#] Sigma) \ (@lim_sup A))) + (Prob . (@lim_sup A)) = 1 by PROB_1:31; hence (Prob . (@lim_inf (Complement A))) + (Prob . (@lim_sup A)) = 1 by A23, SUBSET_1:def_4; ::_thesis: verum end; hence ( lim_inf A = @lim_inf A & @lim_inf (Complement A) = (@lim_sup A) ` & (Prob . (@lim_inf (Complement A))) + (Prob . (@lim_sup A)) = 1 & (Prob . (lim_inf (Complement A))) + (Prob . (lim_sup A)) = 1 ) by A1, A18, A23; ::_thesis: verum end; theorem Th16: :: BOR_CANT:16 for Omega being non empty set for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma holds ( ( Partial_Sums (Prob * A) is convergent implies ( Prob . (lim_sup A) = 0 & Prob . (lim_inf (Complement A)) = 1 ) ) & ( A is_all_independent_wrt Prob & Partial_Sums (Prob * A) is divergent_to+infty implies ( Prob . (lim_inf (Complement A)) = 0 & Prob . (lim_sup A) = 1 ) ) ) proof let Omega be non empty set ; ::_thesis: for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma holds ( ( Partial_Sums (Prob * A) is convergent implies ( Prob . (lim_sup A) = 0 & Prob . (lim_inf (Complement A)) = 1 ) ) & ( A is_all_independent_wrt Prob & Partial_Sums (Prob * A) is divergent_to+infty implies ( Prob . (lim_inf (Complement A)) = 0 & Prob . (lim_sup A) = 1 ) ) ) let Sigma be SigmaField of Omega; ::_thesis: for Prob being Probability of Sigma for A being SetSequence of Sigma holds ( ( Partial_Sums (Prob * A) is convergent implies ( Prob . (lim_sup A) = 0 & Prob . (lim_inf (Complement A)) = 1 ) ) & ( A is_all_independent_wrt Prob & Partial_Sums (Prob * A) is divergent_to+infty implies ( Prob . (lim_inf (Complement A)) = 0 & Prob . (lim_sup A) = 1 ) ) ) let Prob be Probability of Sigma; ::_thesis: for A being SetSequence of Sigma holds ( ( Partial_Sums (Prob * A) is convergent implies ( Prob . (lim_sup A) = 0 & Prob . (lim_inf (Complement A)) = 1 ) ) & ( A is_all_independent_wrt Prob & Partial_Sums (Prob * A) is divergent_to+infty implies ( Prob . (lim_inf (Complement A)) = 0 & Prob . (lim_sup A) = 1 ) ) ) let A be SetSequence of Sigma; ::_thesis: ( ( Partial_Sums (Prob * A) is convergent implies ( Prob . (lim_sup A) = 0 & Prob . (lim_inf (Complement A)) = 1 ) ) & ( A is_all_independent_wrt Prob & Partial_Sums (Prob * A) is divergent_to+infty implies ( Prob . (lim_inf (Complement A)) = 0 & Prob . (lim_sup A) = 1 ) ) ) A1: ( Partial_Sums (Prob * A) is convergent implies Prob . (lim_inf (Complement A)) = 1 ) proof assume A2: Partial_Sums (Prob * A) is convergent ; ::_thesis: Prob . (lim_inf (Complement A)) = 1 A3: Prob . (lim_inf (Complement A)) = Prob . (@lim_inf (Complement A)) by Th15; for A being SetSequence of Sigma st Partial_Sums (Prob * A) is convergent holds ( Prob . (@lim_inf (Complement A)) = 1 & lim (Sum_Shift_Seq (Prob,A)) = 0 & Sum_Shift_Seq (Prob,A) is convergent ) proof let A be SetSequence of Sigma; ::_thesis: ( Partial_Sums (Prob * A) is convergent implies ( Prob . (@lim_inf (Complement A)) = 1 & lim (Sum_Shift_Seq (Prob,A)) = 0 & Sum_Shift_Seq (Prob,A) is convergent ) ) assume A4: Partial_Sums (Prob * A) is convergent ; ::_thesis: ( Prob . (@lim_inf (Complement A)) = 1 & lim (Sum_Shift_Seq (Prob,A)) = 0 & Sum_Shift_Seq (Prob,A) is convergent ) ( (Prob . (@lim_sup A)) + (Prob . (@lim_inf (Complement A))) = 0 + (Prob . (@lim_inf (Complement A))) & lim (Sum_Shift_Seq (Prob,A)) = 0 & Sum_Shift_Seq (Prob,A) is convergent ) by A4, Th13; hence ( Prob . (@lim_inf (Complement A)) = 1 & lim (Sum_Shift_Seq (Prob,A)) = 0 & Sum_Shift_Seq (Prob,A) is convergent ) by Th15; ::_thesis: verum end; hence Prob . (lim_inf (Complement A)) = 1 by A2, A3; ::_thesis: verum end; A5: for A being SetSequence of Sigma st Partial_Sums (Prob * A) is convergent holds Prob . (lim_sup A) = 0 proof let A be SetSequence of Sigma; ::_thesis: ( Partial_Sums (Prob * A) is convergent implies Prob . (lim_sup A) = 0 ) assume A6: Partial_Sums (Prob * A) is convergent ; ::_thesis: Prob . (lim_sup A) = 0 Prob . (lim_sup A) = Prob . (@lim_sup A) by Th15; hence Prob . (lim_sup A) = 0 by A6, Th13; ::_thesis: verum end; for B being SetSequence of Sigma st B is_all_independent_wrt Prob & Partial_Sums (Prob * B) is divergent_to+infty holds ( Prob . (lim_inf (Complement B)) = 0 & Prob . (lim_sup B) = 1 ) proof let B be SetSequence of Sigma; ::_thesis: ( B is_all_independent_wrt Prob & Partial_Sums (Prob * B) is divergent_to+infty implies ( Prob . (lim_inf (Complement B)) = 0 & Prob . (lim_sup B) = 1 ) ) assume that A7: B is_all_independent_wrt Prob and A8: Partial_Sums (Prob * B) is divergent_to+infty ; ::_thesis: ( Prob . (lim_inf (Complement B)) = 0 & Prob . (lim_sup B) = 1 ) A9: Prob . (@lim_sup B) = Prob . (lim_sup B) by Th15; A10: Prob . (@lim_inf (Complement B)) = Prob . (lim_inf (Complement B)) by Th15; for B being SetSequence of Sigma st B is_all_independent_wrt Prob & Partial_Sums (Prob * B) is divergent_to+infty holds ( Prob . (@lim_inf (Complement B)) = 0 & Prob . (@lim_sup B) = 1 ) proof let B be SetSequence of Sigma; ::_thesis: ( B is_all_independent_wrt Prob & Partial_Sums (Prob * B) is divergent_to+infty implies ( Prob . (@lim_inf (Complement B)) = 0 & Prob . (@lim_sup B) = 1 ) ) assume that A11: B is_all_independent_wrt Prob and A12: Partial_Sums (Prob * B) is divergent_to+infty ; ::_thesis: ( Prob . (@lim_inf (Complement B)) = 0 & Prob . (@lim_sup B) = 1 ) A13: for Q being SetSequence of Sigma holds Intersect_Shift_Seq Q is non-descending proof let Q be SetSequence of Sigma; ::_thesis: Intersect_Shift_Seq Q is non-descending inferior_setsequence Q = Intersect_Shift_Seq Q by Th11; hence Intersect_Shift_Seq Q is non-descending ; ::_thesis: verum end; A14: Intersect_Shift_Seq (Complement B) is non-descending by A13; reconsider CB = Complement B as SetSequence of Sigma ; A15: Prob . (@lim_inf CB) = lim (Prob * (Intersect_Shift_Seq (Complement B))) by A14, PROB_2:10; A16: for n being Element of NAT holds (Prob * (Intersect_Shift_Seq (Complement B))) . n = 0 proof let n be Element of NAT ; ::_thesis: (Prob * (Intersect_Shift_Seq (Complement B))) . n = 0 dom (Prob * (Intersect_Shift_Seq (Complement B))) = NAT by FUNCT_2:def_1; then A17: (Prob * (Intersect_Shift_Seq (Complement B))) . n = Prob . ((Intersect_Shift_Seq (Complement B)) . n) by FUNCT_1:12; (Intersect_Shift_Seq (Complement B)) . n = Intersection ((Complement B) ^\ n) by Def9; then A18: (Prob * (Intersect_Shift_Seq (Complement B))) . n = Prob . (Intersection (Partial_Intersection ((Complement B) ^\ n))) by A17, PROB_3:29; Partial_Intersection ((Complement B) ^\ n) is non-ascending by PROB_3:27; then A19: (Prob * (Intersect_Shift_Seq (Complement B))) . n = lim (Prob * (Partial_Intersection ((Complement B) ^\ n))) by A18, PROB_1:def_8; A20: for k being Element of NAT holds (Prob * (Partial_Intersection (Complement (B ^\ n)))) . k <= ((1 + (Partial_Sums (Prob * (B ^\ n)))) ") . k proof let k be Element of NAT ; ::_thesis: (Prob * (Partial_Intersection (Complement (B ^\ n)))) . k <= ((1 + (Partial_Sums (Prob * (B ^\ n)))) ") . k A21: for k being Element of NAT holds B ^\ k is_all_independent_wrt Prob proof let k be Element of NAT ; ::_thesis: B ^\ k is_all_independent_wrt Prob for C being SetSequence of Sigma st ex e being sequence of NAT st ( e is one-to-one & ( for n being Element of NAT holds (B ^\ k) . (e . n) = C . n ) ) holds for n being Element of NAT holds (Partial_Product (Prob * C)) . n = Prob . ((Partial_Intersection C) . n) proof let C be SetSequence of Sigma; ::_thesis: ( ex e being sequence of NAT st ( e is one-to-one & ( for n being Element of NAT holds (B ^\ k) . (e . n) = C . n ) ) implies for n being Element of NAT holds (Partial_Product (Prob * C)) . n = Prob . ((Partial_Intersection C) . n) ) given e being sequence of NAT such that A22: e is one-to-one and A23: for n being Element of NAT holds (B ^\ k) . (e . n) = C . n ; ::_thesis: for n being Element of NAT holds (Partial_Product (Prob * C)) . n = Prob . ((Partial_Intersection C) . n) A24: B ^\ k = B * (Special_Function2 k) proof for n being set st n in NAT holds (B ^\ k) . n = (B * (Special_Function2 k)) . n proof let n be set ; ::_thesis: ( n in NAT implies (B ^\ k) . n = (B * (Special_Function2 k)) . n ) assume n in NAT ; ::_thesis: (B ^\ k) . n = (B * (Special_Function2 k)) . n then reconsider n = n as Element of NAT ; dom (B * (Special_Function2 k)) = NAT by FUNCT_2:def_1; then A25: (B * (Special_Function2 k)) . n = B . ((Special_Function2 k) . n) by FUNCT_1:12; (Special_Function2 k) . n = n + k by Def3; hence (B ^\ k) . n = (B * (Special_Function2 k)) . n by A25, NAT_1:def_3; ::_thesis: verum end; hence B ^\ k = B * (Special_Function2 k) by FUNCT_2:12; ::_thesis: verum end; A26: for n being Element of NAT holds (B * (Special_Function2 k)) . (e . n) = B . (((Special_Function2 k) * e) . n) proof let n be Element of NAT ; ::_thesis: (B * (Special_Function2 k)) . (e . n) = B . (((Special_Function2 k) * e) . n) ( dom (B * (Special_Function2 k)) = NAT & dom ((Special_Function2 k) * e) = NAT ) by FUNCT_2:def_1; then ( (B * (Special_Function2 k)) . (e . n) = B . ((Special_Function2 k) . (e . n)) & ((Special_Function2 k) * e) . n = (Special_Function2 k) . (e . n) ) by FUNCT_1:12; hence (B * (Special_Function2 k)) . (e . n) = B . (((Special_Function2 k) * e) . n) ; ::_thesis: verum end; A27: for n being Element of NAT holds B . (((Special_Function2 k) * e) . n) = C . n proof let n be Element of NAT ; ::_thesis: B . (((Special_Function2 k) * e) . n) = C . n (B * (Special_Function2 k)) . (e . n) = C . n by A24, A23; hence B . (((Special_Function2 k) * e) . n) = C . n by A26; ::_thesis: verum end; (Special_Function2 k) * e is one-to-one by A22, FUNCT_1:24; hence for n being Element of NAT holds (Partial_Product (Prob * C)) . n = Prob . ((Partial_Intersection C) . n) by A11, A27, Def6; ::_thesis: verum end; hence B ^\ k is_all_independent_wrt Prob by Def6; ::_thesis: verum end; A28: for A being SetSequence of Sigma for n being Element of NAT holds (Partial_Product (Prob * (Complement A))) . n <= ((1 + (Partial_Sums (Prob * A))) . n) " proof let A be SetSequence of Sigma; ::_thesis: for n being Element of NAT holds (Partial_Product (Prob * (Complement A))) . n <= ((1 + (Partial_Sums (Prob * A))) . n) " let n be Element of NAT ; ::_thesis: (Partial_Product (Prob * (Complement A))) . n <= ((1 + (Partial_Sums (Prob * A))) . n) " A29: (Partial_Product (Prob * (Complement A))) . n <= 1 / (1 + ((Partial_Sums (Prob * A)) . n)) proof (Partial_Product (Prob * (Complement A))) . n <= (Partial_Product (JSum (Prob * A))) . n by Th4; then A30: (Partial_Product (Prob * (Complement A))) . n <= exp_R . (- ((Partial_Sums (Prob * A)) . n)) by Th3; exp_R . (- ((Partial_Sums (Prob * A)) . n)) <= 1 / (1 + ((Partial_Sums (Prob * A)) . n)) proof A31: for n being Element of NAT holds (Prob * A) . n >= 0 proof let n be Element of NAT ; ::_thesis: (Prob * A) . n >= 0 dom (Prob * A) = NAT by FUNCT_2:def_1; then (Prob * A) . n = Prob . (A . n) by FUNCT_1:12; hence (Prob * A) . n >= 0 by PROB_1:def_8; ::_thesis: verum end; A32: for n being Element of NAT holds (Partial_Sums (Prob * A)) . n >= 0 proof let n be Element of NAT ; ::_thesis: (Partial_Sums (Prob * A)) . n >= 0 defpred S1[ Element of NAT ] means (Partial_Sums (Prob * A)) . \$1 >= 0 ; (Partial_Sums (Prob * A)) . 0 = (Prob * A) . 0 by SERIES_1:def_1; then A33: S1[ 0 ] by A31; A34: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A35: S1[k] ; ::_thesis: S1[k + 1] A36: (Prob * A) . (k + 1) >= 0 by A31; (Partial_Sums (Prob * A)) . (k + 1) = ((Partial_Sums (Prob * A)) . k) + ((Prob * A) . (k + 1)) by SERIES_1:def_1; hence S1[k + 1] by A35, A36; ::_thesis: verum end; for k being Element of NAT holds S1[k] from NAT_1:sch_1(A33, A34); hence (Partial_Sums (Prob * A)) . n >= 0 ; ::_thesis: verum end; for x being Element of REAL st x >= 0 holds exp_R . (- x) <= 1 / (1 + x) proof let x be Element of REAL ; ::_thesis: ( x >= 0 implies exp_R . (- x) <= 1 / (1 + x) ) assume A37: x >= 0 ; ::_thesis: exp_R . (- x) <= 1 / (1 + x) percases ( x > 0 or x <= 0 ) ; supposeA38: x > 0 ; ::_thesis: exp_R . (- x) <= 1 / (1 + x) A39: exp_R . (- x) >= 0 by SIN_COS:54; set z = - x; A40: (exp_R x) * (exp_R (- x)) = exp_R (x + (- x)) by SIN_COS:50; (exp_R . (- x)) * (1 + x) <= 1 by Th2, A39, A40, SIN_COS:51, XREAL_1:64; hence exp_R . (- x) <= 1 / (1 + x) by A38, XREAL_1:77; ::_thesis: verum end; suppose x <= 0 ; ::_thesis: exp_R . (- x) <= 1 / (1 + x) then x = 0 by A37; hence exp_R . (- x) <= 1 / (1 + x) by SIN_COS:51; ::_thesis: verum end; end; end; hence exp_R . (- ((Partial_Sums (Prob * A)) . n)) <= 1 / (1 + ((Partial_Sums (Prob * A)) . n)) by A32; ::_thesis: verum end; hence (Partial_Product (Prob * (Complement A))) . n <= 1 / (1 + ((Partial_Sums (Prob * A)) . n)) by A30, XXREAL_0:2; ::_thesis: verum end; for A being SetSequence of Sigma for n being Element of NAT holds 1 / (1 + ((Partial_Sums (Prob * A)) . n)) = ((1 + (Partial_Sums (Prob * A))) . n) " proof let A be SetSequence of Sigma; ::_thesis: for n being Element of NAT holds 1 / (1 + ((Partial_Sums (Prob * A)) . n)) = ((1 + (Partial_Sums (Prob * A))) . n) " let n be Element of NAT ; ::_thesis: 1 / (1 + ((Partial_Sums (Prob * A)) . n)) = ((1 + (Partial_Sums (Prob * A))) . n) " 1 / (1 + ((Partial_Sums (Prob * A)) . n)) = 1 / ((1 + (Partial_Sums (Prob * A))) . n) by VALUED_1:2; then 1 / (1 + ((Partial_Sums (Prob * A)) . n)) = 1 * (((1 + (Partial_Sums (Prob * A))) . n) ") by XCMPLX_0:def_9; hence 1 / (1 + ((Partial_Sums (Prob * A)) . n)) = ((1 + (Partial_Sums (Prob * A))) . n) " ; ::_thesis: verum end; hence (Partial_Product (Prob * (Complement A))) . n <= ((1 + (Partial_Sums (Prob * A))) . n) " by A29; ::_thesis: verum end; dom (Prob * (Partial_Intersection (Complement (B ^\ n)))) = NAT by FUNCT_2:def_1; then (Prob * (Partial_Intersection (Complement (B ^\ n)))) . k = Prob . ((Partial_Intersection (Complement (B ^\ n))) . k) by FUNCT_1:12; then (Prob * (Partial_Intersection (Complement (B ^\ n)))) . k = (Partial_Product (Prob * (Complement (B ^\ n)))) . k by A21, Th10; then (Prob * (Partial_Intersection (Complement (B ^\ n)))) . k <= ((1 + (Partial_Sums (Prob * (B ^\ n)))) . k) " by A28; hence (Prob * (Partial_Intersection (Complement (B ^\ n)))) . k <= ((1 + (Partial_Sums (Prob * (B ^\ n)))) ") . k by VALUED_1:10; ::_thesis: verum end; A41: Partial_Sums (Prob * (B ^\ n)) is divergent_to+infty proof percases ( n = 0 or n <> 0 ) ; suppose n = 0 ; ::_thesis: Partial_Sums (Prob * (B ^\ n)) is divergent_to+infty hence Partial_Sums (Prob * (B ^\ n)) is divergent_to+infty by A12, NAT_1:47; ::_thesis: verum end; suppose n <> 0 ; ::_thesis: Partial_Sums (Prob * (B ^\ n)) is divergent_to+infty then A42: n - 1 is Element of NAT by NAT_1:20; consider y being Element of NAT such that A43: y = n - 1 by A42; set B2 = NAT --> (- ((Partial_Sums (Prob * B)) . y)); A44: (Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y))) is divergent_to+infty by A12, LIMFUNC1:18; for r being Real ex q being Element of NAT st for m being Element of NAT st q <= m holds r < (Partial_Sums (Prob * (B ^\ n))) . m proof let r be Real; ::_thesis: ex q being Element of NAT st for m being Element of NAT st q <= m holds r < (Partial_Sums (Prob * (B ^\ n))) . m for r being Real ex q being Element of NAT st for m being Element of NAT st q <= m holds r < (Partial_Sums (Prob * (B ^\ n))) . m proof let r be Real; ::_thesis: ex q being Element of NAT st for m being Element of NAT st q <= m holds r < (Partial_Sums (Prob * (B ^\ n))) . m A45: for m being Element of NAT st n <= m holds ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . m = (Partial_Sums (Prob * (B ^\ n))) . (m - n) proof let m be Element of NAT ; ::_thesis: ( n <= m implies ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . m = (Partial_Sums (Prob * (B ^\ n))) . (m - n) ) assume n <= m ; ::_thesis: ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . m = (Partial_Sums (Prob * (B ^\ n))) . (m - n) then consider knat being Nat such that A46: m = n + knat by NAT_1:10; reconsider knat = knat as Element of NAT by ORDINAL1:def_12; defpred S1[ Element of NAT ] means ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . (n + \$1) = (Partial_Sums (Prob * (B ^\ n))) . ((n + \$1) - n); A47: S1[ 0 ] proof dom ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) = NAT by FUNCT_2:def_1; then ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . n = ((Partial_Sums (Prob * B)) . n) + ((NAT --> (- ((Partial_Sums (Prob * B)) . y))) . n) by VALUED_1:def_1; then ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . n = ((Partial_Sums (Prob * B)) . n) + (- ((Partial_Sums (Prob * B)) . (n - 1))) by A43, FUNCOP_1:7; then ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . n = ((Partial_Sums (Prob * B)) . n) - ((Partial_Sums (Prob * B)) . (n - 1)) ; then A48: ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . n = (((Partial_Sums (Prob * B)) . (n - 1)) + ((Prob * B) . ((n - 1) + 1))) - ((Partial_Sums (Prob * B)) . (n - 1)) by A42, SERIES_1:def_1; dom (Prob * (B ^\ n)) = NAT by FUNCT_2:def_1; then A49: (Prob * (B ^\ n)) . 0 = Prob . ((B ^\ n) . 0) by FUNCT_1:12; A50: (B ^\ n) . 0 = B . (0 + n) by NAT_1:def_3; dom (Prob * B) = NAT by FUNCT_2:def_1; then ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . n = (Prob * (B ^\ n)) . 0 by A50, A49, A48, FUNCT_1:12; hence S1[ 0 ] by SERIES_1:def_1; ::_thesis: verum end; A51: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A52: S1[k] ; ::_thesis: S1[k + 1] A53: dom ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) = NAT by FUNCT_2:def_1; ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . ((n + k) + 1) = ((Partial_Sums (Prob * B)) . ((n + k) + 1)) + ((NAT --> (- ((Partial_Sums (Prob * B)) . y))) . ((n + k) + 1)) by A53, VALUED_1:def_1; then ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . ((n + k) + 1) = (((Partial_Sums (Prob * B)) . (n + k)) + ((Prob * B) . ((n + k) + 1))) + ((NAT --> (- ((Partial_Sums (Prob * B)) . y))) . ((n + k) + 1)) by SERIES_1:def_1; then ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . ((n + k) + 1) = (((Partial_Sums (Prob * B)) . (n + k)) + ((Prob * B) . ((n + k) + 1))) + (- ((Partial_Sums (Prob * B)) . (n - 1))) by A43, FUNCOP_1:7; then ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . ((n + k) + 1) = (((Partial_Sums (Prob * B)) . (n + k)) + ((Prob * B) . ((n + k) + 1))) + ((NAT --> (- ((Partial_Sums (Prob * B)) . y))) . (n + k)) by A43, FUNCOP_1:7; then ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . ((n + k) + 1) = (((Partial_Sums (Prob * B)) . (n + k)) + ((NAT --> (- ((Partial_Sums (Prob * B)) . y))) . (n + k))) + ((Prob * B) . ((n + k) + 1)) ; then A54: ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . ((n + k) + 1) = ((Partial_Sums (Prob * (B ^\ n))) . ((n + k) - n)) + ((Prob * B) . ((n + k) + 1)) by A53, A52, VALUED_1:def_1; (Prob * (B ^\ n)) . (((n + k) - n) + 1) = (Prob * B) . ((n + k) + 1) proof dom (Prob * (B ^\ n)) = NAT by FUNCT_2:def_1; then A55: (Prob * (B ^\ n)) . (((n + k) - n) + 1) = Prob . ((B ^\ n) . (k + 1)) by FUNCT_1:12; A56: (B ^\ n) . (k + 1) = B . (n + (k + 1)) by NAT_1:def_3; dom (Prob * B) = NAT by FUNCT_2:def_1; hence (Prob * (B ^\ n)) . (((n + k) - n) + 1) = (Prob * B) . ((n + k) + 1) by A56, A55, FUNCT_1:12; ::_thesis: verum end; hence S1[k + 1] by A54, SERIES_1:def_1; ::_thesis: verum end; for k being Element of NAT holds S1[k] from NAT_1:sch_1(A47, A51); then ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . (n + knat) = (Partial_Sums (Prob * (B ^\ n))) . ((n + knat) - n) ; hence ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . m = (Partial_Sums (Prob * (B ^\ n))) . (m - n) by A46; ::_thesis: verum end; A57: ex q being Element of NAT st for m being Element of NAT st q + n <= m + n holds r < ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . (m + n) proof consider q being Element of NAT such that A58: for m being Element of NAT st q <= m holds r < ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . m by A44, LIMFUNC1:def_4; for m being Element of NAT st q + n <= m + n holds r < ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . (m + n) proof let m be Element of NAT ; ::_thesis: ( q + n <= m + n implies r < ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . (m + n) ) assume q + n <= m + n ; ::_thesis: r < ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . (m + n) then ( q <= q + n & q + n <= m + n ) by NAT_1:11; then q <= m + n by XXREAL_0:2; hence r < ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . (m + n) by A58; ::_thesis: verum end; hence ex q being Element of NAT st for m being Element of NAT st q + n <= m + n holds r < ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . (m + n) ; ::_thesis: verum end; ex s being Element of NAT st for m being Element of NAT st s <= m holds r < (Partial_Sums (Prob * (B ^\ n))) . m proof consider q being Element of NAT such that A59: for m being Element of NAT st q + n <= m + n holds r < ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . (m + n) by A57; take s = q + n; ::_thesis: for m being Element of NAT st s <= m holds r < (Partial_Sums (Prob * (B ^\ n))) . m let m be Element of NAT ; ::_thesis: ( s <= m implies r < (Partial_Sums (Prob * (B ^\ n))) . m ) assume A60: s <= m ; ::_thesis: r < (Partial_Sums (Prob * (B ^\ n))) . m set z = m + n; ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . (m + n) = (Partial_Sums (Prob * (B ^\ n))) . ((m + n) - n) by A45, NAT_1:12; hence r < (Partial_Sums (Prob * (B ^\ n))) . m by A60, A59, NAT_1:12; ::_thesis: verum end; hence ex q being Element of NAT st for m being Element of NAT st q <= m holds r < (Partial_Sums (Prob * (B ^\ n))) . m ; ::_thesis: verum end; hence ex q being Element of NAT st for m being Element of NAT st q <= m holds r < (Partial_Sums (Prob * (B ^\ n))) . m ; ::_thesis: verum end; hence Partial_Sums (Prob * (B ^\ n)) is divergent_to+infty by LIMFUNC1:def_4; ::_thesis: verum end; end; end; A61: for A being SetSequence of Sigma st Partial_Sums (Prob * A) is divergent_to+infty holds ( lim ((1 + (Partial_Sums (Prob * A))) ") = 0 & (1 + (Partial_Sums (Prob * A))) " is convergent ) proof let A be SetSequence of Sigma; ::_thesis: ( Partial_Sums (Prob * A) is divergent_to+infty implies ( lim ((1 + (Partial_Sums (Prob * A))) ") = 0 & (1 + (Partial_Sums (Prob * A))) " is convergent ) ) A62: for A being SetSequence of Sigma st ( for r being Real ex n being Element of NAT st for m being Element of NAT st n <= m holds r < (Partial_Sums (Prob * A)) . m ) holds for r being Real ex n being Element of NAT st for m being Element of NAT st n <= m holds r < (1 + (Partial_Sums (Prob * A))) . m proof let A be SetSequence of Sigma; ::_thesis: ( ( for r being Real ex n being Element of NAT st for m being Element of NAT st n <= m holds r < (Partial_Sums (Prob * A)) . m ) implies for r being Real ex n being Element of NAT st for m being Element of NAT st n <= m holds r < (1 + (Partial_Sums (Prob * A))) . m ) assume A63: for r being Real ex n being Element of NAT st for m being Element of NAT st n <= m holds r < (Partial_Sums (Prob * A)) . m ; ::_thesis: for r being Real ex n being Element of NAT st for m being Element of NAT st n <= m holds r < (1 + (Partial_Sums (Prob * A))) . m let r be Real; ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds r < (1 + (Partial_Sums (Prob * A))) . m consider n being Element of NAT such that A64: for m being Element of NAT st n <= m holds r < (Partial_Sums (Prob * A)) . m by A63; take n ; ::_thesis: for m being Element of NAT st n <= m holds r < (1 + (Partial_Sums (Prob * A))) . m for m being Element of NAT st n <= m holds r < (1 + (Partial_Sums (Prob * A))) . m proof let m be Element of NAT ; ::_thesis: ( n <= m implies r < (1 + (Partial_Sums (Prob * A))) . m ) assume n <= m ; ::_thesis: r < (1 + (Partial_Sums (Prob * A))) . m then A65: r < (Partial_Sums (Prob * A)) . m by A64; A66: (Partial_Sums (Prob * A)) . m < ((Partial_Sums (Prob * A)) . m) + 1 by XREAL_1:29; (1 + (Partial_Sums (Prob * A))) . m = ((Partial_Sums (Prob * A)) . m) + 1 by VALUED_1:2; hence r < (1 + (Partial_Sums (Prob * A))) . m by A65, A66, XXREAL_0:2; ::_thesis: verum end; hence for m being Element of NAT st n <= m holds r < (1 + (Partial_Sums (Prob * A))) . m ; ::_thesis: verum end; assume Partial_Sums (Prob * A) is divergent_to+infty ; ::_thesis: ( lim ((1 + (Partial_Sums (Prob * A))) ") = 0 & (1 + (Partial_Sums (Prob * A))) " is convergent ) then for r being Real ex n being Element of NAT st for m being Element of NAT st n <= m holds r < (Partial_Sums (Prob * A)) . m by LIMFUNC1:def_4; then for r being Real ex n being Element of NAT st for m being Element of NAT st n <= m holds r < (1 + (Partial_Sums (Prob * A))) . m by A62; then 1 + (Partial_Sums (Prob * A)) is divergent_to+infty by LIMFUNC1:def_4; hence ( lim ((1 + (Partial_Sums (Prob * A))) ") = 0 & (1 + (Partial_Sums (Prob * A))) " is convergent ) by LIMFUNC1:34; ::_thesis: verum end; Partial_Intersection (Complement (B ^\ n)) is non-ascending by PROB_3:27; then A67: ( Prob * (Partial_Intersection (Complement (B ^\ n))) is convergent & (1 + (Partial_Sums (Prob * (B ^\ n)))) " is convergent ) by A41, A61, PROB_1:def_8; A68: lim ((1 + (Partial_Sums (Prob * (B ^\ n)))) ") = 0 by A41, A61; A69: for k being Element of NAT holds 0 <= (Prob * (Partial_Intersection (Complement (B ^\ n)))) . k proof let k be Element of NAT ; ::_thesis: 0 <= (Prob * (Partial_Intersection (Complement (B ^\ n)))) . k dom (Prob * (Partial_Intersection (Complement (B ^\ n)))) = NAT by FUNCT_2:def_1; then (Prob * (Partial_Intersection (Complement (B ^\ n)))) . k = Prob . ((Partial_Intersection (Complement (B ^\ n))) . k) by FUNCT_1:12; hence 0 <= (Prob * (Partial_Intersection (Complement (B ^\ n)))) . k by PROB_1:def_8; ::_thesis: verum end; A70: lim (Prob * (Partial_Intersection (Complement (B ^\ n)))) <= 0 by A67, A20, A68, SEQ_2:18; Complement (B ^\ n) = (Complement B) ^\ n proof for k being set st k in NAT holds (Complement (B ^\ n)) . k = ((Complement B) ^\ n) . k proof let k be set ; ::_thesis: ( k in NAT implies (Complement (B ^\ n)) . k = ((Complement B) ^\ n) . k ) assume k in NAT ; ::_thesis: (Complement (B ^\ n)) . k = ((Complement B) ^\ n) . k then reconsider k = k as Element of NAT ; A71: (Complement (B ^\ n)) . k = ((B ^\ n) . k) ` by PROB_1:def_2; ((Complement B) ^\ n) . k = (Complement B) . (k + n) by NAT_1:def_3; then ((Complement B) ^\ n) . k = (B . (k + n)) ` by PROB_1:def_2; hence (Complement (B ^\ n)) . k = ((Complement B) ^\ n) . k by A71, NAT_1:def_3; ::_thesis: verum end; hence Complement (B ^\ n) = (Complement B) ^\ n by FUNCT_2:12; ::_thesis: verum end; hence (Prob * (Intersect_Shift_Seq (Complement B))) . n = 0 by A69, A67, A70, A19, SEQ_2:17; ::_thesis: verum end; set B2 = NAT --> 0; A72: ex n being Element of NAT st (NAT --> 0) . n = 0 proof take 1 ; ::_thesis: (NAT --> 0) . 1 = 0 thus (NAT --> 0) . 1 = 0 by FUNCOP_1:7; ::_thesis: verum end; A73: lim (NAT --> 0) = 0 by A72, SEQ_4:25; A74: ( NAT --> 0 is convergent & ex k being Element of NAT st for n being Element of NAT st k <= n holds (NAT --> 0) . n = (Prob * (Intersect_Shift_Seq (Complement B))) . n ) proof ex k being Element of NAT st for n being Element of NAT st k <= n holds (NAT --> 0) . n = (Prob * (Intersect_Shift_Seq (Complement B))) . n proof A75: for n being Element of NAT st n >= 0 holds (NAT --> 0) . n = (Prob * (Intersect_Shift_Seq (Complement B))) . n proof let n be Element of NAT ; ::_thesis: ( n >= 0 implies (NAT --> 0) . n = (Prob * (Intersect_Shift_Seq (Complement B))) . n ) assume n >= 0 ; ::_thesis: (NAT --> 0) . n = (Prob * (Intersect_Shift_Seq (Complement B))) . n ( (NAT --> 0) . n = 0 & (Prob * (Intersect_Shift_Seq (Complement B))) . n = 0 ) by A16, FUNCOP_1:7; hence (NAT --> 0) . n = (Prob * (Intersect_Shift_Seq (Complement B))) . n ; ::_thesis: verum end; take 0 ; ::_thesis: for n being Element of NAT st 0 <= n holds (NAT --> 0) . n = (Prob * (Intersect_Shift_Seq (Complement B))) . n thus for n being Element of NAT st 0 <= n holds (NAT --> 0) . n = (Prob * (Intersect_Shift_Seq (Complement B))) . n by A75; ::_thesis: verum end; hence ( NAT --> 0 is convergent & ex k being Element of NAT st for n being Element of NAT st k <= n holds (NAT --> 0) . n = (Prob * (Intersect_Shift_Seq (Complement B))) . n ) ; ::_thesis: verum end; ( Prob . (@lim_inf (Complement B)) = 0 & (Prob . (@lim_inf (Complement B))) + (Prob . (@lim_sup B)) = 1 ) by A15, A74, A73, Th15, SEQ_4:19; hence ( Prob . (@lim_inf (Complement B)) = 0 & Prob . (@lim_sup B) = 1 ) ; ::_thesis: verum end; hence ( Prob . (lim_inf (Complement B)) = 0 & Prob . (lim_sup B) = 1 ) by A9, A10, A7, A8; ::_thesis: verum end; hence ( ( Partial_Sums (Prob * A) is convergent implies ( Prob . (lim_sup A) = 0 & Prob . (lim_inf (Complement A)) = 1 ) ) & ( A is_all_independent_wrt Prob & Partial_Sums (Prob * A) is divergent_to+infty implies ( Prob . (lim_inf (Complement A)) = 0 & Prob . (lim_sup A) = 1 ) ) ) by A5, A1; ::_thesis: verum end; theorem Th17: :: BOR_CANT:17 for Omega being non empty set for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma st not Partial_Sums (Prob * A) is convergent & A is_all_independent_wrt Prob holds ( Prob . (lim_inf (Complement A)) = 0 & Prob . (lim_sup A) = 1 ) proof let Omega be non empty set ; ::_thesis: for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma st not Partial_Sums (Prob * A) is convergent & A is_all_independent_wrt Prob holds ( Prob . (lim_inf (Complement A)) = 0 & Prob . (lim_sup A) = 1 ) let Sigma be SigmaField of Omega; ::_thesis: for Prob being Probability of Sigma for A being SetSequence of Sigma st not Partial_Sums (Prob * A) is convergent & A is_all_independent_wrt Prob holds ( Prob . (lim_inf (Complement A)) = 0 & Prob . (lim_sup A) = 1 ) let Prob be Probability of Sigma; ::_thesis: for A being SetSequence of Sigma st not Partial_Sums (Prob * A) is convergent & A is_all_independent_wrt Prob holds ( Prob . (lim_inf (Complement A)) = 0 & Prob . (lim_sup A) = 1 ) let A be SetSequence of Sigma; ::_thesis: ( not Partial_Sums (Prob * A) is convergent & A is_all_independent_wrt Prob implies ( Prob . (lim_inf (Complement A)) = 0 & Prob . (lim_sup A) = 1 ) ) assume A1: not Partial_Sums (Prob * A) is convergent ; ::_thesis: ( not A is_all_independent_wrt Prob or ( Prob . (lim_inf (Complement A)) = 0 & Prob . (lim_sup A) = 1 ) ) assume A2: A is_all_independent_wrt Prob ; ::_thesis: ( Prob . (lim_inf (Complement A)) = 0 & Prob . (lim_sup A) = 1 ) A3: for n being Element of NAT holds (Prob * A) . n >= 0 proof let n be Element of NAT ; ::_thesis: (Prob * A) . n >= 0 dom (Prob * A) = NAT by FUNCT_2:def_1; then (Prob * A) . n = Prob . (A . n) by FUNCT_1:12; hence (Prob * A) . n >= 0 by PROB_1:def_8; ::_thesis: verum end; A4: ( ( not Prob * A is summable implies not Partial_Sums (Prob * A) is bounded_above ) & not Prob * A is summable ) by A3, A1, SERIES_1:17, SERIES_1:def_2; Partial_Sums (Prob * A) is divergent_to+infty by A4, A3, LIMFUNC1:29, SERIES_1:16; hence ( Prob . (lim_inf (Complement A)) = 0 & Prob . (lim_sup A) = 1 ) by A2, Th16; ::_thesis: verum end; theorem :: BOR_CANT:18 for Omega being non empty set for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma st A is_all_independent_wrt Prob holds ( ( Prob . (lim_inf (Complement A)) = 0 or Prob . (lim_inf (Complement A)) = 1 ) & ( Prob . (lim_sup A) = 0 or Prob . (lim_sup A) = 1 ) ) proof let Omega be non empty set ; ::_thesis: for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma st A is_all_independent_wrt Prob holds ( ( Prob . (lim_inf (Complement A)) = 0 or Prob . (lim_inf (Complement A)) = 1 ) & ( Prob . (lim_sup A) = 0 or Prob . (lim_sup A) = 1 ) ) let Sigma be SigmaField of Omega; ::_thesis: for Prob being Probability of Sigma for A being SetSequence of Sigma st A is_all_independent_wrt Prob holds ( ( Prob . (lim_inf (Complement A)) = 0 or Prob . (lim_inf (Complement A)) = 1 ) & ( Prob . (lim_sup A) = 0 or Prob . (lim_sup A) = 1 ) ) let Prob be Probability of Sigma; ::_thesis: for A being SetSequence of Sigma st A is_all_independent_wrt Prob holds ( ( Prob . (lim_inf (Complement A)) = 0 or Prob . (lim_inf (Complement A)) = 1 ) & ( Prob . (lim_sup A) = 0 or Prob . (lim_sup A) = 1 ) ) let A be SetSequence of Sigma; ::_thesis: ( A is_all_independent_wrt Prob implies ( ( Prob . (lim_inf (Complement A)) = 0 or Prob . (lim_inf (Complement A)) = 1 ) & ( Prob . (lim_sup A) = 0 or Prob . (lim_sup A) = 1 ) ) ) assume A1: A is_all_independent_wrt Prob ; ::_thesis: ( ( Prob . (lim_inf (Complement A)) = 0 or Prob . (lim_inf (Complement A)) = 1 ) & ( Prob . (lim_sup A) = 0 or Prob . (lim_sup A) = 1 ) ) percases ( Partial_Sums (Prob * A) is convergent or not Partial_Sums (Prob * A) is convergent ) ; suppose Partial_Sums (Prob * A) is convergent ; ::_thesis: ( ( Prob . (lim_inf (Complement A)) = 0 or Prob . (lim_inf (Complement A)) = 1 ) & ( Prob . (lim_sup A) = 0 or Prob . (lim_sup A) = 1 ) ) hence ( ( Prob . (lim_inf (Complement A)) = 0 or Prob . (lim_inf (Complement A)) = 1 ) & ( Prob . (lim_sup A) = 0 or Prob . (lim_sup A) = 1 ) ) by Th16; ::_thesis: verum end; suppose not Partial_Sums (Prob * A) is convergent ; ::_thesis: ( ( Prob . (lim_inf (Complement A)) = 0 or Prob . (lim_inf (Complement A)) = 1 ) & ( Prob . (lim_sup A) = 0 or Prob . (lim_sup A) = 1 ) ) hence ( ( Prob . (lim_inf (Complement A)) = 0 or Prob . (lim_inf (Complement A)) = 1 ) & ( Prob . (lim_sup A) = 0 or Prob . (lim_sup A) = 1 ) ) by A1, Th17; ::_thesis: verum end; end; end; theorem :: BOR_CANT:19 for Omega being non empty set for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma for n1, n being Element of NAT holds (Partial_Sums (Prob * (A ^\ (n1 + 1)))) . n <= ((Partial_Sums (Prob * A)) . ((n1 + 1) + n)) - ((Partial_Sums (Prob * A)) . n1) proof let Omega be non empty set ; ::_thesis: for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma for n1, n being Element of NAT holds (Partial_Sums (Prob * (A ^\ (n1 + 1)))) . n <= ((Partial_Sums (Prob * A)) . ((n1 + 1) + n)) - ((Partial_Sums (Prob * A)) . n1) let Sigma be SigmaField of Omega; ::_thesis: for Prob being Probability of Sigma for A being SetSequence of Sigma for n1, n being Element of NAT holds (Partial_Sums (Prob * (A ^\ (n1 + 1)))) . n <= ((Partial_Sums (Prob * A)) . ((n1 + 1) + n)) - ((Partial_Sums (Prob * A)) . n1) let Prob be Probability of Sigma; ::_thesis: for A being SetSequence of Sigma for n1, n being Element of NAT holds (Partial_Sums (Prob * (A ^\ (n1 + 1)))) . n <= ((Partial_Sums (Prob * A)) . ((n1 + 1) + n)) - ((Partial_Sums (Prob * A)) . n1) let A be SetSequence of Sigma; ::_thesis: for n1, n being Element of NAT holds (Partial_Sums (Prob * (A ^\ (n1 + 1)))) . n <= ((Partial_Sums (Prob * A)) . ((n1 + 1) + n)) - ((Partial_Sums (Prob * A)) . n1) let n1, n be Element of NAT ; ::_thesis: (Partial_Sums (Prob * (A ^\ (n1 + 1)))) . n <= ((Partial_Sums (Prob * A)) . ((n1 + 1) + n)) - ((Partial_Sums (Prob * A)) . n1) A1: dom (Prob * (A ^\ (n1 + 1))) = NAT by FUNCT_2:def_1; A2: dom (Prob * A) = NAT by FUNCT_2:def_1; defpred S1[ Element of NAT ] means (Partial_Sums (Prob * (A ^\ (n1 + 1)))) . \$1 <= ((Partial_Sums (Prob * A)) . ((\$1 + n1) + 1)) - ((Partial_Sums (Prob * A)) . n1); A3: ((Partial_Sums (Prob * A)) . (n1 + 1)) - ((Partial_Sums (Prob * A)) . n1) = (((Partial_Sums (Prob * A)) . n1) + ((Prob * A) . (n1 + 1))) - ((Partial_Sums (Prob * A)) . n1) by SERIES_1:def_1; A4: Prob . ((A ^\ (n1 + 1)) . 0) = Prob . (A . ((n1 + 1) + 0)) by NAT_1:def_3; A5: Prob . (A . (n1 + 1)) = (Prob * A) . (n1 + 1) by A2, FUNCT_1:12; A6: (Prob * (A ^\ (n1 + 1))) . 0 = (Prob * A) . (n1 + 1) by A1, A4, A5, FUNCT_1:12; A7: S1[ 0 ] by A6, A3, SERIES_1:def_1; A8: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A9: (Partial_Sums (Prob * (A ^\ (n1 + 1)))) . k <= ((Partial_Sums (Prob * A)) . ((k + n1) + 1)) - ((Partial_Sums (Prob * A)) . n1) ; ::_thesis: S1[k + 1] A10: ((Partial_Sums (Prob * (A ^\ (n1 + 1)))) . k) + ((Prob * (A ^\ (n1 + 1))) . (k + 1)) <= (((Partial_Sums (Prob * A)) . ((k + n1) + 1)) - ((Partial_Sums (Prob * A)) . n1)) + ((Prob * (A ^\ (n1 + 1))) . (k + 1)) by A9, XREAL_1:6; A11: (Partial_Sums (Prob * (A ^\ (n1 + 1)))) . (k + 1) <= (((Partial_Sums (Prob * A)) . ((k + n1) + 1)) - ((Partial_Sums (Prob * A)) . n1)) + ((Prob * (A ^\ (n1 + 1))) . (k + 1)) by A10, SERIES_1:def_1; A12: ((Partial_Sums (Prob * (A ^\ (n1 + 1)))) . (k + 1)) + ((Partial_Sums (Prob * A)) . n1) <= ((((Partial_Sums (Prob * A)) . ((k + n1) + 1)) + ((Prob * (A ^\ (n1 + 1))) . (k + 1))) - ((Partial_Sums (Prob * A)) . n1)) + ((Partial_Sums (Prob * A)) . n1) by A11, XREAL_1:6; A13: (((Partial_Sums (Prob * (A ^\ (n1 + 1)))) . (k + 1)) + ((Partial_Sums (Prob * A)) . n1)) - ((Partial_Sums (Prob * A)) . ((k + n1) + 1)) <= (((Prob * (A ^\ (n1 + 1))) . (k + 1)) + ((Partial_Sums (Prob * A)) . ((k + n1) + 1))) - ((Partial_Sums (Prob * A)) . ((k + n1) + 1)) by A12, XREAL_1:9; A14: (A ^\ (n1 + 1)) . (k + 1) = A . ((n1 + 1) + (k + 1)) by NAT_1:def_3; A15: dom (Prob * A) = NAT by FUNCT_2:def_1; A16: dom (Prob * (A ^\ (n1 + 1))) = NAT by FUNCT_2:def_1; A17: Prob . ((A ^\ (n1 + 1)) . (k + 1)) = (Prob * (A ^\ (n1 + 1))) . (k + 1) by A16, FUNCT_1:12; A18: (((Partial_Sums (Prob * (A ^\ (n1 + 1)))) . (k + 1)) + ((Partial_Sums (Prob * A)) . n1)) - ((Partial_Sums (Prob * A)) . ((k + n1) + 1)) <= (Prob * A) . ((n1 + k) + 2) by A13, A17, A14, A15, FUNCT_1:12; A19: ((((Partial_Sums (Prob * (A ^\ (n1 + 1)))) . (k + 1)) + ((Partial_Sums (Prob * A)) . n1)) - ((Partial_Sums (Prob * A)) . ((k + n1) + 1))) + ((Partial_Sums (Prob * A)) . ((k + n1) + 1)) <= ((Prob * A) . ((n1 + k) + 2)) + ((Partial_Sums (Prob * A)) . ((k + n1) + 1)) by A18, XREAL_1:6; A20: (Partial_Sums (Prob * A)) . (((k + n1) + 1) + 1) = ((Partial_Sums (Prob * A)) . ((k + n1) + 1)) + ((Prob * A) . (((k + n1) + 1) + 1)) by SERIES_1:def_1; (((Partial_Sums (Prob * (A ^\ (n1 + 1)))) . (k + 1)) + ((Partial_Sums (Prob * A)) . n1)) - ((Partial_Sums (Prob * A)) . n1) <= ((Partial_Sums (Prob * A)) . ((k + n1) + 2)) - ((Partial_Sums (Prob * A)) . n1) by A19, A20, XREAL_1:9; hence S1[k + 1] ; ::_thesis: verum end; for n being Element of NAT holds S1[n] from NAT_1:sch_1(A7, A8); then S1[n] ; hence (Partial_Sums (Prob * (A ^\ (n1 + 1)))) . n <= ((Partial_Sums (Prob * A)) . ((n1 + 1) + n)) - ((Partial_Sums (Prob * A)) . n1) ; ::_thesis: verum end; theorem :: BOR_CANT:20 for Omega being non empty set for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma for n being Element of NAT holds Prob . ((Intersect_Shift_Seq (Complement A)) . n) = 1 - (Prob . ((Union_Shift_Seq A) . n)) proof let Omega be non empty set ; ::_thesis: for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma for n being Element of NAT holds Prob . ((Intersect_Shift_Seq (Complement A)) . n) = 1 - (Prob . ((Union_Shift_Seq A) . n)) let Sigma be SigmaField of Omega; ::_thesis: for Prob being Probability of Sigma for A being SetSequence of Sigma for n being Element of NAT holds Prob . ((Intersect_Shift_Seq (Complement A)) . n) = 1 - (Prob . ((Union_Shift_Seq A) . n)) let Prob be Probability of Sigma; ::_thesis: for A being SetSequence of Sigma for n being Element of NAT holds Prob . ((Intersect_Shift_Seq (Complement A)) . n) = 1 - (Prob . ((Union_Shift_Seq A) . n)) let A be SetSequence of Sigma; ::_thesis: for n being Element of NAT holds Prob . ((Intersect_Shift_Seq (Complement A)) . n) = 1 - (Prob . ((Union_Shift_Seq A) . n)) let n be Element of NAT ; ::_thesis: Prob . ((Intersect_Shift_Seq (Complement A)) . n) = 1 - (Prob . ((Union_Shift_Seq A) . n)) A1: Prob . ((Intersect_Shift_Seq (Complement A)) . n) = Prob . (((Union_Shift_Seq A) . n) `) by Th9; Prob . (((Union_Shift_Seq A) . n) `) = Prob . (([#] Sigma) \ ((Union_Shift_Seq A) . n)) by SUBSET_1:def_4; hence Prob . ((Intersect_Shift_Seq (Complement A)) . n) = 1 - (Prob . ((Union_Shift_Seq A) . n)) by A1, PROB_1:32; ::_thesis: verum end; theorem :: BOR_CANT:21 for Omega being non empty set for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma for n being Element of NAT holds ( ( Complement A is_all_independent_wrt Prob implies Prob . ((Partial_Intersection A) . n) = (Partial_Product (Prob * A)) . n ) & ( A is_all_independent_wrt Prob implies 1 - (Prob . ((Partial_Union A) . n)) = (Partial_Product (Prob * (Complement A))) . n ) ) proof let Omega be non empty set ; ::_thesis: for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma for n being Element of NAT holds ( ( Complement A is_all_independent_wrt Prob implies Prob . ((Partial_Intersection A) . n) = (Partial_Product (Prob * A)) . n ) & ( A is_all_independent_wrt Prob implies 1 - (Prob . ((Partial_Union A) . n)) = (Partial_Product (Prob * (Complement A))) . n ) ) let Sigma be SigmaField of Omega; ::_thesis: for Prob being Probability of Sigma for A being SetSequence of Sigma for n being Element of NAT holds ( ( Complement A is_all_independent_wrt Prob implies Prob . ((Partial_Intersection A) . n) = (Partial_Product (Prob * A)) . n ) & ( A is_all_independent_wrt Prob implies 1 - (Prob . ((Partial_Union A) . n)) = (Partial_Product (Prob * (Complement A))) . n ) ) let Prob be Probability of Sigma; ::_thesis: for A being SetSequence of Sigma for n being Element of NAT holds ( ( Complement A is_all_independent_wrt Prob implies Prob . ((Partial_Intersection A) . n) = (Partial_Product (Prob * A)) . n ) & ( A is_all_independent_wrt Prob implies 1 - (Prob . ((Partial_Union A) . n)) = (Partial_Product (Prob * (Complement A))) . n ) ) let A be SetSequence of Sigma; ::_thesis: for n being Element of NAT holds ( ( Complement A is_all_independent_wrt Prob implies Prob . ((Partial_Intersection A) . n) = (Partial_Product (Prob * A)) . n ) & ( A is_all_independent_wrt Prob implies 1 - (Prob . ((Partial_Union A) . n)) = (Partial_Product (Prob * (Complement A))) . n ) ) let n be Element of NAT ; ::_thesis: ( ( Complement A is_all_independent_wrt Prob implies Prob . ((Partial_Intersection A) . n) = (Partial_Product (Prob * A)) . n ) & ( A is_all_independent_wrt Prob implies 1 - (Prob . ((Partial_Union A) . n)) = (Partial_Product (Prob * (Complement A))) . n ) ) thus ( Complement A is_all_independent_wrt Prob implies Prob . ((Partial_Intersection A) . n) = (Partial_Product (Prob * A)) . n ) ::_thesis: ( A is_all_independent_wrt Prob implies 1 - (Prob . ((Partial_Union A) . n)) = (Partial_Product (Prob * (Complement A))) . n ) proof assume A1: Complement A is_all_independent_wrt Prob ; ::_thesis: Prob . ((Partial_Intersection A) . n) = (Partial_Product (Prob * A)) . n ( (Partial_Intersection (Complement (Complement A))) . n = (Partial_Intersection A) . n & (Partial_Product (Prob * (Complement (Complement A)))) . n = (Partial_Product (Prob * A)) . n & Prob . ((Partial_Intersection (Complement (Complement A))) . n) = (Partial_Product (Prob * (Complement (Complement A)))) . n ) by A1, Th10; hence Prob . ((Partial_Intersection A) . n) = (Partial_Product (Prob * A)) . n ; ::_thesis: verum end; assume A is_all_independent_wrt Prob ; ::_thesis: 1 - (Prob . ((Partial_Union A) . n)) = (Partial_Product (Prob * (Complement A))) . n then ( Prob . ((Partial_Intersection (Complement A)) . n) = (Partial_Product (Prob * (Complement A))) . n & Prob . ((Partial_Intersection (Complement A)) . n) = 1 - (Prob . ((Partial_Union A) . n)) ) by Th10, Th8; hence 1 - (Prob . ((Partial_Union A) . n)) = (Partial_Product (Prob * (Complement A))) . n ; ::_thesis: verum end;