:: JORDAN1G semantic presentation begin registration cluster trivial V13() V16( NAT ) Function-like V26() FinSequence-like FinSubsequence-like for set ; existence ex b1 being FinSequence st b1 is trivial proof take {} ; ::_thesis: {} is trivial thus {} is trivial ; ::_thesis: verum end; end; theorem Th1: :: JORDAN1G:1 for f being trivial FinSequence holds ( f is empty or ex x being set st f = <*x*> ) proof let f be trivial FinSequence; ::_thesis: ( f is empty or ex x being set st f = <*x*> ) assume not f is empty ; ::_thesis: ex x being set st f = <*x*> then consider x being set such that A1: f = {x} by ZFMISC_1:131; x in {x} by TARSKI:def_1; then consider y, z being set such that A2: x = [y,z] by A1, RELAT_1:def_1; A3: 1 in dom f by A1, FINSEQ_5:6; take z ; ::_thesis: f = <*z*> dom f = {y} by A1, A2, RELAT_1:9; then 1 = y by A3, TARSKI:def_1; hence f = <*z*> by A1, A2, FINSEQ_1:def_5; ::_thesis: verum end; registration let p be non trivial FinSequence; cluster Rev p -> non trivial ; coherence not Rev p is trivial proof assume A1: Rev p is trivial ; ::_thesis: contradiction percases ( Rev p is empty or ex x being set st Rev p = <*x*> ) by A1, Th1; suppose Rev p is empty ; ::_thesis: contradiction hence contradiction ; ::_thesis: verum end; suppose ex x being set st Rev p = <*x*> ; ::_thesis: contradiction then consider x being set such that A2: Rev p = <*x*> ; p = Rev <*x*> by A2 .= <*x*> by FINSEQ_5:60 ; hence contradiction ; ::_thesis: verum end; end; end; end; theorem Th2: :: JORDAN1G:2 for D being non empty set for f being FinSequence of D for G being Matrix of D for p being set st f is_sequence_on G holds f -: p is_sequence_on G proof let D be non empty set ; ::_thesis: for f being FinSequence of D for G being Matrix of D for p being set st f is_sequence_on G holds f -: p is_sequence_on G let f be FinSequence of D; ::_thesis: for G being Matrix of D for p being set st f is_sequence_on G holds f -: p is_sequence_on G let G be Matrix of D; ::_thesis: for p being set st f is_sequence_on G holds f -: p is_sequence_on G let p be set ; ::_thesis: ( f is_sequence_on G implies f -: p is_sequence_on G ) assume f is_sequence_on G ; ::_thesis: f -: p is_sequence_on G then f | (p .. f) is_sequence_on G by GOBOARD1:22; hence f -: p is_sequence_on G by FINSEQ_5:def_1; ::_thesis: verum end; theorem Th3: :: JORDAN1G:3 for D being non empty set for f being FinSequence of D for G being Matrix of D for p being Element of D st p in rng f & f is_sequence_on G holds f :- p is_sequence_on G proof let D be non empty set ; ::_thesis: for f being FinSequence of D for G being Matrix of D for p being Element of D st p in rng f & f is_sequence_on G holds f :- p is_sequence_on G let f be FinSequence of D; ::_thesis: for G being Matrix of D for p being Element of D st p in rng f & f is_sequence_on G holds f :- p is_sequence_on G let G be Matrix of D; ::_thesis: for p being Element of D st p in rng f & f is_sequence_on G holds f :- p is_sequence_on G let p be Element of D; ::_thesis: ( p in rng f & f is_sequence_on G implies f :- p is_sequence_on G ) assume that A1: p in rng f and A2: f is_sequence_on G ; ::_thesis: f :- p is_sequence_on G ex i being Element of NAT st ( i + 1 = p .. f & f :- p = f /^ i ) by A1, FINSEQ_5:49; hence f :- p is_sequence_on G by A2, JORDAN8:2; ::_thesis: verum end; theorem Th4: :: JORDAN1G:4 for n being Element of NAT for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds Upper_Seq (C,n) is_sequence_on Gauge (C,n) proof let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds Upper_Seq (C,n) is_sequence_on Gauge (C,n) let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: Upper_Seq (C,n) is_sequence_on Gauge (C,n) Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def_1; then Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))) is_sequence_on Gauge (C,n) by REVROT_1:34; then (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n)))) is_sequence_on Gauge (C,n) by Th2; hence Upper_Seq (C,n) is_sequence_on Gauge (C,n) by JORDAN1E:def_1; ::_thesis: verum end; theorem Th5: :: JORDAN1G:5 for n being Element of NAT for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds Lower_Seq (C,n) is_sequence_on Gauge (C,n) proof let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds Lower_Seq (C,n) is_sequence_on Gauge (C,n) let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: Lower_Seq (C,n) is_sequence_on Gauge (C,n) Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def_1; then A1: Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))) is_sequence_on Gauge (C,n) by REVROT_1:34; E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46; then E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by FINSEQ_6:90, SPRECT_2:43; then (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) :- (E-max (L~ (Cage (C,n)))) is_sequence_on Gauge (C,n) by A1, Th3; hence Lower_Seq (C,n) is_sequence_on Gauge (C,n) by JORDAN1E:def_2; ::_thesis: verum end; registration let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); let n be Element of NAT ; cluster Upper_Seq (C,n) -> standard ; coherence Upper_Seq (C,n) is standard proof Upper_Seq (C,n) is_sequence_on Gauge (C,n) by Th4; hence Upper_Seq (C,n) is standard by JORDAN8:4; ::_thesis: verum end; cluster Lower_Seq (C,n) -> standard ; coherence Lower_Seq (C,n) is standard proof Lower_Seq (C,n) is_sequence_on Gauge (C,n) by Th5; hence Lower_Seq (C,n) is standard by JORDAN8:4; ::_thesis: verum end; end; theorem Th6: :: JORDAN1G:6 for G being Y_equal-in-column Y_increasing-in-line Matrix of (TOP-REAL 2) for i1, i2, j1, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & (G * (i1,j1)) `2 = (G * (i2,j2)) `2 holds j1 = j2 proof let G be Y_equal-in-column Y_increasing-in-line Matrix of (TOP-REAL 2); ::_thesis: for i1, i2, j1, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & (G * (i1,j1)) `2 = (G * (i2,j2)) `2 holds j1 = j2 let i1, i2, j1, j2 be Element of NAT ; ::_thesis: ( [i1,j1] in Indices G & [i2,j2] in Indices G & (G * (i1,j1)) `2 = (G * (i2,j2)) `2 implies j1 = j2 ) assume that A1: [i1,j1] in Indices G and A2: [i2,j2] in Indices G and A3: (G * (i1,j1)) `2 = (G * (i2,j2)) `2 and A4: j1 <> j2 ; ::_thesis: contradiction A5: ( 1 <= j1 & j1 <= width G ) by A1, MATRIX_1:38; A6: ( j1 < j2 or j1 > j2 ) by A4, XXREAL_0:1; A7: ( 1 <= i2 & i2 <= len G ) by A2, MATRIX_1:38; A8: ( 1 <= j2 & j2 <= width G ) by A2, MATRIX_1:38; A9: ( 1 <= i1 & i1 <= len G ) by A1, MATRIX_1:38; then (G * (i1,j2)) `2 = (G * (1,j2)) `2 by A8, GOBOARD5:1 .= (G * (i2,j2)) `2 by A7, A8, GOBOARD5:1 ; hence contradiction by A3, A9, A5, A8, A6, GOBOARD5:4; ::_thesis: verum end; theorem Th7: :: JORDAN1G:7 for G being X_equal-in-line X_increasing-in-column Matrix of (TOP-REAL 2) for i1, i2, j1, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & (G * (i1,j1)) `1 = (G * (i2,j2)) `1 holds i1 = i2 proof let G be X_equal-in-line X_increasing-in-column Matrix of (TOP-REAL 2); ::_thesis: for i1, i2, j1, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & (G * (i1,j1)) `1 = (G * (i2,j2)) `1 holds i1 = i2 let i1, i2, j1, j2 be Element of NAT ; ::_thesis: ( [i1,j1] in Indices G & [i2,j2] in Indices G & (G * (i1,j1)) `1 = (G * (i2,j2)) `1 implies i1 = i2 ) assume that A1: [i1,j1] in Indices G and A2: [i2,j2] in Indices G and A3: (G * (i1,j1)) `1 = (G * (i2,j2)) `1 and A4: i1 <> i2 ; ::_thesis: contradiction A5: ( 1 <= i1 & i1 <= len G ) by A1, MATRIX_1:38; A6: ( 1 <= i2 & i2 <= len G ) by A2, MATRIX_1:38; A7: ( 1 <= j2 & j2 <= width G ) by A2, MATRIX_1:38; A8: ( i1 < i2 or i1 > i2 ) by A4, XXREAL_0:1; ( 1 <= j1 & j1 <= width G ) by A1, MATRIX_1:38; then (G * (i1,j1)) `1 = (G * (i1,1)) `1 by A5, GOBOARD5:2 .= (G * (i1,j2)) `1 by A5, A7, GOBOARD5:2 ; hence contradiction by A3, A5, A6, A7, A8, GOBOARD5:3; ::_thesis: verum end; theorem Th8: :: JORDAN1G:8 for f being non trivial special unfolded standard FinSequence of (TOP-REAL 2) st ( ( f /. 1 <> N-min (L~ f) & f /. (len f) <> N-min (L~ f) ) or ( f /. 1 <> N-max (L~ f) & f /. (len f) <> N-max (L~ f) ) ) holds (N-min (L~ f)) `1 < (N-max (L~ f)) `1 proof let f be non trivial special unfolded standard FinSequence of (TOP-REAL 2); ::_thesis: ( ( ( f /. 1 <> N-min (L~ f) & f /. (len f) <> N-min (L~ f) ) or ( f /. 1 <> N-max (L~ f) & f /. (len f) <> N-max (L~ f) ) ) implies (N-min (L~ f)) `1 < (N-max (L~ f)) `1 ) set p = N-min (L~ f); set i = (N-min (L~ f)) .. f; assume A1: ( ( f /. 1 <> N-min (L~ f) & f /. (len f) <> N-min (L~ f) ) or ( f /. 1 <> N-max (L~ f) & f /. (len f) <> N-max (L~ f) ) ) ; ::_thesis: (N-min (L~ f)) `1 < (N-max (L~ f)) `1 A2: len f >= 2 by NAT_D:60; A3: (N-min (L~ f)) `2 = N-bound (L~ f) by EUCLID:52; A4: N-min (L~ f) in rng f by SPRECT_2:39; then A5: (N-min (L~ f)) .. f in dom f by FINSEQ_4:20; then A6: ( 1 <= (N-min (L~ f)) .. f & (N-min (L~ f)) .. f <= len f ) by FINSEQ_3:25; A7: N-min (L~ f) = f . ((N-min (L~ f)) .. f) by A4, FINSEQ_4:19 .= f /. ((N-min (L~ f)) .. f) by A5, PARTFUN1:def_6 ; percases ( (N-min (L~ f)) .. f = 1 or (N-min (L~ f)) .. f = len f or ( 1 < (N-min (L~ f)) .. f & (N-min (L~ f)) .. f < len f ) ) by A6, XXREAL_0:1; supposeA8: (N-min (L~ f)) .. f = 1 ; ::_thesis: (N-min (L~ f)) `1 < (N-max (L~ f)) `1 (N-min (L~ f)) `2 = (N-max (L~ f)) `2 by PSCOMP_1:37; then A9: (N-min (L~ f)) `1 <> (N-max (L~ f)) `1 by A1, A7, A8, TOPREAL3:6; (N-min (L~ f)) `1 <= (N-max (L~ f)) `1 by PSCOMP_1:38; hence (N-min (L~ f)) `1 < (N-max (L~ f)) `1 by A9, XXREAL_0:1; ::_thesis: verum end; supposeA10: (N-min (L~ f)) .. f = len f ; ::_thesis: (N-min (L~ f)) `1 < (N-max (L~ f)) `1 (N-min (L~ f)) `2 = (N-max (L~ f)) `2 by PSCOMP_1:37; then A11: (N-min (L~ f)) `1 <> (N-max (L~ f)) `1 by A1, A7, A10, TOPREAL3:6; (N-min (L~ f)) `1 <= (N-max (L~ f)) `1 by PSCOMP_1:38; hence (N-min (L~ f)) `1 < (N-max (L~ f)) `1 by A11, XXREAL_0:1; ::_thesis: verum end; supposethat A12: 1 < (N-min (L~ f)) .. f and A13: (N-min (L~ f)) .. f < len f ; ::_thesis: (N-min (L~ f)) `1 < (N-max (L~ f)) `1 A14: (((N-min (L~ f)) .. f) -' 1) + 1 = (N-min (L~ f)) .. f by A12, XREAL_1:235; then A15: ((N-min (L~ f)) .. f) -' 1 >= 1 by A12, NAT_1:13; then A16: f /. (((N-min (L~ f)) .. f) -' 1) in LSeg (f,(((N-min (L~ f)) .. f) -' 1)) by A13, A14, TOPREAL1:21; ((N-min (L~ f)) .. f) -' 1 <= (N-min (L~ f)) .. f by A14, NAT_1:11; then ((N-min (L~ f)) .. f) -' 1 <= len f by A13, XXREAL_0:2; then A17: ((N-min (L~ f)) .. f) -' 1 in dom f by A15, FINSEQ_3:25; then A18: f /. (((N-min (L~ f)) .. f) -' 1) in L~ f by A2, GOBOARD1:1; A19: ((N-min (L~ f)) .. f) + 1 <= len f by A13, NAT_1:13; then A20: f /. (((N-min (L~ f)) .. f) + 1) in LSeg (f,((N-min (L~ f)) .. f)) by A12, TOPREAL1:21; ((N-min (L~ f)) .. f) + 1 >= 1 by NAT_1:11; then A21: ((N-min (L~ f)) .. f) + 1 in dom f by A19, FINSEQ_3:25; then A22: f /. (((N-min (L~ f)) .. f) + 1) in L~ f by A2, GOBOARD1:1; A23: N-min (L~ f) <> f /. (((N-min (L~ f)) .. f) + 1) by A4, A7, A21, FINSEQ_4:20, GOBOARD7:29; A24: N-min (L~ f) in LSeg (f,((N-min (L~ f)) .. f)) by A7, A12, A19, TOPREAL1:21; A25: N-min (L~ f) in LSeg (f,(((N-min (L~ f)) .. f) -' 1)) by A7, A13, A14, A15, TOPREAL1:21; A26: N-min (L~ f) <> f /. (((N-min (L~ f)) .. f) -' 1) by A5, A7, A14, A17, GOBOARD7:29; A27: ( not LSeg (f,(((N-min (L~ f)) .. f) -' 1)) is vertical or not LSeg (f,((N-min (L~ f)) .. f)) is vertical ) proof assume ( LSeg (f,(((N-min (L~ f)) .. f) -' 1)) is vertical & LSeg (f,((N-min (L~ f)) .. f)) is vertical ) ; ::_thesis: contradiction then A28: ( (N-min (L~ f)) `1 = (f /. (((N-min (L~ f)) .. f) + 1)) `1 & (N-min (L~ f)) `1 = (f /. (((N-min (L~ f)) .. f) -' 1)) `1 ) by A25, A24, A16, A20, SPPOL_1:def_3; A29: ( (f /. (((N-min (L~ f)) .. f) + 1)) `2 <= (f /. (((N-min (L~ f)) .. f) -' 1)) `2 or (f /. (((N-min (L~ f)) .. f) + 1)) `2 >= (f /. (((N-min (L~ f)) .. f) -' 1)) `2 ) ; A30: ( (N-min (L~ f)) `2 >= (f /. (((N-min (L~ f)) .. f) + 1)) `2 & (N-min (L~ f)) `2 >= (f /. (((N-min (L~ f)) .. f) -' 1)) `2 ) by A3, A18, A22, PSCOMP_1:24; ( LSeg (f,((N-min (L~ f)) .. f)) = LSeg ((f /. ((N-min (L~ f)) .. f)),(f /. (((N-min (L~ f)) .. f) + 1))) & LSeg (f,(((N-min (L~ f)) .. f) -' 1)) = LSeg ((f /. ((N-min (L~ f)) .. f)),(f /. (((N-min (L~ f)) .. f) -' 1))) ) by A12, A13, A14, A15, A19, TOPREAL1:def_3; then ( f /. (((N-min (L~ f)) .. f) -' 1) in LSeg (f,((N-min (L~ f)) .. f)) or f /. (((N-min (L~ f)) .. f) + 1) in LSeg (f,(((N-min (L~ f)) .. f) -' 1)) ) by A7, A28, A30, A29, GOBOARD7:7; then ( f /. (((N-min (L~ f)) .. f) -' 1) in (LSeg (f,(((N-min (L~ f)) .. f) -' 1))) /\ (LSeg (f,((N-min (L~ f)) .. f))) or f /. (((N-min (L~ f)) .. f) + 1) in (LSeg (f,(((N-min (L~ f)) .. f) -' 1))) /\ (LSeg (f,((N-min (L~ f)) .. f))) ) by A16, A20, XBOOLE_0:def_4; then ( ((((N-min (L~ f)) .. f) -' 1) + 1) + 1 = (((N-min (L~ f)) .. f) -' 1) + (1 + 1) & (LSeg (f,(((N-min (L~ f)) .. f) -' 1))) /\ (LSeg (f,((N-min (L~ f)) .. f))) <> {(f /. ((N-min (L~ f)) .. f))} ) by A7, A26, A23, TARSKI:def_1; hence contradiction by A14, A15, A19, TOPREAL1:def_6; ::_thesis: verum end; now__::_thesis:_(N-min_(L~_f))_`1_<_(N-max_(L~_f))_`1 percases ( LSeg (f,(((N-min (L~ f)) .. f) -' 1)) is horizontal or LSeg (f,((N-min (L~ f)) .. f)) is horizontal ) by A27, SPPOL_1:19; suppose LSeg (f,(((N-min (L~ f)) .. f) -' 1)) is horizontal ; ::_thesis: (N-min (L~ f)) `1 < (N-max (L~ f)) `1 then A31: (N-min (L~ f)) `2 = (f /. (((N-min (L~ f)) .. f) -' 1)) `2 by A25, A16, SPPOL_1:def_2; then A32: f /. (((N-min (L~ f)) .. f) -' 1) in N-most (L~ f) by A2, A3, A17, GOBOARD1:1, SPRECT_2:10; then A33: (f /. (((N-min (L~ f)) .. f) -' 1)) `1 >= (N-min (L~ f)) `1 by PSCOMP_1:39; (f /. (((N-min (L~ f)) .. f) -' 1)) `1 <> (N-min (L~ f)) `1 by A5, A7, A14, A17, A31, GOBOARD7:29, TOPREAL3:6; then A34: (f /. (((N-min (L~ f)) .. f) -' 1)) `1 > (N-min (L~ f)) `1 by A33, XXREAL_0:1; (f /. (((N-min (L~ f)) .. f) -' 1)) `1 <= (N-max (L~ f)) `1 by A32, PSCOMP_1:39; hence (N-min (L~ f)) `1 < (N-max (L~ f)) `1 by A34, XXREAL_0:2; ::_thesis: verum end; suppose LSeg (f,((N-min (L~ f)) .. f)) is horizontal ; ::_thesis: (N-min (L~ f)) `1 < (N-max (L~ f)) `1 then A35: (N-min (L~ f)) `2 = (f /. (((N-min (L~ f)) .. f) + 1)) `2 by A24, A20, SPPOL_1:def_2; then A36: f /. (((N-min (L~ f)) .. f) + 1) in N-most (L~ f) by A2, A3, A21, GOBOARD1:1, SPRECT_2:10; then A37: (f /. (((N-min (L~ f)) .. f) + 1)) `1 >= (N-min (L~ f)) `1 by PSCOMP_1:39; (f /. (((N-min (L~ f)) .. f) + 1)) `1 <> (N-min (L~ f)) `1 by A5, A7, A21, A35, GOBOARD7:29, TOPREAL3:6; then A38: (f /. (((N-min (L~ f)) .. f) + 1)) `1 > (N-min (L~ f)) `1 by A37, XXREAL_0:1; (f /. (((N-min (L~ f)) .. f) + 1)) `1 <= (N-max (L~ f)) `1 by A36, PSCOMP_1:39; hence (N-min (L~ f)) `1 < (N-max (L~ f)) `1 by A38, XXREAL_0:2; ::_thesis: verum end; end; end; hence (N-min (L~ f)) `1 < (N-max (L~ f)) `1 ; ::_thesis: verum end; end; end; theorem :: JORDAN1G:9 for f being non trivial special unfolded standard FinSequence of (TOP-REAL 2) st ( ( f /. 1 <> N-min (L~ f) & f /. (len f) <> N-min (L~ f) ) or ( f /. 1 <> N-max (L~ f) & f /. (len f) <> N-max (L~ f) ) ) holds N-min (L~ f) <> N-max (L~ f) proof let f be non trivial special unfolded standard FinSequence of (TOP-REAL 2); ::_thesis: ( ( ( f /. 1 <> N-min (L~ f) & f /. (len f) <> N-min (L~ f) ) or ( f /. 1 <> N-max (L~ f) & f /. (len f) <> N-max (L~ f) ) ) implies N-min (L~ f) <> N-max (L~ f) ) assume ( ( f /. 1 <> N-min (L~ f) & f /. (len f) <> N-min (L~ f) ) or ( f /. 1 <> N-max (L~ f) & f /. (len f) <> N-max (L~ f) ) ) ; ::_thesis: N-min (L~ f) <> N-max (L~ f) then (N-min (L~ f)) `1 < (N-max (L~ f)) `1 by Th8; hence N-min (L~ f) <> N-max (L~ f) ; ::_thesis: verum end; theorem Th10: :: JORDAN1G:10 for f being non trivial special unfolded standard FinSequence of (TOP-REAL 2) st ( ( f /. 1 <> S-min (L~ f) & f /. (len f) <> S-min (L~ f) ) or ( f /. 1 <> S-max (L~ f) & f /. (len f) <> S-max (L~ f) ) ) holds (S-min (L~ f)) `1 < (S-max (L~ f)) `1 proof let f be non trivial special unfolded standard FinSequence of (TOP-REAL 2); ::_thesis: ( ( ( f /. 1 <> S-min (L~ f) & f /. (len f) <> S-min (L~ f) ) or ( f /. 1 <> S-max (L~ f) & f /. (len f) <> S-max (L~ f) ) ) implies (S-min (L~ f)) `1 < (S-max (L~ f)) `1 ) set p = S-min (L~ f); set i = (S-min (L~ f)) .. f; assume A1: ( ( f /. 1 <> S-min (L~ f) & f /. (len f) <> S-min (L~ f) ) or ( f /. 1 <> S-max (L~ f) & f /. (len f) <> S-max (L~ f) ) ) ; ::_thesis: (S-min (L~ f)) `1 < (S-max (L~ f)) `1 A2: len f >= 2 by NAT_D:60; A3: (S-min (L~ f)) `2 = S-bound (L~ f) by EUCLID:52; A4: S-min (L~ f) in rng f by SPRECT_2:41; then A5: (S-min (L~ f)) .. f in dom f by FINSEQ_4:20; then A6: ( 1 <= (S-min (L~ f)) .. f & (S-min (L~ f)) .. f <= len f ) by FINSEQ_3:25; A7: S-min (L~ f) = f . ((S-min (L~ f)) .. f) by A4, FINSEQ_4:19 .= f /. ((S-min (L~ f)) .. f) by A5, PARTFUN1:def_6 ; percases ( (S-min (L~ f)) .. f = 1 or (S-min (L~ f)) .. f = len f or ( 1 < (S-min (L~ f)) .. f & (S-min (L~ f)) .. f < len f ) ) by A6, XXREAL_0:1; supposeA8: (S-min (L~ f)) .. f = 1 ; ::_thesis: (S-min (L~ f)) `1 < (S-max (L~ f)) `1 (S-min (L~ f)) `2 = (S-max (L~ f)) `2 by PSCOMP_1:53; then A9: (S-min (L~ f)) `1 <> (S-max (L~ f)) `1 by A1, A7, A8, TOPREAL3:6; (S-min (L~ f)) `1 <= (S-max (L~ f)) `1 by PSCOMP_1:54; hence (S-min (L~ f)) `1 < (S-max (L~ f)) `1 by A9, XXREAL_0:1; ::_thesis: verum end; supposeA10: (S-min (L~ f)) .. f = len f ; ::_thesis: (S-min (L~ f)) `1 < (S-max (L~ f)) `1 (S-min (L~ f)) `2 = (S-max (L~ f)) `2 by PSCOMP_1:53; then A11: (S-min (L~ f)) `1 <> (S-max (L~ f)) `1 by A1, A7, A10, TOPREAL3:6; (S-min (L~ f)) `1 <= (S-max (L~ f)) `1 by PSCOMP_1:54; hence (S-min (L~ f)) `1 < (S-max (L~ f)) `1 by A11, XXREAL_0:1; ::_thesis: verum end; supposethat A12: 1 < (S-min (L~ f)) .. f and A13: (S-min (L~ f)) .. f < len f ; ::_thesis: (S-min (L~ f)) `1 < (S-max (L~ f)) `1 A14: (((S-min (L~ f)) .. f) -' 1) + 1 = (S-min (L~ f)) .. f by A12, XREAL_1:235; then A15: ((S-min (L~ f)) .. f) -' 1 >= 1 by A12, NAT_1:13; then A16: f /. (((S-min (L~ f)) .. f) -' 1) in LSeg (f,(((S-min (L~ f)) .. f) -' 1)) by A13, A14, TOPREAL1:21; ((S-min (L~ f)) .. f) -' 1 <= (S-min (L~ f)) .. f by A14, NAT_1:11; then ((S-min (L~ f)) .. f) -' 1 <= len f by A13, XXREAL_0:2; then A17: ((S-min (L~ f)) .. f) -' 1 in dom f by A15, FINSEQ_3:25; then A18: f /. (((S-min (L~ f)) .. f) -' 1) in L~ f by A2, GOBOARD1:1; A19: ((S-min (L~ f)) .. f) + 1 <= len f by A13, NAT_1:13; then A20: f /. (((S-min (L~ f)) .. f) + 1) in LSeg (f,((S-min (L~ f)) .. f)) by A12, TOPREAL1:21; ((S-min (L~ f)) .. f) + 1 >= 1 by NAT_1:11; then A21: ((S-min (L~ f)) .. f) + 1 in dom f by A19, FINSEQ_3:25; then A22: f /. (((S-min (L~ f)) .. f) + 1) in L~ f by A2, GOBOARD1:1; A23: S-min (L~ f) <> f /. (((S-min (L~ f)) .. f) + 1) by A4, A7, A21, FINSEQ_4:20, GOBOARD7:29; A24: S-min (L~ f) in LSeg (f,((S-min (L~ f)) .. f)) by A7, A12, A19, TOPREAL1:21; A25: S-min (L~ f) in LSeg (f,(((S-min (L~ f)) .. f) -' 1)) by A7, A13, A14, A15, TOPREAL1:21; A26: S-min (L~ f) <> f /. (((S-min (L~ f)) .. f) -' 1) by A5, A7, A14, A17, GOBOARD7:29; A27: ( not LSeg (f,(((S-min (L~ f)) .. f) -' 1)) is vertical or not LSeg (f,((S-min (L~ f)) .. f)) is vertical ) proof assume ( LSeg (f,(((S-min (L~ f)) .. f) -' 1)) is vertical & LSeg (f,((S-min (L~ f)) .. f)) is vertical ) ; ::_thesis: contradiction then A28: ( (S-min (L~ f)) `1 = (f /. (((S-min (L~ f)) .. f) + 1)) `1 & (S-min (L~ f)) `1 = (f /. (((S-min (L~ f)) .. f) -' 1)) `1 ) by A25, A24, A16, A20, SPPOL_1:def_3; A29: ( (f /. (((S-min (L~ f)) .. f) + 1)) `2 <= (f /. (((S-min (L~ f)) .. f) -' 1)) `2 or (f /. (((S-min (L~ f)) .. f) + 1)) `2 >= (f /. (((S-min (L~ f)) .. f) -' 1)) `2 ) ; A30: ( (S-min (L~ f)) `2 <= (f /. (((S-min (L~ f)) .. f) + 1)) `2 & (S-min (L~ f)) `2 <= (f /. (((S-min (L~ f)) .. f) -' 1)) `2 ) by A3, A18, A22, PSCOMP_1:24; ( LSeg (f,((S-min (L~ f)) .. f)) = LSeg ((f /. ((S-min (L~ f)) .. f)),(f /. (((S-min (L~ f)) .. f) + 1))) & LSeg (f,(((S-min (L~ f)) .. f) -' 1)) = LSeg ((f /. ((S-min (L~ f)) .. f)),(f /. (((S-min (L~ f)) .. f) -' 1))) ) by A12, A13, A14, A15, A19, TOPREAL1:def_3; then ( f /. (((S-min (L~ f)) .. f) -' 1) in LSeg (f,((S-min (L~ f)) .. f)) or f /. (((S-min (L~ f)) .. f) + 1) in LSeg (f,(((S-min (L~ f)) .. f) -' 1)) ) by A7, A28, A30, A29, GOBOARD7:7; then ( f /. (((S-min (L~ f)) .. f) -' 1) in (LSeg (f,(((S-min (L~ f)) .. f) -' 1))) /\ (LSeg (f,((S-min (L~ f)) .. f))) or f /. (((S-min (L~ f)) .. f) + 1) in (LSeg (f,(((S-min (L~ f)) .. f) -' 1))) /\ (LSeg (f,((S-min (L~ f)) .. f))) ) by A16, A20, XBOOLE_0:def_4; then ( ((((S-min (L~ f)) .. f) -' 1) + 1) + 1 = (((S-min (L~ f)) .. f) -' 1) + (1 + 1) & (LSeg (f,(((S-min (L~ f)) .. f) -' 1))) /\ (LSeg (f,((S-min (L~ f)) .. f))) <> {(f /. ((S-min (L~ f)) .. f))} ) by A7, A26, A23, TARSKI:def_1; hence contradiction by A14, A15, A19, TOPREAL1:def_6; ::_thesis: verum end; now__::_thesis:_(S-min_(L~_f))_`1_<_(S-max_(L~_f))_`1 percases ( LSeg (f,(((S-min (L~ f)) .. f) -' 1)) is horizontal or LSeg (f,((S-min (L~ f)) .. f)) is horizontal ) by A27, SPPOL_1:19; suppose LSeg (f,(((S-min (L~ f)) .. f) -' 1)) is horizontal ; ::_thesis: (S-min (L~ f)) `1 < (S-max (L~ f)) `1 then A31: (S-min (L~ f)) `2 = (f /. (((S-min (L~ f)) .. f) -' 1)) `2 by A25, A16, SPPOL_1:def_2; then A32: f /. (((S-min (L~ f)) .. f) -' 1) in S-most (L~ f) by A2, A3, A17, GOBOARD1:1, SPRECT_2:11; then A33: (f /. (((S-min (L~ f)) .. f) -' 1)) `1 >= (S-min (L~ f)) `1 by PSCOMP_1:55; (f /. (((S-min (L~ f)) .. f) -' 1)) `1 <> (S-min (L~ f)) `1 by A5, A7, A14, A17, A31, GOBOARD7:29, TOPREAL3:6; then A34: (f /. (((S-min (L~ f)) .. f) -' 1)) `1 > (S-min (L~ f)) `1 by A33, XXREAL_0:1; (f /. (((S-min (L~ f)) .. f) -' 1)) `1 <= (S-max (L~ f)) `1 by A32, PSCOMP_1:55; hence (S-min (L~ f)) `1 < (S-max (L~ f)) `1 by A34, XXREAL_0:2; ::_thesis: verum end; suppose LSeg (f,((S-min (L~ f)) .. f)) is horizontal ; ::_thesis: (S-min (L~ f)) `1 < (S-max (L~ f)) `1 then A35: (S-min (L~ f)) `2 = (f /. (((S-min (L~ f)) .. f) + 1)) `2 by A24, A20, SPPOL_1:def_2; then A36: f /. (((S-min (L~ f)) .. f) + 1) in S-most (L~ f) by A2, A3, A21, GOBOARD1:1, SPRECT_2:11; then A37: (f /. (((S-min (L~ f)) .. f) + 1)) `1 >= (S-min (L~ f)) `1 by PSCOMP_1:55; (f /. (((S-min (L~ f)) .. f) + 1)) `1 <> (S-min (L~ f)) `1 by A5, A7, A21, A35, GOBOARD7:29, TOPREAL3:6; then A38: (f /. (((S-min (L~ f)) .. f) + 1)) `1 > (S-min (L~ f)) `1 by A37, XXREAL_0:1; (f /. (((S-min (L~ f)) .. f) + 1)) `1 <= (S-max (L~ f)) `1 by A36, PSCOMP_1:55; hence (S-min (L~ f)) `1 < (S-max (L~ f)) `1 by A38, XXREAL_0:2; ::_thesis: verum end; end; end; hence (S-min (L~ f)) `1 < (S-max (L~ f)) `1 ; ::_thesis: verum end; end; end; theorem :: JORDAN1G:11 for f being non trivial special unfolded standard FinSequence of (TOP-REAL 2) st ( ( f /. 1 <> S-min (L~ f) & f /. (len f) <> S-min (L~ f) ) or ( f /. 1 <> S-max (L~ f) & f /. (len f) <> S-max (L~ f) ) ) holds S-min (L~ f) <> S-max (L~ f) proof let f be non trivial special unfolded standard FinSequence of (TOP-REAL 2); ::_thesis: ( ( ( f /. 1 <> S-min (L~ f) & f /. (len f) <> S-min (L~ f) ) or ( f /. 1 <> S-max (L~ f) & f /. (len f) <> S-max (L~ f) ) ) implies S-min (L~ f) <> S-max (L~ f) ) assume ( ( f /. 1 <> S-min (L~ f) & f /. (len f) <> S-min (L~ f) ) or ( f /. 1 <> S-max (L~ f) & f /. (len f) <> S-max (L~ f) ) ) ; ::_thesis: S-min (L~ f) <> S-max (L~ f) then (S-min (L~ f)) `1 < (S-max (L~ f)) `1 by Th10; hence S-min (L~ f) <> S-max (L~ f) ; ::_thesis: verum end; theorem Th12: :: JORDAN1G:12 for f being non trivial special unfolded standard FinSequence of (TOP-REAL 2) st ( ( f /. 1 <> W-min (L~ f) & f /. (len f) <> W-min (L~ f) ) or ( f /. 1 <> W-max (L~ f) & f /. (len f) <> W-max (L~ f) ) ) holds (W-min (L~ f)) `2 < (W-max (L~ f)) `2 proof let f be non trivial special unfolded standard FinSequence of (TOP-REAL 2); ::_thesis: ( ( ( f /. 1 <> W-min (L~ f) & f /. (len f) <> W-min (L~ f) ) or ( f /. 1 <> W-max (L~ f) & f /. (len f) <> W-max (L~ f) ) ) implies (W-min (L~ f)) `2 < (W-max (L~ f)) `2 ) set p = W-min (L~ f); set i = (W-min (L~ f)) .. f; assume A1: ( ( f /. 1 <> W-min (L~ f) & f /. (len f) <> W-min (L~ f) ) or ( f /. 1 <> W-max (L~ f) & f /. (len f) <> W-max (L~ f) ) ) ; ::_thesis: (W-min (L~ f)) `2 < (W-max (L~ f)) `2 A2: len f >= 2 by NAT_D:60; A3: (W-min (L~ f)) `1 = W-bound (L~ f) by EUCLID:52; A4: W-min (L~ f) in rng f by SPRECT_2:43; then A5: (W-min (L~ f)) .. f in dom f by FINSEQ_4:20; then A6: ( 1 <= (W-min (L~ f)) .. f & (W-min (L~ f)) .. f <= len f ) by FINSEQ_3:25; A7: W-min (L~ f) = f . ((W-min (L~ f)) .. f) by A4, FINSEQ_4:19 .= f /. ((W-min (L~ f)) .. f) by A5, PARTFUN1:def_6 ; percases ( (W-min (L~ f)) .. f = 1 or (W-min (L~ f)) .. f = len f or ( 1 < (W-min (L~ f)) .. f & (W-min (L~ f)) .. f < len f ) ) by A6, XXREAL_0:1; supposeA8: (W-min (L~ f)) .. f = 1 ; ::_thesis: (W-min (L~ f)) `2 < (W-max (L~ f)) `2 (W-min (L~ f)) `1 = (W-max (L~ f)) `1 by PSCOMP_1:29; then A9: (W-min (L~ f)) `2 <> (W-max (L~ f)) `2 by A1, A7, A8, TOPREAL3:6; (W-min (L~ f)) `2 <= (W-max (L~ f)) `2 by PSCOMP_1:30; hence (W-min (L~ f)) `2 < (W-max (L~ f)) `2 by A9, XXREAL_0:1; ::_thesis: verum end; supposeA10: (W-min (L~ f)) .. f = len f ; ::_thesis: (W-min (L~ f)) `2 < (W-max (L~ f)) `2 (W-min (L~ f)) `1 = (W-max (L~ f)) `1 by PSCOMP_1:29; then A11: (W-min (L~ f)) `2 <> (W-max (L~ f)) `2 by A1, A7, A10, TOPREAL3:6; (W-min (L~ f)) `2 <= (W-max (L~ f)) `2 by PSCOMP_1:30; hence (W-min (L~ f)) `2 < (W-max (L~ f)) `2 by A11, XXREAL_0:1; ::_thesis: verum end; supposethat A12: 1 < (W-min (L~ f)) .. f and A13: (W-min (L~ f)) .. f < len f ; ::_thesis: (W-min (L~ f)) `2 < (W-max (L~ f)) `2 A14: (((W-min (L~ f)) .. f) -' 1) + 1 = (W-min (L~ f)) .. f by A12, XREAL_1:235; then A15: ((W-min (L~ f)) .. f) -' 1 >= 1 by A12, NAT_1:13; then A16: f /. (((W-min (L~ f)) .. f) -' 1) in LSeg (f,(((W-min (L~ f)) .. f) -' 1)) by A13, A14, TOPREAL1:21; ((W-min (L~ f)) .. f) -' 1 <= (W-min (L~ f)) .. f by A14, NAT_1:11; then ((W-min (L~ f)) .. f) -' 1 <= len f by A13, XXREAL_0:2; then A17: ((W-min (L~ f)) .. f) -' 1 in dom f by A15, FINSEQ_3:25; then A18: f /. (((W-min (L~ f)) .. f) -' 1) in L~ f by A2, GOBOARD1:1; A19: ((W-min (L~ f)) .. f) + 1 <= len f by A13, NAT_1:13; then A20: f /. (((W-min (L~ f)) .. f) + 1) in LSeg (f,((W-min (L~ f)) .. f)) by A12, TOPREAL1:21; ((W-min (L~ f)) .. f) + 1 >= 1 by NAT_1:11; then A21: ((W-min (L~ f)) .. f) + 1 in dom f by A19, FINSEQ_3:25; then A22: f /. (((W-min (L~ f)) .. f) + 1) in L~ f by A2, GOBOARD1:1; A23: W-min (L~ f) <> f /. (((W-min (L~ f)) .. f) + 1) by A4, A7, A21, FINSEQ_4:20, GOBOARD7:29; A24: W-min (L~ f) in LSeg (f,((W-min (L~ f)) .. f)) by A7, A12, A19, TOPREAL1:21; A25: W-min (L~ f) in LSeg (f,(((W-min (L~ f)) .. f) -' 1)) by A7, A13, A14, A15, TOPREAL1:21; A26: W-min (L~ f) <> f /. (((W-min (L~ f)) .. f) -' 1) by A5, A7, A14, A17, GOBOARD7:29; A27: ( not LSeg (f,(((W-min (L~ f)) .. f) -' 1)) is horizontal or not LSeg (f,((W-min (L~ f)) .. f)) is horizontal ) proof assume ( LSeg (f,(((W-min (L~ f)) .. f) -' 1)) is horizontal & LSeg (f,((W-min (L~ f)) .. f)) is horizontal ) ; ::_thesis: contradiction then A28: ( (W-min (L~ f)) `2 = (f /. (((W-min (L~ f)) .. f) + 1)) `2 & (W-min (L~ f)) `2 = (f /. (((W-min (L~ f)) .. f) -' 1)) `2 ) by A25, A24, A16, A20, SPPOL_1:def_2; A29: ( (f /. (((W-min (L~ f)) .. f) + 1)) `1 <= (f /. (((W-min (L~ f)) .. f) -' 1)) `1 or (f /. (((W-min (L~ f)) .. f) + 1)) `1 >= (f /. (((W-min (L~ f)) .. f) -' 1)) `1 ) ; A30: ( (W-min (L~ f)) `1 <= (f /. (((W-min (L~ f)) .. f) + 1)) `1 & (W-min (L~ f)) `1 <= (f /. (((W-min (L~ f)) .. f) -' 1)) `1 ) by A3, A18, A22, PSCOMP_1:24; ( LSeg (f,((W-min (L~ f)) .. f)) = LSeg ((f /. ((W-min (L~ f)) .. f)),(f /. (((W-min (L~ f)) .. f) + 1))) & LSeg (f,(((W-min (L~ f)) .. f) -' 1)) = LSeg ((f /. ((W-min (L~ f)) .. f)),(f /. (((W-min (L~ f)) .. f) -' 1))) ) by A12, A13, A14, A15, A19, TOPREAL1:def_3; then ( f /. (((W-min (L~ f)) .. f) -' 1) in LSeg (f,((W-min (L~ f)) .. f)) or f /. (((W-min (L~ f)) .. f) + 1) in LSeg (f,(((W-min (L~ f)) .. f) -' 1)) ) by A7, A28, A30, A29, GOBOARD7:8; then ( f /. (((W-min (L~ f)) .. f) -' 1) in (LSeg (f,(((W-min (L~ f)) .. f) -' 1))) /\ (LSeg (f,((W-min (L~ f)) .. f))) or f /. (((W-min (L~ f)) .. f) + 1) in (LSeg (f,(((W-min (L~ f)) .. f) -' 1))) /\ (LSeg (f,((W-min (L~ f)) .. f))) ) by A16, A20, XBOOLE_0:def_4; then ( ((((W-min (L~ f)) .. f) -' 1) + 1) + 1 = (((W-min (L~ f)) .. f) -' 1) + (1 + 1) & (LSeg (f,(((W-min (L~ f)) .. f) -' 1))) /\ (LSeg (f,((W-min (L~ f)) .. f))) <> {(f /. ((W-min (L~ f)) .. f))} ) by A7, A26, A23, TARSKI:def_1; hence contradiction by A14, A15, A19, TOPREAL1:def_6; ::_thesis: verum end; now__::_thesis:_(W-min_(L~_f))_`2_<_(W-max_(L~_f))_`2 percases ( LSeg (f,(((W-min (L~ f)) .. f) -' 1)) is vertical or LSeg (f,((W-min (L~ f)) .. f)) is vertical ) by A27, SPPOL_1:19; suppose LSeg (f,(((W-min (L~ f)) .. f) -' 1)) is vertical ; ::_thesis: (W-min (L~ f)) `2 < (W-max (L~ f)) `2 then A31: (W-min (L~ f)) `1 = (f /. (((W-min (L~ f)) .. f) -' 1)) `1 by A25, A16, SPPOL_1:def_3; then A32: f /. (((W-min (L~ f)) .. f) -' 1) in W-most (L~ f) by A2, A3, A17, GOBOARD1:1, SPRECT_2:12; then A33: (f /. (((W-min (L~ f)) .. f) -' 1)) `2 >= (W-min (L~ f)) `2 by PSCOMP_1:31; (f /. (((W-min (L~ f)) .. f) -' 1)) `2 <> (W-min (L~ f)) `2 by A5, A7, A14, A17, A31, GOBOARD7:29, TOPREAL3:6; then A34: (f /. (((W-min (L~ f)) .. f) -' 1)) `2 > (W-min (L~ f)) `2 by A33, XXREAL_0:1; (f /. (((W-min (L~ f)) .. f) -' 1)) `2 <= (W-max (L~ f)) `2 by A32, PSCOMP_1:31; hence (W-min (L~ f)) `2 < (W-max (L~ f)) `2 by A34, XXREAL_0:2; ::_thesis: verum end; suppose LSeg (f,((W-min (L~ f)) .. f)) is vertical ; ::_thesis: (W-min (L~ f)) `2 < (W-max (L~ f)) `2 then A35: (W-min (L~ f)) `1 = (f /. (((W-min (L~ f)) .. f) + 1)) `1 by A24, A20, SPPOL_1:def_3; then A36: f /. (((W-min (L~ f)) .. f) + 1) in W-most (L~ f) by A2, A3, A21, GOBOARD1:1, SPRECT_2:12; then A37: (f /. (((W-min (L~ f)) .. f) + 1)) `2 >= (W-min (L~ f)) `2 by PSCOMP_1:31; (f /. (((W-min (L~ f)) .. f) + 1)) `2 <> (W-min (L~ f)) `2 by A5, A7, A21, A35, GOBOARD7:29, TOPREAL3:6; then A38: (f /. (((W-min (L~ f)) .. f) + 1)) `2 > (W-min (L~ f)) `2 by A37, XXREAL_0:1; (f /. (((W-min (L~ f)) .. f) + 1)) `2 <= (W-max (L~ f)) `2 by A36, PSCOMP_1:31; hence (W-min (L~ f)) `2 < (W-max (L~ f)) `2 by A38, XXREAL_0:2; ::_thesis: verum end; end; end; hence (W-min (L~ f)) `2 < (W-max (L~ f)) `2 ; ::_thesis: verum end; end; end; theorem :: JORDAN1G:13 for f being non trivial special unfolded standard FinSequence of (TOP-REAL 2) st ( ( f /. 1 <> W-min (L~ f) & f /. (len f) <> W-min (L~ f) ) or ( f /. 1 <> W-max (L~ f) & f /. (len f) <> W-max (L~ f) ) ) holds W-min (L~ f) <> W-max (L~ f) proof let f be non trivial special unfolded standard FinSequence of (TOP-REAL 2); ::_thesis: ( ( ( f /. 1 <> W-min (L~ f) & f /. (len f) <> W-min (L~ f) ) or ( f /. 1 <> W-max (L~ f) & f /. (len f) <> W-max (L~ f) ) ) implies W-min (L~ f) <> W-max (L~ f) ) assume ( ( f /. 1 <> W-min (L~ f) & f /. (len f) <> W-min (L~ f) ) or ( f /. 1 <> W-max (L~ f) & f /. (len f) <> W-max (L~ f) ) ) ; ::_thesis: W-min (L~ f) <> W-max (L~ f) then (W-min (L~ f)) `2 < (W-max (L~ f)) `2 by Th12; hence W-min (L~ f) <> W-max (L~ f) ; ::_thesis: verum end; theorem Th14: :: JORDAN1G:14 for f being non trivial special unfolded standard FinSequence of (TOP-REAL 2) st ( ( f /. 1 <> E-min (L~ f) & f /. (len f) <> E-min (L~ f) ) or ( f /. 1 <> E-max (L~ f) & f /. (len f) <> E-max (L~ f) ) ) holds (E-min (L~ f)) `2 < (E-max (L~ f)) `2 proof let f be non trivial special unfolded standard FinSequence of (TOP-REAL 2); ::_thesis: ( ( ( f /. 1 <> E-min (L~ f) & f /. (len f) <> E-min (L~ f) ) or ( f /. 1 <> E-max (L~ f) & f /. (len f) <> E-max (L~ f) ) ) implies (E-min (L~ f)) `2 < (E-max (L~ f)) `2 ) set p = E-min (L~ f); set i = (E-min (L~ f)) .. f; assume A1: ( ( f /. 1 <> E-min (L~ f) & f /. (len f) <> E-min (L~ f) ) or ( f /. 1 <> E-max (L~ f) & f /. (len f) <> E-max (L~ f) ) ) ; ::_thesis: (E-min (L~ f)) `2 < (E-max (L~ f)) `2 A2: len f >= 2 by NAT_D:60; A3: (E-min (L~ f)) `1 = E-bound (L~ f) by EUCLID:52; A4: E-min (L~ f) in rng f by SPRECT_2:45; then A5: (E-min (L~ f)) .. f in dom f by FINSEQ_4:20; then A6: ( 1 <= (E-min (L~ f)) .. f & (E-min (L~ f)) .. f <= len f ) by FINSEQ_3:25; A7: E-min (L~ f) = f . ((E-min (L~ f)) .. f) by A4, FINSEQ_4:19 .= f /. ((E-min (L~ f)) .. f) by A5, PARTFUN1:def_6 ; percases ( (E-min (L~ f)) .. f = 1 or (E-min (L~ f)) .. f = len f or ( 1 < (E-min (L~ f)) .. f & (E-min (L~ f)) .. f < len f ) ) by A6, XXREAL_0:1; supposeA8: (E-min (L~ f)) .. f = 1 ; ::_thesis: (E-min (L~ f)) `2 < (E-max (L~ f)) `2 (E-min (L~ f)) `1 = (E-max (L~ f)) `1 by PSCOMP_1:45; then A9: (E-min (L~ f)) `2 <> (E-max (L~ f)) `2 by A1, A7, A8, TOPREAL3:6; (E-min (L~ f)) `2 <= (E-max (L~ f)) `2 by PSCOMP_1:46; hence (E-min (L~ f)) `2 < (E-max (L~ f)) `2 by A9, XXREAL_0:1; ::_thesis: verum end; supposeA10: (E-min (L~ f)) .. f = len f ; ::_thesis: (E-min (L~ f)) `2 < (E-max (L~ f)) `2 (E-min (L~ f)) `1 = (E-max (L~ f)) `1 by PSCOMP_1:45; then A11: (E-min (L~ f)) `2 <> (E-max (L~ f)) `2 by A1, A7, A10, TOPREAL3:6; (E-min (L~ f)) `2 <= (E-max (L~ f)) `2 by PSCOMP_1:46; hence (E-min (L~ f)) `2 < (E-max (L~ f)) `2 by A11, XXREAL_0:1; ::_thesis: verum end; supposethat A12: 1 < (E-min (L~ f)) .. f and A13: (E-min (L~ f)) .. f < len f ; ::_thesis: (E-min (L~ f)) `2 < (E-max (L~ f)) `2 A14: (((E-min (L~ f)) .. f) -' 1) + 1 = (E-min (L~ f)) .. f by A12, XREAL_1:235; then A15: ((E-min (L~ f)) .. f) -' 1 >= 1 by A12, NAT_1:13; then A16: f /. (((E-min (L~ f)) .. f) -' 1) in LSeg (f,(((E-min (L~ f)) .. f) -' 1)) by A13, A14, TOPREAL1:21; ((E-min (L~ f)) .. f) -' 1 <= (E-min (L~ f)) .. f by A14, NAT_1:11; then ((E-min (L~ f)) .. f) -' 1 <= len f by A13, XXREAL_0:2; then A17: ((E-min (L~ f)) .. f) -' 1 in dom f by A15, FINSEQ_3:25; then A18: f /. (((E-min (L~ f)) .. f) -' 1) in L~ f by A2, GOBOARD1:1; A19: ((E-min (L~ f)) .. f) + 1 <= len f by A13, NAT_1:13; then A20: f /. (((E-min (L~ f)) .. f) + 1) in LSeg (f,((E-min (L~ f)) .. f)) by A12, TOPREAL1:21; ((E-min (L~ f)) .. f) + 1 >= 1 by NAT_1:11; then A21: ((E-min (L~ f)) .. f) + 1 in dom f by A19, FINSEQ_3:25; then A22: f /. (((E-min (L~ f)) .. f) + 1) in L~ f by A2, GOBOARD1:1; A23: E-min (L~ f) <> f /. (((E-min (L~ f)) .. f) + 1) by A4, A7, A21, FINSEQ_4:20, GOBOARD7:29; A24: E-min (L~ f) in LSeg (f,((E-min (L~ f)) .. f)) by A7, A12, A19, TOPREAL1:21; A25: E-min (L~ f) in LSeg (f,(((E-min (L~ f)) .. f) -' 1)) by A7, A13, A14, A15, TOPREAL1:21; A26: E-min (L~ f) <> f /. (((E-min (L~ f)) .. f) -' 1) by A5, A7, A14, A17, GOBOARD7:29; A27: ( not LSeg (f,(((E-min (L~ f)) .. f) -' 1)) is horizontal or not LSeg (f,((E-min (L~ f)) .. f)) is horizontal ) proof assume ( LSeg (f,(((E-min (L~ f)) .. f) -' 1)) is horizontal & LSeg (f,((E-min (L~ f)) .. f)) is horizontal ) ; ::_thesis: contradiction then A28: ( (E-min (L~ f)) `2 = (f /. (((E-min (L~ f)) .. f) + 1)) `2 & (E-min (L~ f)) `2 = (f /. (((E-min (L~ f)) .. f) -' 1)) `2 ) by A25, A24, A16, A20, SPPOL_1:def_2; A29: ( (f /. (((E-min (L~ f)) .. f) + 1)) `1 <= (f /. (((E-min (L~ f)) .. f) -' 1)) `1 or (f /. (((E-min (L~ f)) .. f) + 1)) `1 >= (f /. (((E-min (L~ f)) .. f) -' 1)) `1 ) ; A30: ( (E-min (L~ f)) `1 >= (f /. (((E-min (L~ f)) .. f) + 1)) `1 & (E-min (L~ f)) `1 >= (f /. (((E-min (L~ f)) .. f) -' 1)) `1 ) by A3, A18, A22, PSCOMP_1:24; ( LSeg (f,((E-min (L~ f)) .. f)) = LSeg ((f /. ((E-min (L~ f)) .. f)),(f /. (((E-min (L~ f)) .. f) + 1))) & LSeg (f,(((E-min (L~ f)) .. f) -' 1)) = LSeg ((f /. ((E-min (L~ f)) .. f)),(f /. (((E-min (L~ f)) .. f) -' 1))) ) by A12, A13, A14, A15, A19, TOPREAL1:def_3; then ( f /. (((E-min (L~ f)) .. f) -' 1) in LSeg (f,((E-min (L~ f)) .. f)) or f /. (((E-min (L~ f)) .. f) + 1) in LSeg (f,(((E-min (L~ f)) .. f) -' 1)) ) by A7, A28, A30, A29, GOBOARD7:8; then ( f /. (((E-min (L~ f)) .. f) -' 1) in (LSeg (f,(((E-min (L~ f)) .. f) -' 1))) /\ (LSeg (f,((E-min (L~ f)) .. f))) or f /. (((E-min (L~ f)) .. f) + 1) in (LSeg (f,(((E-min (L~ f)) .. f) -' 1))) /\ (LSeg (f,((E-min (L~ f)) .. f))) ) by A16, A20, XBOOLE_0:def_4; then ( ((((E-min (L~ f)) .. f) -' 1) + 1) + 1 = (((E-min (L~ f)) .. f) -' 1) + (1 + 1) & (LSeg (f,(((E-min (L~ f)) .. f) -' 1))) /\ (LSeg (f,((E-min (L~ f)) .. f))) <> {(f /. ((E-min (L~ f)) .. f))} ) by A7, A26, A23, TARSKI:def_1; hence contradiction by A14, A15, A19, TOPREAL1:def_6; ::_thesis: verum end; now__::_thesis:_(E-min_(L~_f))_`2_<_(E-max_(L~_f))_`2 percases ( LSeg (f,(((E-min (L~ f)) .. f) -' 1)) is vertical or LSeg (f,((E-min (L~ f)) .. f)) is vertical ) by A27, SPPOL_1:19; suppose LSeg (f,(((E-min (L~ f)) .. f) -' 1)) is vertical ; ::_thesis: (E-min (L~ f)) `2 < (E-max (L~ f)) `2 then A31: (E-min (L~ f)) `1 = (f /. (((E-min (L~ f)) .. f) -' 1)) `1 by A25, A16, SPPOL_1:def_3; then A32: f /. (((E-min (L~ f)) .. f) -' 1) in E-most (L~ f) by A2, A3, A17, GOBOARD1:1, SPRECT_2:13; then A33: (f /. (((E-min (L~ f)) .. f) -' 1)) `2 >= (E-min (L~ f)) `2 by PSCOMP_1:47; (f /. (((E-min (L~ f)) .. f) -' 1)) `2 <> (E-min (L~ f)) `2 by A5, A7, A14, A17, A31, GOBOARD7:29, TOPREAL3:6; then A34: (f /. (((E-min (L~ f)) .. f) -' 1)) `2 > (E-min (L~ f)) `2 by A33, XXREAL_0:1; (f /. (((E-min (L~ f)) .. f) -' 1)) `2 <= (E-max (L~ f)) `2 by A32, PSCOMP_1:47; hence (E-min (L~ f)) `2 < (E-max (L~ f)) `2 by A34, XXREAL_0:2; ::_thesis: verum end; suppose LSeg (f,((E-min (L~ f)) .. f)) is vertical ; ::_thesis: (E-min (L~ f)) `2 < (E-max (L~ f)) `2 then A35: (E-min (L~ f)) `1 = (f /. (((E-min (L~ f)) .. f) + 1)) `1 by A24, A20, SPPOL_1:def_3; then A36: f /. (((E-min (L~ f)) .. f) + 1) in E-most (L~ f) by A2, A3, A21, GOBOARD1:1, SPRECT_2:13; then A37: (f /. (((E-min (L~ f)) .. f) + 1)) `2 >= (E-min (L~ f)) `2 by PSCOMP_1:47; (f /. (((E-min (L~ f)) .. f) + 1)) `2 <> (E-min (L~ f)) `2 by A5, A7, A21, A35, GOBOARD7:29, TOPREAL3:6; then A38: (f /. (((E-min (L~ f)) .. f) + 1)) `2 > (E-min (L~ f)) `2 by A37, XXREAL_0:1; (f /. (((E-min (L~ f)) .. f) + 1)) `2 <= (E-max (L~ f)) `2 by A36, PSCOMP_1:47; hence (E-min (L~ f)) `2 < (E-max (L~ f)) `2 by A38, XXREAL_0:2; ::_thesis: verum end; end; end; hence (E-min (L~ f)) `2 < (E-max (L~ f)) `2 ; ::_thesis: verum end; end; end; theorem :: JORDAN1G:15 for f being non trivial special unfolded standard FinSequence of (TOP-REAL 2) st ( ( f /. 1 <> E-min (L~ f) & f /. (len f) <> E-min (L~ f) ) or ( f /. 1 <> E-max (L~ f) & f /. (len f) <> E-max (L~ f) ) ) holds E-min (L~ f) <> E-max (L~ f) proof let f be non trivial special unfolded standard FinSequence of (TOP-REAL 2); ::_thesis: ( ( ( f /. 1 <> E-min (L~ f) & f /. (len f) <> E-min (L~ f) ) or ( f /. 1 <> E-max (L~ f) & f /. (len f) <> E-max (L~ f) ) ) implies E-min (L~ f) <> E-max (L~ f) ) assume ( ( f /. 1 <> E-min (L~ f) & f /. (len f) <> E-min (L~ f) ) or ( f /. 1 <> E-max (L~ f) & f /. (len f) <> E-max (L~ f) ) ) ; ::_thesis: E-min (L~ f) <> E-max (L~ f) then (E-min (L~ f)) `2 < (E-max (L~ f)) `2 by Th14; hence E-min (L~ f) <> E-max (L~ f) ; ::_thesis: verum end; theorem Th16: :: JORDAN1G:16 for D being non empty set for f being FinSequence of D for p, q being Element of D st p in rng f & q in rng f & q .. f <= p .. f holds (f -: p) :- q = (f :- q) -: p proof let D be non empty set ; ::_thesis: for f being FinSequence of D for p, q being Element of D st p in rng f & q in rng f & q .. f <= p .. f holds (f -: p) :- q = (f :- q) -: p let f be FinSequence of D; ::_thesis: for p, q being Element of D st p in rng f & q in rng f & q .. f <= p .. f holds (f -: p) :- q = (f :- q) -: p let p, q be Element of D; ::_thesis: ( p in rng f & q in rng f & q .. f <= p .. f implies (f -: p) :- q = (f :- q) -: p ) assume that A1: p in rng f and A2: q in rng f and A3: q .. f <= p .. f ; ::_thesis: (f -: p) :- q = (f :- q) -: p A4: ( f -: p = f | (p .. f) & (f :- q) -: p = (f :- q) | (p .. (f :- q)) ) by FINSEQ_5:def_1; consider i being Element of NAT such that A5: i + 1 = q .. f and A6: f :- q = f /^ i by A2, FINSEQ_5:49; A7: i < p .. f by A3, A5, NAT_1:13; then p .. f = i + (p .. (f /^ i)) by A1, FINSEQ_6:56; then A8: p .. (f /^ i) = (p .. f) - i .= (p .. f) -' i by A7, XREAL_1:233 ; q in rng (f -: p) by A1, A2, A3, FINSEQ_5:46; then A9: ex j being Element of NAT st ( j + 1 = q .. (f -: p) & (f -: p) :- q = (f -: p) /^ j ) by FINSEQ_5:49; q .. (f -: p) = q .. f by A1, A2, A3, SPRECT_5:3; hence (f -: p) :- q = (f :- q) -: p by A5, A6, A9, A4, A8, FINSEQ_5:80; ::_thesis: verum end; theorem Th17: :: JORDAN1G:17 for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) for n being Element of NAT holds (L~ ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) = {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} proof let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for n being Element of NAT holds (L~ ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) = {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} let n be Element of NAT ; ::_thesis: (L~ ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) = {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} set US = (Cage (C,n)) -: (W-min (L~ (Cage (C,n)))); set LS = (Cage (C,n)) :- (W-min (L~ (Cage (C,n)))); set f = Cage (C,n); set pW = W-min (L~ (Cage (C,n))); set pN = N-min (L~ (Cage (C,n))); set pNa = N-max (L~ (Cage (C,n))); set pSa = S-max (L~ (Cage (C,n))); set pSi = S-min (L~ (Cage (C,n))); set pEa = E-max (L~ (Cage (C,n))); set pEi = E-min (L~ (Cage (C,n))); A1: W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43; then A2: (Cage (C,n)) -: (W-min (L~ (Cage (C,n)))) <> {} by FINSEQ_5:47; len ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) = (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A1, FINSEQ_5:42; then ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) /. (len ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) = W-min (L~ (Cage (C,n))) by A1, FINSEQ_5:45; then A3: W-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by A2, REVROT_1:3; A4: (Cage (C,n)) /. 1 = N-min (L~ (Cage (C,n))) by JORDAN9:32; then (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (E-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by SPRECT_2:71; then (N-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (E-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A4, SPRECT_2:70, XXREAL_0:2; then (N-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (S-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by A4, SPRECT_2:72, XXREAL_0:2; then A5: (N-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (S-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A4, SPRECT_2:73, XXREAL_0:2; ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) /. 1 = (Cage (C,n)) /. 1 by A1, FINSEQ_5:44 .= N-min (L~ (Cage (C,n))) by JORDAN9:32 ; then A6: N-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by A2, FINSEQ_6:42; ( N-max (L~ (Cage (C,n))) in rng (Cage (C,n)) & (S-min (L~ (Cage (C,n)))) .. (Cage (C,n)) <= (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) ) by A4, SPRECT_2:40, SPRECT_2:74; then A7: N-max (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by A1, A5, FINSEQ_5:46, XXREAL_0:2; {(N-min (L~ (Cage (C,n)))),(N-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} c= rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(N-min (L~ (Cage (C,n)))),(N-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} or x in rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) ) assume x in {(N-min (L~ (Cage (C,n)))),(N-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} ; ::_thesis: x in rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) hence x in rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by A6, A7, A3, ENUMSET1:def_1; ::_thesis: verum end; then A8: card {(N-min (L~ (Cage (C,n)))),(N-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} c= card (rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) by CARD_1:11; ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) /. 1 = W-min (L~ (Cage (C,n))) by FINSEQ_5:53; then A9: W-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by FINSEQ_6:42; ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) /. (len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) = (Cage (C,n)) /. (len (Cage (C,n))) by A1, FINSEQ_5:54 .= (Cage (C,n)) /. 1 by FINSEQ_6:def_1 .= N-min (L~ (Cage (C,n))) by JORDAN9:32 ; then A10: N-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by REVROT_1:3; {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} c= rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} or x in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) ) assume x in {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} ; ::_thesis: x in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) hence x in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by A10, A9, TARSKI:def_2; ::_thesis: verum end; then A11: card {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} c= card (rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) by CARD_1:11; card (rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) c= card (dom ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) by CARD_2:61; then A12: card (rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) c= len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by CARD_1:62; ( W-max (L~ (Cage (C,n))) in L~ (Cage (C,n)) & (N-min (L~ (Cage (C,n)))) `2 = N-bound (L~ (Cage (C,n))) ) by EUCLID:52, SPRECT_1:13; then (W-max (L~ (Cage (C,n)))) `2 <= (N-min (L~ (Cage (C,n)))) `2 by PSCOMP_1:24; then A13: N-min (L~ (Cage (C,n))) <> W-min (L~ (Cage (C,n))) by SPRECT_2:57; then card {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} = 2 by CARD_2:57; then 2 c= len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by A11, A12, XBOOLE_1:1; then len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) >= 2 by NAT_1:39; then A14: rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) c= L~ ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by SPPOL_2:18; ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) /. (len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) = (Cage (C,n)) /. (len (Cage (C,n))) by A1, FINSEQ_5:54 .= (Cage (C,n)) /. 1 by FINSEQ_6:def_1 .= N-min (L~ (Cage (C,n))) by JORDAN9:32 ; then A15: N-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by REVROT_1:3; (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) <= (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) ; then A16: ( W-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) & W-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) ) by A1, FINSEQ_5:46, FINSEQ_6:61; ( W-max (L~ (Cage (C,n))) in L~ (Cage (C,n)) & (N-max (L~ (Cage (C,n)))) `2 = N-bound (L~ (Cage (C,n))) ) by EUCLID:52, SPRECT_1:13; then (W-max (L~ (Cage (C,n)))) `2 <= (N-max (L~ (Cage (C,n)))) `2 by PSCOMP_1:24; then ( N-min (L~ (Cage (C,n))) <> N-max (L~ (Cage (C,n))) & N-max (L~ (Cage (C,n))) <> W-min (L~ (Cage (C,n))) ) by SPRECT_2:52, SPRECT_2:57; then A17: card {(N-min (L~ (Cage (C,n)))),(N-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} = 3 by A13, CARD_2:58; card (rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) c= card (dom ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) by CARD_2:61; then card (rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) c= len ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by CARD_1:62; then 3 c= len ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by A17, A8, XBOOLE_1:1; then A18: len ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) >= 3 by NAT_1:39; then A19: rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) c= L~ ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by SPPOL_2:18, XXREAL_0:2; thus (L~ ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) c= {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} :: according to XBOOLE_0:def_10 ::_thesis: {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} c= (L~ ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) or x in {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} ) assume A20: x in (L~ ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) ; ::_thesis: x in {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} then reconsider x1 = x as Point of (TOP-REAL 2) ; assume A21: not x in {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} ; ::_thesis: contradiction x in L~ ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by A20, XBOOLE_0:def_4; then consider i1 being Element of NAT such that A22: 1 <= i1 and A23: i1 + 1 <= len ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) and A24: x1 in LSeg (((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))),i1) by SPPOL_2:13; A25: LSeg (((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))),i1) = LSeg ((Cage (C,n)),i1) by A23, SPPOL_2:9; x in L~ ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by A20, XBOOLE_0:def_4; then consider i2 being Element of NAT such that A26: 1 <= i2 and A27: i2 + 1 <= len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) and A28: x1 in LSeg (((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))),i2) by SPPOL_2:13; set i3 = i2 -' 1; A29: (i2 -' 1) + 1 = i2 by A26, XREAL_1:235; then A30: 1 + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) <= ((i2 -' 1) + 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) by A26, XREAL_1:7; A31: len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) = ((len (Cage (C,n))) - ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) + 1 by A1, FINSEQ_5:50; then i2 < ((len (Cage (C,n))) - ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) + 1 by A27, NAT_1:13; then i2 - 1 < (len (Cage (C,n))) - ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) by XREAL_1:19; then A32: (i2 - 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) < len (Cage (C,n)) by XREAL_1:20; i2 - 1 >= 1 - 1 by A26, XREAL_1:9; then A33: (i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) < len (Cage (C,n)) by A32, XREAL_0:def_2; A34: LSeg (((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))),i2) = LSeg ((Cage (C,n)),((i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))))) by A1, A29, SPPOL_2:10; A35: len ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) = (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A1, FINSEQ_5:42; then i1 + 1 < ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) + 1 by A23, NAT_1:13; then i1 + 1 < ((i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) + 1 by A30, XXREAL_0:2; then A36: i1 + 1 <= (i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) by NAT_1:13; A37: (((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) -' 1) + 1 = (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A1, FINSEQ_4:21, XREAL_1:235; (i2 -' 1) + 1 < ((len (Cage (C,n))) - ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) + 1 by A27, A29, A31, NAT_1:13; then i2 -' 1 < (len (Cage (C,n))) - ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) by XREAL_1:7; then A38: (i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) < len (Cage (C,n)) by XREAL_1:20; then A39: ((i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) + 1 <= len (Cage (C,n)) by NAT_1:13; now__::_thesis:_contradiction percases ( ( i1 + 1 < (i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) & i1 > 1 ) or i1 = 1 or i1 + 1 = (i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) ) by A22, A36, XXREAL_0:1; suppose ( i1 + 1 < (i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) & i1 > 1 ) ; ::_thesis: contradiction then LSeg ((Cage (C,n)),i1) misses LSeg ((Cage (C,n)),((i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))))) by A38, GOBOARD5:def_4; then (LSeg ((Cage (C,n)),i1)) /\ (LSeg ((Cage (C,n)),((i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))))) = {} by XBOOLE_0:def_7; hence contradiction by A24, A28, A25, A34, XBOOLE_0:def_4; ::_thesis: verum end; supposeA40: i1 = 1 ; ::_thesis: contradiction (i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) >= 0 + 3 by A18, A35, XREAL_1:7; then A41: i1 + 1 < (i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) by A40, XXREAL_0:2; now__::_thesis:_contradiction percases ( ((i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) + 1 < len (Cage (C,n)) or ((i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) + 1 = len (Cage (C,n)) ) by A39, XXREAL_0:1; suppose ((i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) + 1 < len (Cage (C,n)) ; ::_thesis: contradiction then LSeg ((Cage (C,n)),i1) misses LSeg ((Cage (C,n)),((i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))))) by A41, GOBOARD5:def_4; then (LSeg ((Cage (C,n)),i1)) /\ (LSeg ((Cage (C,n)),((i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))))) = {} by XBOOLE_0:def_7; hence contradiction by A24, A28, A25, A34, XBOOLE_0:def_4; ::_thesis: verum end; suppose ((i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) + 1 = len (Cage (C,n)) ; ::_thesis: contradiction then (i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) = (len (Cage (C,n))) - 1 ; then (i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) = (len (Cage (C,n))) -' 1 by XREAL_0:def_2; then (LSeg ((Cage (C,n)),i1)) /\ (LSeg ((Cage (C,n)),((i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))))) = {((Cage (C,n)) /. 1)} by A40, GOBOARD7:34, REVROT_1:30; then x1 in {((Cage (C,n)) /. 1)} by A24, A28, A25, A34, XBOOLE_0:def_4; then x1 = (Cage (C,n)) /. 1 by TARSKI:def_1 .= N-min (L~ (Cage (C,n))) by JORDAN9:32 ; hence contradiction by A21, TARSKI:def_2; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; supposeA42: i1 + 1 = (i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) ; ::_thesis: contradiction (i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) >= (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by NAT_1:11; then (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) < len (Cage (C,n)) by A33, XXREAL_0:2; then ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) + 1 <= len (Cage (C,n)) by NAT_1:13; then A43: (((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) -' 1) + (1 + 1) <= len (Cage (C,n)) by A37; 0 + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) <= (i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) by XREAL_1:7; then (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) = i1 + 1 by A23, A35, A42, XXREAL_0:1; then (LSeg ((Cage (C,n)),i1)) /\ (LSeg ((Cage (C,n)),((i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))))) = {((Cage (C,n)) /. ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))))} by A22, A37, A42, A43, TOPREAL1:def_6; then x1 in {((Cage (C,n)) /. ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))))} by A24, A28, A25, A34, XBOOLE_0:def_4; then x1 = (Cage (C,n)) /. ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) by TARSKI:def_1 .= W-min (L~ (Cage (C,n))) by A1, FINSEQ_5:38 ; hence contradiction by A21, TARSKI:def_2; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; A44: ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) /. 1 = (Cage (C,n)) /. 1 by A1, FINSEQ_5:44 .= N-min (L~ (Cage (C,n))) by JORDAN9:32 ; not (Cage (C,n)) -: (W-min (L~ (Cage (C,n)))) is empty by A17, A8, NAT_1:39; then A45: N-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by A44, FINSEQ_6:42; thus {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} c= (L~ ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) ::_thesis: verum proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} or x in (L~ ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) ) assume A46: x in {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} ; ::_thesis: x in (L~ ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) percases ( x = N-min (L~ (Cage (C,n))) or x = W-min (L~ (Cage (C,n))) ) by A46, TARSKI:def_2; suppose x = N-min (L~ (Cage (C,n))) ; ::_thesis: x in (L~ ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) hence x in (L~ ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) by A14, A19, A45, A15, XBOOLE_0:def_4; ::_thesis: verum end; suppose x = W-min (L~ (Cage (C,n))) ; ::_thesis: x in (L~ ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) hence x in (L~ ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) by A14, A19, A16, XBOOLE_0:def_4; ::_thesis: verum end; end; end; end; theorem Th18: :: JORDAN1G:18 for n being Element of NAT for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) proof let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) set Nmi = N-min (L~ (Cage (C,n))); set Nma = N-max (L~ (Cage (C,n))); set Wmi = W-min (L~ (Cage (C,n))); set Wma = W-max (L~ (Cage (C,n))); set Ema = E-max (L~ (Cage (C,n))); set Emi = E-min (L~ (Cage (C,n))); set Sma = S-max (L~ (Cage (C,n))); set Smi = S-min (L~ (Cage (C,n))); set RotWmi = Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))); set RotEma = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))); A1: E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46; ( W-max (L~ (Cage (C,n))) in L~ (Cage (C,n)) & (N-min (L~ (Cage (C,n)))) `2 = N-bound (L~ (Cage (C,n))) ) by EUCLID:52, SPRECT_1:13; then (W-max (L~ (Cage (C,n)))) `2 <= (N-min (L~ (Cage (C,n)))) `2 by PSCOMP_1:24; then N-min (L~ (Cage (C,n))) <> W-min (L~ (Cage (C,n))) by SPRECT_2:57; then A2: card {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} = 2 by CARD_2:57; A3: W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43; then A4: (Cage (C,n)) -: (W-min (L~ (Cage (C,n)))) <> {} by FINSEQ_5:47; len ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) = (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A3, FINSEQ_5:42; then ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) /. (len ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) = W-min (L~ (Cage (C,n))) by A3, FINSEQ_5:45; then A5: W-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by A4, REVROT_1:3; ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) /. 1 = (Cage (C,n)) /. 1 by A3, FINSEQ_5:44 .= N-min (L~ (Cage (C,n))) by JORDAN9:32 ; then A6: N-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by A4, FINSEQ_6:42; {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} c= rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} or x in rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) ) assume x in {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} ; ::_thesis: x in rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) hence x in rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by A6, A5, TARSKI:def_2; ::_thesis: verum end; then A7: card {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} c= card (rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) by CARD_1:11; card (rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) c= card (dom ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) by CARD_2:61; then card (rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) c= len ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by CARD_1:62; then 2 c= len ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by A2, A7, XBOOLE_1:1; then len ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) >= 2 by NAT_1:39; then A8: rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) c= L~ ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by SPPOL_2:18; A9: (Cage (C,n)) /. 1 = N-min (L~ (Cage (C,n))) by JORDAN9:32; then (E-min (L~ (Cage (C,n)))) .. (Cage (C,n)) <= (S-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by SPRECT_2:72; then (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (S-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by A9, SPRECT_2:71, XXREAL_0:2; then A10: (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (S-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A9, SPRECT_2:73, XXREAL_0:2; then A11: (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A9, SPRECT_2:74, XXREAL_0:2; A12: (S-min (L~ (Cage (C,n)))) .. (Cage (C,n)) <= (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A9, SPRECT_2:74; then A13: E-max (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by A3, A1, A10, FINSEQ_5:46, XXREAL_0:2; (N-max (L~ (Cage (C,n)))) `1 <= (NE-corner (L~ (Cage (C,n)))) `1 by PSCOMP_1:38; then ( (N-min (L~ (Cage (C,n)))) `1 < (N-max (L~ (Cage (C,n)))) `1 & (N-max (L~ (Cage (C,n)))) `1 <= E-bound (L~ (Cage (C,n))) ) by EUCLID:52, SPRECT_2:51; then A14: N-min (L~ (Cage (C,n))) <> E-max (L~ (Cage (C,n))) by EUCLID:52; A15: not E-max (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) proof ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) /. 1 = W-min (L~ (Cage (C,n))) by FINSEQ_5:53; then A16: W-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by FINSEQ_6:42; ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) /. (len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) = (Cage (C,n)) /. (len (Cage (C,n))) by A3, FINSEQ_5:54 .= (Cage (C,n)) /. 1 by FINSEQ_6:def_1 .= N-min (L~ (Cage (C,n))) by JORDAN9:32 ; then A17: N-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by REVROT_1:3; {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} c= rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} or x in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) ) assume x in {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} ; ::_thesis: x in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) hence x in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by A17, A16, TARSKI:def_2; ::_thesis: verum end; then A18: card {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} c= card (rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) by CARD_1:11; ( W-max (L~ (Cage (C,n))) in L~ (Cage (C,n)) & (N-min (L~ (Cage (C,n)))) `2 = N-bound (L~ (Cage (C,n))) ) by EUCLID:52, SPRECT_1:13; then (W-max (L~ (Cage (C,n)))) `2 <= (N-min (L~ (Cage (C,n)))) `2 by PSCOMP_1:24; then N-min (L~ (Cage (C,n))) <> W-min (L~ (Cage (C,n))) by SPRECT_2:57; then A19: card {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} = 2 by CARD_2:57; card (rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) c= card (dom ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) by CARD_2:61; then card (rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) c= len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by CARD_1:62; then 2 c= len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by A19, A18, XBOOLE_1:1; then len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) >= 2 by NAT_1:39; then A20: rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) c= L~ ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by SPPOL_2:18; assume E-max (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) ; ::_thesis: contradiction then E-max (L~ (Cage (C,n))) in (L~ ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) by A13, A8, A20, XBOOLE_0:def_4; then E-max (L~ (Cage (C,n))) in {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} by Th17; then E-max (L~ (Cage (C,n))) = W-min (L~ (Cage (C,n))) by A14, TARSKI:def_2; hence contradiction by TOPREAL5:19; ::_thesis: verum end; A21: (N-max (L~ (Cage (C,n)))) .. (Cage (C,n)) <= (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by A9, SPRECT_2:70; A22: (N-min (L~ (Cage (C,n)))) .. (Cage (C,n)) < (N-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by A9, SPRECT_2:68; then A23: ( N-min (L~ (Cage (C,n))) in rng (Cage (C,n)) & (N-min (L~ (Cage (C,n)))) .. (Cage (C,n)) < (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) ) by A9, SPRECT_2:39, SPRECT_2:70, XXREAL_0:2; then A24: N-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by A3, A11, FINSEQ_5:46, XXREAL_0:2; A25: (E-max (L~ (Cage (C,n)))) .. ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) <> 1 proof assume A26: (E-max (L~ (Cage (C,n)))) .. ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) = 1 ; ::_thesis: contradiction (N-min (L~ (Cage (C,n)))) .. ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) = (N-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A3, A23, A11, SPRECT_5:3, XXREAL_0:2 .= 1 by A9, FINSEQ_6:43 ; hence contradiction by A22, A21, A13, A24, A26, FINSEQ_5:9; ::_thesis: verum end; then E-max (L~ (Cage (C,n))) in rng (((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) /^ 1) by A13, FINSEQ_6:78; then A27: E-max (L~ (Cage (C,n))) in (rng (((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) /^ 1)) \ (rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) by A15, XBOOLE_0:def_5; A28: W-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by A3, A1, A12, A10, FINSEQ_6:62, XXREAL_0:2; (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) :- (E-max (L~ (Cage (C,n)))) = (((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) ^ (((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) /^ 1)) :- (E-max (L~ (Cage (C,n)))) by A3, FINSEQ_6:def_2 .= (((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) /^ 1) :- (E-max (L~ (Cage (C,n)))) by A27, FINSEQ_6:65 .= ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) :- (E-max (L~ (Cage (C,n)))) by A13, A25, FINSEQ_6:83 .= ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) -: (W-min (L~ (Cage (C,n)))) by A3, A1, A12, A10, Th16, XXREAL_0:2 .= (((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) ^ (((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) /^ 1)) -: (W-min (L~ (Cage (C,n)))) by A28, FINSEQ_6:66 .= (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) by A1, FINSEQ_6:def_2 ; hence Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) by JORDAN1E:def_2; ::_thesis: verum end; theorem Th19: :: JORDAN1G:19 for n being Element of NAT for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds (W-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) = 1 proof let n be Element of NAT ; ::_thesis: for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds (W-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) = 1 let C be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: (W-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) = 1 (Upper_Seq (C,n)) /. 1 = W-min (L~ (Cage (C,n))) by JORDAN1F:5; hence (W-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) = 1 by FINSEQ_6:43; ::_thesis: verum end; theorem Th20: :: JORDAN1G:20 for n being Element of NAT for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds (W-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) < (W-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) proof let n be Element of NAT ; ::_thesis: for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds (W-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) < (W-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) let C be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: (W-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) < (W-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) set Wmi = W-min (L~ (Cage (C,n))); set Wma = W-max (L~ (Cage (C,n))); set Nmi = N-min (L~ (Cage (C,n))); set Nma = N-max (L~ (Cage (C,n))); set Ema = E-max (L~ (Cage (C,n))); set Rot = Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))); A1: L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33; then A2: W-min (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by SPRECT_2:43; A3: W-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A1, SPRECT_2:44; A4: E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A1, SPRECT_2:46; W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43; then A5: (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 = W-min (L~ (Cage (C,n))) by FINSEQ_6:92; then A6: (W-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) < (W-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A1, SPRECT_5:21; A7: ( Upper_Seq (C,n) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n)))) & (N-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) <= (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) ) by A1, A5, JORDAN1E:def_1, SPRECT_5:25; (N-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) < (N-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A1, A5, SPRECT_5:24; then A8: (W-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) < (N-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A1, A5, SPRECT_5:23, XXREAL_0:2; then (W-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) < (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A1, A5, SPRECT_5:25, XXREAL_0:2; then (W-min (L~ (Cage (C,n)))) .. ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n))))) = (W-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A2, A4, A6, SPRECT_5:3, XXREAL_0:2; hence (W-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) < (W-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) by A4, A6, A7, A8, A3, SPRECT_5:3, XXREAL_0:2; ::_thesis: verum end; theorem Th21: :: JORDAN1G:21 for n being Element of NAT for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds (W-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) <= (N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) proof let n be Element of NAT ; ::_thesis: for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds (W-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) <= (N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) let C be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: (W-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) <= (N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) set Wmi = W-min (L~ (Cage (C,n))); set Wma = W-max (L~ (Cage (C,n))); set Nmi = N-min (L~ (Cage (C,n))); set Nma = N-max (L~ (Cage (C,n))); set Ema = E-max (L~ (Cage (C,n))); set Rot = Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))); A1: L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33; then A2: W-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by SPRECT_2:44; A3: N-min (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A1, SPRECT_2:39; A4: E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A1, SPRECT_2:46; W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43; then A5: (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 = W-min (L~ (Cage (C,n))) by FINSEQ_6:92; then A6: (W-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) <= (N-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A1, SPRECT_5:23; A7: ( Upper_Seq (C,n) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n)))) & (N-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) <= (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) ) by A1, A5, JORDAN1E:def_1, SPRECT_5:25; A8: (N-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) < (N-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A1, A5, SPRECT_5:24; then (N-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) < (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A1, A5, SPRECT_5:25, XXREAL_0:2; then (W-max (L~ (Cage (C,n)))) .. ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n))))) = (W-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A2, A4, A6, SPRECT_5:3, XXREAL_0:2; hence (W-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) <= (N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) by A4, A6, A8, A7, A3, SPRECT_5:3, XXREAL_0:2; ::_thesis: verum end; theorem Th22: :: JORDAN1G:22 for n being Element of NAT for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds (N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) < (N-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) proof let n be Element of NAT ; ::_thesis: for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds (N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) < (N-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) let C be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: (N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) < (N-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) set Wmi = W-min (L~ (Cage (C,n))); set Nmi = N-min (L~ (Cage (C,n))); set Nma = N-max (L~ (Cage (C,n))); set Ema = E-max (L~ (Cage (C,n))); set Rot = Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))); A1: L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33; then A2: E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by SPRECT_2:46; W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43; then (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 = W-min (L~ (Cage (C,n))) by FINSEQ_6:92; then A3: ( (N-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) < (N-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) & (N-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) <= (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) ) by A1, SPRECT_5:24, SPRECT_5:25; A4: N-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A1, SPRECT_2:40; N-min (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A1, SPRECT_2:39; then ( Upper_Seq (C,n) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n)))) & (N-min (L~ (Cage (C,n)))) .. ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n))))) = (N-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) ) by A2, A3, JORDAN1E:def_1, SPRECT_5:3, XXREAL_0:2; hence (N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) < (N-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) by A2, A3, A4, SPRECT_5:3; ::_thesis: verum end; theorem Th23: :: JORDAN1G:23 for n being Element of NAT for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds (N-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) <= (E-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) proof let n be Element of NAT ; ::_thesis: for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds (N-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) <= (E-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) let C be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: (N-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) <= (E-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) set Wmi = W-min (L~ (Cage (C,n))); set Nma = N-max (L~ (Cage (C,n))); set Ema = E-max (L~ (Cage (C,n))); set Rot = Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))); A1: L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33; then A2: E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by SPRECT_2:46; W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43; then (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 = W-min (L~ (Cage (C,n))) by FINSEQ_6:92; then A3: (N-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) <= (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A1, SPRECT_5:25; N-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A1, SPRECT_2:40; then ( Upper_Seq (C,n) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n)))) & (N-max (L~ (Cage (C,n)))) .. ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n))))) = (N-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) ) by A2, A3, JORDAN1E:def_1, SPRECT_5:3; hence (N-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) <= (E-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) by A2, A3, SPRECT_5:3; ::_thesis: verum end; theorem Th24: :: JORDAN1G:24 for n being Element of NAT for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds (E-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) = len (Upper_Seq (C,n)) proof let n be Element of NAT ; ::_thesis: for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds (E-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) = len (Upper_Seq (C,n)) let C be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: (E-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) = len (Upper_Seq (C,n)) E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46; then A1: E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by FINSEQ_6:90, SPRECT_2:43; ( Upper_Seq (C,n) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n)))) & (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) <= (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) ) by JORDAN1E:def_1; then E-max (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) by A1, FINSEQ_5:46; then A2: Upper_Seq (C,n) just_once_values E-max (L~ (Cage (C,n))) by FINSEQ_4:8; (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) = E-max (L~ (Cage (C,n))) by JORDAN1F:7; hence (E-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) = len (Upper_Seq (C,n)) by A2, REVROT_1:1; ::_thesis: verum end; theorem Th25: :: JORDAN1G:25 for n being Element of NAT for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds (E-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) = 1 proof let n be Element of NAT ; ::_thesis: for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds (E-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) = 1 let C be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: (E-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) = 1 (Lower_Seq (C,n)) /. 1 = E-max (L~ (Cage (C,n))) by JORDAN1F:6; hence (E-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) = 1 by FINSEQ_6:43; ::_thesis: verum end; theorem Th26: :: JORDAN1G:26 for n being Element of NAT for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds (E-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) < (E-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) proof let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds (E-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) < (E-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: (E-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) < (E-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) set Ema = E-max (L~ (Cage (C,n))); set Emi = E-min (L~ (Cage (C,n))); set Sma = S-max (L~ (Cage (C,n))); set Smi = S-min (L~ (Cage (C,n))); set Wmi = W-min (L~ (Cage (C,n))); set Rot = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))); A1: Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) by Th18; A2: L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33; then A3: E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by SPRECT_2:46; A4: E-min (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A2, SPRECT_2:45; A5: W-min (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A2, SPRECT_2:43; E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46; then A6: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = E-max (L~ (Cage (C,n))) by FINSEQ_6:92; then A7: (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) < (E-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A2, SPRECT_5:37; A8: (S-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) <= (W-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A2, A6, SPRECT_5:41; (S-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) < (S-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A2, A6, SPRECT_5:40; then A9: (E-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) < (S-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A2, A6, SPRECT_5:39, XXREAL_0:2; then (E-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) < (W-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A2, A6, SPRECT_5:41, XXREAL_0:2; then (E-max (L~ (Cage (C,n)))) .. ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n))))) = (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A3, A5, A7, SPRECT_5:3, XXREAL_0:2; hence (E-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) < (E-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) by A1, A5, A7, A8, A9, A4, SPRECT_5:3, XXREAL_0:2; ::_thesis: verum end; theorem Th27: :: JORDAN1G:27 for n being Element of NAT for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds (E-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) <= (S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) proof let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds (E-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) <= (S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: (E-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) <= (S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) set Ema = E-max (L~ (Cage (C,n))); set Emi = E-min (L~ (Cage (C,n))); set Sma = S-max (L~ (Cage (C,n))); set Smi = S-min (L~ (Cage (C,n))); set Wmi = W-min (L~ (Cage (C,n))); set Rot = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))); A1: Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) by Th18; A2: L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33; then A3: E-min (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by SPRECT_2:45; A4: S-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A2, SPRECT_2:42; A5: W-min (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A2, SPRECT_2:43; E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46; then A6: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = E-max (L~ (Cage (C,n))) by FINSEQ_6:92; then A7: (E-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) <= (S-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A2, SPRECT_5:39; A8: (S-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) <= (W-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A2, A6, SPRECT_5:41; A9: (S-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) < (S-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A2, A6, SPRECT_5:40; then (S-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) < (W-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A2, A6, SPRECT_5:41, XXREAL_0:2; then (E-min (L~ (Cage (C,n)))) .. ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n))))) = (E-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A3, A5, A7, SPRECT_5:3, XXREAL_0:2; hence (E-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) <= (S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) by A1, A5, A7, A9, A8, A4, SPRECT_5:3, XXREAL_0:2; ::_thesis: verum end; theorem Th28: :: JORDAN1G:28 for n being Element of NAT for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds (S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) < (S-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) proof let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds (S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) < (S-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: (S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) < (S-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) set Ema = E-max (L~ (Cage (C,n))); set Sma = S-max (L~ (Cage (C,n))); set Smi = S-min (L~ (Cage (C,n))); set Wmi = W-min (L~ (Cage (C,n))); set Rot = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))); A1: Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) by Th18; A2: L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33; then A3: W-min (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by SPRECT_2:43; E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46; then (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = E-max (L~ (Cage (C,n))) by FINSEQ_6:92; then A4: ( (S-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) < (S-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) & (S-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) <= (W-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) ) by A2, SPRECT_5:40, SPRECT_5:41; A5: S-min (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A2, SPRECT_2:41; S-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A2, SPRECT_2:42; then (S-max (L~ (Cage (C,n)))) .. ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n))))) = (S-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A3, A4, SPRECT_5:3, XXREAL_0:2; hence (S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) < (S-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) by A1, A3, A4, A5, SPRECT_5:3; ::_thesis: verum end; theorem Th29: :: JORDAN1G:29 for n being Element of NAT for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds (S-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) <= (W-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) proof let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds (S-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) <= (W-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: (S-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) <= (W-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) set Ema = E-max (L~ (Cage (C,n))); set Smi = S-min (L~ (Cage (C,n))); set Wmi = W-min (L~ (Cage (C,n))); set Rot = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))); A1: Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) by Th18; A2: L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33; then A3: W-min (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by SPRECT_2:43; E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46; then (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = E-max (L~ (Cage (C,n))) by FINSEQ_6:92; then A4: (S-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) <= (W-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A2, SPRECT_5:41; S-min (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A2, SPRECT_2:41; then (S-min (L~ (Cage (C,n)))) .. ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n))))) = (S-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A3, A4, SPRECT_5:3; hence (S-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) <= (W-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) by A1, A3, A4, SPRECT_5:3; ::_thesis: verum end; theorem Th30: :: JORDAN1G:30 for n being Element of NAT for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds (W-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) = len (Lower_Seq (C,n)) proof let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds (W-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) = len (Lower_Seq (C,n)) let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: (W-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) = len (Lower_Seq (C,n)) W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43; then A1: W-min (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by FINSEQ_6:90, SPRECT_2:46; ( Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) & (W-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) <= (W-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) ) by Th18; then W-min (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) by A1, FINSEQ_5:46; then A2: Lower_Seq (C,n) just_once_values W-min (L~ (Cage (C,n))) by FINSEQ_4:8; (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) = W-min (L~ (Cage (C,n))) by JORDAN1F:8; hence (W-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) = len (Lower_Seq (C,n)) by A2, REVROT_1:1; ::_thesis: verum end; theorem Th31: :: JORDAN1G:31 for n being Element of NAT for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds ((Upper_Seq (C,n)) /. 2) `1 = W-bound (L~ (Cage (C,n))) proof let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds ((Upper_Seq (C,n)) /. 2) `1 = W-bound (L~ (Cage (C,n))) let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ((Upper_Seq (C,n)) /. 2) `1 = W-bound (L~ (Cage (C,n))) set Ca = Cage (C,n); set US = Upper_Seq (C,n); set Wmin = W-min (L~ (Cage (C,n))); set Emax = E-max (L~ (Cage (C,n))); set Nmin = N-min (L~ (Cage (C,n))); E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46; then A1: E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by FINSEQ_6:90, SPRECT_2:43; len (Upper_Seq (C,n)) >= 3 by JORDAN1E:15; then len (Upper_Seq (C,n)) >= 2 by XXREAL_0:2; then 2 in Seg (len (Upper_Seq (C,n))) by FINSEQ_1:1; then A2: 2 in Seg ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) by JORDAN1E:8; ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) /. 1 = W-min (L~ (Cage (C,n))) by FINSEQ_5:53; then A3: W-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by FINSEQ_6:42; (Cage (C,n)) /. 1 = N-min (L~ (Cage (C,n))) by JORDAN9:32; then (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) < len (Cage (C,n)) by SPRECT_2:76; then A4: ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) + 1 <= len (Cage (C,n)) by NAT_1:13; ( W-max (L~ (Cage (C,n))) in L~ (Cage (C,n)) & (N-min (L~ (Cage (C,n)))) `2 = N-bound (L~ (Cage (C,n))) ) by EUCLID:52, SPRECT_1:13; then (W-max (L~ (Cage (C,n)))) `2 <= (N-min (L~ (Cage (C,n)))) `2 by PSCOMP_1:24; then N-min (L~ (Cage (C,n))) <> W-min (L~ (Cage (C,n))) by SPRECT_2:57; then A5: card {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} = 2 by CARD_2:57; A6: W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43; then A7: 1 <= (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by FINSEQ_4:21; ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) /. (len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) = (Cage (C,n)) /. (len (Cage (C,n))) by A6, FINSEQ_5:54 .= (Cage (C,n)) /. 1 by FINSEQ_6:def_1 .= N-min (L~ (Cage (C,n))) by JORDAN9:32 ; then A8: N-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by REVROT_1:3; {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} c= rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} or x in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) ) assume x in {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} ; ::_thesis: x in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) hence x in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by A8, A3, TARSKI:def_2; ::_thesis: verum end; then A9: card {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} c= card (rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) by CARD_1:11; card (rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) c= card (dom ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) by CARD_2:61; then card (rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) c= len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by CARD_1:62; then 2 c= len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by A5, A9, XBOOLE_1:1; then A10: len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) >= 2 by NAT_1:39; then A11: len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) >= 1 by XXREAL_0:2; A12: (Upper_Seq (C,n)) /. 1 = ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n))))) /. 1 by JORDAN1E:def_1 .= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 by A1, FINSEQ_5:44 .= (Cage (C,n)) /. ((1 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) by A6, A11, REVROT_1:9 .= (Cage (C,n)) /. (0 + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) by XREAL_1:232 ; (Upper_Seq (C,n)) /. 2 = ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n))))) /. 2 by JORDAN1E:def_1 .= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 2 by A1, A2, FINSEQ_5:43 .= (Cage (C,n)) /. ((2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) by A6, A10, REVROT_1:9 .= (Cage (C,n)) /. ((2 - 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) by XREAL_0:def_2 ; hence ((Upper_Seq (C,n)) /. 2) `1 = W-bound (L~ (Cage (C,n))) by A7, A4, A12, JORDAN1E:22, JORDAN1F:5; ::_thesis: verum end; theorem :: JORDAN1G:32 for n being Element of NAT for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds ((Lower_Seq (C,n)) /. 2) `1 = E-bound (L~ (Cage (C,n))) proof let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds ((Lower_Seq (C,n)) /. 2) `1 = E-bound (L~ (Cage (C,n))) let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ((Lower_Seq (C,n)) /. 2) `1 = E-bound (L~ (Cage (C,n))) set Ca = Cage (C,n); set LS = Lower_Seq (C,n); set Emax = E-max (L~ (Cage (C,n))); set Emin = E-min (L~ (Cage (C,n))); set Smax = S-max (L~ (Cage (C,n))); set Smin = S-min (L~ (Cage (C,n))); set Wmin = W-min (L~ (Cage (C,n))); set Nmin = N-min (L~ (Cage (C,n))); W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43; then A1: W-min (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by FINSEQ_6:90, SPRECT_2:46; len (Lower_Seq (C,n)) >= 3 by JORDAN1E:15; then len (Lower_Seq (C,n)) >= 2 by XXREAL_0:2; then 2 <= (W-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) by Th30; then 2 <= (W-min (L~ (Cage (C,n)))) .. ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n))))) by Th18; then 2 <= (W-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A1, FINSEQ_6:72; then A2: 2 in Seg ((W-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))))) by FINSEQ_1:1; ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) /. 1 = E-max (L~ (Cage (C,n))) by FINSEQ_5:53; then A3: E-max (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by FINSEQ_6:42; ( N-max (L~ (Cage (C,n))) in L~ (Cage (C,n)) & (E-max (L~ (Cage (C,n)))) `1 = E-bound (L~ (Cage (C,n))) ) by EUCLID:52, SPRECT_1:11; then (N-max (L~ (Cage (C,n)))) `1 <= (E-max (L~ (Cage (C,n)))) `1 by PSCOMP_1:24; then N-min (L~ (Cage (C,n))) <> E-max (L~ (Cage (C,n))) by SPRECT_2:51; then A4: card {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} = 2 by CARD_2:57; A5: (Cage (C,n)) /. 1 = N-min (L~ (Cage (C,n))) by JORDAN9:32; then (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (E-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by SPRECT_2:71; then (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (S-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by A5, SPRECT_2:72, XXREAL_0:2; then (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (S-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A5, SPRECT_2:73, XXREAL_0:2; then (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A5, SPRECT_2:74, XXREAL_0:2; then (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < len (Cage (C,n)) by A5, SPRECT_2:76, XXREAL_0:2; then A6: ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) + 1 <= len (Cage (C,n)) by NAT_1:13; A7: E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46; then A8: 1 <= (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by FINSEQ_4:21; ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) /. (len ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) = (Cage (C,n)) /. (len (Cage (C,n))) by A7, FINSEQ_5:54 .= (Cage (C,n)) /. 1 by FINSEQ_6:def_1 .= N-min (L~ (Cage (C,n))) by JORDAN9:32 ; then A9: N-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by REVROT_1:3; {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} c= rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} or x in rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) ) assume x in {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} ; ::_thesis: x in rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) hence x in rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by A9, A3, TARSKI:def_2; ::_thesis: verum end; then A10: card {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} c= card (rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) by CARD_1:11; card (rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) c= card (dom ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) by CARD_2:61; then card (rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) c= len ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by CARD_1:62; then 2 c= len ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by A4, A10, XBOOLE_1:1; then A11: len ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) >= 2 by NAT_1:39; then A12: len ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) >= 1 by XXREAL_0:2; A13: (Lower_Seq (C,n)) /. 1 = ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n))))) /. 1 by Th18 .= (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 by A1, FINSEQ_5:44 .= (Cage (C,n)) /. ((1 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) by A7, A12, REVROT_1:9 .= (Cage (C,n)) /. (0 + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) by XREAL_1:232 ; (Lower_Seq (C,n)) /. 2 = ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n))))) /. 2 by Th18 .= (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 2 by A1, A2, FINSEQ_5:43 .= (Cage (C,n)) /. ((2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) by A7, A11, REVROT_1:9 .= (Cage (C,n)) /. ((2 - 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) by XREAL_0:def_2 ; hence ((Lower_Seq (C,n)) /. 2) `1 = E-bound (L~ (Cage (C,n))) by A8, A6, A13, JORDAN1E:20, JORDAN1F:6; ::_thesis: verum end; theorem Th33: :: JORDAN1G:33 for n being Element of NAT for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds (W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n)))) = (W-bound C) + (E-bound C) proof let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds (W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n)))) = (W-bound C) + (E-bound C) let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: (W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n)))) = (W-bound C) + (E-bound C) thus (W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n)))) = (W-bound (L~ (Cage (C,n)))) + ((E-bound C) + (((E-bound C) - (W-bound C)) / (2 |^ n))) by JORDAN1A:64 .= ((W-bound C) - (((E-bound C) - (W-bound C)) / (2 |^ n))) + ((E-bound C) + (((E-bound C) - (W-bound C)) / (2 |^ n))) by JORDAN1A:62 .= (W-bound C) + (E-bound C) ; ::_thesis: verum end; theorem :: JORDAN1G:34 for n being Element of NAT for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds (S-bound (L~ (Cage (C,n)))) + (N-bound (L~ (Cage (C,n)))) = (S-bound C) + (N-bound C) proof let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds (S-bound (L~ (Cage (C,n)))) + (N-bound (L~ (Cage (C,n)))) = (S-bound C) + (N-bound C) let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: (S-bound (L~ (Cage (C,n)))) + (N-bound (L~ (Cage (C,n)))) = (S-bound C) + (N-bound C) thus (S-bound (L~ (Cage (C,n)))) + (N-bound (L~ (Cage (C,n)))) = (S-bound (L~ (Cage (C,n)))) + ((N-bound C) + (((N-bound C) - (S-bound C)) / (2 |^ n))) by JORDAN10:6 .= ((S-bound C) - (((N-bound C) - (S-bound C)) / (2 |^ n))) + ((N-bound C) + (((N-bound C) - (S-bound C)) / (2 |^ n))) by JORDAN1A:63 .= (S-bound C) + (N-bound C) ; ::_thesis: verum end; theorem Th35: :: JORDAN1G:35 for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) for n being Element of NAT for i being Nat st 1 <= i & i <= width (Gauge (C,n)) & n > 0 holds ((Gauge (C,n)) * ((Center (Gauge (C,n))),i)) `1 = ((W-bound C) + (E-bound C)) / 2 proof let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for n being Element of NAT for i being Nat st 1 <= i & i <= width (Gauge (C,n)) & n > 0 holds ((Gauge (C,n)) * ((Center (Gauge (C,n))),i)) `1 = ((W-bound C) + (E-bound C)) / 2 let n be Element of NAT ; ::_thesis: for i being Nat st 1 <= i & i <= width (Gauge (C,n)) & n > 0 holds ((Gauge (C,n)) * ((Center (Gauge (C,n))),i)) `1 = ((W-bound C) + (E-bound C)) / 2 let i be Nat; ::_thesis: ( 1 <= i & i <= width (Gauge (C,n)) & n > 0 implies ((Gauge (C,n)) * ((Center (Gauge (C,n))),i)) `1 = ((W-bound C) + (E-bound C)) / 2 ) assume A1: ( 1 <= i & i <= width (Gauge (C,n)) ) ; ::_thesis: ( not n > 0 or ((Gauge (C,n)) * ((Center (Gauge (C,n))),i)) `1 = ((W-bound C) + (E-bound C)) / 2 ) reconsider ii = i as Element of NAT by ORDINAL1:def_12; A2: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1; assume A3: n > 0 ; ::_thesis: ((Gauge (C,n)) * ((Center (Gauge (C,n))),i)) `1 = ((W-bound C) + (E-bound C)) / 2 len (Gauge (C,1)) >= 4 by JORDAN8:10; then A4: len (Gauge (C,1)) >= 1 by XXREAL_0:2; thus ((Gauge (C,n)) * ((Center (Gauge (C,n))),i)) `1 = ((Gauge (C,n)) * ((Center (Gauge (C,n))),ii)) `1 .= ((Gauge (C,1)) * ((Center (Gauge (C,1))),1)) `1 by A1, A2, A4, A3, JORDAN1A:36 .= ((W-bound C) + (E-bound C)) / 2 by A4, JORDAN1A:38 ; ::_thesis: verum end; theorem :: JORDAN1G:36 for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) for n, i being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & n > 0 holds ((Gauge (C,n)) * (i,(Center (Gauge (C,n))))) `2 = ((S-bound C) + (N-bound C)) / 2 proof let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for n, i being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & n > 0 holds ((Gauge (C,n)) * (i,(Center (Gauge (C,n))))) `2 = ((S-bound C) + (N-bound C)) / 2 let n, i be Element of NAT ; ::_thesis: ( 1 <= i & i <= len (Gauge (C,n)) & n > 0 implies ((Gauge (C,n)) * (i,(Center (Gauge (C,n))))) `2 = ((S-bound C) + (N-bound C)) / 2 ) assume A1: ( 1 <= i & i <= len (Gauge (C,n)) ) ; ::_thesis: ( not n > 0 or ((Gauge (C,n)) * (i,(Center (Gauge (C,n))))) `2 = ((S-bound C) + (N-bound C)) / 2 ) len (Gauge (C,1)) >= 4 by JORDAN8:10; then A2: len (Gauge (C,1)) >= 1 by XXREAL_0:2; assume n > 0 ; ::_thesis: ((Gauge (C,n)) * (i,(Center (Gauge (C,n))))) `2 = ((S-bound C) + (N-bound C)) / 2 hence ((Gauge (C,n)) * (i,(Center (Gauge (C,n))))) `2 = ((Gauge (C,1)) * (1,(Center (Gauge (C,1))))) `2 by A1, A2, JORDAN1A:37 .= ((S-bound C) + (N-bound C)) / 2 by A2, JORDAN1A:39 ; ::_thesis: verum end; theorem Th37: :: JORDAN1G:37 for f being S-Sequence_in_R2 for k1, k2 being Element of NAT st 1 <= k1 & k1 <= len f & 1 <= k2 & k2 <= len f & f /. 1 in L~ (mid (f,k1,k2)) & not k1 = 1 holds k2 = 1 proof let f be S-Sequence_in_R2; ::_thesis: for k1, k2 being Element of NAT st 1 <= k1 & k1 <= len f & 1 <= k2 & k2 <= len f & f /. 1 in L~ (mid (f,k1,k2)) & not k1 = 1 holds k2 = 1 let k1, k2 be Element of NAT ; ::_thesis: ( 1 <= k1 & k1 <= len f & 1 <= k2 & k2 <= len f & f /. 1 in L~ (mid (f,k1,k2)) & not k1 = 1 implies k2 = 1 ) assume that A1: 1 <= k1 and A2: k1 <= len f and A3: 1 <= k2 and A4: k2 <= len f and A5: f /. 1 in L~ (mid (f,k1,k2)) ; ::_thesis: ( k1 = 1 or k2 = 1 ) assume that A6: k1 <> 1 and A7: k2 <> 1 ; ::_thesis: contradiction A8: len f >= 2 by TOPREAL1:def_8; consider j being Element of NAT such that A9: 1 <= j and A10: j + 1 <= len (mid (f,k1,k2)) and A11: f /. 1 in LSeg ((mid (f,k1,k2)),j) by A5, SPPOL_2:13; percases ( k1 < k2 or k1 > k2 or k1 = k2 ) by XXREAL_0:1; supposeA12: k1 < k2 ; ::_thesis: contradiction then len (mid (f,k1,k2)) = (k2 -' k1) + 1 by A1, A2, A3, A4, FINSEQ_6:118; then j < (k2 -' k1) + 1 by A10, NAT_1:13; then LSeg ((mid (f,k1,k2)),j) = LSeg (f,((j + k1) -' 1)) by A1, A4, A9, A12, JORDAN4:19; then A13: (j + k1) -' 1 = 1 by A11, A8, JORDAN5B:30; j + k1 >= 1 + 1 by A1, A9, XREAL_1:7; then (j + k1) - 1 >= (1 + 1) - 1 by XREAL_1:9; then j + (k1 - 1) = 1 by A13, XREAL_0:def_2; then k1 - 1 = 1 - j ; then k1 - 1 <= 0 by A9, XREAL_1:47; then k1 - 1 = 0 by A1, XREAL_1:48; hence contradiction by A6; ::_thesis: verum end; supposeA14: k1 > k2 ; ::_thesis: contradiction then len (mid (f,k1,k2)) = (k1 -' k2) + 1 by A1, A2, A3, A4, FINSEQ_6:118; then A15: j < (k1 -' k2) + 1 by A10, NAT_1:13; k1 - k2 > 0 by A14, XREAL_1:50; then k1 -' k2 = k1 - k2 by XREAL_0:def_2; then j - 1 < k1 - k2 by A15, XREAL_1:19; then (j - 1) + k2 < k1 by XREAL_1:20; then j + (- (1 - k2)) < k1 ; then A16: k2 - 1 < k1 - j by XREAL_1:20; LSeg ((mid (f,k1,k2)),j) = LSeg (f,(k1 -' j)) by A2, A3, A9, A14, A15, JORDAN4:20; then k1 -' j = 1 by A11, A8, JORDAN5B:30; then k1 - j = 1 by XREAL_0:def_2; then k2 < 1 + 1 by A16, XREAL_1:19; then k2 <= 1 by NAT_1:13; hence contradiction by A3, A7, XXREAL_0:1; ::_thesis: verum end; suppose k1 = k2 ; ::_thesis: contradiction then mid (f,k1,k2) = <*(f /. k1)*> by A1, A2, JORDAN4:15; hence contradiction by A5, SPPOL_2:12; ::_thesis: verum end; end; end; theorem Th38: :: JORDAN1G:38 for f being S-Sequence_in_R2 for k1, k2 being Element of NAT st 1 <= k1 & k1 <= len f & 1 <= k2 & k2 <= len f & f /. (len f) in L~ (mid (f,k1,k2)) & not k1 = len f holds k2 = len f proof let f be S-Sequence_in_R2; ::_thesis: for k1, k2 being Element of NAT st 1 <= k1 & k1 <= len f & 1 <= k2 & k2 <= len f & f /. (len f) in L~ (mid (f,k1,k2)) & not k1 = len f holds k2 = len f let k1, k2 be Element of NAT ; ::_thesis: ( 1 <= k1 & k1 <= len f & 1 <= k2 & k2 <= len f & f /. (len f) in L~ (mid (f,k1,k2)) & not k1 = len f implies k2 = len f ) assume that A1: 1 <= k1 and A2: k1 <= len f and A3: 1 <= k2 and A4: k2 <= len f and A5: f /. (len f) in L~ (mid (f,k1,k2)) ; ::_thesis: ( k1 = len f or k2 = len f ) assume that A6: k1 <> len f and A7: k2 <> len f ; ::_thesis: contradiction consider j being Element of NAT such that A8: 1 <= j and A9: j + 1 <= len (mid (f,k1,k2)) and A10: f /. (len f) in LSeg ((mid (f,k1,k2)),j) by A5, SPPOL_2:13; percases ( k1 < k2 or k1 > k2 or k1 = k2 ) by XXREAL_0:1; supposeA11: k1 < k2 ; ::_thesis: contradiction then A12: len (mid (f,k1,k2)) = (k2 -' k1) + 1 by A1, A2, A3, A4, FINSEQ_6:118; then A13: j < (k2 -' k1) + 1 by A9, NAT_1:13; A14: j + k1 >= 1 + 1 by A1, A8, XREAL_1:7; then A15: (j + k1) - 1 >= (1 + 1) - 1 by XREAL_1:9; then A16: (j + k1) -' 1 = (j + k1) - 1 by XREAL_0:def_2; k2 - k1 > 0 by A11, XREAL_1:50; then A17: k2 -' k1 = k2 - k1 by XREAL_0:def_2; then j - 1 < k2 - k1 by A13, XREAL_1:19; then (j - 1) + k1 < k2 by XREAL_1:20; then A18: (j + k1) - 1 < len f by A4, XXREAL_0:2; then A19: (j + k1) -' 1 in dom f by A15, A16, FINSEQ_3:25; A20: j + k1 >= 1 by A14, XXREAL_0:2; ((j + k1) - 1) + 1 <= len f by A16, A18, NAT_1:13; then j + k1 in Seg (len f) by A20, FINSEQ_1:1; then A21: ((j + k1) -' 1) + 1 in dom f by A16, FINSEQ_1:def_3; LSeg ((mid (f,k1,k2)),j) = LSeg (f,((j + k1) -' 1)) by A1, A4, A8, A11, A13, JORDAN4:19; then A22: ((j + k1) -' 1) + 1 = len f by A10, A19, A21, GOBOARD2:2; A23: (j + k1) -' 1 = (j + k1) - 1 by A15, XREAL_0:def_2; j < (k2 + 1) - k1 by A9, A17, A12, NAT_1:13; then len f < k2 + 1 by A22, A23, XREAL_1:20; then len f <= k2 by NAT_1:13; hence contradiction by A4, A7, XXREAL_0:1; ::_thesis: verum end; supposeA24: k1 > k2 ; ::_thesis: contradiction then len (mid (f,k1,k2)) = (k1 -' k2) + 1 by A1, A2, A3, A4, FINSEQ_6:118; then A25: j < (k1 -' k2) + 1 by A9, NAT_1:13; k1 - k2 > 0 by A24, XREAL_1:50; then k1 -' k2 = k1 - k2 by XREAL_0:def_2; then j - 1 < k1 - k2 by A25, XREAL_1:19; then (j - 1) + k2 < k1 by XREAL_1:20; then A26: j + (- (1 - k2)) < k1 ; then A27: - (1 - k2) < k1 - j by XREAL_1:20; A28: k2 - 1 >= 0 by A3, XREAL_1:48; then A29: (k1 - j) + 1 > 0 + 1 by A27, XREAL_1:6; k2 - 1 < k1 - j by A26, XREAL_1:20; then A30: k1 - j > 0 by A3, XREAL_1:48; then A31: k1 -' j = k1 - j by XREAL_0:def_2; k1 - j <= k1 - 1 by A8, XREAL_1:10; then (k1 - j) + 1 <= (k1 - 1) + 1 by XREAL_1:7; then k1 - j < k1 by A31, NAT_1:13; then A32: k1 - j < len f by A2, XXREAL_0:2; then (k1 - j) + 1 <= len f by A31, NAT_1:13; then A33: (k1 -' j) + 1 in dom f by A31, A29, FINSEQ_3:25; k1 - j >= 0 + 1 by A27, A28, A31, NAT_1:13; then A34: k1 -' j in dom f by A31, A32, FINSEQ_3:25; LSeg ((mid (f,k1,k2)),j) = LSeg (f,(k1 -' j)) by A2, A3, A8, A24, A25, JORDAN4:20; then (k1 -' j) + 1 = len f by A10, A34, A33, GOBOARD2:2; then A35: (k1 - j) + 1 = len f by A30, XREAL_0:def_2; k1 - j <= k1 - 1 by A8, XREAL_1:10; then len f <= (k1 - 1) + 1 by A35, XREAL_1:7; hence contradiction by A2, A6, XXREAL_0:1; ::_thesis: verum end; suppose k1 = k2 ; ::_thesis: contradiction then mid (f,k1,k2) = <*(f /. k1)*> by A1, A2, JORDAN4:15; hence contradiction by A5, SPPOL_2:12; ::_thesis: verum end; end; end; theorem Th39: :: JORDAN1G:39 for C being compact non horizontal non vertical Subset of (TOP-REAL 2) for n being Element of NAT holds ( rng (Upper_Seq (C,n)) c= rng (Cage (C,n)) & rng (Lower_Seq (C,n)) c= rng (Cage (C,n)) ) proof let C be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for n being Element of NAT holds ( rng (Upper_Seq (C,n)) c= rng (Cage (C,n)) & rng (Lower_Seq (C,n)) c= rng (Cage (C,n)) ) let n be Element of NAT ; ::_thesis: ( rng (Upper_Seq (C,n)) c= rng (Cage (C,n)) & rng (Lower_Seq (C,n)) c= rng (Cage (C,n)) ) E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46; then A1: E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by FINSEQ_6:90, SPRECT_2:43; Upper_Seq (C,n) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n)))) by JORDAN1E:def_1; then rng (Upper_Seq (C,n)) c= rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by FINSEQ_5:48; hence rng (Upper_Seq (C,n)) c= rng (Cage (C,n)) by FINSEQ_6:90, SPRECT_2:43; ::_thesis: rng (Lower_Seq (C,n)) c= rng (Cage (C,n)) Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) :- (E-max (L~ (Cage (C,n)))) by JORDAN1E:def_2; then rng (Lower_Seq (C,n)) c= rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A1, FINSEQ_5:55; hence rng (Lower_Seq (C,n)) c= rng (Cage (C,n)) by FINSEQ_6:90, SPRECT_2:43; ::_thesis: verum end; theorem Th40: :: JORDAN1G:40 for n being Element of NAT for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds Upper_Seq (C,n) is_a_h.c._for Cage (C,n) proof let n be Element of NAT ; ::_thesis: for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds Upper_Seq (C,n) is_a_h.c._for Cage (C,n) let C be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: Upper_Seq (C,n) is_a_h.c._for Cage (C,n) A1: ((Upper_Seq (C,n)) /. 1) `1 = (W-min (L~ (Cage (C,n)))) `1 by JORDAN1F:5 .= W-bound (L~ (Cage (C,n))) by EUCLID:52 ; A2: ((Upper_Seq (C,n)) /. (len (Upper_Seq (C,n)))) `1 = (E-max (L~ (Cage (C,n)))) `1 by JORDAN1F:7 .= E-bound (L~ (Cage (C,n))) by EUCLID:52 ; Upper_Seq (C,n) is_in_the_area_of Cage (C,n) by JORDAN1E:17; hence Upper_Seq (C,n) is_a_h.c._for Cage (C,n) by A1, A2, SPRECT_2:def_2; ::_thesis: verum end; theorem Th41: :: JORDAN1G:41 for n being Element of NAT for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds Rev (Lower_Seq (C,n)) is_a_h.c._for Cage (C,n) proof let n be Element of NAT ; ::_thesis: for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds Rev (Lower_Seq (C,n)) is_a_h.c._for Cage (C,n) let C be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: Rev (Lower_Seq (C,n)) is_a_h.c._for Cage (C,n) A1: ((Rev (Lower_Seq (C,n))) /. 1) `1 = ((Lower_Seq (C,n)) /. (len (Lower_Seq (C,n)))) `1 by FINSEQ_5:65 .= (W-min (L~ (Cage (C,n)))) `1 by JORDAN1F:8 .= W-bound (L~ (Cage (C,n))) by EUCLID:52 ; A2: ((Rev (Lower_Seq (C,n))) /. (len (Rev (Lower_Seq (C,n))))) `1 = ((Rev (Lower_Seq (C,n))) /. (len (Lower_Seq (C,n)))) `1 by FINSEQ_5:def_3 .= ((Lower_Seq (C,n)) /. 1) `1 by FINSEQ_5:65 .= (E-max (L~ (Cage (C,n)))) `1 by JORDAN1F:6 .= E-bound (L~ (Cage (C,n))) by EUCLID:52 ; Rev (Lower_Seq (C,n)) is_in_the_area_of Cage (C,n) by JORDAN1E:18, SPRECT_3:51; hence Rev (Lower_Seq (C,n)) is_a_h.c._for Cage (C,n) by A1, A2, SPRECT_2:def_2; ::_thesis: verum end; theorem Th42: :: JORDAN1G:42 for n being Element of NAT for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) for i being Element of NAT st 1 < i & i <= len (Gauge (C,n)) holds not (Gauge (C,n)) * (i,1) in rng (Upper_Seq (C,n)) proof let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) for i being Element of NAT st 1 < i & i <= len (Gauge (C,n)) holds not (Gauge (C,n)) * (i,1) in rng (Upper_Seq (C,n)) let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for i being Element of NAT st 1 < i & i <= len (Gauge (C,n)) holds not (Gauge (C,n)) * (i,1) in rng (Upper_Seq (C,n)) let i be Element of NAT ; ::_thesis: ( 1 < i & i <= len (Gauge (C,n)) implies not (Gauge (C,n)) * (i,1) in rng (Upper_Seq (C,n)) ) assume that A1: ( 1 < i & i <= len (Gauge (C,n)) ) and A2: (Gauge (C,n)) * (i,1) in rng (Upper_Seq (C,n)) ; ::_thesis: contradiction consider i2 being Nat such that A3: i2 in dom (Upper_Seq (C,n)) and A4: (Upper_Seq (C,n)) . i2 = (Gauge (C,n)) * (i,1) by A2, FINSEQ_2:10; reconsider i2 = i2 as Element of NAT by ORDINAL1:def_12; A5: ( 1 <= i2 & i2 <= len (Upper_Seq (C,n)) ) by A3, FINSEQ_3:25; set f = Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))); set i1 = (N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)); A6: ( E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) & rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) = rng (Cage (C,n)) ) by FINSEQ_6:90, SPRECT_2:43, SPRECT_2:46; W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43; then A7: (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 = W-min (L~ (Cage (C,n))) by FINSEQ_6:92; L~ (Cage (C,n)) = L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by REVROT_1:33; then A8: ( (N-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) < (N-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) & (N-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) <= (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) ) by A7, SPRECT_5:24, SPRECT_5:25; (E-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) = len (Upper_Seq (C,n)) by Th24; then (N-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) <= len (Upper_Seq (C,n)) by Th23; then A9: (N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) < len (Upper_Seq (C,n)) by Th22, XXREAL_0:2; 3 <= len (Lower_Seq (C,n)) by JORDAN1E:15; then A10: 2 <= len (Lower_Seq (C,n)) by XXREAL_0:2; A11: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1; 4 <= len (Gauge (C,n)) by JORDAN8:10; then A12: 1 <= len (Gauge (C,n)) by XXREAL_0:2; ( (W-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) = 1 & (W-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) <= (N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) ) by Th19, Th21; then A13: (N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) > 1 by Th20, XXREAL_0:2; then A14: (N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by A9, FINSEQ_3:25; ( Upper_Seq (C,n) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n)))) & N-min (L~ (Cage (C,n))) in rng (Cage (C,n)) ) by JORDAN1E:def_1, SPRECT_2:39; then A15: N-min (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) by A6, A8, FINSEQ_5:46, XXREAL_0:2; then A16: (Upper_Seq (C,n)) /. ((N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n))) = N-min (L~ (Cage (C,n))) by FINSEQ_5:38; A17: (N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) <> i2 proof assume (N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) = i2 ; ::_thesis: contradiction then (Gauge (C,n)) * (i,1) = N-min (L~ (Cage (C,n))) by A4, A14, A16, PARTFUN1:def_6; then ((Gauge (C,n)) * (i,1)) `2 = N-bound (L~ (Cage (C,n))) by EUCLID:52; then S-bound (L~ (Cage (C,n))) = N-bound (L~ (Cage (C,n))) by A1, JORDAN1A:72; hence contradiction by SPRECT_1:16; ::_thesis: verum end; then mid ((Upper_Seq (C,n)),((N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n))),i2) is being_S-Seq by A13, A9, A5, JORDAN3:6; then reconsider h1 = mid ((Upper_Seq (C,n)),((N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n))),i2) as one-to-one special FinSequence of (TOP-REAL 2) ; set h = Rev h1; A18: len h1 = len (Rev h1) by FINSEQ_5:def_3; then A19: not h1 is empty by A3, A14, SPRECT_2:5; then A20: ((Rev h1) /. (len (Rev h1))) `2 = (h1 /. 1) `2 by A18, FINSEQ_5:65 .= ((Upper_Seq (C,n)) /. ((N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)))) `2 by A3, A14, SPRECT_2:8 .= (N-min (L~ (Cage (C,n)))) `2 by A15, FINSEQ_5:38 .= N-bound (L~ (Cage (C,n))) by EUCLID:52 ; h1 is_in_the_area_of Cage (C,n) by A3, A14, JORDAN1E:17, SPRECT_2:22; then A21: Rev h1 is_in_the_area_of Cage (C,n) by SPRECT_3:51; ((Rev h1) /. 1) `2 = (h1 /. (len h1)) `2 by A19, FINSEQ_5:65 .= ((Upper_Seq (C,n)) /. i2) `2 by A3, A14, SPRECT_2:9 .= ((Gauge (C,n)) * (i,1)) `2 by A3, A4, PARTFUN1:def_6 .= S-bound (L~ (Cage (C,n))) by A1, JORDAN1A:72 ; then A22: ( Rev (Lower_Seq (C,n)) is special & Rev h1 is_a_v.c._for Cage (C,n) ) by A21, A20, SPRECT_2:def_3; len (Rev h1) >= 1 by A3, A14, A18, SPRECT_2:5; then len (Rev h1) > 1 by A3, A14, A17, A18, SPRECT_2:6, XXREAL_0:1; then A23: 1 + 1 <= len (Rev h1) by NAT_1:13; ( len (Lower_Seq (C,n)) = len (Rev (Lower_Seq (C,n))) & Rev h1 is special ) by FINSEQ_5:def_3, SPPOL_2:40; then ( L~ (Rev (Lower_Seq (C,n))) = L~ (Lower_Seq (C,n)) & L~ (Rev (Lower_Seq (C,n))) meets L~ (Rev h1) ) by A10, A23, A22, Th41, SPPOL_2:22, SPRECT_2:29; then consider x being set such that A24: x in L~ (Lower_Seq (C,n)) and A25: x in L~ (Rev h1) by XBOOLE_0:3; A26: L~ (Rev h1) = L~ h1 by SPPOL_2:22; L~ (mid ((Upper_Seq (C,n)),((N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n))),i2)) c= L~ (Upper_Seq (C,n)) by A13, A9, A5, JORDAN4:35; then x in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A24, A25, A26, XBOOLE_0:def_4; then A27: x in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16; percases ( x = W-min (L~ (Cage (C,n))) or x = E-max (L~ (Cage (C,n))) ) by A27, TARSKI:def_2; suppose x = W-min (L~ (Cage (C,n))) ; ::_thesis: contradiction then x = (Upper_Seq (C,n)) /. 1 by JORDAN1F:5; then i2 = 1 by A13, A9, A5, A25, A26, Th37; then (Upper_Seq (C,n)) /. 1 = (Gauge (C,n)) * (i,1) by A3, A4, PARTFUN1:def_6; then W-min (L~ (Cage (C,n))) = (Gauge (C,n)) * (i,1) by JORDAN1F:5; then ((Gauge (C,n)) * (i,1)) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52 .= ((Gauge (C,n)) * (1,1)) `1 by A12, JORDAN1A:73 ; hence contradiction by A1, A12, A11, GOBOARD5:3; ::_thesis: verum end; suppose x = E-max (L~ (Cage (C,n))) ; ::_thesis: contradiction then x = (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) by JORDAN1F:7; then i2 = len (Upper_Seq (C,n)) by A13, A9, A5, A25, A26, Th38; then (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) = (Gauge (C,n)) * (i,1) by A3, A4, PARTFUN1:def_6; then A28: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i,1) by JORDAN1F:7; (SE-corner (L~ (Cage (C,n)))) `2 <= (E-min (L~ (Cage (C,n)))) `2 by PSCOMP_1:46; then (SE-corner (L~ (Cage (C,n)))) `2 < (E-max (L~ (Cage (C,n)))) `2 by SPRECT_2:53, XXREAL_0:2; then S-bound (L~ (Cage (C,n))) < ((Gauge (C,n)) * (i,1)) `2 by A28, EUCLID:52; hence contradiction by A1, JORDAN1A:72; ::_thesis: verum end; end; end; theorem Th43: :: JORDAN1G:43 for n being Element of NAT for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) for i being Element of NAT st 1 <= i & i < len (Gauge (C,n)) holds not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in rng (Lower_Seq (C,n)) proof let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) for i being Element of NAT st 1 <= i & i < len (Gauge (C,n)) holds not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in rng (Lower_Seq (C,n)) let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for i being Element of NAT st 1 <= i & i < len (Gauge (C,n)) holds not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in rng (Lower_Seq (C,n)) let i be Element of NAT ; ::_thesis: ( 1 <= i & i < len (Gauge (C,n)) implies not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in rng (Lower_Seq (C,n)) ) set wi = width (Gauge (C,n)); assume that A1: ( 1 <= i & i < len (Gauge (C,n)) ) and A2: (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in rng (Lower_Seq (C,n)) ; ::_thesis: contradiction consider i2 being Nat such that A3: i2 in dom (Lower_Seq (C,n)) and A4: (Lower_Seq (C,n)) . i2 = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) by A2, FINSEQ_2:10; reconsider i2 = i2 as Element of NAT by ORDINAL1:def_12; A5: ( 1 <= i2 & i2 <= len (Lower_Seq (C,n)) ) by A3, FINSEQ_3:25; 3 <= len (Upper_Seq (C,n)) by JORDAN1E:15; then A6: 2 <= len (Upper_Seq (C,n)) by XXREAL_0:2; set f = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))); set i1 = (S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)); A7: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1; E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46; then A8: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = E-max (L~ (Cage (C,n))) by FINSEQ_6:92; L~ (Cage (C,n)) = L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by REVROT_1:33; then A9: ( (S-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) < (S-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) & (S-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) <= (W-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) ) by A8, SPRECT_5:40, SPRECT_5:41; A10: ( W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) & rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = rng (Cage (C,n)) ) by FINSEQ_6:90, SPRECT_2:43, SPRECT_2:46; (W-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) = len (Lower_Seq (C,n)) by Th30; then (S-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) <= len (Lower_Seq (C,n)) by Th29; then A11: (S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) < len (Lower_Seq (C,n)) by Th28, XXREAL_0:2; ( (E-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) = 1 & (E-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) <= (S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) ) by Th25, Th27; then A12: (S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) > 1 by Th26, XXREAL_0:2; then A13: (S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) in dom (Lower_Seq (C,n)) by A11, FINSEQ_3:25; ( Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) & S-max (L~ (Cage (C,n))) in rng (Cage (C,n)) ) by Th18, SPRECT_2:42; then A14: S-max (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) by A10, A9, FINSEQ_5:46, XXREAL_0:2; then A15: (Lower_Seq (C,n)) /. ((S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n))) = S-max (L~ (Cage (C,n))) by FINSEQ_5:38; A16: (S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) <> i2 proof assume (S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) = i2 ; ::_thesis: contradiction then (Gauge (C,n)) * (i,(width (Gauge (C,n)))) = S-max (L~ (Cage (C,n))) by A4, A13, A15, PARTFUN1:def_6; then ((Gauge (C,n)) * (i,(width (Gauge (C,n))))) `2 = S-bound (L~ (Cage (C,n))) by EUCLID:52; then N-bound (L~ (Cage (C,n))) = S-bound (L~ (Cage (C,n))) by A1, A7, JORDAN1A:70; hence contradiction by SPRECT_1:16; ::_thesis: verum end; then mid ((Lower_Seq (C,n)),((S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n))),i2) is being_S-Seq by A12, A11, A5, JORDAN3:6; then reconsider h = mid ((Lower_Seq (C,n)),((S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n))),i2) as one-to-one special FinSequence of (TOP-REAL 2) ; A17: (h /. 1) `2 = ((Lower_Seq (C,n)) /. ((S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)))) `2 by A3, A13, SPRECT_2:8 .= (S-max (L~ (Cage (C,n)))) `2 by A14, FINSEQ_5:38 .= S-bound (L~ (Cage (C,n))) by EUCLID:52 ; len h >= 1 by A3, A13, SPRECT_2:5; then len h > 1 by A3, A13, A16, SPRECT_2:6, XXREAL_0:1; then A18: 1 + 1 <= len h by NAT_1:13; A19: h is_in_the_area_of Cage (C,n) by A3, A13, JORDAN1E:18, SPRECT_2:22; (h /. (len h)) `2 = ((Lower_Seq (C,n)) /. i2) `2 by A3, A13, SPRECT_2:9 .= ((Gauge (C,n)) * (i,(width (Gauge (C,n))))) `2 by A3, A4, PARTFUN1:def_6 .= N-bound (L~ (Cage (C,n))) by A1, A7, JORDAN1A:70 ; then h is_a_v.c._for Cage (C,n) by A19, A17, SPRECT_2:def_3; then L~ (Upper_Seq (C,n)) meets L~ h by A6, A18, Th40, SPRECT_2:29; then consider x being set such that A20: x in L~ (Upper_Seq (C,n)) and A21: x in L~ h by XBOOLE_0:3; L~ (mid ((Lower_Seq (C,n)),((S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n))),i2)) c= L~ (Lower_Seq (C,n)) by A12, A11, A5, JORDAN4:35; then x in (L~ (Lower_Seq (C,n))) /\ (L~ (Upper_Seq (C,n))) by A20, A21, XBOOLE_0:def_4; then A22: x in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16; 4 <= len (Gauge (C,n)) by JORDAN8:10; then A23: 1 <= len (Gauge (C,n)) by XXREAL_0:2; percases ( x = E-max (L~ (Cage (C,n))) or x = W-min (L~ (Cage (C,n))) ) by A22, TARSKI:def_2; suppose x = E-max (L~ (Cage (C,n))) ; ::_thesis: contradiction then x = (Lower_Seq (C,n)) /. 1 by JORDAN1F:6; then i2 = 1 by A12, A11, A5, A21, Th37; then (Lower_Seq (C,n)) /. 1 = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) by A3, A4, PARTFUN1:def_6; then E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) by JORDAN1F:6; then ((Gauge (C,n)) * (i,(width (Gauge (C,n))))) `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52 .= ((Gauge (C,n)) * ((len (Gauge (C,n))),(width (Gauge (C,n))))) `1 by A7, A23, JORDAN1A:71 ; hence contradiction by A1, A7, A23, GOBOARD5:3; ::_thesis: verum end; suppose x = W-min (L~ (Cage (C,n))) ; ::_thesis: contradiction then x = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by JORDAN1F:8; then i2 = len (Lower_Seq (C,n)) by A12, A11, A5, A21, Th38; then (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) by A3, A4, PARTFUN1:def_6; then A24: W-min (L~ (Cage (C,n))) = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) by JORDAN1F:8; (NW-corner (L~ (Cage (C,n)))) `2 >= (W-max (L~ (Cage (C,n)))) `2 by PSCOMP_1:30; then (NW-corner (L~ (Cage (C,n)))) `2 > (W-min (L~ (Cage (C,n)))) `2 by SPRECT_2:57, XXREAL_0:2; then N-bound (L~ (Cage (C,n))) > ((Gauge (C,n)) * (i,(width (Gauge (C,n))))) `2 by A24, EUCLID:52; hence contradiction by A1, A7, JORDAN1A:70; ::_thesis: verum end; end; end; theorem Th44: :: JORDAN1G:44 for n being Element of NAT for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) for i being Element of NAT st 1 < i & i <= len (Gauge (C,n)) holds not (Gauge (C,n)) * (i,1) in L~ (Upper_Seq (C,n)) proof let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) for i being Element of NAT st 1 < i & i <= len (Gauge (C,n)) holds not (Gauge (C,n)) * (i,1) in L~ (Upper_Seq (C,n)) let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for i being Element of NAT st 1 < i & i <= len (Gauge (C,n)) holds not (Gauge (C,n)) * (i,1) in L~ (Upper_Seq (C,n)) let i be Element of NAT ; ::_thesis: ( 1 < i & i <= len (Gauge (C,n)) implies not (Gauge (C,n)) * (i,1) in L~ (Upper_Seq (C,n)) ) assume that A1: ( 1 < i & i <= len (Gauge (C,n)) ) and A2: (Gauge (C,n)) * (i,1) in L~ (Upper_Seq (C,n)) ; ::_thesis: contradiction set Gi1 = (Gauge (C,n)) * (i,1); consider ii being Element of NAT such that A3: 1 <= ii and A4: ii + 1 <= len (Upper_Seq (C,n)) and A5: (Gauge (C,n)) * (i,1) in LSeg ((Upper_Seq (C,n)),ii) by A2, SPPOL_2:13; A6: LSeg ((Upper_Seq (C,n)),ii) = LSeg (((Upper_Seq (C,n)) /. ii),((Upper_Seq (C,n)) /. (ii + 1))) by A3, A4, TOPREAL1:def_3; ii + 1 >= 1 by NAT_1:11; then A7: ii + 1 in dom (Upper_Seq (C,n)) by A4, FINSEQ_3:25; len (Gauge (C,n)) >= 4 by JORDAN8:10; then ( len (Gauge (C,n)) = width (Gauge (C,n)) & len (Gauge (C,n)) > 1 ) by JORDAN8:def_1, XXREAL_0:2; then A8: [i,1] in Indices (Gauge (C,n)) by A1, MATRIX_1:36; ii < len (Upper_Seq (C,n)) by A4, NAT_1:13; then A9: ii in dom (Upper_Seq (C,n)) by A3, FINSEQ_3:25; A10: not (Gauge (C,n)) * (i,1) in rng (Upper_Seq (C,n)) by A1, Th42; Upper_Seq (C,n) is_sequence_on Gauge (C,n) by Th4; then consider i1, j1, i2, j2 being Element of NAT such that A11: [i1,j1] in Indices (Gauge (C,n)) and A12: (Upper_Seq (C,n)) /. ii = (Gauge (C,n)) * (i1,j1) and A13: [i2,j2] in Indices (Gauge (C,n)) and A14: (Upper_Seq (C,n)) /. (ii + 1) = (Gauge (C,n)) * (i2,j2) and A15: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A3, A4, JORDAN8:3; A16: 1 <= i1 by A11, MATRIX_1:38; A17: j2 <= width (Gauge (C,n)) by A13, MATRIX_1:38; A18: 1 <= j1 by A11, MATRIX_1:38; A19: i1 <= len (Gauge (C,n)) by A11, MATRIX_1:38; A20: 1 <= j2 by A13, MATRIX_1:38; A21: i2 <= len (Gauge (C,n)) by A13, MATRIX_1:38; A22: 1 <= i2 by A13, MATRIX_1:38; A23: j1 <= width (Gauge (C,n)) by A11, MATRIX_1:38; percases ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A15; supposeA24: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: contradiction then j1 <= j2 by NAT_1:11; then ((Gauge (C,n)) * (i1,j1)) `2 <= ((Gauge (C,n)) * (i2,j2)) `2 by A16, A19, A18, A17, A24, SPRECT_3:12; then A25: ((Gauge (C,n)) * (i1,j1)) `2 <= ((Gauge (C,n)) * (i,1)) `2 by A5, A6, A12, A14, TOPREAL1:4; ((Gauge (C,n)) * (i1,j1)) `1 = ((Gauge (C,n)) * (i2,1)) `1 by A16, A19, A18, A23, A24, GOBOARD5:2 .= ((Gauge (C,n)) * (i2,j2)) `1 by A22, A21, A20, A17, GOBOARD5:2 ; then LSeg (((Upper_Seq (C,n)) /. ii),((Upper_Seq (C,n)) /. (ii + 1))) is vertical by A12, A14, SPPOL_1:16; then ((Gauge (C,n)) * (i,1)) `1 = ((Gauge (C,n)) * (i1,j1)) `1 by A5, A6, A12, SPPOL_1:41; then A26: i1 = i by A11, A8, Th7; then ((Gauge (C,n)) * (i,1)) `2 <= ((Gauge (C,n)) * (i1,j1)) `2 by A16, A19, A18, A23, SPRECT_3:12; then j1 = 1 by A11, A8, A25, Th6, XXREAL_0:1; hence contradiction by A12, A9, A10, A26, PARTFUN2:2; ::_thesis: verum end; supposeA27: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: contradiction then ((Gauge (C,n)) * (i1,j1)) `2 = ((Gauge (C,n)) * (1,j2)) `2 by A16, A19, A18, A23, GOBOARD5:1 .= ((Gauge (C,n)) * (i2,j2)) `2 by A22, A21, A20, A17, GOBOARD5:1 ; then LSeg (((Upper_Seq (C,n)) /. ii),((Upper_Seq (C,n)) /. (ii + 1))) is horizontal by A12, A14, SPPOL_1:15; then ((Gauge (C,n)) * (i,1)) `2 = ((Gauge (C,n)) * (i1,j1)) `2 by A5, A6, A12, SPPOL_1:40; then A28: j1 = 1 by A11, A8, Th6; i2 > 1 by A16, A27, NAT_1:13; then not (Upper_Seq (C,n)) /. (ii + 1) in rng (Upper_Seq (C,n)) by A14, A21, A27, A28, Th42; hence contradiction by A7, PARTFUN2:2; ::_thesis: verum end; supposeA29: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: contradiction then ((Gauge (C,n)) * (i1,j1)) `2 = ((Gauge (C,n)) * (1,j2)) `2 by A16, A19, A18, A23, GOBOARD5:1 .= ((Gauge (C,n)) * (i2,j2)) `2 by A22, A21, A20, A17, GOBOARD5:1 ; then LSeg (((Upper_Seq (C,n)) /. ii),((Upper_Seq (C,n)) /. (ii + 1))) is horizontal by A12, A14, SPPOL_1:15; then ((Gauge (C,n)) * (i,1)) `2 = ((Gauge (C,n)) * (i1,j1)) `2 by A5, A6, A12, SPPOL_1:40; then A30: j1 = 1 by A11, A8, Th6; i1 > 1 by A22, A29, NAT_1:13; then not (Upper_Seq (C,n)) /. ii in rng (Upper_Seq (C,n)) by A12, A19, A30, Th42; hence contradiction by A9, PARTFUN2:2; ::_thesis: verum end; supposeA31: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: contradiction then j2 <= j1 by NAT_1:11; then ((Gauge (C,n)) * (i2,j2)) `2 <= ((Gauge (C,n)) * (i1,j1)) `2 by A16, A19, A23, A20, A31, SPRECT_3:12; then A32: ((Gauge (C,n)) * (i2,j2)) `2 <= ((Gauge (C,n)) * (i,1)) `2 by A5, A6, A12, A14, TOPREAL1:4; ((Gauge (C,n)) * (i1,j1)) `1 = ((Gauge (C,n)) * (i2,1)) `1 by A16, A19, A18, A23, A31, GOBOARD5:2 .= ((Gauge (C,n)) * (i2,j2)) `1 by A22, A21, A20, A17, GOBOARD5:2 ; then LSeg (((Upper_Seq (C,n)) /. ii),((Upper_Seq (C,n)) /. (ii + 1))) is vertical by A12, A14, SPPOL_1:16; then ((Gauge (C,n)) * (i,1)) `1 = ((Gauge (C,n)) * (i1,j1)) `1 by A5, A6, A12, SPPOL_1:41; then A33: i1 = i by A11, A8, Th7; then ((Gauge (C,n)) * (i,1)) `2 <= ((Gauge (C,n)) * (i2,j2)) `2 by A22, A21, A20, A17, A31, SPRECT_3:12; then j2 = 1 by A13, A8, A32, Th6, XXREAL_0:1; hence contradiction by A14, A7, A10, A31, A33, PARTFUN2:2; ::_thesis: verum end; end; end; theorem :: JORDAN1G:45 for n being Element of NAT for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) for i being Element of NAT st 1 <= i & i < len (Gauge (C,n)) holds not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in L~ (Lower_Seq (C,n)) proof let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) for i being Element of NAT st 1 <= i & i < len (Gauge (C,n)) holds not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in L~ (Lower_Seq (C,n)) let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for i being Element of NAT st 1 <= i & i < len (Gauge (C,n)) holds not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in L~ (Lower_Seq (C,n)) set wi = width (Gauge (C,n)); let i be Element of NAT ; ::_thesis: ( 1 <= i & i < len (Gauge (C,n)) implies not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in L~ (Lower_Seq (C,n)) ) assume that A1: ( 1 <= i & i < len (Gauge (C,n)) ) and A2: (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in L~ (Lower_Seq (C,n)) ; ::_thesis: contradiction set Gi1 = (Gauge (C,n)) * (i,(width (Gauge (C,n)))); consider ii being Element of NAT such that A3: 1 <= ii and A4: ii + 1 <= len (Lower_Seq (C,n)) and A5: (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in LSeg ((Lower_Seq (C,n)),ii) by A2, SPPOL_2:13; A6: LSeg ((Lower_Seq (C,n)),ii) = LSeg (((Lower_Seq (C,n)) /. ii),((Lower_Seq (C,n)) /. (ii + 1))) by A3, A4, TOPREAL1:def_3; ii + 1 >= 1 by NAT_1:11; then A7: ii + 1 in dom (Lower_Seq (C,n)) by A4, FINSEQ_3:25; len (Gauge (C,n)) >= 4 by JORDAN8:10; then ( len (Gauge (C,n)) = width (Gauge (C,n)) & len (Gauge (C,n)) > 1 ) by JORDAN8:def_1, XXREAL_0:2; then A8: [i,(width (Gauge (C,n)))] in Indices (Gauge (C,n)) by A1, MATRIX_1:36; ii < len (Lower_Seq (C,n)) by A4, NAT_1:13; then A9: ii in dom (Lower_Seq (C,n)) by A3, FINSEQ_3:25; A10: not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in rng (Lower_Seq (C,n)) by A1, Th43; Lower_Seq (C,n) is_sequence_on Gauge (C,n) by Th5; then consider i1, j1, i2, j2 being Element of NAT such that A11: [i1,j1] in Indices (Gauge (C,n)) and A12: (Lower_Seq (C,n)) /. ii = (Gauge (C,n)) * (i1,j1) and A13: [i2,j2] in Indices (Gauge (C,n)) and A14: (Lower_Seq (C,n)) /. (ii + 1) = (Gauge (C,n)) * (i2,j2) and A15: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A3, A4, JORDAN8:3; A16: 1 <= i1 by A11, MATRIX_1:38; A17: j2 <= width (Gauge (C,n)) by A13, MATRIX_1:38; A18: 1 <= j1 by A11, MATRIX_1:38; A19: i1 <= len (Gauge (C,n)) by A11, MATRIX_1:38; A20: 1 <= j2 by A13, MATRIX_1:38; A21: i2 <= len (Gauge (C,n)) by A13, MATRIX_1:38; A22: 1 <= i2 by A13, MATRIX_1:38; A23: j1 <= width (Gauge (C,n)) by A11, MATRIX_1:38; percases ( ( i1 = i2 & j2 + 1 = j1 ) or ( i2 + 1 = i1 & j1 = j2 ) or ( i2 = i1 + 1 & j1 = j2 ) or ( i1 = i2 & j2 = j1 + 1 ) ) by A15; supposeA24: ( i1 = i2 & j2 + 1 = j1 ) ; ::_thesis: contradiction then j1 >= j2 by NAT_1:11; then ((Gauge (C,n)) * (i1,j1)) `2 >= ((Gauge (C,n)) * (i2,j2)) `2 by A16, A19, A23, A20, A24, SPRECT_3:12; then A25: ((Gauge (C,n)) * (i1,j1)) `2 >= ((Gauge (C,n)) * (i,(width (Gauge (C,n))))) `2 by A5, A6, A12, A14, TOPREAL1:4; ((Gauge (C,n)) * (i1,j1)) `1 = ((Gauge (C,n)) * (i2,1)) `1 by A16, A19, A18, A23, A24, GOBOARD5:2 .= ((Gauge (C,n)) * (i2,j2)) `1 by A22, A21, A20, A17, GOBOARD5:2 ; then LSeg (((Lower_Seq (C,n)) /. ii),((Lower_Seq (C,n)) /. (ii + 1))) is vertical by A12, A14, SPPOL_1:16; then ((Gauge (C,n)) * (i,(width (Gauge (C,n))))) `1 = ((Gauge (C,n)) * (i1,j1)) `1 by A5, A6, A12, SPPOL_1:41; then A26: i1 = i by A11, A8, Th7; then ((Gauge (C,n)) * (i,(width (Gauge (C,n))))) `2 >= ((Gauge (C,n)) * (i1,j1)) `2 by A16, A19, A18, A23, SPRECT_3:12; then j1 = width (Gauge (C,n)) by A11, A8, A25, Th6, XXREAL_0:1; hence contradiction by A12, A9, A10, A26, PARTFUN2:2; ::_thesis: verum end; supposeA27: ( i2 + 1 = i1 & j1 = j2 ) ; ::_thesis: contradiction then ((Gauge (C,n)) * (i1,j1)) `2 = ((Gauge (C,n)) * (1,j2)) `2 by A16, A19, A18, A23, GOBOARD5:1 .= ((Gauge (C,n)) * (i2,j2)) `2 by A22, A21, A20, A17, GOBOARD5:1 ; then LSeg (((Lower_Seq (C,n)) /. ii),((Lower_Seq (C,n)) /. (ii + 1))) is horizontal by A12, A14, SPPOL_1:15; then ((Gauge (C,n)) * (i,(width (Gauge (C,n))))) `2 = ((Gauge (C,n)) * (i1,j1)) `2 by A5, A6, A12, SPPOL_1:40; then A28: j1 = width (Gauge (C,n)) by A11, A8, Th6; i2 < len (Gauge (C,n)) by A19, A27, NAT_1:13; then not (Lower_Seq (C,n)) /. (ii + 1) in rng (Lower_Seq (C,n)) by A14, A22, A27, A28, Th43; hence contradiction by A7, PARTFUN2:2; ::_thesis: verum end; supposeA29: ( i2 = i1 + 1 & j1 = j2 ) ; ::_thesis: contradiction then ((Gauge (C,n)) * (i1,j1)) `2 = ((Gauge (C,n)) * (1,j2)) `2 by A16, A19, A18, A23, GOBOARD5:1 .= ((Gauge (C,n)) * (i2,j2)) `2 by A22, A21, A20, A17, GOBOARD5:1 ; then LSeg (((Lower_Seq (C,n)) /. ii),((Lower_Seq (C,n)) /. (ii + 1))) is horizontal by A12, A14, SPPOL_1:15; then ((Gauge (C,n)) * (i,(width (Gauge (C,n))))) `2 = ((Gauge (C,n)) * (i1,j1)) `2 by A5, A6, A12, SPPOL_1:40; then A30: j1 = width (Gauge (C,n)) by A11, A8, Th6; i1 < len (Gauge (C,n)) by A21, A29, NAT_1:13; then not (Lower_Seq (C,n)) /. ii in rng (Lower_Seq (C,n)) by A12, A16, A30, Th43; hence contradiction by A9, PARTFUN2:2; ::_thesis: verum end; supposeA31: ( i1 = i2 & j2 = j1 + 1 ) ; ::_thesis: contradiction then j2 >= j1 by NAT_1:11; then ((Gauge (C,n)) * (i2,j2)) `2 >= ((Gauge (C,n)) * (i1,j1)) `2 by A16, A19, A18, A17, A31, SPRECT_3:12; then A32: ((Gauge (C,n)) * (i2,j2)) `2 >= ((Gauge (C,n)) * (i,(width (Gauge (C,n))))) `2 by A5, A6, A12, A14, TOPREAL1:4; ((Gauge (C,n)) * (i1,j1)) `1 = ((Gauge (C,n)) * (i2,1)) `1 by A16, A19, A18, A23, A31, GOBOARD5:2 .= ((Gauge (C,n)) * (i2,j2)) `1 by A22, A21, A20, A17, GOBOARD5:2 ; then LSeg (((Lower_Seq (C,n)) /. ii),((Lower_Seq (C,n)) /. (ii + 1))) is vertical by A12, A14, SPPOL_1:16; then ((Gauge (C,n)) * (i,(width (Gauge (C,n))))) `1 = ((Gauge (C,n)) * (i1,j1)) `1 by A5, A6, A12, SPPOL_1:41; then A33: i1 = i by A11, A8, Th7; then ((Gauge (C,n)) * (i,(width (Gauge (C,n))))) `2 >= ((Gauge (C,n)) * (i2,j2)) `2 by A22, A21, A20, A17, A31, SPRECT_3:12; then j2 = width (Gauge (C,n)) by A13, A8, A32, Th6, XXREAL_0:1; hence contradiction by A14, A7, A10, A31, A33, PARTFUN2:2; ::_thesis: verum end; end; end; theorem Th46: :: JORDAN1G:46 for n being Element of NAT for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) for i, j being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Cage (C,n)) holds LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets L~ (Lower_Seq (C,n)) proof let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) for i, j being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Cage (C,n)) holds LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets L~ (Lower_Seq (C,n)) let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for i, j being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Cage (C,n)) holds LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets L~ (Lower_Seq (C,n)) let i, j be Element of NAT ; ::_thesis: ( 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Cage (C,n)) implies LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets L~ (Lower_Seq (C,n)) ) set Gij = (Gauge (C,n)) * (i,j); assume that A1: 1 <= i and A2: i <= len (Gauge (C,n)) and A3: ( 1 <= j & j <= width (Gauge (C,n)) ) and A4: (Gauge (C,n)) * (i,j) in L~ (Cage (C,n)) ; ::_thesis: LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets L~ (Lower_Seq (C,n)) A5: Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) :- (E-max (L~ (Cage (C,n)))) by JORDAN1E:def_2; set Wmi = W-min (L~ (Cage (C,n))); set h = mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n)))); set v1 = L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))); set NE = NE-corner (L~ (Cage (C,n))); set Gv1 = <*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))); set v = (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>; A6: L~ (Cage (C,n)) = (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n))) by JORDAN1E:13; A7: Upper_Seq (C,n) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n)))) by JORDAN1E:def_1; A8: len (Upper_Seq (C,n)) >= 3 by JORDAN1E:15; then A9: len (Upper_Seq (C,n)) >= 1 by XXREAL_0:2; A10: len (Lower_Seq (C,n)) >= 3 by JORDAN1E:15; then A11: ( len (Lower_Seq (C,n)) >= 2 & len (Lower_Seq (C,n)) >= 1 ) by XXREAL_0:2; A12: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1; A13: ((Gauge (C,n)) * (i,1)) `2 = S-bound (L~ (Cage (C,n))) by A1, A2, JORDAN1A:72; now__::_thesis:_LSeg_(((Gauge_(C,n))_*_(i,1)),((Gauge_(C,n))_*_(i,j)))_meets_L~_(Lower_Seq_(C,n)) percases ( ( (Gauge (C,n)) * (i,j) in L~ (Upper_Seq (C,n)) & i = 1 ) or ( (Gauge (C,n)) * (i,j) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (i,j) <> (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) & E-max (L~ (Cage (C,n))) <> NE-corner (L~ (Cage (C,n))) & i > 1 ) or ( (Gauge (C,n)) * (i,j) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (i,j) <> (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) & E-max (L~ (Cage (C,n))) = NE-corner (L~ (Cage (C,n))) & i > 1 ) or (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) or ( (Gauge (C,n)) * (i,j) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (i,j) = (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) ) ) by A1, A4, A6, XBOOLE_0:def_3, XXREAL_0:1; supposeA14: ( (Gauge (C,n)) * (i,j) in L~ (Upper_Seq (C,n)) & i = 1 ) ; ::_thesis: LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets L~ (Lower_Seq (C,n)) set G11 = (Gauge (C,n)) * (1,1); A15: W-min (L~ (Cage (C,n))) in L~ (Cage (C,n)) by SPRECT_1:13; S-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * (1,1)) `2 by A2, A14, JORDAN1A:72; then A16: ( (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) & ((Gauge (C,n)) * (1,1)) `2 <= (W-min (L~ (Cage (C,n)))) `2 ) by A15, EUCLID:52, PSCOMP_1:24; A17: rng (Lower_Seq (C,n)) c= L~ (Lower_Seq (C,n)) by A10, SPPOL_2:18, XXREAL_0:2; A18: ((Gauge (C,n)) * (i,j)) `1 = W-bound (L~ (Cage (C,n))) by A3, A12, A14, JORDAN1A:73; then (Gauge (C,n)) * (i,j) in W-most (L~ (Cage (C,n))) by A4, SPRECT_2:12; then A19: (W-min (L~ (Cage (C,n)))) `2 <= ((Gauge (C,n)) * (i,j)) `2 by PSCOMP_1:31; (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) = W-min (L~ (Cage (C,n))) by JORDAN1F:8; then A20: W-min (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) by REVROT_1:3; ((Gauge (C,n)) * (1,1)) `1 = W-bound (L~ (Cage (C,n))) by A2, A14, JORDAN1A:73; then W-min (L~ (Cage (C,n))) in LSeg (((Gauge (C,n)) * (1,1)),((Gauge (C,n)) * (1,j))) by A14, A16, A18, A19, GOBOARD7:7; hence LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets L~ (Lower_Seq (C,n)) by A14, A17, A20, XBOOLE_0:3; ::_thesis: verum end; supposeA21: ( (Gauge (C,n)) * (i,j) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (i,j) <> (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) & E-max (L~ (Cage (C,n))) <> NE-corner (L~ (Cage (C,n))) & i > 1 ) ; ::_thesis: LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets L~ (Lower_Seq (C,n)) len (Cage (C,n)) > 4 by GOBOARD7:34; then A22: rng (Cage (C,n)) c= L~ (Cage (C,n)) by SPPOL_2:18, XXREAL_0:2; A23: not NE-corner (L~ (Cage (C,n))) in rng (Cage (C,n)) proof A24: (NE-corner (L~ (Cage (C,n)))) `2 = N-bound (L~ (Cage (C,n))) by EUCLID:52; then ( (NE-corner (L~ (Cage (C,n)))) `1 = E-bound (L~ (Cage (C,n))) & (NE-corner (L~ (Cage (C,n)))) `2 >= S-bound (L~ (Cage (C,n))) ) by EUCLID:52, SPRECT_1:22; then NE-corner (L~ (Cage (C,n))) in { p where p is Point of (TOP-REAL 2) : ( p `1 = E-bound (L~ (Cage (C,n))) & p `2 <= N-bound (L~ (Cage (C,n))) & p `2 >= S-bound (L~ (Cage (C,n))) ) } by A24; then A25: NE-corner (L~ (Cage (C,n))) in LSeg ((SE-corner (L~ (Cage (C,n)))),(NE-corner (L~ (Cage (C,n))))) by SPRECT_1:23; assume NE-corner (L~ (Cage (C,n))) in rng (Cage (C,n)) ; ::_thesis: contradiction then NE-corner (L~ (Cage (C,n))) in (LSeg ((SE-corner (L~ (Cage (C,n)))),(NE-corner (L~ (Cage (C,n)))))) /\ (L~ (Cage (C,n))) by A22, A25, XBOOLE_0:def_4; then A26: (NE-corner (L~ (Cage (C,n)))) `2 <= (E-max (L~ (Cage (C,n)))) `2 by PSCOMP_1:47; A27: (E-max (L~ (Cage (C,n)))) `1 = (NE-corner (L~ (Cage (C,n)))) `1 by PSCOMP_1:45; (E-max (L~ (Cage (C,n)))) `2 <= (NE-corner (L~ (Cage (C,n)))) `2 by PSCOMP_1:46; then (E-max (L~ (Cage (C,n)))) `2 = (NE-corner (L~ (Cage (C,n)))) `2 by A26, XXREAL_0:1; hence contradiction by A21, A27, TOPREAL3:6; ::_thesis: verum end; A28: now__::_thesis:_not_NE-corner_(L~_(Cage_(C,n)))_in_rng_(L_Cut_((Upper_Seq_(C,n)),((Gauge_(C,n))_*_(i,j)))) percases ( (Gauge (C,n)) * (i,j) <> (Upper_Seq (C,n)) . ((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1) or (Gauge (C,n)) * (i,j) = (Upper_Seq (C,n)) . ((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1) ) ; suppose (Gauge (C,n)) * (i,j) <> (Upper_Seq (C,n)) . ((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1) ; ::_thesis: not NE-corner (L~ (Cage (C,n))) in rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) then L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))) = <*((Gauge (C,n)) * (i,j))*> ^ (mid ((Upper_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1),(len (Upper_Seq (C,n))))) by JORDAN3:def_3; then rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) = (rng <*((Gauge (C,n)) * (i,j))*>) \/ (rng (mid ((Upper_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1),(len (Upper_Seq (C,n)))))) by FINSEQ_1:31; then A29: rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) = {((Gauge (C,n)) * (i,j))} \/ (rng (mid ((Upper_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1),(len (Upper_Seq (C,n)))))) by FINSEQ_1:38; not NE-corner (L~ (Cage (C,n))) in L~ (Cage (C,n)) proof assume NE-corner (L~ (Cage (C,n))) in L~ (Cage (C,n)) ; ::_thesis: contradiction then consider i being Element of NAT such that A30: 1 <= i and A31: i + 1 <= len (Cage (C,n)) and A32: NE-corner (L~ (Cage (C,n))) in LSeg (((Cage (C,n)) /. i),((Cage (C,n)) /. (i + 1))) by SPPOL_2:14; percases ( ((Cage (C,n)) /. i) `1 = ((Cage (C,n)) /. (i + 1)) `1 or ((Cage (C,n)) /. i) `2 = ((Cage (C,n)) /. (i + 1)) `2 ) by A30, A31, TOPREAL1:def_5; supposeA33: ((Cage (C,n)) /. i) `1 = ((Cage (C,n)) /. (i + 1)) `1 ; ::_thesis: contradiction ( ((Cage (C,n)) /. i) `2 <= ((Cage (C,n)) /. (i + 1)) `2 or ((Cage (C,n)) /. i) `2 >= ((Cage (C,n)) /. (i + 1)) `2 ) ; then A34: ( (NE-corner (L~ (Cage (C,n)))) `2 <= ((Cage (C,n)) /. (i + 1)) `2 or (NE-corner (L~ (Cage (C,n)))) `2 <= ((Cage (C,n)) /. i) `2 ) by A32, TOPREAL1:4; A35: (NE-corner (L~ (Cage (C,n)))) `1 = ((Cage (C,n)) /. i) `1 by A32, A33, GOBOARD7:5; A36: 1 <= i + 1 by NAT_1:11; then A37: i + 1 in dom (Cage (C,n)) by A31, FINSEQ_3:25; A38: (NE-corner (L~ (Cage (C,n)))) `2 = N-bound (L~ (Cage (C,n))) by EUCLID:52; then A39: ((Cage (C,n)) /. (i + 1)) `2 <= (NE-corner (L~ (Cage (C,n)))) `2 by A31, A36, JORDAN5D:11; A40: i < len (Cage (C,n)) by A31, NAT_1:13; then ((Cage (C,n)) /. i) `2 <= (NE-corner (L~ (Cage (C,n)))) `2 by A30, A38, JORDAN5D:11; then ( (NE-corner (L~ (Cage (C,n)))) `2 = ((Cage (C,n)) /. (i + 1)) `2 or (NE-corner (L~ (Cage (C,n)))) `2 = ((Cage (C,n)) /. i) `2 ) by A39, A34, XXREAL_0:1; then A41: ( NE-corner (L~ (Cage (C,n))) = (Cage (C,n)) /. (i + 1) or NE-corner (L~ (Cage (C,n))) = (Cage (C,n)) /. i ) by A33, A35, TOPREAL3:6; i in dom (Cage (C,n)) by A30, A40, FINSEQ_3:25; hence contradiction by A23, A37, A41, PARTFUN2:2; ::_thesis: verum end; supposeA42: ((Cage (C,n)) /. i) `2 = ((Cage (C,n)) /. (i + 1)) `2 ; ::_thesis: contradiction ( ((Cage (C,n)) /. i) `1 <= ((Cage (C,n)) /. (i + 1)) `1 or ((Cage (C,n)) /. i) `1 >= ((Cage (C,n)) /. (i + 1)) `1 ) ; then A43: ( (NE-corner (L~ (Cage (C,n)))) `1 <= ((Cage (C,n)) /. (i + 1)) `1 or (NE-corner (L~ (Cage (C,n)))) `1 <= ((Cage (C,n)) /. i) `1 ) by A32, TOPREAL1:3; A44: (NE-corner (L~ (Cage (C,n)))) `2 = ((Cage (C,n)) /. i) `2 by A32, A42, GOBOARD7:6; A45: 1 <= i + 1 by NAT_1:11; then A46: i + 1 in dom (Cage (C,n)) by A31, FINSEQ_3:25; A47: (NE-corner (L~ (Cage (C,n)))) `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52; then A48: ((Cage (C,n)) /. (i + 1)) `1 <= (NE-corner (L~ (Cage (C,n)))) `1 by A31, A45, JORDAN5D:12; A49: i < len (Cage (C,n)) by A31, NAT_1:13; then ((Cage (C,n)) /. i) `1 <= (NE-corner (L~ (Cage (C,n)))) `1 by A30, A47, JORDAN5D:12; then ( (NE-corner (L~ (Cage (C,n)))) `1 = ((Cage (C,n)) /. (i + 1)) `1 or (NE-corner (L~ (Cage (C,n)))) `1 = ((Cage (C,n)) /. i) `1 ) by A48, A43, XXREAL_0:1; then A50: ( NE-corner (L~ (Cage (C,n))) = (Cage (C,n)) /. (i + 1) or NE-corner (L~ (Cage (C,n))) = (Cage (C,n)) /. i ) by A42, A44, TOPREAL3:6; i in dom (Cage (C,n)) by A30, A49, FINSEQ_3:25; hence contradiction by A23, A46, A50, PARTFUN2:2; ::_thesis: verum end; end; end; then A51: not NE-corner (L~ (Cage (C,n))) in {((Gauge (C,n)) * (i,j))} by A4, TARSKI:def_1; ( rng (mid ((Upper_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1),(len (Upper_Seq (C,n))))) c= rng (Upper_Seq (C,n)) & rng (Upper_Seq (C,n)) c= rng (Cage (C,n)) ) by Th39, FINSEQ_6:119; then rng (mid ((Upper_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1),(len (Upper_Seq (C,n))))) c= rng (Cage (C,n)) by XBOOLE_1:1; then not NE-corner (L~ (Cage (C,n))) in rng (mid ((Upper_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1),(len (Upper_Seq (C,n))))) by A23; hence not NE-corner (L~ (Cage (C,n))) in rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by A29, A51, XBOOLE_0:def_3; ::_thesis: verum end; suppose (Gauge (C,n)) * (i,j) = (Upper_Seq (C,n)) . ((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1) ; ::_thesis: not NE-corner (L~ (Cage (C,n))) in rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) then L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))) = mid ((Upper_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1),(len (Upper_Seq (C,n)))) by JORDAN3:def_3; then A52: rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) c= rng (Upper_Seq (C,n)) by FINSEQ_6:119; rng (Upper_Seq (C,n)) c= rng (Cage (C,n)) by Th39; then rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) c= rng (Cage (C,n)) by A52, XBOOLE_1:1; hence not NE-corner (L~ (Cage (C,n))) in rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by A23; ::_thesis: verum end; end; end; S-bound (L~ (Cage (C,n))) < N-bound (L~ (Cage (C,n))) by SPRECT_1:32; then NE-corner (L~ (Cage (C,n))) <> (Gauge (C,n)) * (i,1) by A13, EUCLID:52; then not NE-corner (L~ (Cage (C,n))) in {((Gauge (C,n)) * (i,1))} by TARSKI:def_1; then not NE-corner (L~ (Cage (C,n))) in rng <*((Gauge (C,n)) * (i,1))*> by FINSEQ_1:39; then not NE-corner (L~ (Cage (C,n))) in (rng <*((Gauge (C,n)) * (i,1))*>) \/ (rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) by A28, XBOOLE_0:def_3; then not NE-corner (L~ (Cage (C,n))) in rng (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) by FINSEQ_1:31; then rng (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) misses {(NE-corner (L~ (Cage (C,n))))} by ZFMISC_1:50; then A53: rng (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) misses rng <*(NE-corner (L~ (Cage (C,n))))*> by FINSEQ_1:38; A54: len ((<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) = (len (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))))) + 1 by FINSEQ_2:16 .= (1 + (len (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))))) + 1 by FINSEQ_5:8 ; A55: not L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))) is empty by A21, JORDAN1E:3; then A56: 0 + 1 <= len (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by NAT_1:13; then 1 in dom (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by FINSEQ_3:25; then A57: (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) /. 1 = (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) . 1 by PARTFUN1:def_6 .= (Gauge (C,n)) * (i,j) by A21, JORDAN3:23 ; then A58: ((L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) /. 1 = (Gauge (C,n)) * (i,j) by A56, BOOLMARK:7; 1 + (len (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) >= 1 + 1 by A56, XREAL_1:7; then A59: 2 < len ((<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) by A54, NAT_1:13; A60: L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))) is being_S-Seq by A21, JORDAN3:34; (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(NE-corner (L~ (Cage (C,n))))*> = <*((Gauge (C,n)) * (i,1))*> ^ ((L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) by FINSEQ_1:32; then ((<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) /. 1 = (Gauge (C,n)) * (i,1) by FINSEQ_5:15; then A61: (((<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) /. 1) `2 = S-bound (L~ (Cage (C,n))) by A1, A2, JORDAN1A:72; len ((<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) = (len (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))))) + 1 by FINSEQ_2:16; then ((<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) /. (len ((<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>)) = NE-corner (L~ (Cage (C,n))) by FINSEQ_4:67; then A62: (((<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) /. (len ((<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>))) `2 = N-bound (L~ (Cage (C,n))) by EUCLID:52; A63: (Cage (C,n)) /. 1 = N-min (L~ (Cage (C,n))) by JORDAN9:32; then (N-max (L~ (Cage (C,n)))) .. (Cage (C,n)) <= (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by SPRECT_2:70; then A64: (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) > 1 by A63, SPRECT_2:69, XXREAL_0:2; (E-min (L~ (Cage (C,n)))) .. (Cage (C,n)) <= (S-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by A63, SPRECT_2:72; then (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (S-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by A63, SPRECT_2:71, XXREAL_0:2; then (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (S-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A63, SPRECT_2:73, XXREAL_0:2; then A65: (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A63, SPRECT_2:74, XXREAL_0:2; then (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < len (Cage (C,n)) by A63, SPRECT_2:76, XXREAL_0:2; then A66: ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) + 1 <= len (Cage (C,n)) by NAT_1:13; A67: E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46; then (Cage (C,n)) /. ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) = E-max (L~ (Cage (C,n))) by FINSEQ_5:38; then A68: ((Cage (C,n)) /. (((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) + 1)) `1 = E-bound (L~ (Cage (C,n))) by A64, A66, JORDAN1E:20; A69: W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43; then A70: (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) = ((len (Cage (C,n))) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) - ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) by A67, A65, SPRECT_5:9; now__::_thesis:_for_m_being_Element_of_NAT_st_m_in_dom_<*((Gauge_(C,n))_*_(i,1))*>_holds_ (_W-bound_(L~_(Cage_(C,n)))_<=_(<*((Gauge_(C,n))_*_(i,1))*>_/._m)_`1_&_(<*((Gauge_(C,n))_*_(i,1))*>_/._m)_`1_<=_E-bound_(L~_(Cage_(C,n)))_&_S-bound_(L~_(Cage_(C,n)))_<=_(<*((Gauge_(C,n))_*_(i,1))*>_/._m)_`2_&_(<*((Gauge_(C,n))_*_(i,1))*>_/._m)_`2_<=_N-bound_(L~_(Cage_(C,n)))_) let m be Element of NAT ; ::_thesis: ( m in dom <*((Gauge (C,n)) * (i,1))*> implies ( W-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,1))*> /. m) `1 & (<*((Gauge (C,n)) * (i,1))*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,1))*> /. m) `2 & (<*((Gauge (C,n)) * (i,1))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) ) assume A71: m in dom <*((Gauge (C,n)) * (i,1))*> ; ::_thesis: ( W-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,1))*> /. m) `1 & (<*((Gauge (C,n)) * (i,1))*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,1))*> /. m) `2 & (<*((Gauge (C,n)) * (i,1))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) then m in Seg 1 by FINSEQ_1:38; then m = 1 by FINSEQ_1:2, TARSKI:def_1; then <*((Gauge (C,n)) * (i,1))*> . m = (Gauge (C,n)) * (i,1) by FINSEQ_1:40; then A72: <*((Gauge (C,n)) * (i,1))*> /. m = (Gauge (C,n)) * (i,1) by A71, PARTFUN1:def_6; width (Gauge (C,n)) >= 4 by A12, JORDAN8:10; then A73: 1 <= width (Gauge (C,n)) by XXREAL_0:2; then ((Gauge (C,n)) * (1,1)) `1 <= ((Gauge (C,n)) * (i,1)) `1 by A1, A2, SPRECT_3:13; hence W-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,1))*> /. m) `1 by A12, A72, A73, JORDAN1A:73; ::_thesis: ( (<*((Gauge (C,n)) * (i,1))*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,1))*> /. m) `2 & (<*((Gauge (C,n)) * (i,1))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) ((Gauge (C,n)) * (i,1)) `1 <= ((Gauge (C,n)) * ((len (Gauge (C,n))),1)) `1 by A1, A2, A73, SPRECT_3:13; hence (<*((Gauge (C,n)) * (i,1))*> /. m) `1 <= E-bound (L~ (Cage (C,n))) by A12, A72, A73, JORDAN1A:71; ::_thesis: ( S-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,1))*> /. m) `2 & (<*((Gauge (C,n)) * (i,1))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) thus S-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,1))*> /. m) `2 by A1, A2, A72, JORDAN1A:72; ::_thesis: (<*((Gauge (C,n)) * (i,1))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) S-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * (i,1)) `2 by A1, A2, JORDAN1A:72; hence (<*((Gauge (C,n)) * (i,1))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) by A72, SPRECT_1:22; ::_thesis: verum end; then A74: <*((Gauge (C,n)) * (i,1))*> is_in_the_area_of Cage (C,n) by SPRECT_2:def_1; A75: <*(NE-corner (L~ (Cage (C,n))))*> is_in_the_area_of Cage (C,n) by SPRECT_2:25; 3 <= len (Lower_Seq (C,n)) by JORDAN1E:15; then 2 <= len (Lower_Seq (C,n)) by XXREAL_0:2; then A76: 2 in dom (Lower_Seq (C,n)) by FINSEQ_3:25; <*((Gauge (C,n)) * (i,j))*> is_in_the_area_of Cage (C,n) by A21, JORDAN1E:17, SPRECT_3:46; then L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))) is_in_the_area_of Cage (C,n) by A21, JORDAN1E:17, SPRECT_3:56; then <*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) is_in_the_area_of Cage (C,n) by A74, SPRECT_2:24; then (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(NE-corner (L~ (Cage (C,n))))*> is_in_the_area_of Cage (C,n) by A75, SPRECT_2:24; then A77: (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(NE-corner (L~ (Cage (C,n))))*> is_a_v.c._for Cage (C,n) by A61, A62, SPRECT_2:def_3; A78: ((1 + (((len (Cage (C,n))) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) - ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))))) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) - (len (Cage (C,n))) = 1 + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) ; A79: len (Lower_Seq (C,n)) in dom (Lower_Seq (C,n)) by FINSEQ_5:6; then mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n)))) is_in_the_area_of Cage (C,n) by A76, JORDAN1E:18, SPRECT_2:22; then A80: Rev (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))) is_in_the_area_of Cage (C,n) by SPRECT_3:51; 1 + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) <= 0 + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) by A65, NAT_1:13; then (1 + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) - ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) <= 0 by XREAL_1:20; then A81: (len (Cage (C,n))) + ((1 + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) - ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) <= (len (Cage (C,n))) + 0 by XREAL_1:6; A82: len (Lower_Seq (C,n)) >= 2 + 1 by JORDAN1E:15; then A83: len (Lower_Seq (C,n)) > 2 by NAT_1:13; (len (Cage (C,n))) + 0 <= (len (Cage (C,n))) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) by XREAL_1:6; then (len (Cage (C,n))) - ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) <= (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A70, XREAL_1:9; then ((len (Cage (C,n))) - ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) + 1 <= 1 + ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) by XREAL_1:6; then A84: len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) <= 1 + ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) by A69, FINSEQ_5:50; E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46; then A85: E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by FINSEQ_6:90, SPRECT_2:43; A86: L~ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) c= L~ (Upper_Seq (C,n)) by A21, JORDAN3:42; A87: len (Lower_Seq (C,n)) > 1 by A82, XXREAL_0:2; then A88: not mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n)))) is empty by A83, JORDAN1B:2; A89: E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A67, FINSEQ_6:90, SPRECT_2:43; then (Lower_Seq (C,n)) /. (1 + 1) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. (1 + ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) by A5, A76, FINSEQ_5:52 .= (Cage (C,n)) /. (((1 + ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) -' (len (Cage (C,n)))) by A69, A70, A84, A81, REVROT_1:17 .= (Cage (C,n)) /. (((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) + 1) by A70, A78, XREAL_0:def_2 ; then ((mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))) /. 1) `1 = E-bound (L~ (Cage (C,n))) by A76, A79, A68, SPRECT_2:8; then ((Rev (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n)))))) /. (len (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))))) `1 = E-bound (L~ (Cage (C,n))) by A88, FINSEQ_5:65; then A90: ((Rev (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n)))))) /. (len (Rev (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n)))))))) `1 = E-bound (L~ (Cage (C,n))) by FINSEQ_5:def_3; (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. (len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) by A5, A89, FINSEQ_5:54 .= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 by FINSEQ_6:def_1 .= W-min (L~ (Cage (C,n))) by A69, FINSEQ_6:92 ; then ((Lower_Seq (C,n)) /. (len (Lower_Seq (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52; then ((mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))) /. (len (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))))) `1 = W-bound (L~ (Cage (C,n))) by A76, A79, SPRECT_2:9; then ((Rev (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n)))))) /. 1) `1 = W-bound (L~ (Cage (C,n))) by A88, FINSEQ_5:65; then A91: Rev (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))) is_a_h.c._for Cage (C,n) by A80, A90, SPRECT_2:def_2; A92: len (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by A9, FINSEQ_3:25; set ci = mid ((Upper_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1),(len (Upper_Seq (C,n)))); rng (Upper_Seq (C,n)) c= L~ (Upper_Seq (C,n)) by A8, SPPOL_2:18, XXREAL_0:2; then A93: not (Gauge (C,n)) * (i,1) in rng (Upper_Seq (C,n)) by A2, A21, Th44; not (Gauge (C,n)) * (i,1) in L~ (Upper_Seq (C,n)) by A2, A21, Th44; then not (Gauge (C,n)) * (i,1) in {((Gauge (C,n)) * (i,j))} by A21, TARSKI:def_1; then A94: not (Gauge (C,n)) * (i,1) in rng <*((Gauge (C,n)) * (i,j))*> by FINSEQ_1:38; now__::_thesis:_not_(Gauge_(C,n))_*_(i,1)_in_rng_(L_Cut_((Upper_Seq_(C,n)),((Gauge_(C,n))_*_(i,j)))) percases ( (Gauge (C,n)) * (i,j) <> (Upper_Seq (C,n)) . ((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1) or (Gauge (C,n)) * (i,j) = (Upper_Seq (C,n)) . ((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1) ) ; supposeA95: (Gauge (C,n)) * (i,j) <> (Upper_Seq (C,n)) . ((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1) ; ::_thesis: not (Gauge (C,n)) * (i,1) in rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) rng (mid ((Upper_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1),(len (Upper_Seq (C,n))))) c= rng (Upper_Seq (C,n)) by FINSEQ_6:119; then not (Gauge (C,n)) * (i,1) in rng (mid ((Upper_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1),(len (Upper_Seq (C,n))))) by A93; then not (Gauge (C,n)) * (i,1) in (rng <*((Gauge (C,n)) * (i,j))*>) \/ (rng (mid ((Upper_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1),(len (Upper_Seq (C,n)))))) by A94, XBOOLE_0:def_3; then not (Gauge (C,n)) * (i,1) in rng (<*((Gauge (C,n)) * (i,j))*> ^ (mid ((Upper_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1),(len (Upper_Seq (C,n)))))) by FINSEQ_1:31; hence not (Gauge (C,n)) * (i,1) in rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by A95, JORDAN3:def_3; ::_thesis: verum end; suppose (Gauge (C,n)) * (i,j) = (Upper_Seq (C,n)) . ((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1) ; ::_thesis: not (Gauge (C,n)) * (i,1) in rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) then L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))) = mid ((Upper_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1),(len (Upper_Seq (C,n)))) by JORDAN3:def_3; then rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) c= rng (Upper_Seq (C,n)) by FINSEQ_6:119; hence not (Gauge (C,n)) * (i,1) in rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by A93; ::_thesis: verum end; end; end; then {((Gauge (C,n)) * (i,1))} misses rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by ZFMISC_1:50; then A96: rng <*((Gauge (C,n)) * (i,1))*> misses rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by FINSEQ_1:38; A97: <*(NE-corner (L~ (Cage (C,n))))*> is one-to-one by FINSEQ_3:93; (Lower_Seq (C,n)) /. 1 = ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) :- (E-max (L~ (Cage (C,n))))) /. 1 by JORDAN1E:def_2 .= E-max (L~ (Cage (C,n))) by FINSEQ_5:53 ; then A98: not E-max (L~ (Cage (C,n))) in L~ (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))) by A83, JORDAN5B:16; <*((Gauge (C,n)) * (i,1))*> is one-to-one by FINSEQ_3:93; then <*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) is one-to-one by A96, A60, FINSEQ_3:91; then A99: (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(NE-corner (L~ (Cage (C,n))))*> is one-to-one by A53, A97, FINSEQ_3:91; A100: L~ (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))) c= L~ (Lower_Seq (C,n)) by A11, JORDAN4:35; (<*((Gauge (C,n)) * (i,1))*> /. (len <*((Gauge (C,n)) * (i,1))*>)) `1 = (<*((Gauge (C,n)) * (i,1))*> /. 1) `1 by FINSEQ_1:39 .= ((Gauge (C,n)) * (i,1)) `1 by FINSEQ_4:16 .= ((L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) /. 1) `1 by A1, A2, A3, A57, GOBOARD5:2 ; then A101: <*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) is special by A60, GOBOARD2:8; len (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) in dom (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by A56, FINSEQ_3:25; then A102: (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) /. (len (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) = (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) . (len (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) by PARTFUN1:def_6 .= (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) by A21, JORDAN1B:4 .= (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) by A92, PARTFUN1:def_6 .= ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n))))) /. ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) by A7, A85, FINSEQ_5:42 .= E-max (L~ (Cage (C,n))) by A85, FINSEQ_5:45 ; then (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) /. (len (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))))) = E-max (L~ (Cage (C,n))) by A55, SPRECT_3:1; then ((<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) /. (len (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))))) `1 = (NE-corner (L~ (Cage (C,n)))) `1 by PSCOMP_1:45 .= (<*(NE-corner (L~ (Cage (C,n))))*> /. 1) `1 by FINSEQ_4:16 ; then A103: (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(NE-corner (L~ (Cage (C,n))))*> is special by A101, GOBOARD2:8; mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n)))) is S-Sequence_in_R2 by A83, A87, JORDAN3:6; then A104: Rev (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))) is S-Sequence_in_R2 ; then 2 <= len (Rev (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n)))))) by TOPREAL1:def_8; then L~ (Rev (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n)))))) meets L~ ((<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) by A59, A99, A103, A104, A91, A77, SPRECT_2:29; then L~ (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))) meets L~ ((<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) by SPPOL_2:22; then consider x being set such that A105: x in L~ (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))) and A106: x in L~ ((<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) by XBOOLE_0:3; A107: W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43; L~ ((<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) = L~ (<*((Gauge (C,n)) * (i,1))*> ^ ((L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) ^ <*(NE-corner (L~ (Cage (C,n))))*>)) by FINSEQ_1:32 .= (LSeg (((Gauge (C,n)) * (i,1)),(((L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) /. 1))) \/ (L~ ((L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) ^ <*(NE-corner (L~ (Cage (C,n))))*>)) by SPPOL_2:20 .= (LSeg (((Gauge (C,n)) * (i,1)),(((L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) /. 1))) \/ ((L~ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) \/ (LSeg (((L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) /. (len (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))))),(NE-corner (L~ (Cage (C,n))))))) by A55, SPPOL_2:19 ; then A108: ( x in LSeg (((Gauge (C,n)) * (i,1)),(((L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) /. 1)) or x in (L~ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) \/ (LSeg (((L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) /. (len (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))))),(NE-corner (L~ (Cage (C,n)))))) ) by A106, XBOOLE_0:def_3; now__::_thesis:_L~_(Lower_Seq_(C,n))_meets_L~_<*((Gauge_(C,n))_*_(i,1)),((Gauge_(C,n))_*_(i,j))*> percases ( x in LSeg (((Gauge (C,n)) * (i,1)),(((L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) /. 1)) or x in L~ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) or x in LSeg (((L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) /. (len (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))))),(NE-corner (L~ (Cage (C,n))))) ) by A108, XBOOLE_0:def_3; suppose x in LSeg (((Gauge (C,n)) * (i,1)),(((L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) /. 1)) ; ::_thesis: L~ (Lower_Seq (C,n)) meets L~ <*((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))*> then x in L~ <*((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))*> by A58, SPPOL_2:21; hence L~ (Lower_Seq (C,n)) meets L~ <*((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))*> by A105, A100, XBOOLE_0:3; ::_thesis: verum end; supposeA109: x in L~ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) ; ::_thesis: L~ (Lower_Seq (C,n)) meets L~ <*((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))*> then x in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A105, A100, A86, XBOOLE_0:def_4; then x in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16; then A110: x = W-min (L~ (Cage (C,n))) by A105, A98, TARSKI:def_2; 1 in dom (Upper_Seq (C,n)) by A9, FINSEQ_3:25; then (Upper_Seq (C,n)) . 1 = (Upper_Seq (C,n)) /. 1 by PARTFUN1:def_6 .= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 by A7, A85, FINSEQ_5:44 .= W-min (L~ (Cage (C,n))) by A107, FINSEQ_6:92 ; then x = (Gauge (C,n)) * (i,j) by A21, A109, A110, JORDAN1E:7; then x in LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) by RLTOPSP1:68; then x in L~ <*((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))*> by SPPOL_2:21; hence L~ (Lower_Seq (C,n)) meets L~ <*((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))*> by A105, A100, XBOOLE_0:3; ::_thesis: verum end; supposeA111: x in LSeg (((L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) /. (len (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))))),(NE-corner (L~ (Cage (C,n))))) ; ::_thesis: L~ (Lower_Seq (C,n)) meets L~ <*((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))*> x in L~ (Cage (C,n)) by A6, A105, A100, XBOOLE_0:def_3; then x in (LSeg ((E-max (L~ (Cage (C,n)))),(NE-corner (L~ (Cage (C,n)))))) /\ (L~ (Cage (C,n))) by A102, A111, XBOOLE_0:def_4; then x in {(E-max (L~ (Cage (C,n))))} by PSCOMP_1:51; hence L~ (Lower_Seq (C,n)) meets L~ <*((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))*> by A105, A98, TARSKI:def_1; ::_thesis: verum end; end; end; then L~ <*((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))*> meets L~ (Lower_Seq (C,n)) ; hence LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets L~ (Lower_Seq (C,n)) by SPPOL_2:21; ::_thesis: verum end; supposeA112: ( (Gauge (C,n)) * (i,j) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (i,j) <> (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) & E-max (L~ (Cage (C,n))) = NE-corner (L~ (Cage (C,n))) & i > 1 ) ; ::_thesis: LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets L~ (Lower_Seq (C,n)) now__::_thesis:_for_m_being_Element_of_NAT_st_m_in_dom_<*((Gauge_(C,n))_*_(i,1))*>_holds_ (_W-bound_(L~_(Cage_(C,n)))_<=_(<*((Gauge_(C,n))_*_(i,1))*>_/._m)_`1_&_(<*((Gauge_(C,n))_*_(i,1))*>_/._m)_`1_<=_E-bound_(L~_(Cage_(C,n)))_&_S-bound_(L~_(Cage_(C,n)))_<=_(<*((Gauge_(C,n))_*_(i,1))*>_/._m)_`2_&_(<*((Gauge_(C,n))_*_(i,1))*>_/._m)_`2_<=_N-bound_(L~_(Cage_(C,n)))_) let m be Element of NAT ; ::_thesis: ( m in dom <*((Gauge (C,n)) * (i,1))*> implies ( W-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,1))*> /. m) `1 & (<*((Gauge (C,n)) * (i,1))*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,1))*> /. m) `2 & (<*((Gauge (C,n)) * (i,1))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) ) assume A113: m in dom <*((Gauge (C,n)) * (i,1))*> ; ::_thesis: ( W-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,1))*> /. m) `1 & (<*((Gauge (C,n)) * (i,1))*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,1))*> /. m) `2 & (<*((Gauge (C,n)) * (i,1))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) then m in Seg 1 by FINSEQ_1:38; then m = 1 by FINSEQ_1:2, TARSKI:def_1; then <*((Gauge (C,n)) * (i,1))*> . m = (Gauge (C,n)) * (i,1) by FINSEQ_1:40; then A114: <*((Gauge (C,n)) * (i,1))*> /. m = (Gauge (C,n)) * (i,1) by A113, PARTFUN1:def_6; width (Gauge (C,n)) >= 4 by A12, JORDAN8:10; then A115: 1 <= width (Gauge (C,n)) by XXREAL_0:2; then ((Gauge (C,n)) * (1,1)) `1 <= ((Gauge (C,n)) * (i,1)) `1 by A1, A2, SPRECT_3:13; hence W-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,1))*> /. m) `1 by A12, A114, A115, JORDAN1A:73; ::_thesis: ( (<*((Gauge (C,n)) * (i,1))*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,1))*> /. m) `2 & (<*((Gauge (C,n)) * (i,1))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) ((Gauge (C,n)) * (i,1)) `1 <= ((Gauge (C,n)) * ((len (Gauge (C,n))),1)) `1 by A1, A2, A115, SPRECT_3:13; hence (<*((Gauge (C,n)) * (i,1))*> /. m) `1 <= E-bound (L~ (Cage (C,n))) by A12, A114, A115, JORDAN1A:71; ::_thesis: ( S-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,1))*> /. m) `2 & (<*((Gauge (C,n)) * (i,1))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) thus S-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,1))*> /. m) `2 by A1, A2, A114, JORDAN1A:72; ::_thesis: (<*((Gauge (C,n)) * (i,1))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) S-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * (i,1)) `2 by A1, A2, JORDAN1A:72; hence (<*((Gauge (C,n)) * (i,1))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) by A114, SPRECT_1:22; ::_thesis: verum end; then A116: <*((Gauge (C,n)) * (i,1))*> is_in_the_area_of Cage (C,n) by SPRECT_2:def_1; <*((Gauge (C,n)) * (i,j))*> is_in_the_area_of Cage (C,n) by A112, JORDAN1E:17, SPRECT_3:46; then L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))) is_in_the_area_of Cage (C,n) by A112, JORDAN1E:17, SPRECT_3:56; then A117: <*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) is_in_the_area_of Cage (C,n) by A116, SPRECT_2:24; A118: len (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by A9, FINSEQ_3:25; E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46; then A119: E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by FINSEQ_6:90, SPRECT_2:43; A120: not L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))) is empty by A112, JORDAN1E:3; then A121: 0 + 1 <= len (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by NAT_1:13; then 1 in dom (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by FINSEQ_3:25; then A122: (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) /. 1 = (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) . 1 by PARTFUN1:def_6 .= (Gauge (C,n)) * (i,j) by A112, JORDAN3:23 ; len (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) in dom (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by A121, FINSEQ_3:25; then (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) /. (len (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) = (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) . (len (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) by PARTFUN1:def_6 .= (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) by A112, JORDAN1B:4 .= (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) by A118, PARTFUN1:def_6 .= ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n))))) /. ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) by A7, A119, FINSEQ_5:42 .= E-max (L~ (Cage (C,n))) by A119, FINSEQ_5:45 ; then (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) /. (len (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))))) = E-max (L~ (Cage (C,n))) by A120, SPRECT_3:1; then A123: ((<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) /. (len (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))))) `2 = N-bound (L~ (Cage (C,n))) by A112, EUCLID:52; (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) /. 1 = (Gauge (C,n)) * (i,1) by FINSEQ_5:15; then ((<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) /. 1) `2 = S-bound (L~ (Cage (C,n))) by A1, A2, JORDAN1A:72; then A124: <*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) is_a_v.c._for Cage (C,n) by A117, A123, SPRECT_2:def_3; A125: (Cage (C,n)) /. 1 = N-min (L~ (Cage (C,n))) by JORDAN9:32; then (N-max (L~ (Cage (C,n)))) .. (Cage (C,n)) <= (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by SPRECT_2:70; then A126: (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) > 1 by A125, SPRECT_2:69, XXREAL_0:2; (E-min (L~ (Cage (C,n)))) .. (Cage (C,n)) <= (S-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by A125, SPRECT_2:72; then (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (S-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by A125, SPRECT_2:71, XXREAL_0:2; then (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (S-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A125, SPRECT_2:73, XXREAL_0:2; then A127: (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A125, SPRECT_2:74, XXREAL_0:2; then (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < len (Cage (C,n)) by A125, SPRECT_2:76, XXREAL_0:2; then A128: ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) + 1 <= len (Cage (C,n)) by NAT_1:13; A129: E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46; then (Cage (C,n)) /. ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) = E-max (L~ (Cage (C,n))) by FINSEQ_5:38; then A130: ((Cage (C,n)) /. (((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) + 1)) `1 = E-bound (L~ (Cage (C,n))) by A126, A128, JORDAN1E:20; 1 + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) <= 0 + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) by A127, NAT_1:13; then (1 + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) - ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) <= 0 by XREAL_1:20; then A131: (len (Cage (C,n))) + ((1 + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) - ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) <= (len (Cage (C,n))) + 0 by XREAL_1:6; A132: len (Lower_Seq (C,n)) >= 2 + 1 by JORDAN1E:15; then A133: len (Lower_Seq (C,n)) > 2 by NAT_1:13; set ci = mid ((Upper_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1),(len (Upper_Seq (C,n)))); rng (Upper_Seq (C,n)) c= L~ (Upper_Seq (C,n)) by A8, SPPOL_2:18, XXREAL_0:2; then A134: not (Gauge (C,n)) * (i,1) in rng (Upper_Seq (C,n)) by A2, A112, Th44; not (Gauge (C,n)) * (i,1) in L~ (Upper_Seq (C,n)) by A2, A112, Th44; then not (Gauge (C,n)) * (i,1) in {((Gauge (C,n)) * (i,j))} by A112, TARSKI:def_1; then A135: not (Gauge (C,n)) * (i,1) in rng <*((Gauge (C,n)) * (i,j))*> by FINSEQ_1:38; now__::_thesis:_not_(Gauge_(C,n))_*_(i,1)_in_rng_(L_Cut_((Upper_Seq_(C,n)),((Gauge_(C,n))_*_(i,j)))) percases ( (Gauge (C,n)) * (i,j) <> (Upper_Seq (C,n)) . ((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1) or (Gauge (C,n)) * (i,j) = (Upper_Seq (C,n)) . ((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1) ) ; supposeA136: (Gauge (C,n)) * (i,j) <> (Upper_Seq (C,n)) . ((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1) ; ::_thesis: not (Gauge (C,n)) * (i,1) in rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) rng (mid ((Upper_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1),(len (Upper_Seq (C,n))))) c= rng (Upper_Seq (C,n)) by FINSEQ_6:119; then not (Gauge (C,n)) * (i,1) in rng (mid ((Upper_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1),(len (Upper_Seq (C,n))))) by A134; then not (Gauge (C,n)) * (i,1) in (rng <*((Gauge (C,n)) * (i,j))*>) \/ (rng (mid ((Upper_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1),(len (Upper_Seq (C,n)))))) by A135, XBOOLE_0:def_3; then not (Gauge (C,n)) * (i,1) in rng (<*((Gauge (C,n)) * (i,j))*> ^ (mid ((Upper_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1),(len (Upper_Seq (C,n)))))) by FINSEQ_1:31; hence not (Gauge (C,n)) * (i,1) in rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by A136, JORDAN3:def_3; ::_thesis: verum end; suppose (Gauge (C,n)) * (i,j) = (Upper_Seq (C,n)) . ((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1) ; ::_thesis: not (Gauge (C,n)) * (i,1) in rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) then L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))) = mid ((Upper_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1),(len (Upper_Seq (C,n)))) by JORDAN3:def_3; then rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) c= rng (Upper_Seq (C,n)) by FINSEQ_6:119; hence not (Gauge (C,n)) * (i,1) in rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by A134; ::_thesis: verum end; end; end; then {((Gauge (C,n)) * (i,1))} misses rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by ZFMISC_1:50; then A137: rng <*((Gauge (C,n)) * (i,1))*> misses rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by FINSEQ_1:38; A138: ((1 + (((len (Cage (C,n))) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) - ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))))) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) - (len (Cage (C,n))) = 1 + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) ; 1 + (len (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) >= 1 + 1 by A121, XREAL_1:7; then A139: len (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) >= 2 by FINSEQ_5:8; 3 <= len (Lower_Seq (C,n)) by JORDAN1E:15; then 2 <= len (Lower_Seq (C,n)) by XXREAL_0:2; then A140: 2 in dom (Lower_Seq (C,n)) by FINSEQ_3:25; (Lower_Seq (C,n)) /. 1 = ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) :- (E-max (L~ (Cage (C,n))))) /. 1 by JORDAN1E:def_2 .= E-max (L~ (Cage (C,n))) by FINSEQ_5:53 ; then A141: not E-max (L~ (Cage (C,n))) in L~ (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))) by A133, JORDAN5B:16; A142: L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))) is being_S-Seq by A112, JORDAN3:34; (<*((Gauge (C,n)) * (i,1))*> /. (len <*((Gauge (C,n)) * (i,1))*>)) `1 = (<*((Gauge (C,n)) * (i,1))*> /. 1) `1 by FINSEQ_1:39 .= ((Gauge (C,n)) * (i,1)) `1 by FINSEQ_4:16 .= ((L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) /. 1) `1 by A1, A2, A3, A122, GOBOARD5:2 ; then A143: <*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) is special by A142, GOBOARD2:8; A144: L~ (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) = (LSeg (((Gauge (C,n)) * (i,1)),((L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) /. 1))) \/ (L~ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) by A120, SPPOL_2:20; A145: W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43; then A146: (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) = ((len (Cage (C,n))) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) - ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) by A129, A127, SPRECT_5:9; (len (Cage (C,n))) + 0 <= (len (Cage (C,n))) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) by XREAL_1:6; then (len (Cage (C,n))) - ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) <= (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A146, XREAL_1:9; then ((len (Cage (C,n))) - ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) + 1 <= 1 + ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) by XREAL_1:6; then A147: len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) <= 1 + ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) by A145, FINSEQ_5:50; A148: len (Lower_Seq (C,n)) > 1 by A132, XXREAL_0:2; then A149: not mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n)))) is empty by A133, JORDAN1B:2; A150: len (Lower_Seq (C,n)) in dom (Lower_Seq (C,n)) by FINSEQ_5:6; then mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n)))) is_in_the_area_of Cage (C,n) by A140, JORDAN1E:18, SPRECT_2:22; then A151: Rev (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))) is_in_the_area_of Cage (C,n) by SPRECT_3:51; A152: E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A129, FINSEQ_6:90, SPRECT_2:43; then (Lower_Seq (C,n)) /. (1 + 1) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. (1 + ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) by A5, A140, FINSEQ_5:52 .= (Cage (C,n)) /. (((1 + ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) -' (len (Cage (C,n)))) by A145, A146, A147, A131, REVROT_1:17 .= (Cage (C,n)) /. (((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) + 1) by A146, A138, XREAL_0:def_2 ; then ((mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))) /. 1) `1 = E-bound (L~ (Cage (C,n))) by A140, A150, A130, SPRECT_2:8; then ((Rev (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n)))))) /. (len (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))))) `1 = E-bound (L~ (Cage (C,n))) by A149, FINSEQ_5:65; then A153: ((Rev (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n)))))) /. (len (Rev (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n)))))))) `1 = E-bound (L~ (Cage (C,n))) by FINSEQ_5:def_3; (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. (len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) by A5, A152, FINSEQ_5:54 .= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 by FINSEQ_6:def_1 .= W-min (L~ (Cage (C,n))) by A145, FINSEQ_6:92 ; then ((Lower_Seq (C,n)) /. (len (Lower_Seq (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52; then ((mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))) /. (len (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))))) `1 = W-bound (L~ (Cage (C,n))) by A140, A150, SPRECT_2:9; then ((Rev (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n)))))) /. 1) `1 = W-bound (L~ (Cage (C,n))) by A149, FINSEQ_5:65; then A154: Rev (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))) is_a_h.c._for Cage (C,n) by A151, A153, SPRECT_2:def_2; <*((Gauge (C,n)) * (i,1))*> is one-to-one by FINSEQ_3:93; then A155: <*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) is one-to-one by A137, A142, FINSEQ_3:91; A156: L~ (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))) c= L~ (Lower_Seq (C,n)) by A11, JORDAN4:35; mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n)))) is S-Sequence_in_R2 by A133, A148, JORDAN3:6; then A157: Rev (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))) is S-Sequence_in_R2 ; then 2 <= len (Rev (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n)))))) by TOPREAL1:def_8; then L~ (Rev (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n)))))) meets L~ (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) by A139, A155, A143, A157, A154, A124, SPRECT_2:29; then L~ (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))) meets L~ (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) by SPPOL_2:22; then consider x being set such that A158: x in L~ (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))) and A159: x in L~ (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) by XBOOLE_0:3; A160: W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43; A161: L~ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) c= L~ (Upper_Seq (C,n)) by A112, JORDAN3:42; now__::_thesis:_L~_(Lower_Seq_(C,n))_meets_L~_<*((Gauge_(C,n))_*_(i,1)),((Gauge_(C,n))_*_(i,j))*> percases ( x in LSeg (((Gauge (C,n)) * (i,1)),((L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) /. 1)) or x in L~ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) ) by A159, A144, XBOOLE_0:def_3; suppose x in LSeg (((Gauge (C,n)) * (i,1)),((L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) /. 1)) ; ::_thesis: L~ (Lower_Seq (C,n)) meets L~ <*((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))*> then x in L~ <*((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))*> by A122, SPPOL_2:21; hence L~ (Lower_Seq (C,n)) meets L~ <*((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))*> by A158, A156, XBOOLE_0:3; ::_thesis: verum end; supposeA162: x in L~ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) ; ::_thesis: L~ (Lower_Seq (C,n)) meets L~ <*((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))*> then x in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A158, A156, A161, XBOOLE_0:def_4; then x in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16; then A163: x = W-min (L~ (Cage (C,n))) by A158, A141, TARSKI:def_2; 1 in dom (Upper_Seq (C,n)) by A9, FINSEQ_3:25; then (Upper_Seq (C,n)) . 1 = (Upper_Seq (C,n)) /. 1 by PARTFUN1:def_6 .= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 by A7, A119, FINSEQ_5:44 .= W-min (L~ (Cage (C,n))) by A160, FINSEQ_6:92 ; then x = (Gauge (C,n)) * (i,j) by A112, A162, A163, JORDAN1E:7; then x in LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) by RLTOPSP1:68; then x in L~ <*((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))*> by SPPOL_2:21; hence L~ (Lower_Seq (C,n)) meets L~ <*((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))*> by A158, A156, XBOOLE_0:3; ::_thesis: verum end; end; end; then L~ <*((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))*> meets L~ (Lower_Seq (C,n)) ; hence LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets L~ (Lower_Seq (C,n)) by SPPOL_2:21; ::_thesis: verum end; supposeA164: (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) ; ::_thesis: LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets L~ (Lower_Seq (C,n)) (Gauge (C,n)) * (i,j) in LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) by RLTOPSP1:68; hence LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets L~ (Lower_Seq (C,n)) by A164, XBOOLE_0:3; ::_thesis: verum end; supposeA165: ( (Gauge (C,n)) * (i,j) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (i,j) = (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) ) ; ::_thesis: LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets L~ (Lower_Seq (C,n)) A166: (Gauge (C,n)) * (i,j) in LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) by RLTOPSP1:68; A167: ( rng (Lower_Seq (C,n)) c= L~ (Lower_Seq (C,n)) & E-max (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) ) by A5, A10, FINSEQ_6:61, SPPOL_2:18, XXREAL_0:2; E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46; then A168: E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by FINSEQ_6:90, SPRECT_2:43; len (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by A9, FINSEQ_3:25; then (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) = (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) by PARTFUN1:def_6 .= ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n))))) /. ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) by A7, A168, FINSEQ_5:42 .= E-max (L~ (Cage (C,n))) by A168, FINSEQ_5:45 ; hence LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets L~ (Lower_Seq (C,n)) by A165, A167, A166, XBOOLE_0:3; ::_thesis: verum end; end; end; hence LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets L~ (Lower_Seq (C,n)) ; ::_thesis: verum end; theorem Th47: :: JORDAN1G:47 for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) for n being Element of NAT st n > 0 holds First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Upper_Seq (C,n)) proof let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for n being Element of NAT st n > 0 holds First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Upper_Seq (C,n)) let n be Element of NAT ; ::_thesis: ( n > 0 implies First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Upper_Seq (C,n)) ) assume A1: n > 0 ; ::_thesis: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Upper_Seq (C,n)) set sr = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2; set Ebo = E-bound (L~ (Cage (C,n))); set Wbo = W-bound (L~ (Cage (C,n))); set Emax = E-max (L~ (Cage (C,n))); set Wmin = W-min (L~ (Cage (C,n))); set FiP = First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))); A2: 1 <= Center (Gauge (C,n)) by JORDAN1B:11; A3: ( (Upper_Seq (C,n)) /. 1 = W-min (L~ (Cage (C,n))) & (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) = E-max (L~ (Cage (C,n))) ) by JORDAN1F:5, JORDAN1F:7; then A4: L~ (Upper_Seq (C,n)) is_an_arc_of W-min (L~ (Cage (C,n))), E-max (L~ (Cage (C,n))) by TOPREAL1:25; A5: W-bound (L~ (Cage (C,n))) < E-bound (L~ (Cage (C,n))) by SPRECT_1:31; then W-bound (L~ (Cage (C,n))) < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by XREAL_1:226; then A6: (W-min (L~ (Cage (C,n)))) `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by EUCLID:52; A7: Center (Gauge (C,n)) <= len (Gauge (C,n)) by JORDAN1B:13; ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 < E-bound (L~ (Cage (C,n))) by A5, XREAL_1:226; then A8: ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 <= (E-max (L~ (Cage (C,n)))) `1 by EUCLID:52; then A9: L~ (Upper_Seq (C,n)) meets Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) by A4, A6, JORDAN6:49; (L~ (Upper_Seq (C,n))) /\ (Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)) is closed by A4, A6, A8, JORDAN6:49; then A10: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in (L~ (Upper_Seq (C,n))) /\ (Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)) by A4, A9, JORDAN5C:def_1; then First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in L~ (Upper_Seq (C,n)) by XBOOLE_0:def_4; then consider t being Element of NAT such that A11: 1 <= t and A12: t + 1 <= len (Upper_Seq (C,n)) and A13: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in LSeg ((Upper_Seq (C,n)),t) by SPPOL_2:13; A14: LSeg ((Upper_Seq (C,n)),t) = LSeg (((Upper_Seq (C,n)) /. t),((Upper_Seq (C,n)) /. (t + 1))) by A11, A12, TOPREAL1:def_3; t < len (Upper_Seq (C,n)) by A12, NAT_1:13; then A15: t in dom (Upper_Seq (C,n)) by A11, FINSEQ_3:25; 1 <= t + 1 by A11, NAT_1:13; then A16: t + 1 in dom (Upper_Seq (C,n)) by A12, FINSEQ_3:25; First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) by A10, XBOOLE_0:def_4; then A17: (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by JORDAN6:31; A18: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) = First_Point ((LSeg ((Upper_Seq (C,n)),t)),((Upper_Seq (C,n)) /. t),((Upper_Seq (C,n)) /. (t + 1)),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A3, A9, A11, A12, A13, JORDAN5C:19, JORDAN6:30; now__::_thesis:_First_Point_((L~_(Upper_Seq_(C,n))),(W-min_(L~_(Cage_(C,n)))),(E-max_(L~_(Cage_(C,n)))),(Vertical_Line_(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2)))_in_rng_(Upper_Seq_(C,n)) percases ( LSeg ((Upper_Seq (C,n)),t) is vertical or LSeg ((Upper_Seq (C,n)),t) is horizontal ) by SPPOL_1:19; supposeA19: LSeg ((Upper_Seq (C,n)),t) is vertical ; ::_thesis: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Upper_Seq (C,n)) then ((Upper_Seq (C,n)) /. (t + 1)) `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A13, A14, A17, SPPOL_1:41; then (Upper_Seq (C,n)) /. (t + 1) in { p where p is Point of (TOP-REAL 2) : p `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 } ; then A20: (Upper_Seq (C,n)) /. (t + 1) in Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) by JORDAN6:def_6; A21: ( LSeg ((Upper_Seq (C,n)),t) is closed & LSeg ((Upper_Seq (C,n)),t) is_an_arc_of (Upper_Seq (C,n)) /. t,(Upper_Seq (C,n)) /. (t + 1) ) by A14, A15, A16, GOBOARD7:29, TOPREAL1:9; ((Upper_Seq (C,n)) /. t) `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A13, A14, A17, A19, SPPOL_1:41; then (Upper_Seq (C,n)) /. t in { p where p is Point of (TOP-REAL 2) : p `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 } ; then (Upper_Seq (C,n)) /. t in Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) by JORDAN6:def_6; then LSeg ((Upper_Seq (C,n)),t) c= Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) by A14, A20, JORDAN1A:13; then First_Point ((LSeg ((Upper_Seq (C,n)),t)),((Upper_Seq (C,n)) /. t),((Upper_Seq (C,n)) /. (t + 1)),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) = (Upper_Seq (C,n)) /. t by A21, JORDAN5C:7; hence First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Upper_Seq (C,n)) by A18, A15, PARTFUN2:2; ::_thesis: verum end; suppose LSeg ((Upper_Seq (C,n)),t) is horizontal ; ::_thesis: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Upper_Seq (C,n)) then A22: ((Upper_Seq (C,n)) /. t) `2 = ((Upper_Seq (C,n)) /. (t + 1)) `2 by A14, SPPOL_1:15; then A23: (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 = ((Upper_Seq (C,n)) /. t) `2 by A13, A14, GOBOARD7:6; Upper_Seq (C,n) is_sequence_on Gauge (C,n) by Th4; then consider i1, j1, i2, j2 being Element of NAT such that A24: [i1,j1] in Indices (Gauge (C,n)) and A25: (Upper_Seq (C,n)) /. t = (Gauge (C,n)) * (i1,j1) and A26: [i2,j2] in Indices (Gauge (C,n)) and A27: (Upper_Seq (C,n)) /. (t + 1) = (Gauge (C,n)) * (i2,j2) and A28: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A11, A12, JORDAN8:3; A29: 1 <= i1 by A24, MATRIX_1:38; A30: 1 <= i2 by A26, MATRIX_1:38; A31: i1 <= len (Gauge (C,n)) by A24, MATRIX_1:38; A32: j1 = j2 by A22, A24, A25, A26, A27, Th6; A33: i2 <= len (Gauge (C,n)) by A26, MATRIX_1:38; A34: ( 1 <= j1 & j1 <= width (Gauge (C,n)) ) by A24, MATRIX_1:38; then A35: ((Gauge (C,n)) * ((Center (Gauge (C,n))),j1)) `1 = ((W-bound C) + (E-bound C)) / 2 by A1, Th35 .= (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 by A17, Th33 ; ((Gauge (C,n)) * ((Center (Gauge (C,n))),j1)) `2 = ((Gauge (C,n)) * (1,j1)) `2 by A2, A7, A34, GOBOARD5:1 .= (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 by A23, A25, A29, A31, A34, GOBOARD5:1 ; then A36: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) = (Gauge (C,n)) * ((Center (Gauge (C,n))),j1) by A35, TOPREAL3:6; now__::_thesis:_First_Point_((L~_(Upper_Seq_(C,n))),(W-min_(L~_(Cage_(C,n)))),(E-max_(L~_(Cage_(C,n)))),(Vertical_Line_(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2)))_in_rng_(Upper_Seq_(C,n)) percases ( i1 + 1 = i2 or i1 = i2 + 1 ) by A28, A32; supposeA37: i1 + 1 = i2 ; ::_thesis: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Upper_Seq (C,n)) i1 < i1 + 1 by NAT_1:13; then A38: ((Gauge (C,n)) * (i1,j1)) `1 <= ((Gauge (C,n)) * ((i1 + 1),j1)) `1 by A29, A34, A33, A37, SPRECT_3:13; then ((Gauge (C,n)) * (i1,j1)) `1 <= (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 by A13, A14, A25, A27, A32, A37, TOPREAL1:3; then i1 <= Center (Gauge (C,n)) by A2, A31, A34, A35, GOBOARD5:3; then ( i1 = Center (Gauge (C,n)) or i1 < Center (Gauge (C,n)) ) by XXREAL_0:1; then A39: ( i1 = Center (Gauge (C,n)) or i1 + 1 <= Center (Gauge (C,n)) ) by NAT_1:13; (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 <= ((Gauge (C,n)) * ((i1 + 1),j1)) `1 by A13, A14, A25, A27, A32, A37, A38, TOPREAL1:3; then Center (Gauge (C,n)) <= i1 + 1 by A7, A34, A30, A35, A37, GOBOARD5:3; then ( i1 = Center (Gauge (C,n)) or i1 + 1 = Center (Gauge (C,n)) ) by A39, XXREAL_0:1; hence First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Upper_Seq (C,n)) by A15, A16, A25, A27, A32, A36, A37, PARTFUN2:2; ::_thesis: verum end; supposeA40: i1 = i2 + 1 ; ::_thesis: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Upper_Seq (C,n)) i2 < i2 + 1 by NAT_1:13; then A41: ((Gauge (C,n)) * (i2,j1)) `1 <= ((Gauge (C,n)) * ((i2 + 1),j1)) `1 by A31, A34, A30, A40, SPRECT_3:13; then ((Gauge (C,n)) * (i2,j1)) `1 <= (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 by A13, A14, A25, A27, A32, A40, TOPREAL1:3; then i2 <= Center (Gauge (C,n)) by A2, A34, A33, A35, GOBOARD5:3; then ( i2 = Center (Gauge (C,n)) or i2 < Center (Gauge (C,n)) ) by XXREAL_0:1; then A42: ( i2 = Center (Gauge (C,n)) or i2 + 1 <= Center (Gauge (C,n)) ) by NAT_1:13; (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 <= ((Gauge (C,n)) * ((i2 + 1),j1)) `1 by A13, A14, A25, A27, A32, A40, A41, TOPREAL1:3; then Center (Gauge (C,n)) <= i2 + 1 by A7, A29, A34, A35, A40, GOBOARD5:3; then ( i2 = Center (Gauge (C,n)) or i2 + 1 = Center (Gauge (C,n)) ) by A42, XXREAL_0:1; hence First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Upper_Seq (C,n)) by A15, A16, A25, A27, A32, A36, A40, PARTFUN2:2; ::_thesis: verum end; end; end; hence First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Upper_Seq (C,n)) ; ::_thesis: verum end; end; end; hence First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Upper_Seq (C,n)) ; ::_thesis: verum end; theorem Th48: :: JORDAN1G:48 for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) for n being Element of NAT st n > 0 holds Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Lower_Seq (C,n)) proof let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for n being Element of NAT st n > 0 holds Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Lower_Seq (C,n)) let n be Element of NAT ; ::_thesis: ( n > 0 implies Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Lower_Seq (C,n)) ) assume A1: n > 0 ; ::_thesis: Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Lower_Seq (C,n)) set sr = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2; set Ebo = E-bound (L~ (Cage (C,n))); set Wbo = W-bound (L~ (Cage (C,n))); set Emax = E-max (L~ (Cage (C,n))); set Wmin = W-min (L~ (Cage (C,n))); set LaP = Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))); A2: ( (Lower_Seq (C,n)) /. 1 = E-max (L~ (Cage (C,n))) & (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) = W-min (L~ (Cage (C,n))) ) by JORDAN1F:6, JORDAN1F:8; then A3: L~ (Lower_Seq (C,n)) is_an_arc_of E-max (L~ (Cage (C,n))), W-min (L~ (Cage (C,n))) by TOPREAL1:25; A4: W-bound (L~ (Cage (C,n))) <= E-bound (L~ (Cage (C,n))) by SPRECT_1:21; then W-bound (L~ (Cage (C,n))) <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by JORDAN6:1; then A5: (W-min (L~ (Cage (C,n)))) `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by EUCLID:52; ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 <= E-bound (L~ (Cage (C,n))) by A4, JORDAN6:1; then A6: ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 <= (E-max (L~ (Cage (C,n)))) `1 by EUCLID:52; A7: L~ (Lower_Seq (C,n)) is_an_arc_of W-min (L~ (Cage (C,n))), E-max (L~ (Cage (C,n))) by A2, JORDAN5B:14, TOPREAL1:25; then A8: L~ (Lower_Seq (C,n)) meets Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) by A5, A6, JORDAN6:49; (L~ (Lower_Seq (C,n))) /\ (Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)) is closed by A7, A5, A6, JORDAN6:49; then A9: Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in (L~ (Lower_Seq (C,n))) /\ (Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)) by A3, A8, JORDAN5C:def_2; then Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in L~ (Lower_Seq (C,n)) by XBOOLE_0:def_4; then consider t being Element of NAT such that A10: 1 <= t and A11: t + 1 <= len (Lower_Seq (C,n)) and A12: Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in LSeg ((Lower_Seq (C,n)),t) by SPPOL_2:13; A13: LSeg ((Lower_Seq (C,n)),t) = LSeg (((Lower_Seq (C,n)) /. t),((Lower_Seq (C,n)) /. (t + 1))) by A10, A11, TOPREAL1:def_3; 1 <= t + 1 by A10, NAT_1:13; then A14: t + 1 in dom (Lower_Seq (C,n)) by A11, FINSEQ_3:25; t < len (Lower_Seq (C,n)) by A11, NAT_1:13; then A15: t in dom (Lower_Seq (C,n)) by A10, FINSEQ_3:25; Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) by A9, XBOOLE_0:def_4; then A16: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by JORDAN6:31; A17: Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) = Last_Point ((LSeg ((Lower_Seq (C,n)),t)),((Lower_Seq (C,n)) /. t),((Lower_Seq (C,n)) /. (t + 1)),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A2, A8, A10, A11, A12, JORDAN5C:20, JORDAN6:30; now__::_thesis:_Last_Point_((L~_(Lower_Seq_(C,n))),(E-max_(L~_(Cage_(C,n)))),(W-min_(L~_(Cage_(C,n)))),(Vertical_Line_(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2)))_in_rng_(Lower_Seq_(C,n)) percases ( LSeg ((Lower_Seq (C,n)),t) is vertical or LSeg ((Lower_Seq (C,n)),t) is horizontal ) by SPPOL_1:19; supposeA18: LSeg ((Lower_Seq (C,n)),t) is vertical ; ::_thesis: Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Lower_Seq (C,n)) then ((Lower_Seq (C,n)) /. (t + 1)) `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A12, A13, A16, SPPOL_1:41; then (Lower_Seq (C,n)) /. (t + 1) in { p where p is Point of (TOP-REAL 2) : p `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 } ; then A19: (Lower_Seq (C,n)) /. (t + 1) in Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) by JORDAN6:def_6; A20: ( LSeg ((Lower_Seq (C,n)),t) is closed & LSeg ((Lower_Seq (C,n)),t) is_an_arc_of (Lower_Seq (C,n)) /. t,(Lower_Seq (C,n)) /. (t + 1) ) by A13, A15, A14, GOBOARD7:29, TOPREAL1:9; ((Lower_Seq (C,n)) /. t) `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A12, A13, A16, A18, SPPOL_1:41; then (Lower_Seq (C,n)) /. t in { p where p is Point of (TOP-REAL 2) : p `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 } ; then (Lower_Seq (C,n)) /. t in Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) by JORDAN6:def_6; then LSeg ((Lower_Seq (C,n)),t) c= Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) by A13, A19, JORDAN1A:13; then Last_Point ((LSeg ((Lower_Seq (C,n)),t)),((Lower_Seq (C,n)) /. t),((Lower_Seq (C,n)) /. (t + 1)),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) = (Lower_Seq (C,n)) /. (t + 1) by A20, JORDAN5C:7; hence Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Lower_Seq (C,n)) by A17, A14, PARTFUN2:2; ::_thesis: verum end; suppose LSeg ((Lower_Seq (C,n)),t) is horizontal ; ::_thesis: Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Lower_Seq (C,n)) then A21: ((Lower_Seq (C,n)) /. t) `2 = ((Lower_Seq (C,n)) /. (t + 1)) `2 by A13, SPPOL_1:15; then A22: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 = ((Lower_Seq (C,n)) /. t) `2 by A12, A13, GOBOARD7:6; Lower_Seq (C,n) is_sequence_on Gauge (C,n) by Th5; then consider i1, j1, i2, j2 being Element of NAT such that A23: [i1,j1] in Indices (Gauge (C,n)) and A24: (Lower_Seq (C,n)) /. t = (Gauge (C,n)) * (i1,j1) and A25: [i2,j2] in Indices (Gauge (C,n)) and A26: (Lower_Seq (C,n)) /. (t + 1) = (Gauge (C,n)) * (i2,j2) and A27: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A10, A11, JORDAN8:3; A28: 1 <= i1 by A23, MATRIX_1:38; A29: j1 = j2 by A21, A23, A24, A25, A26, Th6; A30: i2 <= len (Gauge (C,n)) by A25, MATRIX_1:38; A31: i1 <= len (Gauge (C,n)) by A23, MATRIX_1:38; A32: 1 <= i2 by A25, MATRIX_1:38; A33: Center (Gauge (C,n)) <= len (Gauge (C,n)) by JORDAN1B:13; A34: ( 1 <= j1 & j1 <= width (Gauge (C,n)) ) by A23, MATRIX_1:38; then A35: ((Gauge (C,n)) * ((Center (Gauge (C,n))),j1)) `1 = ((W-bound C) + (E-bound C)) / 2 by A1, Th35 .= (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 by A16, Th33 ; A36: 1 <= Center (Gauge (C,n)) by JORDAN1B:11; then ((Gauge (C,n)) * ((Center (Gauge (C,n))),j1)) `2 = ((Gauge (C,n)) * (1,j1)) `2 by A34, A33, GOBOARD5:1 .= (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 by A22, A24, A28, A31, A34, GOBOARD5:1 ; then A37: Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) = (Gauge (C,n)) * ((Center (Gauge (C,n))),j1) by A35, TOPREAL3:6; now__::_thesis:_Last_Point_((L~_(Lower_Seq_(C,n))),(E-max_(L~_(Cage_(C,n)))),(W-min_(L~_(Cage_(C,n)))),(Vertical_Line_(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2)))_in_rng_(Lower_Seq_(C,n)) percases ( i1 + 1 = i2 or i1 = i2 + 1 ) by A27, A29; supposeA38: i1 + 1 = i2 ; ::_thesis: Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Lower_Seq (C,n)) i1 < i1 + 1 by NAT_1:13; then A39: ((Gauge (C,n)) * (i1,j1)) `1 <= ((Gauge (C,n)) * ((i1 + 1),j1)) `1 by A28, A34, A30, A38, SPRECT_3:13; then ((Gauge (C,n)) * (i1,j1)) `1 <= (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 by A12, A13, A24, A26, A29, A38, TOPREAL1:3; then i1 <= Center (Gauge (C,n)) by A31, A34, A36, A35, GOBOARD5:3; then ( i1 = Center (Gauge (C,n)) or i1 < Center (Gauge (C,n)) ) by XXREAL_0:1; then A40: ( i1 = Center (Gauge (C,n)) or i1 + 1 <= Center (Gauge (C,n)) ) by NAT_1:13; (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 <= ((Gauge (C,n)) * ((i1 + 1),j1)) `1 by A12, A13, A24, A26, A29, A38, A39, TOPREAL1:3; then Center (Gauge (C,n)) <= i1 + 1 by A34, A32, A33, A35, A38, GOBOARD5:3; then ( i1 = Center (Gauge (C,n)) or i1 + 1 = Center (Gauge (C,n)) ) by A40, XXREAL_0:1; hence Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Lower_Seq (C,n)) by A15, A14, A24, A26, A29, A37, A38, PARTFUN2:2; ::_thesis: verum end; supposeA41: i1 = i2 + 1 ; ::_thesis: Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Lower_Seq (C,n)) i2 < i2 + 1 by NAT_1:13; then A42: ((Gauge (C,n)) * (i2,j1)) `1 <= ((Gauge (C,n)) * ((i2 + 1),j1)) `1 by A31, A34, A32, A41, SPRECT_3:13; then ((Gauge (C,n)) * (i2,j1)) `1 <= (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 by A12, A13, A24, A26, A29, A41, TOPREAL1:3; then i2 <= Center (Gauge (C,n)) by A34, A30, A36, A35, GOBOARD5:3; then ( i2 = Center (Gauge (C,n)) or i2 < Center (Gauge (C,n)) ) by XXREAL_0:1; then A43: ( i2 = Center (Gauge (C,n)) or i2 + 1 <= Center (Gauge (C,n)) ) by NAT_1:13; (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 <= ((Gauge (C,n)) * ((i2 + 1),j1)) `1 by A12, A13, A24, A26, A29, A41, A42, TOPREAL1:3; then Center (Gauge (C,n)) <= i2 + 1 by A28, A34, A33, A35, A41, GOBOARD5:3; then ( i2 = Center (Gauge (C,n)) or i2 + 1 = Center (Gauge (C,n)) ) by A43, XXREAL_0:1; hence Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Lower_Seq (C,n)) by A15, A14, A24, A26, A29, A37, A41, PARTFUN2:2; ::_thesis: verum end; end; end; hence Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Lower_Seq (C,n)) ; ::_thesis: verum end; end; end; hence Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Lower_Seq (C,n)) ; ::_thesis: verum end; theorem Th49: :: JORDAN1G:49 for f being S-Sequence_in_R2 for p being Point of (TOP-REAL 2) st p in rng f holds R_Cut (f,p) = mid (f,1,(p .. f)) proof let f be S-Sequence_in_R2; ::_thesis: for p being Point of (TOP-REAL 2) st p in rng f holds R_Cut (f,p) = mid (f,1,(p .. f)) let p be Point of (TOP-REAL 2); ::_thesis: ( p in rng f implies R_Cut (f,p) = mid (f,1,(p .. f)) ) assume A1: p in rng f ; ::_thesis: R_Cut (f,p) = mid (f,1,(p .. f)) then consider i being Nat such that A2: i in dom f and A3: f . i = p by FINSEQ_2:10; reconsider i = i as Element of NAT by ORDINAL1:def_12; A4: i <= len f by A2, FINSEQ_3:25; len f >= 2 by TOPREAL1:def_8; then A5: rng f c= L~ f by SPPOL_2:18; then A6: 1 <= Index (p,f) by A1, JORDAN3:8; A7: Index (p,f) < len f by A1, A5, JORDAN3:8; A8: 0 + 1 <= i by A2, FINSEQ_3:25; then A9: i - 1 >= 0 by XREAL_1:19; percases ( 1 < i or 1 = i ) by A8, XXREAL_0:1; supposeA10: 1 < i ; ::_thesis: R_Cut (f,p) = mid (f,1,(p .. f)) 1 <= len f by A8, A4, XXREAL_0:2; then 1 in dom f by FINSEQ_3:25; then p <> f . 1 by A2, A3, A10, FUNCT_1:def_4; then A11: R_Cut (f,p) = (mid (f,1,(Index (p,f)))) ^ <*p*> by JORDAN3:def_4; A12: (Index (p,f)) + 1 = i by A3, A4, A10, JORDAN3:12; A13: len (mid (f,1,(Index (p,f)))) = ((Index (p,f)) -' 1) + 1 by A6, A7, JORDAN4:8 .= i -' 1 by A1, A5, A12, JORDAN3:8, NAT_D:38 ; A14: len (mid (f,1,i)) = (i -' 1) + 1 by A8, A4, JORDAN4:8 .= i by A8, XREAL_1:235 ; then A15: dom (mid (f,1,i)) = Seg i by FINSEQ_1:def_3; A16: now__::_thesis:_for_j_being_Nat_st_j_in_dom_(mid_(f,1,i))_holds_ (mid_(f,1,i))_._j_=_((mid_(f,1,(Index_(p,f))))_^_<*p*>)_._j let j be Nat; ::_thesis: ( j in dom (mid (f,1,i)) implies (mid (f,1,i)) . j = ((mid (f,1,(Index (p,f)))) ^ <*p*>) . j ) reconsider a = j as Element of NAT by ORDINAL1:def_12; assume A17: j in dom (mid (f,1,i)) ; ::_thesis: (mid (f,1,i)) . j = ((mid (f,1,(Index (p,f)))) ^ <*p*>) . j then A18: 1 <= j by A15, FINSEQ_1:1; A19: j <= i by A15, A17, FINSEQ_1:1; now__::_thesis:_(mid_(f,1,i))_._j_=_((mid_(f,1,(Index_(p,f))))_^_<*p*>)_._j percases ( j < i or j = i ) by A19, XXREAL_0:1; suppose j < i ; ::_thesis: (mid (f,1,i)) . j = ((mid (f,1,(Index (p,f)))) ^ <*p*>) . j then A20: j <= Index (p,f) by A12, NAT_1:13; then j <= i -' 1 by A9, A12, XREAL_0:def_2; then A21: j in dom (mid (f,1,(Index (p,f)))) by A13, A18, FINSEQ_3:25; thus (mid (f,1,i)) . j = f . a by A4, A18, A19, FINSEQ_6:123 .= (mid (f,1,(Index (p,f)))) . a by A7, A18, A20, FINSEQ_6:123 .= ((mid (f,1,(Index (p,f)))) ^ <*p*>) . j by A21, FINSEQ_1:def_7 ; ::_thesis: verum end; supposeA22: j = i ; ::_thesis: (mid (f,1,i)) . j = ((mid (f,1,(Index (p,f)))) ^ <*p*>) . j A23: (i -' 1) + 1 = i by A8, XREAL_1:235; thus (mid (f,1,i)) . j = f . a by A4, A18, A19, FINSEQ_6:123 .= ((mid (f,1,(Index (p,f)))) ^ <*p*>) . j by A3, A13, A22, A23, FINSEQ_1:42 ; ::_thesis: verum end; end; end; hence (mid (f,1,i)) . j = ((mid (f,1,(Index (p,f)))) ^ <*p*>) . j ; ::_thesis: verum end; len ((mid (f,1,(Index (p,f)))) ^ <*p*>) = (i -' 1) + 1 by A13, FINSEQ_2:16 .= i by A8, XREAL_1:235 ; then mid (f,1,i) = R_Cut (f,p) by A11, A14, A16, FINSEQ_2:9; hence R_Cut (f,p) = mid (f,1,(p .. f)) by A2, A3, FINSEQ_5:11; ::_thesis: verum end; supposeA24: 1 = i ; ::_thesis: R_Cut (f,p) = mid (f,1,(p .. f)) then A25: R_Cut (f,p) = <*p*> by A3, JORDAN3:def_4; A26: p = f /. 1 by A2, A3, A24, PARTFUN1:def_6; then p .. f = 1 by FINSEQ_6:43; hence R_Cut (f,p) = mid (f,1,(p .. f)) by A4, A24, A25, A26, JORDAN4:15; ::_thesis: verum end; end; end; theorem Th50: :: JORDAN1G:50 for f being S-Sequence_in_R2 for Q being closed Subset of (TOP-REAL 2) st L~ f meets Q & not f /. 1 in Q & First_Point ((L~ f),(f /. 1),(f /. (len f)),Q) in rng f holds (L~ (mid (f,1,((First_Point ((L~ f),(f /. 1),(f /. (len f)),Q)) .. f)))) /\ Q = {(First_Point ((L~ f),(f /. 1),(f /. (len f)),Q))} proof let f be S-Sequence_in_R2; ::_thesis: for Q being closed Subset of (TOP-REAL 2) st L~ f meets Q & not f /. 1 in Q & First_Point ((L~ f),(f /. 1),(f /. (len f)),Q) in rng f holds (L~ (mid (f,1,((First_Point ((L~ f),(f /. 1),(f /. (len f)),Q)) .. f)))) /\ Q = {(First_Point ((L~ f),(f /. 1),(f /. (len f)),Q))} let Q be closed Subset of (TOP-REAL 2); ::_thesis: ( L~ f meets Q & not f /. 1 in Q & First_Point ((L~ f),(f /. 1),(f /. (len f)),Q) in rng f implies (L~ (mid (f,1,((First_Point ((L~ f),(f /. 1),(f /. (len f)),Q)) .. f)))) /\ Q = {(First_Point ((L~ f),(f /. 1),(f /. (len f)),Q))} ) assume that A1: ( L~ f meets Q & not f /. 1 in Q ) and A2: First_Point ((L~ f),(f /. 1),(f /. (len f)),Q) in rng f ; ::_thesis: (L~ (mid (f,1,((First_Point ((L~ f),(f /. 1),(f /. (len f)),Q)) .. f)))) /\ Q = {(First_Point ((L~ f),(f /. 1),(f /. (len f)),Q))} (L~ (R_Cut (f,(First_Point ((L~ f),(f /. 1),(f /. (len f)),Q))))) /\ Q = {(First_Point ((L~ f),(f /. 1),(f /. (len f)),Q))} by A1, SPRECT_4:1; hence (L~ (mid (f,1,((First_Point ((L~ f),(f /. 1),(f /. (len f)),Q)) .. f)))) /\ Q = {(First_Point ((L~ f),(f /. 1),(f /. (len f)),Q))} by A2, Th49; ::_thesis: verum end; theorem Th51: :: JORDAN1G:51 for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) for n being Element of NAT st n > 0 holds for k being Element of NAT st 1 <= k & k < (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) holds ((Upper_Seq (C,n)) /. k) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 proof let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for n being Element of NAT st n > 0 holds for k being Element of NAT st 1 <= k & k < (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) holds ((Upper_Seq (C,n)) /. k) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 let n be Element of NAT ; ::_thesis: ( n > 0 implies for k being Element of NAT st 1 <= k & k < (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) holds ((Upper_Seq (C,n)) /. k) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 ) assume A1: n > 0 ; ::_thesis: for k being Element of NAT st 1 <= k & k < (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) holds ((Upper_Seq (C,n)) /. k) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 set US = Upper_Seq (C,n); set sr = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2; set Ebo = E-bound (L~ (Cage (C,n))); set Wbo = W-bound (L~ (Cage (C,n))); set Emax = E-max (L~ (Cage (C,n))); set Wmin = W-min (L~ (Cage (C,n))); set FiP = First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))); defpred S1[ Nat] means ( 1 <= $1 & $1 < (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) implies ((Upper_Seq (C,n)) /. $1) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 ); A2: W-bound (L~ (Cage (C,n))) < E-bound (L~ (Cage (C,n))) by SPRECT_1:31; then A3: W-bound (L~ (Cage (C,n))) < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by XREAL_1:226; A4: ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 < E-bound (L~ (Cage (C,n))) by A2, XREAL_1:226; A5: for k being non empty Nat st S1[k] holds S1[k + 1] proof set GC1 = (Gauge (C,n)) * ((Center (Gauge (C,n))),1); let k be non empty Nat; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A6: ( 1 <= k & k < (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) implies ((Upper_Seq (C,n)) /. k) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 ) ; ::_thesis: S1[k + 1] 4 <= len (Gauge (C,n)) by JORDAN8:10; then 1 <= len (Gauge (C,n)) by XXREAL_0:2; then A7: 1 <= width (Gauge (C,n)) by JORDAN8:def_1; then A8: ((Gauge (C,n)) * ((Center (Gauge (C,n))),1)) `1 = ((W-bound C) + (E-bound C)) / 2 by A1, Th35 .= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by Th33 ; A9: k >= 1 by NAT_1:14; A10: (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) = E-max (L~ (Cage (C,n))) by JORDAN1F:7; A11: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Upper_Seq (C,n)) by A1, Th47; then A12: (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by FINSEQ_4:20; then A13: 1 <= (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) by FINSEQ_3:25; A14: 1 <= Center (Gauge (C,n)) by JORDAN1B:11; A15: (Upper_Seq (C,n)) /. 1 = W-min (L~ (Cage (C,n))) by JORDAN1F:5; reconsider kk = k as Element of NAT by ORDINAL1:def_12; assume that A16: 1 <= k + 1 and A17: k + 1 < (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) ; ::_thesis: ((Upper_Seq (C,n)) /. (k + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 A18: (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) <= len (Upper_Seq (C,n)) by A12, FINSEQ_3:25; then A19: k + 1 <= len (Upper_Seq (C,n)) by A17, XXREAL_0:2; Upper_Seq (C,n) is_sequence_on Gauge (C,n) by Th4; then consider i1, j1, i2, j2 being Element of NAT such that A20: [i1,j1] in Indices (Gauge (C,n)) and A21: (Upper_Seq (C,n)) /. kk = (Gauge (C,n)) * (i1,j1) and A22: [i2,j2] in Indices (Gauge (C,n)) and A23: (Upper_Seq (C,n)) /. (kk + 1) = (Gauge (C,n)) * (i2,j2) and A24: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A9, A19, JORDAN8:3; A25: 1 <= i1 by A20, MATRIX_1:38; A26: ( 1 <= j1 & j1 <= width (Gauge (C,n)) ) by A20, MATRIX_1:38; A27: j2 <= width (Gauge (C,n)) by A22, MATRIX_1:38; A28: ( 1 <= i2 & 1 <= j2 ) by A22, MATRIX_1:38; A29: i2 <= len (Gauge (C,n)) by A22, MATRIX_1:38; A30: i1 <= len (Gauge (C,n)) by A20, MATRIX_1:38; A31: ( Center (Gauge (C,n)) <= len (Gauge (C,n)) & i1 + 1 >= 1 ) by JORDAN1B:13, NAT_1:11; now__::_thesis:_((Upper_Seq_(C,n))_/._(k_+_1))_`1_<_((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2 percases ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A24; suppose ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: ((Upper_Seq (C,n)) /. (k + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 then ((Upper_Seq (C,n)) /. k) `1 = ((Gauge (C,n)) * (i2,1)) `1 by A21, A25, A30, A26, GOBOARD5:2 .= ((Upper_Seq (C,n)) /. (k + 1)) `1 by A23, A29, A28, A27, GOBOARD5:2 ; hence ((Upper_Seq (C,n)) /. (k + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A6, A17, NAT_1:13, NAT_1:14; ::_thesis: verum end; supposeA32: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: ((Upper_Seq (C,n)) /. (k + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 A33: now__::_thesis:_not_((Upper_Seq_(C,n))_/._(k_+_1))_`1_=_((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2 A34: k + 1 >= 1 + 1 by A9, XREAL_1:7; len (mid ((Upper_Seq (C,n)),1,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) = (((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) -' 1) + 1 by A13, A18, JORDAN4:8 .= (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) by A13, XREAL_1:235 ; then A35: rng (mid ((Upper_Seq (C,n)),1,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) c= L~ (mid ((Upper_Seq (C,n)),1,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) by A17, A34, SPPOL_2:18, XXREAL_0:2; A36: (Upper_Seq (C,n)) /. ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) = First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A11, FINSEQ_5:38; A37: now__::_thesis:_not_(Upper_Seq_(C,n))_/._1_in_Vertical_Line_(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2) assume (Upper_Seq (C,n)) /. 1 in Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) ; ::_thesis: contradiction then (W-min (L~ (Cage (C,n)))) `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A15, JORDAN6:31; hence contradiction by A3, EUCLID:52; ::_thesis: verum end; A38: ( (W-min (L~ (Cage (C,n)))) `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 & ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 <= (E-max (L~ (Cage (C,n)))) `1 ) by A3, A4, EUCLID:52; A39: ( Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) is closed & L~ (Upper_Seq (C,n)) is_an_arc_of W-min (L~ (Cage (C,n))), E-max (L~ (Cage (C,n))) ) by A15, A10, JORDAN6:30, TOPREAL1:25; First_Point ((L~ (Upper_Seq (C,n))),((Upper_Seq (C,n)) /. 1),((Upper_Seq (C,n)) /. (len (Upper_Seq (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Upper_Seq (C,n)) by A1, A15, A10, Th47; then A40: (L~ (mid ((Upper_Seq (C,n)),1,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))))) /\ (Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)) = {(First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))} by A15, A10, A39, A38, A37, Th50, JORDAN6:49; A41: ( mid ((Upper_Seq (C,n)),1,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))) = (Upper_Seq (C,n)) | ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) & (Upper_Seq (C,n)) | (Seg ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))) = (Upper_Seq (C,n)) | ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) ) by A13, FINSEQ_1:def_15, FINSEQ_6:116; assume ((Upper_Seq (C,n)) /. (k + 1)) `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 ; ::_thesis: contradiction then (Upper_Seq (C,n)) /. (k + 1) in { p where p is Point of (TOP-REAL 2) : p `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 } ; then A42: (Upper_Seq (C,n)) /. (k + 1) in Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) by JORDAN6:def_6; A43: k + 1 in dom (Upper_Seq (C,n)) by A16, A19, FINSEQ_3:25; k + 1 in Seg ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) by A16, A17, FINSEQ_1:1; then (Upper_Seq (C,n)) /. (k + 1) in rng (mid ((Upper_Seq (C,n)),1,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) by A41, A43, PARTFUN2:18; then (Upper_Seq (C,n)) /. (k + 1) in {(First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))} by A42, A35, A40, XBOOLE_0:def_4; then (Upper_Seq (C,n)) /. (k + 1) = First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by TARSKI:def_1; hence contradiction by A17, A12, A43, A36, PARTFUN2:10; ::_thesis: verum end; i1 < Center (Gauge (C,n)) by A6, A17, A21, A30, A26, A14, A7, A8, JORDAN1A:18, NAT_1:13, NAT_1:14; then i1 + 1 <= Center (Gauge (C,n)) by NAT_1:13; then ((Upper_Seq (C,n)) /. (k + 1)) `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A23, A26, A7, A8, A31, A32, JORDAN1A:18; hence ((Upper_Seq (C,n)) /. (k + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A33, XXREAL_0:1; ::_thesis: verum end; suppose ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: ((Upper_Seq (C,n)) /. (k + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 then i2 < i1 by NAT_1:13; then ((Upper_Seq (C,n)) /. (k + 1)) `1 <= ((Upper_Seq (C,n)) /. k) `1 by A21, A23, A30, A26, A28, A27, JORDAN1A:18; hence ((Upper_Seq (C,n)) /. (k + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A6, A17, NAT_1:13, NAT_1:14, XXREAL_0:2; ::_thesis: verum end; suppose ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: ((Upper_Seq (C,n)) /. (k + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 then ((Upper_Seq (C,n)) /. k) `1 = ((Gauge (C,n)) * (i2,1)) `1 by A21, A25, A30, A26, GOBOARD5:2 .= ((Upper_Seq (C,n)) /. (k + 1)) `1 by A23, A29, A28, A27, GOBOARD5:2 ; hence ((Upper_Seq (C,n)) /. (k + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A6, A17, NAT_1:13, NAT_1:14; ::_thesis: verum end; end; end; hence ((Upper_Seq (C,n)) /. (k + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 ; ::_thesis: verum end; A44: S1[1] proof assume that 1 <= 1 and 1 < (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) ; ::_thesis: ((Upper_Seq (C,n)) /. 1) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 (Upper_Seq (C,n)) /. 1 = W-min (L~ (Cage (C,n))) by JORDAN1F:5; hence ((Upper_Seq (C,n)) /. 1) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A3, EUCLID:52; ::_thesis: verum end; A45: for k being non empty Nat holds S1[k] from NAT_1:sch_10(A44, A5); let k be Element of NAT ; ::_thesis: ( 1 <= k & k < (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) implies ((Upper_Seq (C,n)) /. k) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 ) assume ( 1 <= k & k < (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) ) ; ::_thesis: ((Upper_Seq (C,n)) /. k) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 hence ((Upper_Seq (C,n)) /. k) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A45; ::_thesis: verum end; theorem Th52: :: JORDAN1G:52 for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) for n being Element of NAT st n > 0 holds for k being Nat st 1 <= k & k < (First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))) holds ((Rev (Lower_Seq (C,n))) /. k) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 proof let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for n being Element of NAT st n > 0 holds for k being Nat st 1 <= k & k < (First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))) holds ((Rev (Lower_Seq (C,n))) /. k) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 let n be Element of NAT ; ::_thesis: ( n > 0 implies for k being Nat st 1 <= k & k < (First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))) holds ((Rev (Lower_Seq (C,n))) /. k) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 ) assume A1: n > 0 ; ::_thesis: for k being Nat st 1 <= k & k < (First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))) holds ((Rev (Lower_Seq (C,n))) /. k) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 set LS = Lower_Seq (C,n); set sr = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2; set Ebo = E-bound (L~ (Cage (C,n))); set Wbo = W-bound (L~ (Cage (C,n))); set Emax = E-max (L~ (Cage (C,n))); set Wmin = W-min (L~ (Cage (C,n))); set RLS = Rev (Lower_Seq (C,n)); set FiP = First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))); set LaP = Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))); A2: L~ (Rev (Lower_Seq (C,n))) = L~ (Lower_Seq (C,n)) by SPPOL_2:22; A3: len (Rev (Lower_Seq (C,n))) = len (Lower_Seq (C,n)) by FINSEQ_5:def_3; defpred S1[ Nat] means ( 1 <= $1 & $1 < (First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))) implies ((Rev (Lower_Seq (C,n))) /. $1) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 ); A4: rng (Rev (Lower_Seq (C,n))) = rng (Lower_Seq (C,n)) by FINSEQ_5:57; A5: W-bound (L~ (Cage (C,n))) < E-bound (L~ (Cage (C,n))) by SPRECT_1:31; then A6: W-bound (L~ (Cage (C,n))) < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by XREAL_1:226; A7: ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 < E-bound (L~ (Cage (C,n))) by A5, XREAL_1:226; A8: for k being non empty Nat st S1[k] holds S1[k + 1] proof A9: W-bound (L~ (Cage (C,n))) <= E-bound (L~ (Cage (C,n))) by SPRECT_1:21; then W-bound (L~ (Cage (C,n))) <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by JORDAN6:1; then A10: (W-min (L~ (Cage (C,n)))) `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by EUCLID:52; ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 <= E-bound (L~ (Cage (C,n))) by A9, JORDAN6:1; then A11: ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 <= (E-max (L~ (Cage (C,n)))) `1 by EUCLID:52; A12: (Rev (Lower_Seq (C,n))) /. (len (Rev (Lower_Seq (C,n)))) = (Lower_Seq (C,n)) /. 1 by A3, FINSEQ_5:65 .= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ; set GC1 = (Gauge (C,n)) * ((Center (Gauge (C,n))),1); let k be non empty Nat; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A13: ( 1 <= k & k < (First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))) implies ((Rev (Lower_Seq (C,n))) /. k) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 ) ; ::_thesis: S1[k + 1] 4 <= len (Gauge (C,n)) by JORDAN8:10; then 1 <= len (Gauge (C,n)) by XXREAL_0:2; then A14: 1 <= width (Gauge (C,n)) by JORDAN8:def_1; then A15: ((Gauge (C,n)) * ((Center (Gauge (C,n))),1)) `1 = ((W-bound C) + (E-bound C)) / 2 by A1, Th35 .= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by Th33 ; A16: ( (Lower_Seq (C,n)) /. 1 = E-max (L~ (Cage (C,n))) & (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) = W-min (L~ (Cage (C,n))) ) by JORDAN1F:6, JORDAN1F:8; then A17: L~ (Lower_Seq (C,n)) is_an_arc_of E-max (L~ (Cage (C,n))), W-min (L~ (Cage (C,n))) by TOPREAL1:25; A18: 1 <= Center (Gauge (C,n)) by JORDAN1B:11; A19: (Rev (Lower_Seq (C,n))) /. 1 = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by FINSEQ_5:65 .= W-min (L~ (Cage (C,n))) by JORDAN1F:8 ; L~ (Lower_Seq (C,n)) is_an_arc_of W-min (L~ (Cage (C,n))), E-max (L~ (Cage (C,n))) by A16, JORDAN5B:14, TOPREAL1:25; then ( L~ (Lower_Seq (C,n)) meets Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) & (L~ (Lower_Seq (C,n))) /\ (Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)) is closed ) by A10, A11, JORDAN6:49; then A20: First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) = Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A2, A17, JORDAN5C:18; then A21: (First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))) in dom (Rev (Lower_Seq (C,n))) by A1, A4, Th48, FINSEQ_4:20; then A22: 1 <= (First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))) by FINSEQ_3:25; A23: k >= 1 by NAT_1:14; reconsider kk = k as Element of NAT by ORDINAL1:def_12; assume that A24: 1 <= k + 1 and A25: k + 1 < (First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))) ; ::_thesis: ((Rev (Lower_Seq (C,n))) /. (k + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 A26: (First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))) <= len (Rev (Lower_Seq (C,n))) by A21, FINSEQ_3:25; then A27: k + 1 <= len (Rev (Lower_Seq (C,n))) by A25, XXREAL_0:2; Lower_Seq (C,n) is_sequence_on Gauge (C,n) by Th5; then Rev (Lower_Seq (C,n)) is_sequence_on Gauge (C,n) by JORDAN9:5; then consider i1, j1, i2, j2 being Element of NAT such that A28: [i1,j1] in Indices (Gauge (C,n)) and A29: (Rev (Lower_Seq (C,n))) /. kk = (Gauge (C,n)) * (i1,j1) and A30: [i2,j2] in Indices (Gauge (C,n)) and A31: (Rev (Lower_Seq (C,n))) /. (kk + 1) = (Gauge (C,n)) * (i2,j2) and A32: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A23, A27, JORDAN8:3; A33: 1 <= i1 by A28, MATRIX_1:38; A34: ( 1 <= j1 & j1 <= width (Gauge (C,n)) ) by A28, MATRIX_1:38; A35: i2 <= len (Gauge (C,n)) by A30, MATRIX_1:38; A36: i1 <= len (Gauge (C,n)) by A28, MATRIX_1:38; A37: j2 <= width (Gauge (C,n)) by A30, MATRIX_1:38; A38: ( 1 <= i2 & 1 <= j2 ) by A30, MATRIX_1:38; A39: ( Center (Gauge (C,n)) <= len (Gauge (C,n)) & i1 + 1 >= 1 ) by JORDAN1B:13, NAT_1:11; now__::_thesis:_((Rev_(Lower_Seq_(C,n)))_/._(k_+_1))_`1_<_((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2 percases ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A32; suppose ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: ((Rev (Lower_Seq (C,n))) /. (k + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 then ((Rev (Lower_Seq (C,n))) /. k) `1 = ((Gauge (C,n)) * (i2,1)) `1 by A29, A33, A36, A34, GOBOARD5:2 .= ((Rev (Lower_Seq (C,n))) /. (k + 1)) `1 by A31, A35, A38, A37, GOBOARD5:2 ; hence ((Rev (Lower_Seq (C,n))) /. (k + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A13, A25, NAT_1:13, NAT_1:14; ::_thesis: verum end; supposeA40: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: ((Rev (Lower_Seq (C,n))) /. (k + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 A41: now__::_thesis:_not_((Rev_(Lower_Seq_(C,n)))_/._(k_+_1))_`1_=_((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2 A42: now__::_thesis:_not_(Rev_(Lower_Seq_(C,n)))_/._1_in_Vertical_Line_(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2) assume (Rev (Lower_Seq (C,n))) /. 1 in Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) ; ::_thesis: contradiction then (W-min (L~ (Cage (C,n)))) `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A19, JORDAN6:31; hence contradiction by A6, EUCLID:52; ::_thesis: verum end; assume ((Rev (Lower_Seq (C,n))) /. (k + 1)) `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 ; ::_thesis: contradiction then (Rev (Lower_Seq (C,n))) /. (k + 1) in { p where p is Point of (TOP-REAL 2) : p `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 } ; then A43: (Rev (Lower_Seq (C,n))) /. (k + 1) in Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) by JORDAN6:def_6; A44: ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 <= (E-max (L~ (Cage (C,n)))) `1 by A7, EUCLID:52; ( L~ (Rev (Lower_Seq (C,n))) is_an_arc_of W-min (L~ (Cage (C,n))), E-max (L~ (Cage (C,n))) & (W-min (L~ (Cage (C,n)))) `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 ) by A6, A19, A12, EUCLID:52, TOPREAL1:25; then A45: L~ (Rev (Lower_Seq (C,n))) meets Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) by A44, JORDAN6:49; A46: (Rev (Lower_Seq (C,n))) /. ((First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n)))) = First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A1, A4, A20, Th48, FINSEQ_5:38; A47: k + 1 >= 1 + 1 by A23, XREAL_1:7; len (mid ((Rev (Lower_Seq (C,n))),1,((First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n)))))) = (((First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n)))) -' 1) + 1 by A22, A26, JORDAN4:8 .= (First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))) by A22, XREAL_1:235 ; then A48: rng (mid ((Rev (Lower_Seq (C,n))),1,((First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n)))))) c= L~ (mid ((Rev (Lower_Seq (C,n))),1,((First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n)))))) by A25, A47, SPPOL_2:18, XXREAL_0:2; A49: k + 1 in dom (Rev (Lower_Seq (C,n))) by A24, A27, FINSEQ_3:25; ( Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) is closed & Rev (Lower_Seq (C,n)) is being_S-Seq ) by JORDAN6:30; then A50: (L~ (mid ((Rev (Lower_Seq (C,n))),1,((First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))))))) /\ (Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)) = {(First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))} by A1, A4, A20, A19, A12, A45, A42, Th48, Th50; A51: ( mid ((Rev (Lower_Seq (C,n))),1,((First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))))) = (Rev (Lower_Seq (C,n))) | ((First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n)))) & (Rev (Lower_Seq (C,n))) | (Seg ((First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))))) = (Rev (Lower_Seq (C,n))) | ((First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n)))) ) by A22, FINSEQ_1:def_15, FINSEQ_6:116; k + 1 in Seg ((First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n)))) by A24, A25, FINSEQ_1:1; then (Rev (Lower_Seq (C,n))) /. (k + 1) in rng (mid ((Rev (Lower_Seq (C,n))),1,((First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n)))))) by A51, A49, PARTFUN2:18; then (Rev (Lower_Seq (C,n))) /. (k + 1) in {(First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))} by A43, A48, A50, XBOOLE_0:def_4; then (Rev (Lower_Seq (C,n))) /. (k + 1) = First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by TARSKI:def_1; hence contradiction by A25, A21, A49, A46, PARTFUN2:10; ::_thesis: verum end; i1 < Center (Gauge (C,n)) by A13, A25, A29, A36, A34, A18, A14, A15, JORDAN1A:18, NAT_1:13, NAT_1:14; then i1 + 1 <= Center (Gauge (C,n)) by NAT_1:13; then ((Rev (Lower_Seq (C,n))) /. (k + 1)) `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A31, A34, A14, A15, A39, A40, JORDAN1A:18; hence ((Rev (Lower_Seq (C,n))) /. (k + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A41, XXREAL_0:1; ::_thesis: verum end; suppose ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: ((Rev (Lower_Seq (C,n))) /. (k + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 then i2 < i1 by NAT_1:13; then ((Rev (Lower_Seq (C,n))) /. (k + 1)) `1 <= ((Rev (Lower_Seq (C,n))) /. k) `1 by A29, A31, A36, A34, A38, A37, JORDAN1A:18; hence ((Rev (Lower_Seq (C,n))) /. (k + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A13, A25, NAT_1:13, NAT_1:14, XXREAL_0:2; ::_thesis: verum end; suppose ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: ((Rev (Lower_Seq (C,n))) /. (k + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 then ((Rev (Lower_Seq (C,n))) /. k) `1 = ((Gauge (C,n)) * (i2,1)) `1 by A29, A33, A36, A34, GOBOARD5:2 .= ((Rev (Lower_Seq (C,n))) /. (k + 1)) `1 by A31, A35, A38, A37, GOBOARD5:2 ; hence ((Rev (Lower_Seq (C,n))) /. (k + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A13, A25, NAT_1:13, NAT_1:14; ::_thesis: verum end; end; end; hence ((Rev (Lower_Seq (C,n))) /. (k + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 ; ::_thesis: verum end; A52: S1[1] proof assume that 1 <= 1 and 1 < (First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))) ; ::_thesis: ((Rev (Lower_Seq (C,n))) /. 1) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 (Rev (Lower_Seq (C,n))) /. 1 = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by FINSEQ_5:65 .= W-min (L~ (Cage (C,n))) by JORDAN1F:8 ; hence ((Rev (Lower_Seq (C,n))) /. 1) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A6, EUCLID:52; ::_thesis: verum end; A53: for k being non empty Nat holds S1[k] from NAT_1:sch_10(A52, A8); let k be Nat; ::_thesis: ( 1 <= k & k < (First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))) implies ((Rev (Lower_Seq (C,n))) /. k) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 ) assume ( 1 <= k & k < (First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))) ) ; ::_thesis: ((Rev (Lower_Seq (C,n))) /. k) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 hence ((Rev (Lower_Seq (C,n))) /. k) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A53; ::_thesis: verum end; theorem Th53: :: JORDAN1G:53 for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) for n being Element of NAT st n > 0 holds for q being Point of (TOP-REAL 2) st q in rng (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) holds q `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 proof let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for n being Element of NAT st n > 0 holds for q being Point of (TOP-REAL 2) st q in rng (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) holds q `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 let n be Element of NAT ; ::_thesis: ( n > 0 implies for q being Point of (TOP-REAL 2) st q in rng (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) holds q `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 ) set Wmin = W-min (L~ (Cage (C,n))); set Emax = E-max (L~ (Cage (C,n))); set Wbo = W-bound (L~ (Cage (C,n))); set Ebo = E-bound (L~ (Cage (C,n))); set sr = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2; set US = Upper_Seq (C,n); set FiP = First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))); A1: (Upper_Seq (C,n)) /. 1 = W-min (L~ (Cage (C,n))) by JORDAN1F:5; (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) = E-max (L~ (Cage (C,n))) by JORDAN1F:7; then A2: L~ (Upper_Seq (C,n)) is_an_arc_of W-min (L~ (Cage (C,n))), E-max (L~ (Cage (C,n))) by A1, TOPREAL1:25; assume A3: n > 0 ; ::_thesis: for q being Point of (TOP-REAL 2) st q in rng (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) holds q `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 then A4: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Upper_Seq (C,n)) by Th47; then A5: (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by FINSEQ_4:20; then A6: (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) <= len (Upper_Seq (C,n)) by FINSEQ_3:25; A7: W-bound (L~ (Cage (C,n))) < E-bound (L~ (Cage (C,n))) by SPRECT_1:31; then A8: W-bound (L~ (Cage (C,n))) < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by XREAL_1:226; ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 < E-bound (L~ (Cage (C,n))) by A7, XREAL_1:226; then A9: ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 <= (E-max (L~ (Cage (C,n)))) `1 by EUCLID:52; (W-min (L~ (Cage (C,n)))) `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A8, EUCLID:52; then ( L~ (Upper_Seq (C,n)) meets Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) & (L~ (Upper_Seq (C,n))) /\ (Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)) is closed ) by A2, A9, JORDAN6:49; then First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in (L~ (Upper_Seq (C,n))) /\ (Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)) by A2, JORDAN5C:def_1; then First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) by XBOOLE_0:def_4; then A10: (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by JORDAN6:31; A11: W-min (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) by A1, FINSEQ_6:42; A12: now__::_thesis:_not_(First_Point_((L~_(Upper_Seq_(C,n))),(W-min_(L~_(Cage_(C,n)))),(E-max_(L~_(Cage_(C,n)))),(Vertical_Line_(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2))))_.._(Upper_Seq_(C,n))_=_1 assume (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) = 1 ; ::_thesis: contradiction then (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) = ((Upper_Seq (C,n)) /. 1) .. (Upper_Seq (C,n)) by FINSEQ_6:43 .= (W-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) by JORDAN1F:5 ; then First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) = W-min (L~ (Cage (C,n))) by A4, A11, FINSEQ_5:9; hence contradiction by A8, A10, EUCLID:52; ::_thesis: verum end; 1 <= (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) by A5, FINSEQ_3:25; then (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) > 1 by A12, XXREAL_0:1; then A13: (1 + 1) + 0 <= (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) by NAT_1:13; then ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) - 2 >= 0 by XREAL_1:19; then ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) -' 2 = ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) - 2 by XREAL_0:def_2; then A14: len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) = (((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) - 2) + 1 by A6, A13, JORDAN4:8; let q be Point of (TOP-REAL 2); ::_thesis: ( q in rng (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) implies q `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 ) assume q in rng (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) ; ::_thesis: q `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 then consider k being Element of NAT such that A15: k in dom (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) and A16: q = (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. k by PARTFUN2:2; k + 2 >= 1 + 1 by NAT_1:11; then A17: (k + 2) - 1 >= (1 + 1) - 1 by XREAL_1:9; len (Upper_Seq (C,n)) >= 3 by JORDAN1E:15; then len (Upper_Seq (C,n)) >= 2 by XXREAL_0:2; then 2 in dom (Upper_Seq (C,n)) by FINSEQ_3:25; then A18: (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. k = (Upper_Seq (C,n)) /. ((k + 2) -' 1) by A15, A5, A13, SPRECT_2:3 .= (Upper_Seq (C,n)) /. (k + (2 - 1)) by A17, XREAL_0:def_2 ; k <= len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) by A15, FINSEQ_3:25; then k < ((((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) - 2) + 1) + 1 by A14, NAT_1:13; then A19: k + 1 <= (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) by NAT_1:13; percases ( k + 1 < (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) or k + 1 = (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) ) by A19, XXREAL_0:1; suppose k + 1 < (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) ; ::_thesis: q `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 hence q `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A3, A16, A18, Th51, NAT_1:11; ::_thesis: verum end; suppose k + 1 = (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) ; ::_thesis: q `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 hence q `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A16, A4, A10, A18, FINSEQ_5:38; ::_thesis: verum end; end; end; theorem Th54: :: JORDAN1G:54 for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) for n being Element of NAT st n > 0 holds (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 > (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 proof let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for n being Element of NAT st n > 0 holds (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 > (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 let n be Element of NAT ; ::_thesis: ( n > 0 implies (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 > (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 ) set sr = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2; set Wmin = W-min (L~ (Cage (C,n))); set Emax = E-max (L~ (Cage (C,n))); set Nbo = N-bound (L~ (Cage (C,n))); set Ebo = E-bound (L~ (Cage (C,n))); set Sbo = S-bound (L~ (Cage (C,n))); set Wbo = W-bound (L~ (Cage (C,n))); set SW = SW-corner (L~ (Cage (C,n))); set FiP = First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))); set LaP = Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))); set g = (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) ^ <*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*>; set h = (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) ^ <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*>; A1: (Upper_Seq (C,n)) /. 1 = W-min (L~ (Cage (C,n))) by JORDAN1F:5; A2: W-bound (L~ (Cage (C,n))) <= E-bound (L~ (Cage (C,n))) by SPRECT_1:21; then W-bound (L~ (Cage (C,n))) <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by JORDAN6:1; then A3: (W-min (L~ (Cage (C,n)))) `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by EUCLID:52; ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 <= E-bound (L~ (Cage (C,n))) by A2, JORDAN6:1; then A4: ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 <= (E-max (L~ (Cage (C,n)))) `1 by EUCLID:52; set GCw = (Gauge (C,n)) * ((Center (Gauge (C,n))),(width (Gauge (C,n)))); A5: 1 <= Center (Gauge (C,n)) by JORDAN1B:11; len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1; then A6: ((Gauge (C,n)) * ((Center (Gauge (C,n))),(width (Gauge (C,n))))) `2 = N-bound (L~ (Cage (C,n))) by A5, JORDAN1A:70, JORDAN1B:13; A7: (SW-corner (L~ (Cage (C,n)))) `2 <= (W-min (L~ (Cage (C,n)))) `2 by PSCOMP_1:30; A8: |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| `2 = N-bound (L~ (Cage (C,n))) by EUCLID:52; set RevL = (Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))); A9: ( <*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> is one-to-one & <*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> is special ) by FINSEQ_3:93; A10: rng ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) c= rng (Rev (Lower_Seq (C,n))) by FINSEQ_5:48; A11: ( (Lower_Seq (C,n)) /. 1 = E-max (L~ (Cage (C,n))) & (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) = W-min (L~ (Cage (C,n))) ) by JORDAN1F:6, JORDAN1F:8; then A12: L~ (Lower_Seq (C,n)) is_an_arc_of E-max (L~ (Cage (C,n))), W-min (L~ (Cage (C,n))) by TOPREAL1:25; A13: 4 <= len (Gauge (C,n)) by JORDAN8:10; then A14: len (Gauge (C,n)) >= 3 by XXREAL_0:2; A15: W-bound (L~ (Cage (C,n))) < E-bound (L~ (Cage (C,n))) by SPRECT_1:31; then A16: W-bound (L~ (Cage (C,n))) < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by XREAL_1:226; L~ (Lower_Seq (C,n)) is_an_arc_of W-min (L~ (Cage (C,n))), E-max (L~ (Cage (C,n))) by A11, JORDAN5B:14, TOPREAL1:25; then A17: ( L~ (Lower_Seq (C,n)) meets Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) & (L~ (Lower_Seq (C,n))) /\ (Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)) is closed ) by A3, A4, JORDAN6:49; then A18: Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in (L~ (Lower_Seq (C,n))) /\ (Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)) by A12, JORDAN5C:def_2; then A19: Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in L~ (Lower_Seq (C,n)) by XBOOLE_0:def_4; then Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n))) by XBOOLE_0:def_3; then A20: Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in L~ (Cage (C,n)) by JORDAN1E:13; assume A21: n > 0 ; ::_thesis: (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 > (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 then A22: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Upper_Seq (C,n)) by Th47; then A23: (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by FINSEQ_4:20; then A24: 1 <= (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) by FINSEQ_3:25; 1 <= len (Gauge (C,n)) by A13, XXREAL_0:2; then 1 <= width (Gauge (C,n)) by JORDAN8:def_1; then ((Gauge (C,n)) * ((Center (Gauge (C,n))),(width (Gauge (C,n))))) `1 = ((W-bound C) + (E-bound C)) / 2 by A21, Th35 .= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by Th33 ; then (Gauge (C,n)) * ((Center (Gauge (C,n))),(width (Gauge (C,n)))) = |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| by A6, EUCLID:53; then not |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| in rng (Lower_Seq (C,n)) by A5, A14, Th43, JORDAN1B:15; then not |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| in rng (Rev (Lower_Seq (C,n))) by FINSEQ_5:57; then A25: not |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| in rng ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) by A10; (SW-corner (L~ (Cage (C,n)))) `2 = S-bound (L~ (Cage (C,n))) by EUCLID:52; then |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| <> SW-corner (L~ (Cage (C,n))) by A8, SPRECT_1:32; then not |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| in {(SW-corner (L~ (Cage (C,n))))} by TARSKI:def_1; then not |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| in rng <*(SW-corner (L~ (Cage (C,n))))*> by FINSEQ_1:38; then not |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| in (rng <*(SW-corner (L~ (Cage (C,n))))*>) \/ (rng ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) by A25, XBOOLE_0:def_3; then not |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| in rng (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) by FINSEQ_1:31; then rng (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) misses {|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|} by ZFMISC_1:50; then (rng (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))))) /\ {|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|} = {} by XBOOLE_0:def_7; then (rng (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))))) /\ (rng <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*>) = {} by FINSEQ_1:38; then A26: rng (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) misses rng <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> by XBOOLE_0:def_7; Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Lower_Seq (C,n)) by A21, Th48; then A27: Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Rev (Lower_Seq (C,n))) by FINSEQ_5:57; then A28: not (Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) is empty by FINSEQ_5:47; A29: len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) = (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))) by A27, FINSEQ_5:42; A30: (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) = E-max (L~ (Cage (C,n))) by JORDAN1F:7; then A31: L~ (Upper_Seq (C,n)) is_an_arc_of W-min (L~ (Cage (C,n))), E-max (L~ (Cage (C,n))) by A1, TOPREAL1:25; A32: ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 < E-bound (L~ (Cage (C,n))) by A15, XREAL_1:226; then A33: ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 <= (E-max (L~ (Cage (C,n)))) `1 by EUCLID:52; (W-min (L~ (Cage (C,n)))) `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A16, EUCLID:52; then ( L~ (Upper_Seq (C,n)) meets Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) & (L~ (Upper_Seq (C,n))) /\ (Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)) is closed ) by A31, A33, JORDAN6:49; then A34: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in (L~ (Upper_Seq (C,n))) /\ (Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)) by A31, JORDAN5C:def_1; then A35: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in L~ (Upper_Seq (C,n)) by XBOOLE_0:def_4; then First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n))) by XBOOLE_0:def_3; then A36: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in L~ (Cage (C,n)) by JORDAN1E:13; now__::_thesis:_for_m_being_Element_of_NAT_st_m_in_dom_<*|[(E-bound_(L~_(Cage_(C,n)))),((First_Point_((L~_(Upper_Seq_(C,n))),(W-min_(L~_(Cage_(C,n)))),(E-max_(L~_(Cage_(C,n)))),(Vertical_Line_(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2))))_`2)]|*>_holds_ (_W-bound_(L~_(Cage_(C,n)))_<=_(<*|[(E-bound_(L~_(Cage_(C,n)))),((First_Point_((L~_(Upper_Seq_(C,n))),(W-min_(L~_(Cage_(C,n)))),(E-max_(L~_(Cage_(C,n)))),(Vertical_Line_(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2))))_`2)]|*>_/._m)_`1_&_(<*|[(E-bound_(L~_(Cage_(C,n)))),((First_Point_((L~_(Upper_Seq_(C,n))),(W-min_(L~_(Cage_(C,n)))),(E-max_(L~_(Cage_(C,n)))),(Vertical_Line_(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2))))_`2)]|*>_/._m)_`1_<=_E-bound_(L~_(Cage_(C,n)))_&_S-bound_(L~_(Cage_(C,n)))_<=_(<*|[(E-bound_(L~_(Cage_(C,n)))),((First_Point_((L~_(Upper_Seq_(C,n))),(W-min_(L~_(Cage_(C,n)))),(E-max_(L~_(Cage_(C,n)))),(Vertical_Line_(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2))))_`2)]|*>_/._m)_`2_&_(<*|[(E-bound_(L~_(Cage_(C,n)))),((First_Point_((L~_(Upper_Seq_(C,n))),(W-min_(L~_(Cage_(C,n)))),(E-max_(L~_(Cage_(C,n)))),(Vertical_Line_(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2))))_`2)]|*>_/._m)_`2_<=_N-bound_(L~_(Cage_(C,n)))_) let m be Element of NAT ; ::_thesis: ( m in dom <*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> implies ( W-bound (L~ (Cage (C,n))) <= (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `1 & (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `2 & (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) ) assume m in dom <*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> ; ::_thesis: ( W-bound (L~ (Cage (C,n))) <= (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `1 & (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `2 & (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) then m in Seg 1 by FINSEQ_1:38; then m = 1 by FINSEQ_1:2, TARSKI:def_1; then A37: <*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m = |[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]| by FINSEQ_4:16; then (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52; hence ( W-bound (L~ (Cage (C,n))) <= (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `1 & (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `1 <= E-bound (L~ (Cage (C,n))) ) by SPRECT_1:21; ::_thesis: ( S-bound (L~ (Cage (C,n))) <= (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `2 & (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `2 = (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 by A37, EUCLID:52; hence ( S-bound (L~ (Cage (C,n))) <= (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `2 & (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) by A36, PSCOMP_1:24; ::_thesis: verum end; then A38: <*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> is_in_the_area_of Cage (C,n) by SPRECT_2:def_1; A39: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) by A34, XBOOLE_0:def_4; then A40: (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by JORDAN6:31; now__::_thesis:_not_(rng_(mid_((Upper_Seq_(C,n)),2,((First_Point_((L~_(Upper_Seq_(C,n))),(W-min_(L~_(Cage_(C,n)))),(E-max_(L~_(Cage_(C,n)))),(Vertical_Line_(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2))))_.._(Upper_Seq_(C,n))))))_/\_{|[(E-bound_(L~_(Cage_(C,n)))),((First_Point_((L~_(Upper_Seq_(C,n))),(W-min_(L~_(Cage_(C,n)))),(E-max_(L~_(Cage_(C,n)))),(Vertical_Line_(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2))))_`2)]|}_<>_{} assume (rng (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))))) /\ {|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|} <> {} ; ::_thesis: contradiction then consider x being set such that A41: x in (rng (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))))) /\ {|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|} by XBOOLE_0:def_1; ( x in rng (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) & x in {|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|} ) by A41, XBOOLE_0:def_4; then |[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]| in rng (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) by TARSKI:def_1; then |[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]| `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A21, Th53; hence contradiction by A32, EUCLID:52; ::_thesis: verum end; then rng (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) misses {|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|} by XBOOLE_0:def_7; then A42: rng (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) misses rng <*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> by FINSEQ_1:38; A43: (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) <= len (Upper_Seq (C,n)) by A23, FINSEQ_3:25; Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) by A18, XBOOLE_0:def_4; then A44: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by JORDAN6:31; A45: now__::_thesis:_not_(First_Point_((L~_(Upper_Seq_(C,n))),(W-min_(L~_(Cage_(C,n)))),(E-max_(L~_(Cage_(C,n)))),(Vertical_Line_(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2))))_`2_=_(Last_Point_((L~_(Lower_Seq_(C,n))),(E-max_(L~_(Cage_(C,n)))),(W-min_(L~_(Cage_(C,n)))),(Vertical_Line_(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2))))_`2 assume (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 = (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 ; ::_thesis: contradiction then First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) = Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A40, A44, TOPREAL3:6; then First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A35, A19, XBOOLE_0:def_4; then First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16; then ( First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) = W-min (L~ (Cage (C,n))) or First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) = E-max (L~ (Cage (C,n))) ) by TARSKI:def_2; hence contradiction by A16, A32, A40, EUCLID:52; ::_thesis: verum end; len (Upper_Seq (C,n)) >= 3 by JORDAN1E:15; then A46: len (Upper_Seq (C,n)) > 2 by XXREAL_0:2; then A47: 2 in dom (Upper_Seq (C,n)) by FINSEQ_3:25; then A48: ((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))))) `2 = ((Upper_Seq (C,n)) /. ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))) `2 by A23, SPRECT_2:9 .= (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 by A22, FINSEQ_5:38 .= |[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]| `2 by EUCLID:52 .= (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. 1) `2 by FINSEQ_4:16 ; 2 <> (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) proof assume 2 = (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) ; ::_thesis: contradiction then (Upper_Seq (C,n)) /. 2 = First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A22, FINSEQ_5:38; then (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 = W-bound (L~ (Cage (C,n))) by Th31; then W-bound (L~ (Cage (C,n))) = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A39, JORDAN6:31; hence contradiction by SPRECT_1:31; ::_thesis: verum end; then mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))) is being_S-Seq by A46, A24, A43, JORDAN3:6; then reconsider g = (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) ^ <*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> as one-to-one special FinSequence of (TOP-REAL 2) by A42, A48, A9, FINSEQ_3:91, GOBOARD2:8; mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))) is_in_the_area_of Cage (C,n) by A47, A23, JORDAN1E:17, SPRECT_2:22; then A49: g is_in_the_area_of Cage (C,n) by A38, SPRECT_2:24; A50: (g /. (len g)) `1 = (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. (len <*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*>)) `1 by SPRECT_3:1 .= (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. 1) `1 by FINSEQ_1:39 .= |[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]| `1 by FINSEQ_4:16 .= E-bound (L~ (Cage (C,n))) by EUCLID:52 ; A51: 1 <= len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) by A47, A23, SPRECT_2:5; then 1 in dom (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) by FINSEQ_3:25; then (g /. 1) `1 = ((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. 1) `1 by FINSEQ_4:68 .= ((Upper_Seq (C,n)) /. 2) `1 by A47, A23, SPRECT_2:8 .= W-bound (L~ (Cage (C,n))) by Th31 ; then A52: g is_a_h.c._for Cage (C,n) by A49, A50, SPRECT_2:def_2; assume (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 <= (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 ; ::_thesis: contradiction then A53: (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 < (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 by A45, XXREAL_0:1; A54: rng (Lower_Seq (C,n)) c= rng (Cage (C,n)) by Th39; now__::_thesis:_contradiction percases ( SW-corner (L~ (Cage (C,n))) <> W-min (L~ (Cage (C,n))) or SW-corner (L~ (Cage (C,n))) = W-min (L~ (Cage (C,n))) ) ; supposeA55: SW-corner (L~ (Cage (C,n))) <> W-min (L~ (Cage (C,n))) ; ::_thesis: contradiction not SW-corner (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) proof (SW-corner (L~ (Cage (C,n)))) `1 = (W-min (L~ (Cage (C,n)))) `1 by PSCOMP_1:29; then A56: (SW-corner (L~ (Cage (C,n)))) `2 <> (W-min (L~ (Cage (C,n)))) `2 by A55, TOPREAL3:6; assume SW-corner (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) ; ::_thesis: contradiction then A57: SW-corner (L~ (Cage (C,n))) in rng (Cage (C,n)) by A54; len (Cage (C,n)) > 4 by GOBOARD7:34; then A58: rng (Cage (C,n)) c= L~ (Cage (C,n)) by SPPOL_2:18, XXREAL_0:2; (SW-corner (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52; then SW-corner (L~ (Cage (C,n))) in W-most (L~ (Cage (C,n))) by A57, A58, SPRECT_2:12; then (W-min (L~ (Cage (C,n)))) `2 <= (SW-corner (L~ (Cage (C,n)))) `2 by PSCOMP_1:31; hence contradiction by A7, A56, XXREAL_0:1; ::_thesis: verum end; then not SW-corner (L~ (Cage (C,n))) in rng (Rev (Lower_Seq (C,n))) by FINSEQ_5:57; then not SW-corner (L~ (Cage (C,n))) in rng ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) by A10; then {(SW-corner (L~ (Cage (C,n))))} misses rng ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) by ZFMISC_1:50; then {(SW-corner (L~ (Cage (C,n))))} /\ (rng ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) = {} by XBOOLE_0:def_7; then (rng <*(SW-corner (L~ (Cage (C,n))))*>) /\ (rng ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) = {} by FINSEQ_1:38; then A59: rng <*(SW-corner (L~ (Cage (C,n))))*> misses rng ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) by XBOOLE_0:def_7; <*(SW-corner (L~ (Cage (C,n))))*> is one-to-one by FINSEQ_3:93; then A60: <*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) is one-to-one by A59, FINSEQ_3:91; set FiP2 = First_Point ((L~ (Lower_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))); set midU = mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))); reconsider RevLS = Rev (Lower_Seq (C,n)) as special FinSequence of (TOP-REAL 2) ; (<*(SW-corner (L~ (Cage (C,n))))*> /. (len <*(SW-corner (L~ (Cage (C,n))))*>)) `1 = (<*(SW-corner (L~ (Cage (C,n))))*> /. 1) `1 by FINSEQ_1:39 .= (SW-corner (L~ (Cage (C,n)))) `1 by FINSEQ_4:16 .= W-bound (L~ (Cage (C,n))) by EUCLID:52 .= (W-min (L~ (Cage (C,n)))) `1 by EUCLID:52 .= ((Lower_Seq (C,n)) /. (len (Lower_Seq (C,n)))) `1 by JORDAN1F:8 .= ((Rev (Lower_Seq (C,n))) /. 1) `1 by FINSEQ_5:65 .= (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. 1) `1 by A27, FINSEQ_5:44 ; then A61: <*(SW-corner (L~ (Cage (C,n))))*> ^ (RevLS -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) is special by GOBOARD2:8; not (Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) is empty by A27, FINSEQ_5:47; then A62: ((<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) /. (len (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))))) `1 = (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))))) `1 by SPRECT_3:1 .= (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. ((Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))))) `1 by A27, FINSEQ_5:42 .= (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 by A27, FINSEQ_5:45 .= |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| `1 by A44, EUCLID:52 .= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. 1) `1 by FINSEQ_4:16 ; ( <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> is one-to-one & <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> is special ) by FINSEQ_3:93; then reconsider h = (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) ^ <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> as one-to-one special FinSequence of (TOP-REAL 2) by A26, A60, A62, A61, FINSEQ_3:91, GOBOARD2:8; A63: |[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]| `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52; now__::_thesis:_for_m_being_Element_of_NAT_st_m_in_dom_<*(SW-corner_(L~_(Cage_(C,n))))*>_holds_ (_W-bound_(L~_(Cage_(C,n)))_<=_(<*(SW-corner_(L~_(Cage_(C,n))))*>_/._m)_`1_&_(<*(SW-corner_(L~_(Cage_(C,n))))*>_/._m)_`1_<=_E-bound_(L~_(Cage_(C,n)))_&_S-bound_(L~_(Cage_(C,n)))_<=_(<*(SW-corner_(L~_(Cage_(C,n))))*>_/._m)_`2_&_(<*(SW-corner_(L~_(Cage_(C,n))))*>_/._m)_`2_<=_N-bound_(L~_(Cage_(C,n)))_) let m be Element of NAT ; ::_thesis: ( m in dom <*(SW-corner (L~ (Cage (C,n))))*> implies ( W-bound (L~ (Cage (C,n))) <= (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `1 & (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `2 & (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) ) assume m in dom <*(SW-corner (L~ (Cage (C,n))))*> ; ::_thesis: ( W-bound (L~ (Cage (C,n))) <= (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `1 & (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `2 & (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) then m in Seg 1 by FINSEQ_1:38; then m = 1 by FINSEQ_1:2, TARSKI:def_1; then A64: <*(SW-corner (L~ (Cage (C,n))))*> /. m = SW-corner (L~ (Cage (C,n))) by FINSEQ_4:16; then (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52; hence ( W-bound (L~ (Cage (C,n))) <= (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `1 & (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `1 <= E-bound (L~ (Cage (C,n))) ) by SPRECT_1:21; ::_thesis: ( S-bound (L~ (Cage (C,n))) <= (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `2 & (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `2 = S-bound (L~ (Cage (C,n))) by A64, EUCLID:52; hence ( S-bound (L~ (Cage (C,n))) <= (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `2 & (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) by SPRECT_1:22; ::_thesis: verum end; then A65: <*(SW-corner (L~ (Cage (C,n))))*> is_in_the_area_of Cage (C,n) by SPRECT_2:def_1; A66: ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) = ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. ((Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n)))) by A27, FINSEQ_5:42 .= Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A27, FINSEQ_5:45 ; now__::_thesis:_for_m_being_Element_of_NAT_st_m_in_dom_<*|[(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2),(N-bound_(L~_(Cage_(C,n))))]|*>_holds_ (_W-bound_(L~_(Cage_(C,n)))_<=_(<*|[(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2),(N-bound_(L~_(Cage_(C,n))))]|*>_/._m)_`1_&_(<*|[(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2),(N-bound_(L~_(Cage_(C,n))))]|*>_/._m)_`1_<=_E-bound_(L~_(Cage_(C,n)))_&_S-bound_(L~_(Cage_(C,n)))_<=_(<*|[(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2),(N-bound_(L~_(Cage_(C,n))))]|*>_/._m)_`2_&_(<*|[(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2),(N-bound_(L~_(Cage_(C,n))))]|*>_/._m)_`2_<=_N-bound_(L~_(Cage_(C,n)))_) let m be Element of NAT ; ::_thesis: ( m in dom <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> implies ( W-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) ) A67: W-bound (L~ (Cage (C,n))) <= E-bound (L~ (Cage (C,n))) by SPRECT_1:21; assume m in dom <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> ; ::_thesis: ( W-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) then m in Seg 1 by FINSEQ_1:38; then m = 1 by FINSEQ_1:2, TARSKI:def_1; then A68: <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m = |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| by FINSEQ_4:16; then (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by EUCLID:52; hence ( W-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 <= E-bound (L~ (Cage (C,n))) ) by A67, JORDAN6:1; ::_thesis: ( S-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 = N-bound (L~ (Cage (C,n))) by A68, EUCLID:52; hence ( S-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) by SPRECT_1:22; ::_thesis: verum end; then A69: <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> is_in_the_area_of Cage (C,n) by SPRECT_2:def_1; A70: ( L~ (Rev (Lower_Seq (C,n))) = L~ (Lower_Seq (C,n)) & First_Point ((L~ (Lower_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) = Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) ) by A12, A17, JORDAN5C:18, SPPOL_2:22; Rev (Lower_Seq (C,n)) is_in_the_area_of Cage (C,n) by JORDAN1E:18, SPRECT_3:51; then (Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) is_in_the_area_of Cage (C,n) by A27, JORDAN1E:1; then <*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) is_in_the_area_of Cage (C,n) by A65, SPRECT_2:24; then A71: h is_in_the_area_of Cage (C,n) by A69, SPRECT_2:24; len (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) = 1 + (len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) by FINSEQ_5:8; then A72: len (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) >= 1 by NAT_1:11; 1 in dom h by FINSEQ_5:6; then h /. 1 = h . 1 by PARTFUN1:def_6; then A73: (h /. 1) `2 = ((<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) /. 1) `2 by A72, FINSEQ_6:109 .= (SW-corner (L~ (Cage (C,n)))) `2 by FINSEQ_5:15 .= S-bound (L~ (Cage (C,n))) by EUCLID:52 ; A74: len h = (len (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))))) + 1 by FINSEQ_2:16; then A75: 1 + 1 <= len h by A72, XREAL_1:7; L~ (Cage (C,n)) = (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n))) by JORDAN1E:13; then A76: L~ (Upper_Seq (C,n)) c= L~ (Cage (C,n)) by XBOOLE_1:7; A77: (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))))) = (Upper_Seq (C,n)) /. ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) by A47, A23, SPRECT_2:9 .= First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A22, FINSEQ_5:38 ; A78: W-min (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) by A1, FINSEQ_6:42; now__::_thesis:_not_(First_Point_((L~_(Upper_Seq_(C,n))),(W-min_(L~_(Cage_(C,n)))),(E-max_(L~_(Cage_(C,n)))),(Vertical_Line_(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2))))_.._(Upper_Seq_(C,n))_=_1 assume (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) = 1 ; ::_thesis: contradiction then (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) = ((Upper_Seq (C,n)) /. 1) .. (Upper_Seq (C,n)) by FINSEQ_6:43 .= (W-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) by JORDAN1F:5 ; then First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) = W-min (L~ (Cage (C,n))) by A22, A78, FINSEQ_5:9; hence contradiction by A16, A40, EUCLID:52; ::_thesis: verum end; then (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) > 1 by A24, XXREAL_0:1; then A79: (1 + 1) + 0 <= (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) by NAT_1:13; then ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) - 2 >= 0 by XREAL_1:19; then ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) -' 2 = ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) - 2 by XREAL_0:def_2; then A80: len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) = (((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) - 2) + 1 by A43, A79, JORDAN4:8 .= ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) - (2 - 1) ; 1 in dom ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) by A28, FINSEQ_5:6; then A81: (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ^ <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*>) /. 1 = ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. 1 by FINSEQ_4:68 .= (Rev (Lower_Seq (C,n))) /. 1 by A27, FINSEQ_5:44 .= (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by FINSEQ_5:65 .= W-min (L~ (Cage (C,n))) by JORDAN1F:8 ; A82: (SW-corner (L~ (Cage (C,n)))) `2 <= (W-min (L~ (Cage (C,n)))) `2 by PSCOMP_1:30; len g = (len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))))) + 1 by FINSEQ_2:16; then A83: 1 + 1 <= len g by A51, XREAL_1:7; A84: L~ g = (L~ (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))))) \/ (LSeg (((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))))),|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|)) by A47, A23, SPPOL_2:19, SPRECT_2:7; L~ (Rev (Lower_Seq (C,n))) = (L~ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) \/ (L~ ((Rev (Lower_Seq (C,n))) :- (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) by A27, SPPOL_2:24; then L~ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) c= L~ (Rev (Lower_Seq (C,n))) by XBOOLE_1:7; then A85: L~ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) c= L~ (Lower_Seq (C,n)) by SPPOL_2:22; A86: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 <= N-bound (L~ (Cage (C,n))) by A20, PSCOMP_1:24; A87: |[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]| `2 = (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 by EUCLID:52; then A88: LSeg ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|) is horizontal by SPPOL_1:15; (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 = |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| `1 by A44, EUCLID:52; then A89: LSeg ((Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|) is vertical by SPPOL_1:16; A90: L~ (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) c= L~ (Upper_Seq (C,n)) by A47, A23, SPRECT_3:18; (h /. (len h)) `2 = |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| `2 by A74, FINSEQ_4:67 .= N-bound (L~ (Cage (C,n))) by EUCLID:52 ; then h is_a_v.c._for Cage (C,n) by A71, A73, SPRECT_2:def_3; then L~ g meets L~ h by A52, A75, A83, SPRECT_2:29; then consider x being set such that A91: x in L~ g and A92: x in L~ h by XBOOLE_0:3; reconsider x = x as Point of (TOP-REAL 2) by A91; L~ h = L~ (<*(SW-corner (L~ (Cage (C,n))))*> ^ (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ^ <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*>)) by FINSEQ_1:32 .= (LSeg ((SW-corner (L~ (Cage (C,n)))),((((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ^ <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*>) /. 1))) \/ (L~ (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ^ <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*>)) by SPPOL_2:20 .= (LSeg ((SW-corner (L~ (Cage (C,n)))),((((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ^ <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*>) /. 1))) \/ ((L~ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) \/ (LSeg ((((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))))),|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|))) by A27, FINSEQ_5:47, SPPOL_2:19 ; then A93: ( x in LSeg ((SW-corner (L~ (Cage (C,n)))),((((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ^ <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*>) /. 1)) or x in (L~ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) \/ (LSeg ((((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))))),|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|)) ) by A92, XBOOLE_0:def_3; A94: (SW-corner (L~ (Cage (C,n)))) `1 = (W-min (L~ (Cage (C,n)))) `1 by PSCOMP_1:29; then A95: LSeg ((SW-corner (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))) is vertical by SPPOL_1:16; now__::_thesis:_contradiction percases ( x in LSeg ((SW-corner (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))) or x in L~ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) or x in LSeg ((Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|) ) by A93, A81, A66, XBOOLE_0:def_3; supposeA96: x in LSeg ((SW-corner (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))) ; ::_thesis: contradiction then A97: x `2 <= (W-min (L~ (Cage (C,n)))) `2 by A82, TOPREAL1:4; A98: x `1 = (SW-corner (L~ (Cage (C,n)))) `1 by A95, A96, SPPOL_1:41; then A99: x `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52; now__::_thesis:_contradiction percases ( x in L~ (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) or x in LSeg ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|) ) by A91, A84, A77, XBOOLE_0:def_3; supposeA100: x in L~ (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) ; ::_thesis: contradiction then x in L~ (Upper_Seq (C,n)) by A90; then x in W-most (L~ (Cage (C,n))) by A76, A98, EUCLID:52, SPRECT_2:12; then x `2 >= (W-min (L~ (Cage (C,n)))) `2 by PSCOMP_1:31; then x `2 = (W-min (L~ (Cage (C,n)))) `2 by A97, XXREAL_0:1; then x = W-min (L~ (Cage (C,n))) by A94, A98, TOPREAL3:6; then (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) = 1 by A46, A1, A24, A43, A100, Th37; then W-min (L~ (Cage (C,n))) = First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A1, A22, FINSEQ_5:38; hence contradiction by A16, A40, EUCLID:52; ::_thesis: verum end; suppose x in LSeg ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|) ; ::_thesis: contradiction hence contradiction by A16, A32, A40, A63, A99, TOPREAL1:3; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; supposeA101: x in L~ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ; ::_thesis: contradiction now__::_thesis:_contradiction percases ( x in L~ (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) or x in LSeg ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|) ) by A91, A84, A77, XBOOLE_0:def_3; supposeA102: x in L~ (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) ; ::_thesis: contradiction then x in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A90, A85, A101, XBOOLE_0:def_4; then A103: x in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16; now__::_thesis:_contradiction percases ( x = W-min (L~ (Cage (C,n))) or x = E-max (L~ (Cage (C,n))) ) by A103, TARSKI:def_2; suppose x = W-min (L~ (Cage (C,n))) ; ::_thesis: contradiction then (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) = 1 by A46, A1, A24, A43, A102, Th37; then W-min (L~ (Cage (C,n))) = First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A1, A22, FINSEQ_5:38; hence contradiction by A16, A40, EUCLID:52; ::_thesis: verum end; suppose x = E-max (L~ (Cage (C,n))) ; ::_thesis: contradiction then (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) = len (Upper_Seq (C,n)) by A46, A30, A24, A43, A102, Th38; then E-max (L~ (Cage (C,n))) = First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A30, A22, FINSEQ_5:38; hence contradiction by A32, A40, EUCLID:52; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; supposeA104: x in LSeg ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|) ; ::_thesis: contradiction LSeg ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|) is horizontal by A87, SPPOL_1:15; then A105: x `2 = (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 by A104, SPPOL_1:40; consider i being Element of NAT such that A106: 1 <= i and A107: i + 1 <= len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) and A108: x in LSeg ((((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. i),(((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (i + 1))) by A101, SPPOL_2:14; A109: i < len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) by A107, NAT_1:13; then A110: ((Rev (Lower_Seq (C,n))) /. i) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A21, A29, A70, A106, Th52; i in Seg ((Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n)))) by A29, A106, A109, FINSEQ_1:1; then A111: ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. i = (Rev (Lower_Seq (C,n))) /. i by A27, FINSEQ_5:43; i + 1 >= 1 by NAT_1:11; then i + 1 in Seg ((Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n)))) by A29, A107, FINSEQ_1:1; then A112: ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (i + 1) = (Rev (Lower_Seq (C,n))) /. (i + 1) by A27, FINSEQ_5:43; A113: (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 <= x `1 by A32, A40, A63, A104, TOPREAL1:3; now__::_thesis:_contradiction percases ( i + 1 < len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) or i + 1 = len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ) by A107, XXREAL_0:1; supposeA114: i + 1 < len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ; ::_thesis: contradiction ( (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. i) `1 <= (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (i + 1)) `1 or (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (i + 1)) `1 <= (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. i) `1 ) ; then A115: ( x `1 <= (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (i + 1)) `1 or x `1 <= (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. i) `1 ) by A108, TOPREAL1:3; ((Rev (Lower_Seq (C,n))) /. (i + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A21, A29, A70, A114, Th52, NAT_1:11; hence contradiction by A40, A113, A111, A112, A110, A115, XXREAL_0:2; ::_thesis: verum end; supposeA116: i + 1 = len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ; ::_thesis: contradiction then i + 1 <= len (Rev (Lower_Seq (C,n))) by A27, A29, FINSEQ_4:21; then LSeg (((Rev (Lower_Seq (C,n))) /. i),((Rev (Lower_Seq (C,n))) /. (i + 1))) = LSeg ((Rev (Lower_Seq (C,n))),i) by A106, TOPREAL1:def_3; then ( LSeg ((((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. i),(((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (i + 1))) is vertical or LSeg ((((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. i),(((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (i + 1))) is horizontal ) by A111, A112, SPPOL_1:19; hence contradiction by A44, A45, A66, A105, A108, A111, A110, A116, SPPOL_1:16, SPPOL_1:40; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; supposeA117: x in LSeg ((Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|) ; ::_thesis: contradiction then A118: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 <= x `2 by A8, A86, TOPREAL1:4; A119: x `1 = (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 by A89, A117, SPPOL_1:41; now__::_thesis:_contradiction percases ( x in L~ (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) or x in LSeg ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|) ) by A91, A84, A77, XBOOLE_0:def_3; suppose x in L~ (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) ; ::_thesis: contradiction then consider i being Element of NAT such that A120: 1 <= i and A121: i + 1 <= len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) and A122: x in LSeg (((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. i),((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (i + 1))) by SPPOL_2:14; i + 2 >= 1 + 1 by NAT_1:11; then A123: (i + 2) - 1 >= (1 + 1) - 1 by XREAL_1:9; i < len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) by A121, NAT_1:13; then i in dom (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) by A120, FINSEQ_3:25; then A124: (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. i = (Upper_Seq (C,n)) /. ((i + 2) -' 1) by A47, A23, A79, SPRECT_2:3 .= (Upper_Seq (C,n)) /. (i + (2 - 1)) by A123, XREAL_0:def_2 ; (i + 1) + 2 >= 1 + 1 by NAT_1:11; then A125: ((i + 1) + 2) - 1 >= (1 + 1) - 1 by XREAL_1:9; A126: 1 <= i + 1 by NAT_1:11; then i + 1 in dom (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) by A121, FINSEQ_3:25; then A127: (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (i + 1) = (Upper_Seq (C,n)) /. (((i + 1) + 2) -' 1) by A47, A23, A79, SPRECT_2:3 .= (Upper_Seq (C,n)) /. ((i + 1) + (2 - 1)) by A125, XREAL_0:def_2 ; A128: (i + 1) + 1 <= (((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) - 1) + 1 by A80, A121, XREAL_1:7; then i + 1 < (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) by NAT_1:13; then A129: ((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. i) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A21, A124, Th51, NAT_1:11; (i + 1) + 1 <= len (Upper_Seq (C,n)) by A43, A128, XXREAL_0:2; then LSeg (((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. i),((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (i + 1))) = LSeg ((Upper_Seq (C,n)),(i + 1)) by A124, A126, A127, TOPREAL1:def_3; then A130: ( LSeg (((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. i),((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (i + 1))) is vertical or LSeg (((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. i),((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (i + 1))) is horizontal ) by SPPOL_1:19; now__::_thesis:_contradiction percases ( i + 1 < len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) or i + 1 = len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) ) by A121, XXREAL_0:1; suppose i + 1 < len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) ; ::_thesis: contradiction then (i + 1) + 1 <= len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) by NAT_1:13; then ((i + 1) + 1) + 1 <= (((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) - 1) + 1 by A80, XREAL_1:7; then (i + 1) + 1 < (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) by NAT_1:13; then A131: ((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (i + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A21, A127, Th51, NAT_1:11; ( ((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. i) `1 <= ((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (i + 1)) `1 or ((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (i + 1)) `1 <= ((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. i) `1 ) ; hence contradiction by A44, A119, A122, A129, A131, TOPREAL1:3; ::_thesis: verum end; supposeA132: i + 1 = len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) ; ::_thesis: contradiction then ((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. i) `2 = ((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (i + 1)) `2 by A40, A77, A129, A130, SPPOL_1:15, SPPOL_1:16; hence contradiction by A53, A77, A118, A122, A132, GOBOARD7:6; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; suppose x in LSeg ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|) ; ::_thesis: contradiction hence contradiction by A53, A88, A118, SPPOL_1:40; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; supposeA133: SW-corner (L~ (Cage (C,n))) = W-min (L~ (Cage (C,n))) ; ::_thesis: contradiction reconsider RevLS = Rev (Lower_Seq (C,n)) as special FinSequence of (TOP-REAL 2) ; set h = ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ^ <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*>; A134: ( <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> is one-to-one & RevLS -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) is special ) by FINSEQ_3:93; rng ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) misses {|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|} by A25, ZFMISC_1:50; then (rng ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) /\ {|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|} = {} by XBOOLE_0:def_7; then (rng ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) /\ (rng <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*>) = {} by FINSEQ_1:38; then A135: rng ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) misses rng <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> by XBOOLE_0:def_7; (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))))) `1 = (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. ((Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))))) `1 by A27, FINSEQ_5:42 .= (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 by A27, FINSEQ_5:45 .= |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| `1 by A44, EUCLID:52 .= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. 1) `1 by FINSEQ_4:16 ; then reconsider h = ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ^ <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> as one-to-one special FinSequence of (TOP-REAL 2) by A135, A134, FINSEQ_3:91, GOBOARD2:8; now__::_thesis:_for_m_being_Element_of_NAT_st_m_in_dom_<*|[(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2),(N-bound_(L~_(Cage_(C,n))))]|*>_holds_ (_W-bound_(L~_(Cage_(C,n)))_<=_(<*|[(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2),(N-bound_(L~_(Cage_(C,n))))]|*>_/._m)_`1_&_(<*|[(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2),(N-bound_(L~_(Cage_(C,n))))]|*>_/._m)_`1_<=_E-bound_(L~_(Cage_(C,n)))_&_S-bound_(L~_(Cage_(C,n)))_<=_(<*|[(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2),(N-bound_(L~_(Cage_(C,n))))]|*>_/._m)_`2_&_(<*|[(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2),(N-bound_(L~_(Cage_(C,n))))]|*>_/._m)_`2_<=_N-bound_(L~_(Cage_(C,n)))_) let m be Element of NAT ; ::_thesis: ( m in dom <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> implies ( W-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) ) A136: W-bound (L~ (Cage (C,n))) <= E-bound (L~ (Cage (C,n))) by SPRECT_1:21; assume m in dom <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> ; ::_thesis: ( W-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) then m in Seg 1 by FINSEQ_1:38; then m = 1 by FINSEQ_1:2, TARSKI:def_1; then A137: <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m = |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| by FINSEQ_4:16; then (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by EUCLID:52; hence ( W-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 <= E-bound (L~ (Cage (C,n))) ) by A136, JORDAN6:1; ::_thesis: ( S-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 = N-bound (L~ (Cage (C,n))) by A137, EUCLID:52; hence ( S-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) by SPRECT_1:22; ::_thesis: verum end; then A138: <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> is_in_the_area_of Cage (C,n) by SPRECT_2:def_1; Rev (Lower_Seq (C,n)) is_in_the_area_of Cage (C,n) by JORDAN1E:18, SPRECT_3:51; then (Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) is_in_the_area_of Cage (C,n) by A27, JORDAN1E:1; then A139: h is_in_the_area_of Cage (C,n) by A138, SPRECT_2:24; A140: len h = (len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) + 1 by FINSEQ_2:16; then A141: (h /. (len h)) `2 = |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| `2 by FINSEQ_4:67 .= N-bound (L~ (Cage (C,n))) by EUCLID:52 ; L~ (Rev (Lower_Seq (C,n))) = (L~ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) \/ (L~ ((Rev (Lower_Seq (C,n))) :- (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) by A27, SPPOL_2:24; then L~ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) c= L~ (Rev (Lower_Seq (C,n))) by XBOOLE_1:7; then A142: L~ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) c= L~ (Lower_Seq (C,n)) by SPPOL_2:22; A143: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 <= N-bound (L~ (Cage (C,n))) by A20, PSCOMP_1:24; (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))) >= 0 + 1 by A28, A29, NAT_1:13; then A144: len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) >= 1 by A27, FINSEQ_5:42; 1 in dom h by FINSEQ_5:6; then h /. 1 = h . 1 by PARTFUN1:def_6; then (h /. 1) `2 = (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. 1) `2 by A144, FINSEQ_6:109 .= ((Rev (Lower_Seq (C,n))) /. 1) `2 by A27, FINSEQ_5:44 .= ((Lower_Seq (C,n)) /. (len (Lower_Seq (C,n)))) `2 by FINSEQ_5:65 .= (W-min (L~ (Cage (C,n)))) `2 by JORDAN1F:8 .= S-bound (L~ (Cage (C,n))) by A133, EUCLID:52 ; then A145: h is_a_v.c._for Cage (C,n) by A139, A141, SPRECT_2:def_3; set FiP2 = First_Point ((L~ (Lower_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))); set midU = mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))); A146: |[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]| `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52; A147: L~ g = (L~ (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))))) \/ (LSeg (((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))))),|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|)) by A47, A23, SPPOL_2:19, SPRECT_2:7; A148: W-min (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) by A1, FINSEQ_6:42; now__::_thesis:_not_(First_Point_((L~_(Upper_Seq_(C,n))),(W-min_(L~_(Cage_(C,n)))),(E-max_(L~_(Cage_(C,n)))),(Vertical_Line_(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2))))_.._(Upper_Seq_(C,n))_=_1 assume (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) = 1 ; ::_thesis: contradiction then (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) = ((Upper_Seq (C,n)) /. 1) .. (Upper_Seq (C,n)) by FINSEQ_6:43 .= (W-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) by JORDAN1F:5 ; then First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) = W-min (L~ (Cage (C,n))) by A22, A148, FINSEQ_5:9; hence contradiction by A16, A40, EUCLID:52; ::_thesis: verum end; then (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) > 1 by A24, XXREAL_0:1; then A149: (1 + 1) + 0 <= (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) by NAT_1:13; then ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) - 2 >= 0 by XREAL_1:19; then ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) -' 2 = ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) - 2 by XREAL_0:def_2; then A150: len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) = (((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) - 2) + 1 by A43, A149, JORDAN4:8 .= ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) - (2 - 1) ; (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 = |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| `1 by A44, EUCLID:52; then A151: LSeg ((Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|) is vertical by SPPOL_1:16; len g = (len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))))) + 1 by FINSEQ_2:16; then A152: 1 + 1 <= len g by A51, XREAL_1:7; A153: ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) = ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. ((Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n)))) by A27, FINSEQ_5:42 .= Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A27, FINSEQ_5:45 ; 1 + 1 <= len h by A144, A140, XREAL_1:7; then L~ g meets L~ h by A52, A145, A152, SPRECT_2:29; then consider x being set such that A154: x in L~ g and A155: x in L~ h by XBOOLE_0:3; reconsider x = x as Point of (TOP-REAL 2) by A154; A156: L~ h = (L~ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) \/ (LSeg ((((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))))),|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|)) by A27, FINSEQ_5:47, SPPOL_2:19; A157: (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))))) = (Upper_Seq (C,n)) /. ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) by A47, A23, SPRECT_2:9 .= First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A22, FINSEQ_5:38 ; A158: L~ (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) c= L~ (Upper_Seq (C,n)) by A47, A23, SPRECT_3:18; A159: ( L~ (Rev (Lower_Seq (C,n))) = L~ (Lower_Seq (C,n)) & First_Point ((L~ (Lower_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) = Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) ) by A12, A17, JORDAN5C:18, SPPOL_2:22; A160: |[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]| `2 = (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 by EUCLID:52; then A161: LSeg ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|) is horizontal by SPPOL_1:15; now__::_thesis:_contradiction percases ( x in L~ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) or x in LSeg ((Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|) ) by A155, A156, A153, XBOOLE_0:def_3; supposeA162: x in L~ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ; ::_thesis: contradiction now__::_thesis:_contradiction percases ( x in L~ (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) or x in LSeg ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|) ) by A154, A147, A157, XBOOLE_0:def_3; supposeA163: x in L~ (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) ; ::_thesis: contradiction then x in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A158, A142, A162, XBOOLE_0:def_4; then A164: x in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16; now__::_thesis:_contradiction percases ( x = W-min (L~ (Cage (C,n))) or x = E-max (L~ (Cage (C,n))) ) by A164, TARSKI:def_2; suppose x = W-min (L~ (Cage (C,n))) ; ::_thesis: contradiction then (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) = 1 by A46, A1, A24, A43, A163, Th37; then W-min (L~ (Cage (C,n))) = First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A1, A22, FINSEQ_5:38; hence contradiction by A16, A40, EUCLID:52; ::_thesis: verum end; suppose x = E-max (L~ (Cage (C,n))) ; ::_thesis: contradiction then (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) = len (Upper_Seq (C,n)) by A46, A30, A24, A43, A163, Th38; then E-max (L~ (Cage (C,n))) = First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A30, A22, FINSEQ_5:38; hence contradiction by A32, A40, EUCLID:52; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; supposeA165: x in LSeg ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|) ; ::_thesis: contradiction LSeg ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|) is horizontal by A160, SPPOL_1:15; then A166: x `2 = (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 by A165, SPPOL_1:40; consider i being Element of NAT such that A167: 1 <= i and A168: i + 1 <= len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) and A169: x in LSeg ((((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. i),(((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (i + 1))) by A162, SPPOL_2:14; A170: i < len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) by A168, NAT_1:13; then A171: ((Rev (Lower_Seq (C,n))) /. i) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A21, A29, A159, A167, Th52; i in Seg ((Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n)))) by A29, A167, A170, FINSEQ_1:1; then A172: ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. i = (Rev (Lower_Seq (C,n))) /. i by A27, FINSEQ_5:43; i + 1 >= 1 by NAT_1:11; then i + 1 in Seg ((Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n)))) by A29, A168, FINSEQ_1:1; then A173: ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (i + 1) = (Rev (Lower_Seq (C,n))) /. (i + 1) by A27, FINSEQ_5:43; A174: (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 <= x `1 by A32, A40, A146, A165, TOPREAL1:3; now__::_thesis:_contradiction percases ( i + 1 < len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) or i + 1 = len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ) by A168, XXREAL_0:1; supposeA175: i + 1 < len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ; ::_thesis: contradiction ( (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. i) `1 <= (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (i + 1)) `1 or (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (i + 1)) `1 <= (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. i) `1 ) ; then A176: ( x `1 <= (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (i + 1)) `1 or x `1 <= (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. i) `1 ) by A169, TOPREAL1:3; ((Rev (Lower_Seq (C,n))) /. (i + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A21, A29, A159, A175, Th52, NAT_1:11; hence contradiction by A40, A174, A172, A173, A171, A176, XXREAL_0:2; ::_thesis: verum end; supposeA177: i + 1 = len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ; ::_thesis: contradiction then i + 1 <= len (Rev (Lower_Seq (C,n))) by A27, A29, FINSEQ_4:21; then LSeg (((Rev (Lower_Seq (C,n))) /. i),((Rev (Lower_Seq (C,n))) /. (i + 1))) = LSeg ((Rev (Lower_Seq (C,n))),i) by A167, TOPREAL1:def_3; then ( LSeg ((((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. i),(((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (i + 1))) is vertical or LSeg ((((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. i),(((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (i + 1))) is horizontal ) by A172, A173, SPPOL_1:19; hence contradiction by A44, A45, A153, A166, A169, A172, A171, A177, SPPOL_1:16, SPPOL_1:40; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; supposeA178: x in LSeg ((Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|) ; ::_thesis: contradiction then A179: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 <= x `2 by A8, A143, TOPREAL1:4; A180: x `1 = (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 by A151, A178, SPPOL_1:41; now__::_thesis:_contradiction percases ( x in L~ (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) or x in LSeg ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|) ) by A154, A147, A157, XBOOLE_0:def_3; suppose x in L~ (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) ; ::_thesis: contradiction then consider i being Element of NAT such that A181: 1 <= i and A182: i + 1 <= len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) and A183: x in LSeg (((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. i),((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (i + 1))) by SPPOL_2:14; i + 2 >= 1 + 1 by NAT_1:11; then A184: (i + 2) - 1 >= (1 + 1) - 1 by XREAL_1:9; i < len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) by A182, NAT_1:13; then i in dom (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) by A181, FINSEQ_3:25; then A185: (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. i = (Upper_Seq (C,n)) /. ((i + 2) -' 1) by A47, A23, A149, SPRECT_2:3 .= (Upper_Seq (C,n)) /. (i + (2 - 1)) by A184, XREAL_0:def_2 ; (i + 1) + 2 >= 1 + 1 by NAT_1:11; then A186: ((i + 1) + 2) - 1 >= (1 + 1) - 1 by XREAL_1:9; A187: 1 <= i + 1 by NAT_1:11; then i + 1 in dom (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) by A182, FINSEQ_3:25; then A188: (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (i + 1) = (Upper_Seq (C,n)) /. (((i + 1) + 2) -' 1) by A47, A23, A149, SPRECT_2:3 .= (Upper_Seq (C,n)) /. ((i + 1) + (2 - 1)) by A186, XREAL_0:def_2 ; A189: (i + 1) + 1 <= (((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) - 1) + 1 by A150, A182, XREAL_1:7; then i + 1 < (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) by NAT_1:13; then A190: ((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. i) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A21, A185, Th51, NAT_1:11; (i + 1) + 1 <= len (Upper_Seq (C,n)) by A43, A189, XXREAL_0:2; then LSeg (((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. i),((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (i + 1))) = LSeg ((Upper_Seq (C,n)),(i + 1)) by A185, A187, A188, TOPREAL1:def_3; then A191: ( LSeg (((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. i),((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (i + 1))) is vertical or LSeg (((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. i),((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (i + 1))) is horizontal ) by SPPOL_1:19; now__::_thesis:_contradiction percases ( i + 1 < len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) or i + 1 = len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) ) by A182, XXREAL_0:1; suppose i + 1 < len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) ; ::_thesis: contradiction then (i + 1) + 1 <= len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) by NAT_1:13; then ((i + 1) + 1) + 1 <= (((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) - 1) + 1 by A150, XREAL_1:7; then (i + 1) + 1 < (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) by NAT_1:13; then A192: ((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (i + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A21, A188, Th51, NAT_1:11; ( ((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. i) `1 <= ((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (i + 1)) `1 or ((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (i + 1)) `1 <= ((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. i) `1 ) ; hence contradiction by A44, A180, A183, A190, A192, TOPREAL1:3; ::_thesis: verum end; supposeA193: i + 1 = len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) ; ::_thesis: contradiction then ((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. i) `2 = ((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (i + 1)) `2 by A40, A157, A190, A191, SPPOL_1:15, SPPOL_1:16; hence contradiction by A53, A157, A179, A183, A193, GOBOARD7:6; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; suppose x in LSeg ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|) ; ::_thesis: contradiction hence contradiction by A53, A161, A179, SPPOL_1:40; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; theorem Th55: :: JORDAN1G:55 for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) for n being Element of NAT st n > 0 holds L~ (Upper_Seq (C,n)) = Upper_Arc (L~ (Cage (C,n))) proof let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for n being Element of NAT st n > 0 holds L~ (Upper_Seq (C,n)) = Upper_Arc (L~ (Cage (C,n))) let n be Element of NAT ; ::_thesis: ( n > 0 implies L~ (Upper_Seq (C,n)) = Upper_Arc (L~ (Cage (C,n))) ) A1: W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43; E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46; then A2: E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by FINSEQ_6:90, SPRECT_2:43; A3: Upper_Seq (C,n) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n)))) by JORDAN1E:def_1; then (Upper_Seq (C,n)) /. 1 = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 by A2, FINSEQ_5:44; then A4: (Upper_Seq (C,n)) /. 1 = W-min (L~ (Cage (C,n))) by A1, FINSEQ_6:92; (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) = ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n))))) /. ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) by A3, A2, FINSEQ_5:42 .= E-max (L~ (Cage (C,n))) by A2, FINSEQ_5:45 ; then A5: L~ (Upper_Seq (C,n)) is_an_arc_of W-min (L~ (Cage (C,n))), E-max (L~ (Cage (C,n))) by A4, TOPREAL1:25; assume n > 0 ; ::_thesis: L~ (Upper_Seq (C,n)) = Upper_Arc (L~ (Cage (C,n))) then A6: (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 > (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 by Th54; A7: (Lower_Seq (C,n)) /. 1 = ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) :- (E-max (L~ (Cage (C,n))))) /. 1 by JORDAN1E:def_2 .= E-max (L~ (Cage (C,n))) by FINSEQ_5:53 ; Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) :- (E-max (L~ (Cage (C,n)))) by JORDAN1E:def_2; then (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. (len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) by A2, FINSEQ_5:54 .= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 by FINSEQ_6:def_1 .= W-min (L~ (Cage (C,n))) by A1, FINSEQ_6:92 ; then A8: L~ (Lower_Seq (C,n)) is_an_arc_of E-max (L~ (Cage (C,n))), W-min (L~ (Cage (C,n))) by A7, TOPREAL1:25; ( (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) = {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} & (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n))) = L~ (Cage (C,n)) ) by JORDAN1E:13, JORDAN1E:16; hence L~ (Upper_Seq (C,n)) = Upper_Arc (L~ (Cage (C,n))) by A5, A8, A6, JORDAN6:def_8; ::_thesis: verum end; theorem Th56: :: JORDAN1G:56 for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) for n being Element of NAT st n > 0 holds L~ (Lower_Seq (C,n)) = Lower_Arc (L~ (Cage (C,n))) proof let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for n being Element of NAT st n > 0 holds L~ (Lower_Seq (C,n)) = Lower_Arc (L~ (Cage (C,n))) let n be Element of NAT ; ::_thesis: ( n > 0 implies L~ (Lower_Seq (C,n)) = Lower_Arc (L~ (Cage (C,n))) ) A1: W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43; A2: (Lower_Seq (C,n)) /. 1 = ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) :- (E-max (L~ (Cage (C,n))))) /. 1 by JORDAN1E:def_2 .= E-max (L~ (Cage (C,n))) by FINSEQ_5:53 ; E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46; then ( Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) :- (E-max (L~ (Cage (C,n)))) & E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) ) by FINSEQ_6:90, JORDAN1E:def_2, SPRECT_2:43; then (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. (len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) by FINSEQ_5:54 .= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 by FINSEQ_6:def_1 .= W-min (L~ (Cage (C,n))) by A1, FINSEQ_6:92 ; then A3: L~ (Lower_Seq (C,n)) is_an_arc_of E-max (L~ (Cage (C,n))), W-min (L~ (Cage (C,n))) by A2, TOPREAL1:25; assume n > 0 ; ::_thesis: L~ (Lower_Seq (C,n)) = Lower_Arc (L~ (Cage (C,n))) then A4: ( L~ (Upper_Seq (C,n)) = Upper_Arc (L~ (Cage (C,n))) & (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 > (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 ) by Th54, Th55; ( (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) = {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} & (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n))) = L~ (Cage (C,n)) ) by JORDAN1E:13, JORDAN1E:16; hence L~ (Lower_Seq (C,n)) = Lower_Arc (L~ (Cage (C,n))) by A3, A4, JORDAN6:def_9; ::_thesis: verum end; theorem :: JORDAN1G:57 for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) for n being Element of NAT st n > 0 holds for i, j being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Cage (C,n)) holds LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets Lower_Arc (L~ (Cage (C,n))) proof let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for n being Element of NAT st n > 0 holds for i, j being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Cage (C,n)) holds LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets Lower_Arc (L~ (Cage (C,n))) let n be Element of NAT ; ::_thesis: ( n > 0 implies for i, j being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Cage (C,n)) holds LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets Lower_Arc (L~ (Cage (C,n))) ) assume n > 0 ; ::_thesis: for i, j being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Cage (C,n)) holds LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets Lower_Arc (L~ (Cage (C,n))) then A1: L~ (Lower_Seq (C,n)) = Lower_Arc (L~ (Cage (C,n))) by Th56; let i, j be Element of NAT ; ::_thesis: ( 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Cage (C,n)) implies LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets Lower_Arc (L~ (Cage (C,n))) ) assume ( 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Cage (C,n)) ) ; ::_thesis: LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets Lower_Arc (L~ (Cage (C,n))) hence LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets Lower_Arc (L~ (Cage (C,n))) by A1, Th46; ::_thesis: verum end;