:: JORDAN1J semantic presentation begin theorem Th1: :: JORDAN1J:1 for G being Go-board for i1, i2, j1, j2 being Element of NAT st 1 <= j1 & j1 <= width G & 1 <= j2 & j2 <= width G & 1 <= i1 & i1 < i2 & i2 <= len G holds (G * (i1,j1)) `1 < (G * (i2,j2)) `1 proof let G be Go-board; ::_thesis: for i1, i2, j1, j2 being Element of NAT st 1 <= j1 & j1 <= width G & 1 <= j2 & j2 <= width G & 1 <= i1 & i1 < i2 & i2 <= len G holds (G * (i1,j1)) `1 < (G * (i2,j2)) `1 let i1, i2, j1, j2 be Element of NAT ; ::_thesis: ( 1 <= j1 & j1 <= width G & 1 <= j2 & j2 <= width G & 1 <= i1 & i1 < i2 & i2 <= len G implies (G * (i1,j1)) `1 < (G * (i2,j2)) `1 ) assume that A1: 1 <= j1 and A2: j1 <= width G and A3: 1 <= j2 and A4: j2 <= width G and A5: 1 <= i1 and A6: i1 < i2 and A7: i2 <= len G ; ::_thesis: (G * (i1,j1)) `1 < (G * (i2,j2)) `1 A8: 1 <= i2 by A5, A6, XXREAL_0:2; then (G * (i2,j1)) `1 = (G * (i2,1)) `1 by A1, A2, A7, GOBOARD5:2 .= (G * (i2,j2)) `1 by A3, A4, A7, A8, GOBOARD5:2 ; hence (G * (i1,j1)) `1 < (G * (i2,j2)) `1 by A1, A2, A5, A6, A7, GOBOARD5:3; ::_thesis: verum end; theorem Th2: :: JORDAN1J:2 for G being Go-board for i1, i2, j1, j2 being Element of NAT st 1 <= i1 & i1 <= len G & 1 <= i2 & i2 <= len G & 1 <= j1 & j1 < j2 & j2 <= width G holds (G * (i1,j1)) `2 < (G * (i2,j2)) `2 proof let G be Go-board; ::_thesis: for i1, i2, j1, j2 being Element of NAT st 1 <= i1 & i1 <= len G & 1 <= i2 & i2 <= len G & 1 <= j1 & j1 < j2 & j2 <= width G holds (G * (i1,j1)) `2 < (G * (i2,j2)) `2 let i1, i2, j1, j2 be Element of NAT ; ::_thesis: ( 1 <= i1 & i1 <= len G & 1 <= i2 & i2 <= len G & 1 <= j1 & j1 < j2 & j2 <= width G implies (G * (i1,j1)) `2 < (G * (i2,j2)) `2 ) assume that A1: 1 <= i1 and A2: i1 <= len G and A3: 1 <= i2 and A4: i2 <= len G and A5: 1 <= j1 and A6: j1 < j2 and A7: j2 <= width G ; ::_thesis: (G * (i1,j1)) `2 < (G * (i2,j2)) `2 A8: 1 <= j2 by A5, A6, XXREAL_0:2; then (G * (i1,j2)) `2 = (G * (1,j2)) `2 by A1, A2, A7, GOBOARD5:1 .= (G * (i2,j2)) `2 by A3, A4, A7, A8, GOBOARD5:1 ; hence (G * (i1,j1)) `2 < (G * (i2,j2)) `2 by A1, A2, A5, A6, A7, GOBOARD5:4; ::_thesis: verum end; registration let f be non empty FinSequence; let g be FinSequence; clusterf ^' g -> non empty ; coherence not f ^' g is empty proof f ^' g = f ^ ((2,(len g)) -cut g) by GRAPH_2:def_2; hence not f ^' g is empty ; ::_thesis: verum end; end; theorem Th3: :: JORDAN1J:3 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)) -: (E-max (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) = {(N-min (L~ (Cage (C,n)))),(E-max (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)) -: (E-max (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) = {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} let n be Element of NAT ; ::_thesis: (L~ ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) = {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} set US = (Cage (C,n)) -: (E-max (L~ (Cage (C,n)))); set LS = (Cage (C,n)) :- (E-max (L~ (Cage (C,n)))); set f = Cage (C,n); set pW = E-max (L~ (Cage (C,n))); set pN = N-min (L~ (Cage (C,n))); A1: (E-max (L~ (Cage (C,n)))) `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52; A2: E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46; then A3: (Cage (C,n)) -: (E-max (L~ (Cage (C,n)))) <> {} by FINSEQ_5:47; ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) /. 1 = E-max (L~ (Cage (C,n))) by FINSEQ_5:53; then A4: E-max (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by FINSEQ_6:42; ((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 A2, FINSEQ_5:54 .= (Cage (C,n)) /. 1 by FINSEQ_6:def_1 .= N-min (L~ (Cage (C,n))) by JORDAN9:32 ; then A5: 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 A5, A4, TARSKI:def_2; ::_thesis: verum end; then A6: 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 A7: 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; N-max (L~ (Cage (C,n))) in L~ (Cage (C,n)) by SPRECT_1:11; then (N-max (L~ (Cage (C,n)))) `1 <= (E-max (L~ (Cage (C,n)))) `1 by A1, PSCOMP_1:24; then A8: N-min (L~ (Cage (C,n))) <> E-max (L~ (Cage (C,n))) by SPRECT_2:51; then card {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} = 2 by CARD_2:57; then 2 c= len ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by A6, A7, XBOOLE_1:1; then len ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) >= 2 by NAT_1:39; then A9: rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) c= L~ ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by SPPOL_2:18; len ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) = (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by A2, FINSEQ_5:42; then ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) /. (len ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) = E-max (L~ (Cage (C,n))) by A2, FINSEQ_5:45; then A10: E-max (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) by A3, REVROT_1:3; ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) /. 1 = (Cage (C,n)) /. 1 by A2, FINSEQ_5:44 .= N-min (L~ (Cage (C,n))) by JORDAN9:32 ; then A11: N-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) by A3, FINSEQ_6:42; {(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 A11, A10, TARSKI:def_2; ::_thesis: verum end; then A12: 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 A13: 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; ((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 A2, FINSEQ_5:54 .= (Cage (C,n)) /. 1 by FINSEQ_6:def_1 .= N-min (L~ (Cage (C,n))) by JORDAN9:32 ; then A14: N-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by REVROT_1:3; (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) <= (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) ; then A15: E-max (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) by A2, FINSEQ_5:46; card {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} = 2 by A8, CARD_2:57; then A16: 2 c= len ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) by A12, A13, XBOOLE_1:1; then A17: len ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) >= 2 by NAT_1:39; then A18: rng ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) c= L~ ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) by SPPOL_2:18; thus (L~ ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) c= {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} :: according to XBOOLE_0:def_10 ::_thesis: {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} c= (L~ ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) or x in {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} ) assume A19: x in (L~ ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) ; ::_thesis: x in {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} then reconsider x1 = x as Point of (TOP-REAL 2) ; assume A20: not x in {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} ; ::_thesis: contradiction x in L~ ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) by A19, XBOOLE_0:def_4; then consider i1 being Element of NAT such that A21: 1 <= i1 and A22: i1 + 1 <= len ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) and A23: x1 in LSeg (((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))),i1) by SPPOL_2:13; A24: LSeg (((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))),i1) = LSeg ((Cage (C,n)),i1) by A22, SPPOL_2:9; x in L~ ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by A19, XBOOLE_0:def_4; then consider i2 being Element of NAT such that A25: 1 <= i2 and A26: i2 + 1 <= len ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) and A27: x1 in LSeg (((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))),i2) by SPPOL_2:13; set i3 = i2 -' 1; A28: (i2 -' 1) + 1 = i2 by A25, XREAL_1:235; then A29: 1 + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) <= ((i2 -' 1) + 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) by A25, XREAL_1:7; A30: len ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) = ((len (Cage (C,n))) - ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) + 1 by A2, FINSEQ_5:50; then i2 < ((len (Cage (C,n))) - ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) + 1 by A26, NAT_1:13; then i2 - 1 < (len (Cage (C,n))) - ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) by XREAL_1:19; then A31: (i2 - 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) < len (Cage (C,n)) by XREAL_1:20; i2 - 1 >= 1 - 1 by A25, XREAL_1:9; then A32: (i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) < len (Cage (C,n)) by A31, XREAL_0:def_2; A33: LSeg (((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))),i2) = LSeg ((Cage (C,n)),((i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))))) by A2, A28, SPPOL_2:10; A34: len ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) = (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by A2, FINSEQ_5:42; then i1 + 1 < ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) + 1 by A22, NAT_1:13; then i1 + 1 < ((i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) + 1 by A29, XXREAL_0:2; then A35: i1 + 1 <= (i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) by NAT_1:13; A36: (((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) -' 1) + 1 = (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by A2, FINSEQ_4:21, XREAL_1:235; (i2 -' 1) + 1 < ((len (Cage (C,n))) - ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) + 1 by A26, A28, A30, NAT_1:13; then i2 -' 1 < (len (Cage (C,n))) - ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) by XREAL_1:7; then A37: (i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) < len (Cage (C,n)) by XREAL_1:20; then A38: ((i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) + 1 <= len (Cage (C,n)) by NAT_1:13; now__::_thesis:_contradiction percases ( ( i1 + 1 < (i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) & i1 > 1 ) or i1 = 1 or i1 + 1 = (i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) ) by A21, A35, XXREAL_0:1; suppose ( i1 + 1 < (i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) & i1 > 1 ) ; ::_thesis: contradiction then LSeg ((Cage (C,n)),i1) misses LSeg ((Cage (C,n)),((i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))))) by A37, GOBOARD5:def_4; then (LSeg ((Cage (C,n)),i1)) /\ (LSeg ((Cage (C,n)),((i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))))) = {} by XBOOLE_0:def_7; hence contradiction by A23, A27, A24, A33, XBOOLE_0:def_4; ::_thesis: verum end; supposeA39: i1 = 1 ; ::_thesis: contradiction A40: (i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) >= 0 + 2 by A17, A34, XREAL_1:7; now__::_thesis:_contradiction percases ( (i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) > 2 or (i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) = 2 ) by A40, XXREAL_0:1; suppose (i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) > 2 ; ::_thesis: contradiction then A41: i1 + 1 < (i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) by A39; now__::_thesis:_contradiction percases ( ((i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) + 1 < len (Cage (C,n)) or ((i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) + 1 = len (Cage (C,n)) ) by A38, XXREAL_0:1; suppose ((i2 -' 1) + ((E-max (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) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))))) by A41, GOBOARD5:def_4; then (LSeg ((Cage (C,n)),i1)) /\ (LSeg ((Cage (C,n)),((i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))))) = {} by XBOOLE_0:def_7; hence contradiction by A23, A27, A24, A33, XBOOLE_0:def_4; ::_thesis: verum end; suppose ((i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) + 1 = len (Cage (C,n)) ; ::_thesis: contradiction then (i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) = (len (Cage (C,n))) - 1 ; then (i2 -' 1) + ((E-max (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) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))))) = {((Cage (C,n)) /. 1)} by A39, GOBOARD7:34, REVROT_1:30; then x1 in {((Cage (C,n)) /. 1)} by A23, A27, A24, A33, 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 A20, TARSKI:def_2; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; supposeA42: (i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) = 2 ; ::_thesis: contradiction A43: 1 + 2 <= len (Cage (C,n)) by GOBOARD7:34, XXREAL_0:2; x1 in (LSeg ((Cage (C,n)),1)) /\ (LSeg ((Cage (C,n)),(1 + 1))) by A23, A27, A24, A33, A39, A42, XBOOLE_0:def_4; then x1 in {((Cage (C,n)) /. (1 + 1))} by A43, TOPREAL1:def_6; then A44: x1 = (Cage (C,n)) /. (1 + 1) by TARSKI:def_1; 0 + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) >= (i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) by A16, A34, A42, NAT_1:39; then A45: i2 -' 1 = 0 by XREAL_1:6; 0 + 1 in dom ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by FINSEQ_5:6; then ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) /. 1 = x1 by A2, A42, A44, A45, FINSEQ_5:52; then x1 = E-max (L~ (Cage (C,n))) by FINSEQ_5:53; hence contradiction by A20, TARSKI:def_2; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; supposeA46: i1 + 1 = (i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) ; ::_thesis: contradiction (i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) >= (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by NAT_1:11; then (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < len (Cage (C,n)) by A32, XXREAL_0:2; then ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) + 1 <= len (Cage (C,n)) by NAT_1:13; then A47: (((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) -' 1) + (1 + 1) <= len (Cage (C,n)) by A36; 0 + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) <= (i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) by XREAL_1:7; then (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) = i1 + 1 by A22, A34, A46, XXREAL_0:1; then (LSeg ((Cage (C,n)),i1)) /\ (LSeg ((Cage (C,n)),((i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))))) = {((Cage (C,n)) /. ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))))} by A21, A36, A46, A47, TOPREAL1:def_6; then x1 in {((Cage (C,n)) /. ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))))} by A23, A27, A24, A33, XBOOLE_0:def_4; then x1 = (Cage (C,n)) /. ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) by TARSKI:def_1 .= E-max (L~ (Cage (C,n))) by A2, FINSEQ_5:38 ; hence contradiction by A20, TARSKI:def_2; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; A48: ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) /. 1 = (Cage (C,n)) /. 1 by A2, FINSEQ_5:44 .= N-min (L~ (Cage (C,n))) by JORDAN9:32 ; not (Cage (C,n)) -: (E-max (L~ (Cage (C,n)))) is empty by A16, NAT_1:39; then A49: N-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) by A48, FINSEQ_6:42; A50: E-max (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by FINSEQ_6:61; thus {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} c= (L~ ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (E-max (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)))),(E-max (L~ (Cage (C,n))))} or x in (L~ ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) ) assume A51: x in {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} ; ::_thesis: x in (L~ ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) percases ( x = N-min (L~ (Cage (C,n))) or x = E-max (L~ (Cage (C,n))) ) by A51, TARSKI:def_2; suppose x = N-min (L~ (Cage (C,n))) ; ::_thesis: x in (L~ ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) hence x in (L~ ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) by A9, A18, A49, A14, XBOOLE_0:def_4; ::_thesis: verum end; suppose x = E-max (L~ (Cage (C,n))) ; ::_thesis: x in (L~ ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) hence x in (L~ ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) by A9, A50, A18, A15, XBOOLE_0:def_4; ::_thesis: verum end; end; end; end; theorem Th4: :: JORDAN1J: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) = (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 Upper_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: Upper_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: N-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:39; A2: (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 A2, SPRECT_2:71, XXREAL_0:2; then A3: (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (S-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A2, SPRECT_2:73, XXREAL_0:2; then A4: (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A2, SPRECT_2:74, XXREAL_0:2; ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) /. 1 = E-max (L~ (Cage (C,n))) by FINSEQ_5:53; then A5: E-max (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by FINSEQ_6:42; A6: 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))))) /. (len ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) = (Cage (C,n)) /. (len (Cage (C,n))) by FINSEQ_5:54 .= (Cage (C,n)) /. 1 by FINSEQ_6:def_1 .= N-min (L~ (Cage (C,n))) by JORDAN9:32 ; then A7: 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 A7, A5, TARSKI:def_2; ::_thesis: verum end; then A8: 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; A9: W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43; then A10: (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 A9, 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 A9, FINSEQ_5:45; then A11: W-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by A10, REVROT_1:3; ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) /. 1 = (Cage (C,n)) /. 1 by A9, FINSEQ_5:44 .= N-min (L~ (Cage (C,n))) by JORDAN9:32 ; then A12: N-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by A10, 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 A12, A11, TARSKI:def_2; ::_thesis: verum end; then A13: 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 A14: 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; A15: (N-min (L~ (Cage (C,n)))) `2 = N-bound (L~ (Cage (C,n))) by EUCLID:52; (N-max (L~ (Cage (C,n)))) `1 <= (NE-corner (L~ (Cage (C,n)))) `1 by PSCOMP_1:38; then A16: (N-max (L~ (Cage (C,n)))) `1 <= E-bound (L~ (Cage (C,n))) by EUCLID:52; (N-min (L~ (Cage (C,n)))) `1 < (N-max (L~ (Cage (C,n)))) `1 by SPRECT_2:51; then A17: N-min (L~ (Cage (C,n))) <> E-max (L~ (Cage (C,n))) by A16, EUCLID:52; then A18: card {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} = 2 by CARD_2:57; A19: (S-min (L~ (Cage (C,n)))) .. (Cage (C,n)) <= (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A2, SPRECT_2:74; then A20: E-max (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by A9, A6, A3, FINSEQ_5:46, XXREAL_0:2; A21: W-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by A9, A6, A19, A3, FINSEQ_6:62, XXREAL_0:2; W-max (L~ (Cage (C,n))) in L~ (Cage (C,n)) by SPRECT_1:13; then (W-max (L~ (Cage (C,n)))) `2 <= (N-min (L~ (Cage (C,n)))) `2 by A15, PSCOMP_1:24; then A22: 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 A13, A14, XBOOLE_1:1; then len ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) >= 2 by NAT_1:39; then A23: 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; A24: 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 A25: 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 A9, FINSEQ_5:54 .= (Cage (C,n)) /. 1 by FINSEQ_6:def_1 .= N-min (L~ (Cage (C,n))) by JORDAN9:32 ; then A26: 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 A26, A25, TARSKI:def_2; ::_thesis: verum end; then A27: 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; A28: (N-min (L~ (Cage (C,n)))) `2 = N-bound (L~ (Cage (C,n))) by EUCLID:52; W-max (L~ (Cage (C,n))) in L~ (Cage (C,n)) by SPRECT_1:13; then (W-max (L~ (Cage (C,n)))) `2 <= (N-min (L~ (Cage (C,n)))) `2 by A28, PSCOMP_1:24; then N-min (L~ (Cage (C,n))) <> W-min (L~ (Cage (C,n))) by SPRECT_2:57; then A29: 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 A29, A27, XBOOLE_1:1; then len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) >= 2 by NAT_1:39; then A30: 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 A20, A23, A30, 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 JORDAN1G:17; then E-max (L~ (Cage (C,n))) = W-min (L~ (Cage (C,n))) by A17, TARSKI:def_2; hence contradiction by TOPREAL5:19; ::_thesis: verum end; A31: (N-max (L~ (Cage (C,n)))) .. (Cage (C,n)) <= (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by A2, SPRECT_2:70; A32: (N-min (L~ (Cage (C,n)))) .. (Cage (C,n)) < (N-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by A2, SPRECT_2:68; then A33: (N-min (L~ (Cage (C,n)))) .. (Cage (C,n)) < (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by A2, SPRECT_2:70, XXREAL_0:2; then A34: N-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by A1, A9, A4, FINSEQ_5:46, XXREAL_0:2; A35: (E-max (L~ (Cage (C,n)))) .. ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) <> 1 proof assume A36: (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 A1, A9, A33, A4, SPRECT_5:3, XXREAL_0:2 .= 1 by A2, FINSEQ_6:43 ; hence contradiction by A32, A31, A20, A34, A36, 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 A20, FINSEQ_6:78; then A37: 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 A24, XBOOLE_0:def_5; 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 A18, A8, XBOOLE_1:1; then len ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) >= 2 by NAT_1:39; then A38: rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) c= L~ ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by SPPOL_2:18; not W-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) proof assume A39: W-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) ; ::_thesis: contradiction ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) /. (len ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) = ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) /. ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) by A6, FINSEQ_5:42 .= E-max (L~ (Cage (C,n))) by A6, FINSEQ_5:45 ; then A40: E-max (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) by A39, RELAT_1:38, REVROT_1:3; ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) /. 1 = (Cage (C,n)) /. 1 by A6, FINSEQ_5:44 .= N-min (L~ (Cage (C,n))) by JORDAN9:32 ; then A41: N-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) by A39, FINSEQ_6:42, RELAT_1:38; {(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 A41, A40, TARSKI:def_2; ::_thesis: verum end; then A42: 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 A18, A42, XBOOLE_1:1; then len ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) >= 2 by NAT_1:39; then rng ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) c= L~ ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) by SPPOL_2:18; then W-min (L~ (Cage (C,n))) in (L~ ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) by A21, A38, A39, XBOOLE_0:def_4; then W-min (L~ (Cage (C,n))) in {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by Th3; then W-min (L~ (Cage (C,n))) = E-max (L~ (Cage (C,n))) by A22, TARSKI:def_2; hence contradiction by TOPREAL5:19; ::_thesis: verum end; then A43: W-min (L~ (Cage (C,n))) in (rng (Cage (C,n))) \ (rng ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) by A9, XBOOLE_0:def_5; (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 A9, FINSEQ_6:def_2 .= ((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 A37, FINSEQ_6:67 .= ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) ^ ((((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) -: (E-max (L~ (Cage (C,n))))) /^ 1) by A20, A35, FINSEQ_6:60 .= (((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) :- (W-min (L~ (Cage (C,n))))) ^ ((((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) -: (E-max (L~ (Cage (C,n))))) /^ 1) by A43, FINSEQ_6:71, SPRECT_2:46 .= (((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) :- (W-min (L~ (Cage (C,n))))) ^ (((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) /^ 1) by A9, A20, FINSEQ_6:75 .= (((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 A21, FINSEQ_6:64 .= (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) :- (W-min (L~ (Cage (C,n)))) by A6, FINSEQ_6:def_2 ; hence Upper_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) :- (W-min (L~ (Cage (C,n)))) by JORDAN1E:def_1; ::_thesis: verum end; theorem :: JORDAN1J:5 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))) in rng (Upper_Seq (C,n)) & W-min (L~ (Cage (C,n))) in L~ (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))) in rng (Upper_Seq (C,n)) & W-min (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) ) let C be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( W-min (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) & W-min (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) ) set p = W-min (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; 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 (Upper_Seq (C,n)) /. 1 = W-min (L~ (Cage (C,n))) by A1, FINSEQ_6:92; hence A3: W-min (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) by FINSEQ_6:42; ::_thesis: W-min (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) len (Upper_Seq (C,n)) >= 2 by TOPREAL1:def_8; then rng (Upper_Seq (C,n)) c= L~ (Upper_Seq (C,n)) by SPPOL_2:18; hence W-min (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) by A3; ::_thesis: verum end; theorem :: JORDAN1J:6 for n being Element of NAT for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds ( W-max (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) & W-max (L~ (Cage (C,n))) 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) holds ( W-max (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) & W-max (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) ) let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( W-max (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) & W-max (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) ) set x = W-max (L~ (Cage (C,n))); set p = W-min (L~ (Cage (C,n))); set f = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))); A1: rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = rng (Cage (C,n)) by FINSEQ_6:90, SPRECT_2:46; A2: W-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:44; A3: L~ (Cage (C,n)) = L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by REVROT_1:33; W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43; then A4: 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; A5: W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43; Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) by JORDAN1G:18; then (Lower_Seq (C,n)) /. 1 = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 by A4, FINSEQ_5:44; then (W-min (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) < (W-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A3, JORDAN1F:6, SPRECT_5:42; then W-max (L~ (Cage (C,n))) in rng ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) :- (W-min (L~ (Cage (C,n))))) by A1, A2, A5, A3, FINSEQ_6:62; hence A6: W-max (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) by Th4; ::_thesis: W-max (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) len (Upper_Seq (C,n)) >= 2 by TOPREAL1:def_8; then rng (Upper_Seq (C,n)) c= L~ (Upper_Seq (C,n)) by SPPOL_2:18; hence W-max (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) by A6; ::_thesis: verum end; theorem Th7: :: JORDAN1J:7 for n being Element of NAT for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds ( N-min (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) & N-min (L~ (Cage (C,n))) 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) holds ( N-min (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) & N-min (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) ) let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( N-min (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) & N-min (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) ) set x = N-min (L~ (Cage (C,n))); set p = W-min (L~ (Cage (C,n))); set f = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))); A1: rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = rng (Cage (C,n)) by FINSEQ_6:90, SPRECT_2:46; A2: N-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:39; A3: L~ (Cage (C,n)) = L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by REVROT_1:33; W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43; then A4: 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; A5: W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43; Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) by JORDAN1G:18; then A6: (Lower_Seq (C,n)) /. 1 = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 by A4, FINSEQ_5:44; then A7: (W-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) <= (N-min (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A3, JORDAN1F:6, SPRECT_5:43; (W-min (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) < (W-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A6, A3, JORDAN1F:6, SPRECT_5:42; then N-min (L~ (Cage (C,n))) in rng ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) :- (W-min (L~ (Cage (C,n))))) by A1, A2, A5, A3, A7, FINSEQ_6:62, XXREAL_0:2; hence A8: N-min (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) by Th4; ::_thesis: N-min (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) len (Upper_Seq (C,n)) >= 2 by TOPREAL1:def_8; then rng (Upper_Seq (C,n)) c= L~ (Upper_Seq (C,n)) by SPPOL_2:18; hence N-min (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) by A8; ::_thesis: verum end; theorem :: JORDAN1J:8 for n being Element of NAT for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds ( N-max (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) & N-max (L~ (Cage (C,n))) 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) holds ( N-max (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) & N-max (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) ) let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( N-max (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) & N-max (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) ) set x = N-max (L~ (Cage (C,n))); set p = W-min (L~ (Cage (C,n))); set f = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))); A1: rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = rng (Cage (C,n)) by FINSEQ_6:90, SPRECT_2:46; A2: N-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:40; A3: (Lower_Seq (C,n)) /. 1 = E-max (L~ (Cage (C,n))) by JORDAN1F:6; A4: L~ (Cage (C,n)) = L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by REVROT_1:33; W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43; then A5: 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; A6: W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43; Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) by JORDAN1G:18; then A7: (Lower_Seq (C,n)) /. 1 = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 by A5, FINSEQ_5:44; then A8: (W-min (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) < (W-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A4, JORDAN1F:6, SPRECT_5:42; A9: (W-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) <= (N-min (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A7, A4, JORDAN1F:6, SPRECT_5:43; percases ( N-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))))) <> E-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))))) or N-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))))) = E-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))))) ) ; suppose N-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))))) <> E-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))))) ; ::_thesis: ( N-max (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) & N-max (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) ) then (W-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) < (N-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A7, A3, A4, A9, SPRECT_5:44, XXREAL_0:2; then N-max (L~ (Cage (C,n))) in rng ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) :- (W-min (L~ (Cage (C,n))))) by A1, A2, A6, A4, A8, FINSEQ_6:62, XXREAL_0:2; hence A10: N-max (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) by Th4; ::_thesis: N-max (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) len (Upper_Seq (C,n)) >= 2 by TOPREAL1:def_8; then rng (Upper_Seq (C,n)) c= L~ (Upper_Seq (C,n)) by SPPOL_2:18; hence N-max (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) by A10; ::_thesis: verum end; supposeA11: N-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))))) = E-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))))) ; ::_thesis: ( N-max (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) & N-max (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) ) A12: (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)))))) ; E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46; then A13: 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; hence A14: N-max (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) by A4, A11, A13, A12, FINSEQ_5:46; ::_thesis: N-max (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) len (Upper_Seq (C,n)) >= 2 by TOPREAL1:def_8; then rng (Upper_Seq (C,n)) c= L~ (Upper_Seq (C,n)) by SPPOL_2:18; hence N-max (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) by A14; ::_thesis: verum end; end; end; theorem :: JORDAN1J:9 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))) in rng (Upper_Seq (C,n)) & E-max (L~ (Cage (C,n))) in L~ (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))) in rng (Upper_Seq (C,n)) & E-max (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) ) let C be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( E-max (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) & E-max (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) ) set x = E-max (L~ (Cage (C,n))); set p = W-min (L~ (Cage (C,n))); A1: (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)))))) ; 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; Upper_Seq (C,n) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n)))) by JORDAN1E:def_1; hence A3: E-max (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) by A2, A1, FINSEQ_5:46; ::_thesis: E-max (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) len (Upper_Seq (C,n)) >= 2 by TOPREAL1:def_8; then rng (Upper_Seq (C,n)) c= L~ (Upper_Seq (C,n)) by SPPOL_2:18; hence E-max (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) by A3; ::_thesis: verum end; theorem :: JORDAN1J:10 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))) in rng (Lower_Seq (C,n)) & E-max (L~ (Cage (C,n))) in L~ (Lower_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))) in rng (Lower_Seq (C,n)) & E-max (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) ) let C be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( E-max (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) & E-max (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) ) set p = E-max (L~ (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; hence A1: E-max (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) by FINSEQ_6:61; ::_thesis: E-max (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) len (Lower_Seq (C,n)) >= 2 by TOPREAL1:def_8; then rng (Lower_Seq (C,n)) c= L~ (Lower_Seq (C,n)) by SPPOL_2:18; hence E-max (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) by A1; ::_thesis: verum end; theorem :: JORDAN1J:11 for n being Element of NAT for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds ( E-min (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) & E-min (L~ (Cage (C,n))) in L~ (Lower_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-min (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) & E-min (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) ) let C be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( E-min (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) & E-min (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) ) set x = E-min (L~ (Cage (C,n))); set p = E-max (L~ (Cage (C,n))); set f = Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))); A1: rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) = rng (Cage (C,n)) by FINSEQ_6:90, SPRECT_2:43; A2: E-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:45; A3: L~ (Cage (C,n)) = L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by REVROT_1:33; E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46; then A4: 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; A5: E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46; 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 A4, FINSEQ_5:44; then (E-max (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) < (E-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A3, JORDAN1F:5, SPRECT_5:26; then E-min (L~ (Cage (C,n))) in rng ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) :- (E-max (L~ (Cage (C,n))))) by A1, A2, A5, A3, FINSEQ_6:62; hence A6: E-min (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) by JORDAN1E:def_2; ::_thesis: E-min (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) len (Lower_Seq (C,n)) >= 2 by TOPREAL1:def_8; then rng (Lower_Seq (C,n)) c= L~ (Lower_Seq (C,n)) by SPPOL_2:18; hence E-min (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) by A6; ::_thesis: verum end; theorem Th12: :: JORDAN1J:12 for n being Element of NAT for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds ( S-max (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) & S-max (L~ (Cage (C,n))) in L~ (Lower_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 ( S-max (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) & S-max (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) ) let C be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( S-max (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) & S-max (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) ) set x = S-max (L~ (Cage (C,n))); set p = E-max (L~ (Cage (C,n))); set f = Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))); A1: rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) = rng (Cage (C,n)) by FINSEQ_6:90, SPRECT_2:43; A2: S-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:42; A3: L~ (Cage (C,n)) = L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by REVROT_1:33; E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46; then A4: 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; A5: E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46; 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 A6: (Upper_Seq (C,n)) /. 1 = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 by A4, FINSEQ_5:44; then A7: (E-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) <= (S-max (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A3, JORDAN1F:5, SPRECT_5:27; (E-max (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) < (E-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A6, A3, JORDAN1F:5, SPRECT_5:26; then S-max (L~ (Cage (C,n))) in rng ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) :- (E-max (L~ (Cage (C,n))))) by A1, A2, A5, A3, A7, FINSEQ_6:62, XXREAL_0:2; hence A8: S-max (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) by JORDAN1E:def_2; ::_thesis: S-max (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) len (Lower_Seq (C,n)) >= 2 by TOPREAL1:def_8; then rng (Lower_Seq (C,n)) c= L~ (Lower_Seq (C,n)) by SPPOL_2:18; hence S-max (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) by A8; ::_thesis: verum end; theorem :: JORDAN1J:13 for n being Element of NAT for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds ( S-min (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) & S-min (L~ (Cage (C,n))) in L~ (Lower_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 ( S-min (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) & S-min (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) ) let C be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( S-min (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) & S-min (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) ) set x = S-min (L~ (Cage (C,n))); set p = E-max (L~ (Cage (C,n))); set f = Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))); A1: rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) = rng (Cage (C,n)) by FINSEQ_6:90, SPRECT_2:43; A2: S-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:41; A3: (Upper_Seq (C,n)) /. 1 = W-min (L~ (Cage (C,n))) by JORDAN1F:5; A4: L~ (Cage (C,n)) = L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by REVROT_1:33; E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46; then A5: 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; A6: E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46; 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 A7: (Upper_Seq (C,n)) /. 1 = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 by A5, FINSEQ_5:44; then A8: (E-max (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) < (E-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A4, JORDAN1F:5, SPRECT_5:26; A9: (E-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) <= (S-max (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A7, A4, JORDAN1F:5, SPRECT_5:27; percases ( S-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) <> W-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) or S-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) = W-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) ) ; suppose S-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) <> W-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) ; ::_thesis: ( S-min (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) & S-min (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) ) then (E-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) < (S-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A7, A3, A4, A9, SPRECT_5:28, XXREAL_0:2; then S-min (L~ (Cage (C,n))) in rng ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) :- (E-max (L~ (Cage (C,n))))) by A1, A2, A6, A4, A8, FINSEQ_6:62, XXREAL_0:2; hence A10: S-min (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) by JORDAN1E:def_2; ::_thesis: S-min (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) len (Lower_Seq (C,n)) >= 2 by TOPREAL1:def_8; then rng (Lower_Seq (C,n)) c= L~ (Lower_Seq (C,n)) by SPPOL_2:18; hence S-min (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) by A10; ::_thesis: verum end; supposeA11: S-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) = W-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) ; ::_thesis: ( S-min (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) & S-min (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) ) (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) = W-min (L~ (Cage (C,n))) by JORDAN1F:8; hence A12: S-min (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) by A4, A11, REVROT_1:3; ::_thesis: S-min (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) len (Lower_Seq (C,n)) >= 2 by TOPREAL1:def_8; then rng (Lower_Seq (C,n)) c= L~ (Lower_Seq (C,n)) by SPPOL_2:18; hence S-min (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) by A12; ::_thesis: verum end; end; end; theorem :: JORDAN1J:14 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))) in rng (Lower_Seq (C,n)) & W-min (L~ (Cage (C,n))) in L~ (Lower_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))) in rng (Lower_Seq (C,n)) & W-min (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) ) let C be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( W-min (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) & W-min (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) ) set p = W-min (L~ (Cage (C,n))); (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) = W-min (L~ (Cage (C,n))) by JORDAN1F:8; hence A1: W-min (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) by REVROT_1:3; ::_thesis: W-min (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) len (Lower_Seq (C,n)) >= 2 by TOPREAL1:def_8; then rng (Lower_Seq (C,n)) c= L~ (Lower_Seq (C,n)) by SPPOL_2:18; hence W-min (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) by A1; ::_thesis: verum end; theorem Th15: :: JORDAN1J:15 for X, Y being non empty compact Subset of (TOP-REAL 2) st X c= Y & N-min Y in X holds N-min X = N-min Y proof let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( X c= Y & N-min Y in X implies N-min X = N-min Y ) assume that A1: X c= Y and A2: N-min Y in X ; ::_thesis: N-min X = N-min Y A3: N-bound X >= (N-min Y) `2 by A2, PSCOMP_1:24; A4: (N-min X) `2 = N-bound X by EUCLID:52; A5: (N-min Y) `2 = N-bound Y by EUCLID:52; A6: N-bound X <= N-bound Y by A1, PSCOMP_1:66; then A7: N-bound X = N-bound Y by A5, A3, XXREAL_0:1; N-min Y in N-most X by A2, A6, A5, A3, SPRECT_2:10, XXREAL_0:1; then A8: (N-min X) `1 <= (N-min Y) `1 by PSCOMP_1:39; N-min X in X by SPRECT_1:11; then N-min X in N-most Y by A1, A6, A4, A5, A3, SPRECT_2:10, XXREAL_0:1; then (N-min X) `1 >= (N-min Y) `1 by PSCOMP_1:39; then (N-min X) `1 = (N-min Y) `1 by A8, XXREAL_0:1; hence N-min X = N-min Y by A4, A5, A7, TOPREAL3:6; ::_thesis: verum end; theorem Th16: :: JORDAN1J:16 for X, Y being non empty compact Subset of (TOP-REAL 2) st X c= Y & N-max Y in X holds N-max X = N-max Y proof let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( X c= Y & N-max Y in X implies N-max X = N-max Y ) assume that A1: X c= Y and A2: N-max Y in X ; ::_thesis: N-max X = N-max Y A3: N-bound X >= (N-max Y) `2 by A2, PSCOMP_1:24; A4: (N-max X) `2 = N-bound X by EUCLID:52; A5: (N-max Y) `2 = N-bound Y by EUCLID:52; A6: N-bound X <= N-bound Y by A1, PSCOMP_1:66; then A7: N-bound X = N-bound Y by A5, A3, XXREAL_0:1; N-max Y in N-most X by A2, A6, A5, A3, SPRECT_2:10, XXREAL_0:1; then A8: (N-max X) `1 >= (N-max Y) `1 by PSCOMP_1:39; N-max X in X by SPRECT_1:11; then N-max X in N-most Y by A1, A6, A4, A5, A3, SPRECT_2:10, XXREAL_0:1; then (N-max X) `1 <= (N-max Y) `1 by PSCOMP_1:39; then (N-max X) `1 = (N-max Y) `1 by A8, XXREAL_0:1; hence N-max X = N-max Y by A4, A5, A7, TOPREAL3:6; ::_thesis: verum end; theorem Th17: :: JORDAN1J:17 for X, Y being non empty compact Subset of (TOP-REAL 2) st X c= Y & E-min Y in X holds E-min X = E-min Y proof let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( X c= Y & E-min Y in X implies E-min X = E-min Y ) assume that A1: X c= Y and A2: E-min Y in X ; ::_thesis: E-min X = E-min Y A3: E-bound X >= (E-min Y) `1 by A2, PSCOMP_1:24; A4: (E-min X) `1 = E-bound X by EUCLID:52; A5: (E-min Y) `1 = E-bound Y by EUCLID:52; A6: E-bound X <= E-bound Y by A1, PSCOMP_1:67; then A7: E-bound X = E-bound Y by A5, A3, XXREAL_0:1; E-min Y in E-most X by A2, A6, A5, A3, SPRECT_2:13, XXREAL_0:1; then A8: (E-min X) `2 <= (E-min Y) `2 by PSCOMP_1:47; E-min X in X by SPRECT_1:14; then E-min X in E-most Y by A1, A6, A4, A5, A3, SPRECT_2:13, XXREAL_0:1; then (E-min X) `2 >= (E-min Y) `2 by PSCOMP_1:47; then (E-min X) `2 = (E-min Y) `2 by A8, XXREAL_0:1; hence E-min X = E-min Y by A4, A5, A7, TOPREAL3:6; ::_thesis: verum end; theorem Th18: :: JORDAN1J:18 for X, Y being non empty compact Subset of (TOP-REAL 2) st X c= Y & E-max Y in X holds E-max X = E-max Y proof let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( X c= Y & E-max Y in X implies E-max X = E-max Y ) assume that A1: X c= Y and A2: E-max Y in X ; ::_thesis: E-max X = E-max Y A3: E-bound X >= (E-max Y) `1 by A2, PSCOMP_1:24; A4: (E-max X) `1 = E-bound X by EUCLID:52; A5: (E-max Y) `1 = E-bound Y by EUCLID:52; A6: E-bound X <= E-bound Y by A1, PSCOMP_1:67; then A7: E-bound X = E-bound Y by A5, A3, XXREAL_0:1; E-max Y in E-most X by A2, A6, A5, A3, SPRECT_2:13, XXREAL_0:1; then A8: (E-max X) `2 >= (E-max Y) `2 by PSCOMP_1:47; E-max X in X by SPRECT_1:14; then E-max X in E-most Y by A1, A6, A4, A5, A3, SPRECT_2:13, XXREAL_0:1; then (E-max X) `2 <= (E-max Y) `2 by PSCOMP_1:47; then (E-max X) `2 = (E-max Y) `2 by A8, XXREAL_0:1; hence E-max X = E-max Y by A4, A5, A7, TOPREAL3:6; ::_thesis: verum end; theorem Th19: :: JORDAN1J:19 for X, Y being non empty compact Subset of (TOP-REAL 2) st X c= Y & S-min Y in X holds S-min X = S-min Y proof let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( X c= Y & S-min Y in X implies S-min X = S-min Y ) assume that A1: X c= Y and A2: S-min Y in X ; ::_thesis: S-min X = S-min Y A3: S-bound X <= (S-min Y) `2 by A2, PSCOMP_1:24; A4: (S-min X) `2 = S-bound X by EUCLID:52; A5: (S-min Y) `2 = S-bound Y by EUCLID:52; A6: S-bound X >= S-bound Y by A1, PSCOMP_1:68; then A7: S-bound X = S-bound Y by A5, A3, XXREAL_0:1; S-min Y in S-most X by A2, A6, A5, A3, SPRECT_2:11, XXREAL_0:1; then A8: (S-min X) `1 <= (S-min Y) `1 by PSCOMP_1:55; S-min X in X by SPRECT_1:12; then S-min X in S-most Y by A1, A6, A4, A5, A3, SPRECT_2:11, XXREAL_0:1; then (S-min X) `1 >= (S-min Y) `1 by PSCOMP_1:55; then (S-min X) `1 = (S-min Y) `1 by A8, XXREAL_0:1; hence S-min X = S-min Y by A4, A5, A7, TOPREAL3:6; ::_thesis: verum end; theorem Th20: :: JORDAN1J:20 for X, Y being non empty compact Subset of (TOP-REAL 2) st X c= Y & S-max Y in X holds S-max X = S-max Y proof let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( X c= Y & S-max Y in X implies S-max X = S-max Y ) assume that A1: X c= Y and A2: S-max Y in X ; ::_thesis: S-max X = S-max Y A3: S-bound X <= (S-max Y) `2 by A2, PSCOMP_1:24; A4: (S-max X) `2 = S-bound X by EUCLID:52; A5: (S-max Y) `2 = S-bound Y by EUCLID:52; A6: S-bound X >= S-bound Y by A1, PSCOMP_1:68; then A7: S-bound X = S-bound Y by A5, A3, XXREAL_0:1; S-max Y in S-most X by A2, A6, A5, A3, SPRECT_2:11, XXREAL_0:1; then A8: (S-max X) `1 >= (S-max Y) `1 by PSCOMP_1:55; S-max X in X by SPRECT_1:12; then S-max X in S-most Y by A1, A6, A4, A5, A3, SPRECT_2:11, XXREAL_0:1; then (S-max X) `1 <= (S-max Y) `1 by PSCOMP_1:55; then (S-max X) `1 = (S-max Y) `1 by A8, XXREAL_0:1; hence S-max X = S-max Y by A4, A5, A7, TOPREAL3:6; ::_thesis: verum end; theorem Th21: :: JORDAN1J:21 for X, Y being non empty compact Subset of (TOP-REAL 2) st X c= Y & W-min Y in X holds W-min X = W-min Y proof let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( X c= Y & W-min Y in X implies W-min X = W-min Y ) assume that A1: X c= Y and A2: W-min Y in X ; ::_thesis: W-min X = W-min Y A3: W-bound X <= (W-min Y) `1 by A2, PSCOMP_1:24; A4: (W-min X) `1 = W-bound X by EUCLID:52; A5: (W-min Y) `1 = W-bound Y by EUCLID:52; A6: W-bound X >= W-bound Y by A1, PSCOMP_1:69; then A7: W-bound X = W-bound Y by A5, A3, XXREAL_0:1; W-min Y in W-most X by A2, A6, A5, A3, SPRECT_2:12, XXREAL_0:1; then A8: (W-min X) `2 <= (W-min Y) `2 by PSCOMP_1:31; W-min X in X by SPRECT_1:13; then W-min X in W-most Y by A1, A6, A4, A5, A3, SPRECT_2:12, XXREAL_0:1; then (W-min X) `2 >= (W-min Y) `2 by PSCOMP_1:31; then (W-min X) `2 = (W-min Y) `2 by A8, XXREAL_0:1; hence W-min X = W-min Y by A4, A5, A7, TOPREAL3:6; ::_thesis: verum end; theorem Th22: :: JORDAN1J:22 for X, Y being non empty compact Subset of (TOP-REAL 2) st X c= Y & W-max Y in X holds W-max X = W-max Y proof let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( X c= Y & W-max Y in X implies W-max X = W-max Y ) assume that A1: X c= Y and A2: W-max Y in X ; ::_thesis: W-max X = W-max Y A3: W-bound X <= (W-max Y) `1 by A2, PSCOMP_1:24; A4: (W-max X) `1 = W-bound X by EUCLID:52; A5: (W-max Y) `1 = W-bound Y by EUCLID:52; A6: W-bound X >= W-bound Y by A1, PSCOMP_1:69; then A7: W-bound X = W-bound Y by A5, A3, XXREAL_0:1; W-max Y in W-most X by A2, A6, A5, A3, SPRECT_2:12, XXREAL_0:1; then A8: (W-max X) `2 >= (W-max Y) `2 by PSCOMP_1:31; W-max X in X by SPRECT_1:13; then W-max X in W-most Y by A1, A6, A4, A5, A3, SPRECT_2:12, XXREAL_0:1; then (W-max X) `2 <= (W-max Y) `2 by PSCOMP_1:31; then (W-max X) `2 = (W-max Y) `2 by A8, XXREAL_0:1; hence W-max X = W-max Y by A4, A5, A7, TOPREAL3:6; ::_thesis: verum end; theorem Th23: :: JORDAN1J:23 for X, Y being non empty compact Subset of (TOP-REAL 2) st N-bound X <= N-bound Y holds N-bound (X \/ Y) = N-bound Y proof let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( N-bound X <= N-bound Y implies N-bound (X \/ Y) = N-bound Y ) assume N-bound X <= N-bound Y ; ::_thesis: N-bound (X \/ Y) = N-bound Y then max ((N-bound X),(N-bound Y)) = N-bound Y by XXREAL_0:def_10; hence N-bound (X \/ Y) = N-bound Y by SPRECT_1:49; ::_thesis: verum end; theorem Th24: :: JORDAN1J:24 for X, Y being non empty compact Subset of (TOP-REAL 2) st E-bound X <= E-bound Y holds E-bound (X \/ Y) = E-bound Y proof let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( E-bound X <= E-bound Y implies E-bound (X \/ Y) = E-bound Y ) assume E-bound X <= E-bound Y ; ::_thesis: E-bound (X \/ Y) = E-bound Y then max ((E-bound X),(E-bound Y)) = E-bound Y by XXREAL_0:def_10; hence E-bound (X \/ Y) = E-bound Y by SPRECT_1:50; ::_thesis: verum end; theorem Th25: :: JORDAN1J:25 for X, Y being non empty compact Subset of (TOP-REAL 2) st S-bound X <= S-bound Y holds S-bound (X \/ Y) = S-bound X proof let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( S-bound X <= S-bound Y implies S-bound (X \/ Y) = S-bound X ) assume S-bound X <= S-bound Y ; ::_thesis: S-bound (X \/ Y) = S-bound X then min ((S-bound X),(S-bound Y)) = S-bound X by XXREAL_0:def_9; hence S-bound (X \/ Y) = S-bound X by SPRECT_1:48; ::_thesis: verum end; theorem Th26: :: JORDAN1J:26 for X, Y being non empty compact Subset of (TOP-REAL 2) st W-bound X <= W-bound Y holds W-bound (X \/ Y) = W-bound X proof let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( W-bound X <= W-bound Y implies W-bound (X \/ Y) = W-bound X ) assume W-bound X <= W-bound Y ; ::_thesis: W-bound (X \/ Y) = W-bound X then min ((W-bound X),(W-bound Y)) = W-bound X by XXREAL_0:def_9; hence W-bound (X \/ Y) = W-bound X by SPRECT_1:47; ::_thesis: verum end; theorem :: JORDAN1J:27 for X, Y being non empty compact Subset of (TOP-REAL 2) st N-bound X < N-bound Y holds N-min (X \/ Y) = N-min Y proof let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( N-bound X < N-bound Y implies N-min (X \/ Y) = N-min Y ) A1: (N-min (X \/ Y)) `2 = N-bound (X \/ Y) by EUCLID:52; A2: X \/ Y is compact by COMPTS_1:10; then A3: N-min (X \/ Y) in X \/ Y by SPRECT_1:11; A4: N-min Y in Y by SPRECT_1:11; A5: (N-min Y) `2 = N-bound Y by EUCLID:52; assume A6: N-bound X < N-bound Y ; ::_thesis: N-min (X \/ Y) = N-min Y then A7: N-bound (X \/ Y) = N-bound Y by Th23; Y c= X \/ Y by XBOOLE_1:7; then N-min Y in N-most (X \/ Y) by A2, A7, A5, A4, SPRECT_2:10; then A8: (N-min (X \/ Y)) `1 <= (N-min Y) `1 by A2, PSCOMP_1:39; percases ( N-min (X \/ Y) in Y or N-min (X \/ Y) in X ) by A3, XBOOLE_0:def_3; suppose N-min (X \/ Y) in Y ; ::_thesis: N-min (X \/ Y) = N-min Y then N-min (X \/ Y) in N-most Y by A6, A1, Th23, SPRECT_2:10; then (N-min (X \/ Y)) `1 >= (N-min Y) `1 by PSCOMP_1:39; then (N-min (X \/ Y)) `1 = (N-min Y) `1 by A8, XXREAL_0:1; hence N-min (X \/ Y) = N-min Y by A6, A1, A5, Th23, TOPREAL3:6; ::_thesis: verum end; suppose N-min (X \/ Y) in X ; ::_thesis: N-min (X \/ Y) = N-min Y hence N-min (X \/ Y) = N-min Y by A6, A7, A1, PSCOMP_1:24; ::_thesis: verum end; end; end; theorem :: JORDAN1J:28 for X, Y being non empty compact Subset of (TOP-REAL 2) st N-bound X < N-bound Y holds N-max (X \/ Y) = N-max Y proof let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( N-bound X < N-bound Y implies N-max (X \/ Y) = N-max Y ) A1: (N-max (X \/ Y)) `2 = N-bound (X \/ Y) by EUCLID:52; A2: X \/ Y is compact by COMPTS_1:10; then A3: N-max (X \/ Y) in X \/ Y by SPRECT_1:11; A4: N-max Y in Y by SPRECT_1:11; A5: (N-max Y) `2 = N-bound Y by EUCLID:52; assume A6: N-bound X < N-bound Y ; ::_thesis: N-max (X \/ Y) = N-max Y then A7: N-bound (X \/ Y) = N-bound Y by Th23; Y c= X \/ Y by XBOOLE_1:7; then N-max Y in N-most (X \/ Y) by A2, A7, A5, A4, SPRECT_2:10; then A8: (N-max (X \/ Y)) `1 >= (N-max Y) `1 by A2, PSCOMP_1:39; percases ( N-max (X \/ Y) in Y or N-max (X \/ Y) in X ) by A3, XBOOLE_0:def_3; suppose N-max (X \/ Y) in Y ; ::_thesis: N-max (X \/ Y) = N-max Y then N-max (X \/ Y) in N-most Y by A6, A1, Th23, SPRECT_2:10; then (N-max (X \/ Y)) `1 <= (N-max Y) `1 by PSCOMP_1:39; then (N-max (X \/ Y)) `1 = (N-max Y) `1 by A8, XXREAL_0:1; hence N-max (X \/ Y) = N-max Y by A6, A1, A5, Th23, TOPREAL3:6; ::_thesis: verum end; suppose N-max (X \/ Y) in X ; ::_thesis: N-max (X \/ Y) = N-max Y hence N-max (X \/ Y) = N-max Y by A6, A7, A1, PSCOMP_1:24; ::_thesis: verum end; end; end; theorem :: JORDAN1J:29 for X, Y being non empty compact Subset of (TOP-REAL 2) st E-bound X < E-bound Y holds E-min (X \/ Y) = E-min Y proof let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( E-bound X < E-bound Y implies E-min (X \/ Y) = E-min Y ) A1: (E-min (X \/ Y)) `1 = E-bound (X \/ Y) by EUCLID:52; A2: X \/ Y is compact by COMPTS_1:10; then A3: E-min (X \/ Y) in X \/ Y by SPRECT_1:14; A4: E-min Y in Y by SPRECT_1:14; A5: (E-min Y) `1 = E-bound Y by EUCLID:52; assume A6: E-bound X < E-bound Y ; ::_thesis: E-min (X \/ Y) = E-min Y then A7: E-bound (X \/ Y) = E-bound Y by Th24; Y c= X \/ Y by XBOOLE_1:7; then E-min Y in E-most (X \/ Y) by A2, A7, A5, A4, SPRECT_2:13; then A8: (E-min (X \/ Y)) `2 <= (E-min Y) `2 by A2, PSCOMP_1:47; percases ( E-min (X \/ Y) in Y or E-min (X \/ Y) in X ) by A3, XBOOLE_0:def_3; suppose E-min (X \/ Y) in Y ; ::_thesis: E-min (X \/ Y) = E-min Y then E-min (X \/ Y) in E-most Y by A6, A1, Th24, SPRECT_2:13; then (E-min (X \/ Y)) `2 >= (E-min Y) `2 by PSCOMP_1:47; then (E-min (X \/ Y)) `2 = (E-min Y) `2 by A8, XXREAL_0:1; hence E-min (X \/ Y) = E-min Y by A6, A1, A5, Th24, TOPREAL3:6; ::_thesis: verum end; suppose E-min (X \/ Y) in X ; ::_thesis: E-min (X \/ Y) = E-min Y hence E-min (X \/ Y) = E-min Y by A6, A7, A1, PSCOMP_1:24; ::_thesis: verum end; end; end; theorem :: JORDAN1J:30 for X, Y being non empty compact Subset of (TOP-REAL 2) st E-bound X < E-bound Y holds E-max (X \/ Y) = E-max Y proof let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( E-bound X < E-bound Y implies E-max (X \/ Y) = E-max Y ) A1: (E-max (X \/ Y)) `1 = E-bound (X \/ Y) by EUCLID:52; A2: X \/ Y is compact by COMPTS_1:10; then A3: E-max (X \/ Y) in X \/ Y by SPRECT_1:14; A4: E-max Y in Y by SPRECT_1:14; A5: (E-max Y) `1 = E-bound Y by EUCLID:52; assume A6: E-bound X < E-bound Y ; ::_thesis: E-max (X \/ Y) = E-max Y then A7: E-bound (X \/ Y) = E-bound Y by Th24; Y c= X \/ Y by XBOOLE_1:7; then E-max Y in E-most (X \/ Y) by A2, A7, A5, A4, SPRECT_2:13; then A8: (E-max (X \/ Y)) `2 >= (E-max Y) `2 by A2, PSCOMP_1:47; percases ( E-max (X \/ Y) in Y or E-max (X \/ Y) in X ) by A3, XBOOLE_0:def_3; suppose E-max (X \/ Y) in Y ; ::_thesis: E-max (X \/ Y) = E-max Y then E-max (X \/ Y) in E-most Y by A6, A1, Th24, SPRECT_2:13; then (E-max (X \/ Y)) `2 <= (E-max Y) `2 by PSCOMP_1:47; then (E-max (X \/ Y)) `2 = (E-max Y) `2 by A8, XXREAL_0:1; hence E-max (X \/ Y) = E-max Y by A6, A1, A5, Th24, TOPREAL3:6; ::_thesis: verum end; suppose E-max (X \/ Y) in X ; ::_thesis: E-max (X \/ Y) = E-max Y hence E-max (X \/ Y) = E-max Y by A6, A7, A1, PSCOMP_1:24; ::_thesis: verum end; end; end; theorem :: JORDAN1J:31 for X, Y being non empty compact Subset of (TOP-REAL 2) st S-bound X < S-bound Y holds S-min (X \/ Y) = S-min X proof let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( S-bound X < S-bound Y implies S-min (X \/ Y) = S-min X ) A1: (S-min (X \/ Y)) `2 = S-bound (X \/ Y) by EUCLID:52; A2: X \/ Y is compact by COMPTS_1:10; then A3: S-min (X \/ Y) in X \/ Y by SPRECT_1:12; A4: S-min X in X by SPRECT_1:12; A5: (S-min X) `2 = S-bound X by EUCLID:52; assume A6: S-bound X < S-bound Y ; ::_thesis: S-min (X \/ Y) = S-min X then A7: S-bound (X \/ Y) = S-bound X by Th25; X c= X \/ Y by XBOOLE_1:7; then S-min X in S-most (X \/ Y) by A2, A7, A5, A4, SPRECT_2:11; then A8: (S-min (X \/ Y)) `1 <= (S-min X) `1 by A2, PSCOMP_1:55; percases ( S-min (X \/ Y) in X or S-min (X \/ Y) in Y ) by A3, XBOOLE_0:def_3; suppose S-min (X \/ Y) in X ; ::_thesis: S-min (X \/ Y) = S-min X then S-min (X \/ Y) in S-most X by A6, A1, Th25, SPRECT_2:11; then (S-min (X \/ Y)) `1 >= (S-min X) `1 by PSCOMP_1:55; then (S-min (X \/ Y)) `1 = (S-min X) `1 by A8, XXREAL_0:1; hence S-min (X \/ Y) = S-min X by A6, A1, A5, Th25, TOPREAL3:6; ::_thesis: verum end; suppose S-min (X \/ Y) in Y ; ::_thesis: S-min (X \/ Y) = S-min X hence S-min (X \/ Y) = S-min X by A6, A7, A1, PSCOMP_1:24; ::_thesis: verum end; end; end; theorem :: JORDAN1J:32 for X, Y being non empty compact Subset of (TOP-REAL 2) st S-bound X < S-bound Y holds S-max (X \/ Y) = S-max X proof let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( S-bound X < S-bound Y implies S-max (X \/ Y) = S-max X ) A1: (S-max (X \/ Y)) `2 = S-bound (X \/ Y) by EUCLID:52; A2: X \/ Y is compact by COMPTS_1:10; then A3: S-max (X \/ Y) in X \/ Y by SPRECT_1:12; A4: S-max X in X by SPRECT_1:12; A5: (S-max X) `2 = S-bound X by EUCLID:52; assume A6: S-bound X < S-bound Y ; ::_thesis: S-max (X \/ Y) = S-max X then A7: S-bound (X \/ Y) = S-bound X by Th25; X c= X \/ Y by XBOOLE_1:7; then S-max X in S-most (X \/ Y) by A2, A7, A5, A4, SPRECT_2:11; then A8: (S-max (X \/ Y)) `1 >= (S-max X) `1 by A2, PSCOMP_1:55; percases ( S-max (X \/ Y) in X or S-max (X \/ Y) in Y ) by A3, XBOOLE_0:def_3; suppose S-max (X \/ Y) in X ; ::_thesis: S-max (X \/ Y) = S-max X then S-max (X \/ Y) in S-most X by A6, A1, Th25, SPRECT_2:11; then (S-max (X \/ Y)) `1 <= (S-max X) `1 by PSCOMP_1:55; then (S-max (X \/ Y)) `1 = (S-max X) `1 by A8, XXREAL_0:1; hence S-max (X \/ Y) = S-max X by A6, A1, A5, Th25, TOPREAL3:6; ::_thesis: verum end; suppose S-max (X \/ Y) in Y ; ::_thesis: S-max (X \/ Y) = S-max X hence S-max (X \/ Y) = S-max X by A6, A7, A1, PSCOMP_1:24; ::_thesis: verum end; end; end; theorem Th33: :: JORDAN1J:33 for X, Y being non empty compact Subset of (TOP-REAL 2) st W-bound X < W-bound Y holds W-min (X \/ Y) = W-min X proof let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( W-bound X < W-bound Y implies W-min (X \/ Y) = W-min X ) A1: (W-min (X \/ Y)) `1 = W-bound (X \/ Y) by EUCLID:52; A2: X \/ Y is compact by COMPTS_1:10; then A3: W-min (X \/ Y) in X \/ Y by SPRECT_1:13; A4: W-min X in X by SPRECT_1:13; A5: (W-min X) `1 = W-bound X by EUCLID:52; assume A6: W-bound X < W-bound Y ; ::_thesis: W-min (X \/ Y) = W-min X then A7: W-bound (X \/ Y) = W-bound X by Th26; X c= X \/ Y by XBOOLE_1:7; then W-min X in W-most (X \/ Y) by A2, A7, A5, A4, SPRECT_2:12; then A8: (W-min (X \/ Y)) `2 <= (W-min X) `2 by A2, PSCOMP_1:31; percases ( W-min (X \/ Y) in X or W-min (X \/ Y) in Y ) by A3, XBOOLE_0:def_3; suppose W-min (X \/ Y) in X ; ::_thesis: W-min (X \/ Y) = W-min X then W-min (X \/ Y) in W-most X by A6, A1, Th26, SPRECT_2:12; then (W-min (X \/ Y)) `2 >= (W-min X) `2 by PSCOMP_1:31; then (W-min (X \/ Y)) `2 = (W-min X) `2 by A8, XXREAL_0:1; hence W-min (X \/ Y) = W-min X by A6, A1, A5, Th26, TOPREAL3:6; ::_thesis: verum end; suppose W-min (X \/ Y) in Y ; ::_thesis: W-min (X \/ Y) = W-min X hence W-min (X \/ Y) = W-min X by A6, A7, A1, PSCOMP_1:24; ::_thesis: verum end; end; end; theorem :: JORDAN1J:34 for X, Y being non empty compact Subset of (TOP-REAL 2) st W-bound X < W-bound Y holds W-max (X \/ Y) = W-max X proof let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( W-bound X < W-bound Y implies W-max (X \/ Y) = W-max X ) A1: (W-max (X \/ Y)) `1 = W-bound (X \/ Y) by EUCLID:52; A2: X \/ Y is compact by COMPTS_1:10; then A3: W-max (X \/ Y) in X \/ Y by SPRECT_1:13; A4: W-max X in X by SPRECT_1:13; A5: (W-max X) `1 = W-bound X by EUCLID:52; assume A6: W-bound X < W-bound Y ; ::_thesis: W-max (X \/ Y) = W-max X then A7: W-bound (X \/ Y) = W-bound X by Th26; X c= X \/ Y by XBOOLE_1:7; then W-max X in W-most (X \/ Y) by A2, A7, A5, A4, SPRECT_2:12; then A8: (W-max (X \/ Y)) `2 >= (W-max X) `2 by A2, PSCOMP_1:31; percases ( W-max (X \/ Y) in X or W-max (X \/ Y) in Y ) by A3, XBOOLE_0:def_3; suppose W-max (X \/ Y) in X ; ::_thesis: W-max (X \/ Y) = W-max X then W-max (X \/ Y) in W-most X by A6, A1, Th26, SPRECT_2:12; then (W-max (X \/ Y)) `2 <= (W-max X) `2 by PSCOMP_1:31; then (W-max (X \/ Y)) `2 = (W-max X) `2 by A8, XXREAL_0:1; hence W-max (X \/ Y) = W-max X by A6, A1, A5, Th26, TOPREAL3:6; ::_thesis: verum end; suppose W-max (X \/ Y) in Y ; ::_thesis: W-max (X \/ Y) = W-max X hence W-max (X \/ Y) = W-max X by A6, A7, A1, PSCOMP_1:24; ::_thesis: verum end; end; end; theorem Th35: :: JORDAN1J:35 for f being FinSequence of (TOP-REAL 2) for p being Point of (TOP-REAL 2) st f is being_S-Seq & p in L~ f holds (L_Cut (f,p)) /. (len (L_Cut (f,p))) = f /. (len f) proof let f be FinSequence of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st f is being_S-Seq & p in L~ f holds (L_Cut (f,p)) /. (len (L_Cut (f,p))) = f /. (len f) let p be Point of (TOP-REAL 2); ::_thesis: ( f is being_S-Seq & p in L~ f implies (L_Cut (f,p)) /. (len (L_Cut (f,p))) = f /. (len f) ) assume that A1: f is being_S-Seq and A2: p in L~ f ; ::_thesis: (L_Cut (f,p)) /. (len (L_Cut (f,p))) = f /. (len f) A3: len f in dom f by A1, FINSEQ_5:6; L_Cut (f,p) <> {} by A2, JORDAN1E:3; then len (L_Cut (f,p)) in dom (L_Cut (f,p)) by FINSEQ_5:6; hence (L_Cut (f,p)) /. (len (L_Cut (f,p))) = (L_Cut (f,p)) . (len (L_Cut (f,p))) by PARTFUN1:def_6 .= f . (len f) by A1, A2, JORDAN1B:4 .= f /. (len f) by A3, PARTFUN1:def_6 ; ::_thesis: verum end; theorem Th36: :: JORDAN1J:36 for f being non constant standard special_circular_sequence for p, q being Point of (TOP-REAL 2) for g being connected Subset of (TOP-REAL 2) st p in RightComp f & q in LeftComp f & p in g & q in g holds g meets L~ f proof let f be non constant standard special_circular_sequence; ::_thesis: for p, q being Point of (TOP-REAL 2) for g being connected Subset of (TOP-REAL 2) st p in RightComp f & q in LeftComp f & p in g & q in g holds g meets L~ f let p, q be Point of (TOP-REAL 2); ::_thesis: for g being connected Subset of (TOP-REAL 2) st p in RightComp f & q in LeftComp f & p in g & q in g holds g meets L~ f let g be connected Subset of (TOP-REAL 2); ::_thesis: ( p in RightComp f & q in LeftComp f & p in g & q in g implies g meets L~ f ) assume that A1: p in RightComp f and A2: q in LeftComp f and A3: p in g and A4: q in g ; ::_thesis: g meets L~ f assume g misses L~ f ; ::_thesis: contradiction then g c= (L~ f) ` by TDLAT_1:2; then reconsider A = g as Subset of ((TOP-REAL 2) | ((L~ f) `)) by PRE_TOPC:8; RightComp f is_a_component_of (L~ f) ` by GOBOARD9:def_2; then consider R being Subset of ((TOP-REAL 2) | ((L~ f) `)) such that A5: R = RightComp f and A6: R is a_component by CONNSP_1:def_6; R /\ A <> {} by A1, A3, A5, XBOOLE_0:def_4; then A7: R meets A by XBOOLE_0:def_7; LeftComp f is_a_component_of (L~ f) ` by GOBOARD9:def_1; then consider L being Subset of ((TOP-REAL 2) | ((L~ f) `)) such that A8: L = LeftComp f and A9: L is a_component by CONNSP_1:def_6; L /\ A <> {} by A2, A4, A8, XBOOLE_0:def_4; then A10: L meets A by XBOOLE_0:def_7; A is connected by CONNSP_1:23; hence contradiction by A5, A6, A8, A9, A7, A10, JORDAN2C:92, SPRECT_4:6; ::_thesis: verum end; registration cluster non empty non trivial V13() V16( NAT ) V17( the U1 of (TOP-REAL 2)) Function-like one-to-one non constant V26() FinSequence-like FinSubsequence-like special unfolded s.n.c. being_S-Seq s.c.c. standard for FinSequence of the U1 of (TOP-REAL 2); existence ex b1 being being_S-Seq FinSequence of (TOP-REAL 2) st ( not b1 is constant & b1 is standard & b1 is s.c.c. ) proof set n = the Element of NAT ; set C = the Simple_closed_curve; A1: Upper_Seq ( the Simple_closed_curve, the Element of NAT ) is_sequence_on Gauge ( the Simple_closed_curve, the Element of NAT ) by JORDAN1G:4; take Upper_Seq ( the Simple_closed_curve, the Element of NAT ) ; ::_thesis: ( not Upper_Seq ( the Simple_closed_curve, the Element of NAT ) is constant & Upper_Seq ( the Simple_closed_curve, the Element of NAT ) is standard & Upper_Seq ( the Simple_closed_curve, the Element of NAT ) is s.c.c. ) len (Upper_Seq ( the Simple_closed_curve, the Element of NAT )) >= 2 by TOPREAL1:def_8; hence ( not Upper_Seq ( the Simple_closed_curve, the Element of NAT ) is constant & Upper_Seq ( the Simple_closed_curve, the Element of NAT ) is standard & Upper_Seq ( the Simple_closed_curve, the Element of NAT ) is s.c.c. ) by A1, JGRAPH_1:12, JORDAN8:5; ::_thesis: verum end; end; theorem Th37: :: JORDAN1J:37 for f being S-Sequence_in_R2 for p being Point of (TOP-REAL 2) st p in rng f holds L_Cut (f,p) = mid (f,(p .. f),(len f)) proof let f be S-Sequence_in_R2; ::_thesis: for p being Point of (TOP-REAL 2) st p in rng f holds L_Cut (f,p) = mid (f,(p .. f),(len f)) let p be Point of (TOP-REAL 2); ::_thesis: ( p in rng f implies L_Cut (f,p) = mid (f,(p .. f),(len f)) ) A1: len f >= 2 by TOPREAL1:def_8; assume p in rng f ; ::_thesis: L_Cut (f,p) = mid (f,(p .. f),(len f)) then consider i being Nat such that A2: i in dom f and A3: f . i = p by FINSEQ_2:10; A4: 0 + 1 <= i by A2, FINSEQ_3:25; A5: i <= len f by A2, FINSEQ_3:25; percases ( i > 1 or i = 1 ) by A4, XXREAL_0:1; suppose i > 1 ; ::_thesis: L_Cut (f,p) = mid (f,(p .. f),(len f)) then A6: (Index (p,f)) + 1 = i by A3, A5, JORDAN3:12; then L_Cut (f,p) = mid (f,((Index (p,f)) + 1),(len f)) by A3, JORDAN3:def_3; hence L_Cut (f,p) = mid (f,(p .. f),(len f)) by A2, A3, A6, FINSEQ_5:11; ::_thesis: verum end; supposeA7: i = 1 ; ::_thesis: L_Cut (f,p) = mid (f,(p .. f),(len f)) thus L_Cut (f,p) = L_Cut (f,(f /. i)) by A2, A3, PARTFUN1:def_6 .= f by A7, JORDAN5B:27 .= mid (f,1,(len f)) by A1, FINSEQ_6:120, XXREAL_0:2 .= mid (f,(p .. f),(len f)) by A2, A3, A7, FINSEQ_5:11 ; ::_thesis: verum end; end; end; theorem Th38: :: JORDAN1J:38 for M being Go-board for f being S-Sequence_in_R2 st f is_sequence_on M holds for p being Point of (TOP-REAL 2) st p in rng f holds R_Cut (f,p) is_sequence_on M proof let M be Go-board; ::_thesis: for f being S-Sequence_in_R2 st f is_sequence_on M holds for p being Point of (TOP-REAL 2) st p in rng f holds R_Cut (f,p) is_sequence_on M let f be S-Sequence_in_R2; ::_thesis: ( f is_sequence_on M implies for p being Point of (TOP-REAL 2) st p in rng f holds R_Cut (f,p) is_sequence_on M ) assume A1: f is_sequence_on M ; ::_thesis: for p being Point of (TOP-REAL 2) st p in rng f holds R_Cut (f,p) is_sequence_on M let p be Point of (TOP-REAL 2); ::_thesis: ( p in rng f implies R_Cut (f,p) is_sequence_on M ) assume p in rng f ; ::_thesis: R_Cut (f,p) is_sequence_on M then R_Cut (f,p) = mid (f,1,(p .. f)) by JORDAN1G:49; hence R_Cut (f,p) is_sequence_on M by A1, JORDAN1H:27; ::_thesis: verum end; theorem Th39: :: JORDAN1J:39 for M being Go-board for f being S-Sequence_in_R2 st f is_sequence_on M holds for p being Point of (TOP-REAL 2) st p in rng f holds L_Cut (f,p) is_sequence_on M proof let M be Go-board; ::_thesis: for f being S-Sequence_in_R2 st f is_sequence_on M holds for p being Point of (TOP-REAL 2) st p in rng f holds L_Cut (f,p) is_sequence_on M let f be S-Sequence_in_R2; ::_thesis: ( f is_sequence_on M implies for p being Point of (TOP-REAL 2) st p in rng f holds L_Cut (f,p) is_sequence_on M ) assume A1: f is_sequence_on M ; ::_thesis: for p being Point of (TOP-REAL 2) st p in rng f holds L_Cut (f,p) is_sequence_on M let p be Point of (TOP-REAL 2); ::_thesis: ( p in rng f implies L_Cut (f,p) is_sequence_on M ) assume p in rng f ; ::_thesis: L_Cut (f,p) is_sequence_on M then L_Cut (f,p) = mid (f,(p .. f),(len f)) by Th37; hence L_Cut (f,p) is_sequence_on M by A1, JORDAN1H:27; ::_thesis: verum end; theorem Th40: :: JORDAN1J:40 for G being Go-board for f being FinSequence of (TOP-REAL 2) st f is_sequence_on G holds for i, j being Element of NAT st 1 <= i & i <= len G & 1 <= j & j <= width G & G * (i,j) in L~ f holds G * (i,j) in rng f proof let G be Go-board; ::_thesis: for f being FinSequence of (TOP-REAL 2) st f is_sequence_on G holds for i, j being Element of NAT st 1 <= i & i <= len G & 1 <= j & j <= width G & G * (i,j) in L~ f holds G * (i,j) in rng f let f be FinSequence of (TOP-REAL 2); ::_thesis: ( f is_sequence_on G implies for i, j being Element of NAT st 1 <= i & i <= len G & 1 <= j & j <= width G & G * (i,j) in L~ f holds G * (i,j) in rng f ) assume A1: f is_sequence_on G ; ::_thesis: for i, j being Element of NAT st 1 <= i & i <= len G & 1 <= j & j <= width G & G * (i,j) in L~ f holds G * (i,j) in rng f let i, j be Element of NAT ; ::_thesis: ( 1 <= i & i <= len G & 1 <= j & j <= width G & G * (i,j) in L~ f implies G * (i,j) in rng f ) assume that A2: 1 <= i and A3: i <= len G and A4: 1 <= j and A5: j <= width G ; ::_thesis: ( not G * (i,j) in L~ f or G * (i,j) in rng f ) assume G * (i,j) in L~ f ; ::_thesis: G * (i,j) in rng f then consider k being Element of NAT such that A6: 1 <= k and A7: k + 1 <= len f and A8: G * (i,j) in LSeg ((f /. k),(f /. (k + 1))) by SPPOL_2:14; consider i1, j1, i2, j2 being Element of NAT such that A9: [i1,j1] in Indices G and A10: f /. k = G * (i1,j1) and A11: [i2,j2] in Indices G and A12: f /. (k + 1) = G * (i2,j2) and A13: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A1, A6, A7, JORDAN8:3; A14: 1 <= i1 by A9, MATRIX_1:38; A15: 1 <= j2 by A11, MATRIX_1:38; A16: i2 <= len G by A11, MATRIX_1:38; k + 1 >= 1 by NAT_1:11; then A17: k + 1 in dom f by A7, FINSEQ_3:25; A18: 1 <= j1 by A9, MATRIX_1:38; k < len f by A7, NAT_1:13; then A19: k in dom f by A6, FINSEQ_3:25; A20: i1 <= len G by A9, MATRIX_1:38; A21: j2 <= width G by A11, MATRIX_1:38; A22: 1 <= i2 by A11, MATRIX_1:38; A23: j1 <= width G by A9, 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 A13; supposeA24: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: G * (i,j) in rng f j1 <= j1 + 1 by NAT_1:11; then A25: (G * (i1,j1)) `2 <= (G * (i1,(j1 + 1))) `2 by A14, A20, A18, A21, A24, JORDAN1A:19; then (G * (i1,j1)) `2 <= (G * (i,j)) `2 by A8, A10, A12, A24, TOPREAL1:4; then A26: j1 <= j by A2, A3, A4, A14, A20, A23, Th2; A27: (G * (i1,j1)) `1 <= (G * (i1,(j1 + 1))) `1 by A14, A20, A18, A23, A15, A21, A24, JORDAN1A:18; then (G * (i1,j1)) `1 <= (G * (i,j)) `1 by A8, A10, A12, A24, TOPREAL1:3; then A28: i1 <= i by A2, A4, A5, A20, A18, A23, Th1; (G * (i,j)) `2 <= (G * (i1,(j1 + 1))) `2 by A8, A10, A12, A24, A25, TOPREAL1:4; then j <= j1 + 1 by A2, A3, A5, A14, A20, A15, A24, Th2; then A29: ( j = j1 or j = j1 + 1 ) by A26, NAT_1:9; (G * (i,j)) `1 <= (G * (i1,(j1 + 1))) `1 by A8, A10, A12, A24, A27, TOPREAL1:3; then i <= i1 by A3, A4, A5, A14, A15, A21, A24, Th1; then i = i1 by A28, XXREAL_0:1; hence G * (i,j) in rng f by A10, A12, A19, A17, A24, A29, PARTFUN2:2; ::_thesis: verum end; supposeA30: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: G * (i,j) in rng f i1 <= i1 + 1 by NAT_1:11; then A31: (G * (i1,j1)) `1 <= (G * ((i1 + 1),j1)) `1 by A14, A18, A23, A16, A30, JORDAN1A:18; then (G * (i1,j1)) `1 <= (G * (i,j)) `1 by A8, A10, A12, A30, TOPREAL1:3; then A32: i1 <= i by A2, A4, A5, A20, A18, A23, Th1; A33: (G * (i1,j1)) `2 <= (G * ((i1 + 1),j1)) `2 by A14, A20, A18, A23, A22, A16, A30, JORDAN1A:19; then (G * (i1,j1)) `2 <= (G * (i,j)) `2 by A8, A10, A12, A30, TOPREAL1:4; then A34: j1 <= j by A2, A3, A4, A14, A20, A23, Th2; (G * (i,j)) `1 <= (G * ((i1 + 1),j1)) `1 by A8, A10, A12, A30, A31, TOPREAL1:3; then i <= i1 + 1 by A3, A4, A5, A18, A23, A22, A30, Th1; then A35: ( i = i1 or i = i1 + 1 ) by A32, NAT_1:9; (G * (i,j)) `2 <= (G * ((i1 + 1),j1)) `2 by A8, A10, A12, A30, A33, TOPREAL1:4; then j <= j1 by A2, A3, A5, A18, A22, A16, A30, Th2; then j = j1 by A34, XXREAL_0:1; hence G * (i,j) in rng f by A10, A12, A19, A17, A30, A35, PARTFUN2:2; ::_thesis: verum end; supposeA36: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: G * (i,j) in rng f i2 <= i2 + 1 by NAT_1:11; then A37: (G * (i2,j1)) `1 <= (G * ((i2 + 1),j1)) `1 by A20, A18, A23, A22, A36, JORDAN1A:18; then (G * (i2,j1)) `1 <= (G * (i,j)) `1 by A8, A10, A12, A36, TOPREAL1:3; then A38: i2 <= i by A2, A4, A5, A18, A23, A16, Th1; A39: (G * (i2,j1)) `2 <= (G * ((i2 + 1),j1)) `2 by A14, A20, A18, A23, A22, A16, A36, JORDAN1A:19; then (G * (i2,j1)) `2 <= (G * (i,j)) `2 by A8, A10, A12, A36, TOPREAL1:4; then A40: j1 <= j by A2, A3, A4, A23, A22, A16, Th2; (G * (i,j)) `1 <= (G * ((i2 + 1),j1)) `1 by A8, A10, A12, A36, A37, TOPREAL1:3; then i <= i2 + 1 by A3, A4, A5, A14, A18, A23, A36, Th1; then A41: ( i = i2 or i = i2 + 1 ) by A38, NAT_1:9; (G * (i,j)) `2 <= (G * ((i2 + 1),j1)) `2 by A8, A10, A12, A36, A39, TOPREAL1:4; then j <= j1 by A2, A3, A5, A14, A20, A18, A36, Th2; then j = j1 by A40, XXREAL_0:1; hence G * (i,j) in rng f by A10, A12, A19, A17, A36, A41, PARTFUN2:2; ::_thesis: verum end; supposeA42: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: G * (i,j) in rng f j2 <= j2 + 1 by NAT_1:11; then A43: (G * (i1,j2)) `2 <= (G * (i1,(j2 + 1))) `2 by A14, A20, A23, A15, A42, JORDAN1A:19; then (G * (i1,j2)) `2 <= (G * (i,j)) `2 by A8, A10, A12, A42, TOPREAL1:4; then A44: j2 <= j by A2, A3, A4, A14, A20, A21, Th2; A45: (G * (i1,j2)) `1 <= (G * (i1,(j2 + 1))) `1 by A14, A20, A18, A23, A15, A21, A42, JORDAN1A:18; then (G * (i1,j2)) `1 <= (G * (i,j)) `1 by A8, A10, A12, A42, TOPREAL1:3; then A46: i1 <= i by A2, A4, A5, A20, A15, A21, Th1; (G * (i,j)) `2 <= (G * (i1,(j2 + 1))) `2 by A8, A10, A12, A42, A43, TOPREAL1:4; then j <= j2 + 1 by A2, A3, A5, A14, A20, A18, A42, Th2; then A47: ( j = j2 or j = j2 + 1 ) by A44, NAT_1:9; (G * (i,j)) `1 <= (G * (i1,(j2 + 1))) `1 by A8, A10, A12, A42, A45, TOPREAL1:3; then i <= i1 by A3, A4, A5, A14, A18, A23, A42, Th1; then i = i1 by A46, XXREAL_0:1; hence G * (i,j) in rng f by A10, A12, A19, A17, A42, A47, PARTFUN2:2; ::_thesis: verum end; end; end; theorem :: JORDAN1J:41 for f being S-Sequence_in_R2 for g being FinSequence of (TOP-REAL 2) st g is unfolded & g is s.n.c. & g is one-to-one & (L~ f) /\ (L~ g) = {(f /. 1)} & f /. 1 = g /. (len g) & ( for i being Element of NAT st 1 <= i & i + 2 <= len f holds (LSeg (f,i)) /\ (LSeg ((f /. (len f)),(g /. 1))) = {} ) & ( for i being Element of NAT st 2 <= i & i + 1 <= len g holds (LSeg (g,i)) /\ (LSeg ((f /. (len f)),(g /. 1))) = {} ) holds f ^ g is s.c.c. proof let f be S-Sequence_in_R2; ::_thesis: for g being FinSequence of (TOP-REAL 2) st g is unfolded & g is s.n.c. & g is one-to-one & (L~ f) /\ (L~ g) = {(f /. 1)} & f /. 1 = g /. (len g) & ( for i being Element of NAT st 1 <= i & i + 2 <= len f holds (LSeg (f,i)) /\ (LSeg ((f /. (len f)),(g /. 1))) = {} ) & ( for i being Element of NAT st 2 <= i & i + 1 <= len g holds (LSeg (g,i)) /\ (LSeg ((f /. (len f)),(g /. 1))) = {} ) holds f ^ g is s.c.c. let g be FinSequence of (TOP-REAL 2); ::_thesis: ( g is unfolded & g is s.n.c. & g is one-to-one & (L~ f) /\ (L~ g) = {(f /. 1)} & f /. 1 = g /. (len g) & ( for i being Element of NAT st 1 <= i & i + 2 <= len f holds (LSeg (f,i)) /\ (LSeg ((f /. (len f)),(g /. 1))) = {} ) & ( for i being Element of NAT st 2 <= i & i + 1 <= len g holds (LSeg (g,i)) /\ (LSeg ((f /. (len f)),(g /. 1))) = {} ) implies f ^ g is s.c.c. ) assume that A1: ( g is unfolded & g is s.n.c. & g is one-to-one ) and A2: (L~ f) /\ (L~ g) = {(f /. 1)} and A3: f /. 1 = g /. (len g) and A4: for i being Element of NAT st 1 <= i & i + 2 <= len f holds (LSeg (f,i)) /\ (LSeg ((f /. (len f)),(g /. 1))) = {} and A5: for i being Element of NAT st 2 <= i & i + 1 <= len g holds (LSeg (g,i)) /\ (LSeg ((f /. (len f)),(g /. 1))) = {} ; ::_thesis: f ^ g is s.c.c. let i, j be Element of NAT ; :: according to GOBOARD5:def_4 ::_thesis: ( j <= i + 1 or ( ( i <= 1 or len (f ^ g) <= j ) & len (f ^ g) <= j + 1 ) or LSeg ((f ^ g),i) misses LSeg ((f ^ g),j) ) assume that A6: i + 1 < j and A7: ( ( i > 1 & j < len (f ^ g) ) or j + 1 < len (f ^ g) ) ; ::_thesis: LSeg ((f ^ g),i) misses LSeg ((f ^ g),j) A8: j + 1 <= len (f ^ g) by A7, NAT_1:13; A9: now__::_thesis:_for_i_being_Element_of_NAT_st_2_<=_i_&_i_+_1_<=_len_g_holds_ LSeg_(g,i)_misses_LSeg_((f_/._(len_f)),(g_/._1)) let i be Element of NAT ; ::_thesis: ( 2 <= i & i + 1 <= len g implies LSeg (g,i) misses LSeg ((f /. (len f)),(g /. 1)) ) assume that A10: 2 <= i and A11: i + 1 <= len g ; ::_thesis: LSeg (g,i) misses LSeg ((f /. (len f)),(g /. 1)) (LSeg (g,i)) /\ (LSeg ((f /. (len f)),(g /. 1))) = {} by A5, A10, A11; hence LSeg (g,i) misses LSeg ((f /. (len f)),(g /. 1)) by XBOOLE_0:def_7; ::_thesis: verum end; A12: now__::_thesis:_for_i_being_Element_of_NAT_st_1_<=_i_&_i_+_2_<=_len_f_holds_ LSeg_(f,i)_misses_LSeg_((f_/._(len_f)),(g_/._1)) let i be Element of NAT ; ::_thesis: ( 1 <= i & i + 2 <= len f implies LSeg (f,i) misses LSeg ((f /. (len f)),(g /. 1)) ) assume that A13: 1 <= i and A14: i + 2 <= len f ; ::_thesis: LSeg (f,i) misses LSeg ((f /. (len f)),(g /. 1)) (LSeg (f,i)) /\ (LSeg ((f /. (len f)),(g /. 1))) = {} by A4, A13, A14; hence LSeg (f,i) misses LSeg ((f /. (len f)),(g /. 1)) by XBOOLE_0:def_7; ::_thesis: verum end; percases ( i = 0 or i <> 0 ) ; suppose i = 0 ; ::_thesis: LSeg ((f ^ g),i) misses LSeg ((f ^ g),j) then LSeg ((f ^ g),i) = {} by TOPREAL1:def_3; then (LSeg ((f ^ g),i)) /\ (LSeg ((f ^ g),j)) = {} ; hence LSeg ((f ^ g),i) misses LSeg ((f ^ g),j) by XBOOLE_0:def_7; ::_thesis: verum end; supposeA15: i <> 0 ; ::_thesis: LSeg ((f ^ g),i) misses LSeg ((f ^ g),j) A16: len (f ^ g) = (len f) + (len g) by FINSEQ_1:22; i <= i + 1 by NAT_1:11; then A17: i < j by A6, XXREAL_0:2; now__::_thesis:_LSeg_((f_^_g),i)_misses_LSeg_((f_^_g),j) percases ( j + 1 <= len f or j + 1 > len f ) ; supposeA18: j + 1 <= len f ; ::_thesis: LSeg ((f ^ g),i) misses LSeg ((f ^ g),j) j <= j + 1 by NAT_1:11; then i < j + 1 by A17, XXREAL_0:2; then i < len f by A18, XXREAL_0:2; then i + 1 <= len f by NAT_1:13; then A19: LSeg ((f ^ g),i) = LSeg (f,i) by SPPOL_2:6; LSeg ((f ^ g),j) = LSeg (f,j) by A18, SPPOL_2:6; hence LSeg ((f ^ g),i) misses LSeg ((f ^ g),j) by A6, A19, TOPREAL1:def_7; ::_thesis: verum end; suppose j + 1 > len f ; ::_thesis: LSeg ((f ^ g),i) misses LSeg ((f ^ g),j) then A20: len f <= j by NAT_1:13; then reconsider j9 = j - (len f) as Element of NAT by INT_1:5; A21: (j + 1) - (len f) <= len g by A8, A16, XREAL_1:20; then A22: j9 + 1 <= len g ; A23: (len f) + j9 = j ; now__::_thesis:_LSeg_((f_^_g),i)_misses_LSeg_((f_^_g),j) percases ( i <= len f or i > len f ) ; supposeA24: i <= len f ; ::_thesis: LSeg ((f ^ g),i) misses LSeg ((f ^ g),j) now__::_thesis:_LSeg_((f_^_g),i)_misses_LSeg_((f_^_g),j) percases ( i = len f or i <> len f ) ; supposeA25: i = len f ; ::_thesis: LSeg ((f ^ g),i) misses LSeg ((f ^ g),j) not g is empty by A22; then A26: LSeg ((f ^ g),i) = LSeg ((f /. (len f)),(g /. 1)) by A25, SPPOL_2:8; ((len f) + 1) + 1 <= j by A6, A25, NAT_1:13; then (len f) + (1 + 1) <= j ; then A27: 1 + 1 <= j9 by XREAL_1:19; then LSeg ((f ^ g),j) = LSeg (g,j9) by A23, SPPOL_2:7, XXREAL_0:2; hence LSeg ((f ^ g),i) misses LSeg ((f ^ g),j) by A9, A22, A27, A26; ::_thesis: verum end; suppose i <> len f ; ::_thesis: LSeg ((f ^ g),i) misses LSeg ((f ^ g),j) then i < len f by A24, XXREAL_0:1; then i + 1 <= len f by NAT_1:13; then A28: LSeg ((f ^ g),i) = LSeg (f,i) by SPPOL_2:6; now__::_thesis:_LSeg_((f_^_g),i)_misses_LSeg_((f_^_g),j) percases ( j = len f or j <> len f ) ; supposeA29: j = len f ; ::_thesis: LSeg ((f ^ g),i) misses LSeg ((f ^ g),j) then (i + 1) + 1 <= len f by A6, NAT_1:13; then A30: i + (1 + 1) <= len f ; not g is empty by A8, A16, A29, XREAL_1:6; then LSeg ((f ^ g),j) = LSeg ((f /. (len f)),(g /. 1)) by A29, SPPOL_2:8; hence LSeg ((f ^ g),i) misses LSeg ((f ^ g),j) by A12, A15, A28, A30, NAT_1:14; ::_thesis: verum end; supposeA31: j <> len f ; ::_thesis: LSeg ((f ^ g),i) misses LSeg ((f ^ g),j) A32: len f >= 2 by TOPREAL1:def_8; A33: LSeg ((f ^ g),i) c= L~ f by A28, TOPREAL3:19; len f < j by A20, A31, XXREAL_0:1; then (len f) + 1 <= j by NAT_1:13; then A34: 1 <= j9 by XREAL_1:19; then A35: LSeg ((f ^ g),((len f) + j9)) = LSeg (g,j9) by SPPOL_2:7; then LSeg ((f ^ g),j) c= L~ g by TOPREAL3:19; then A36: (LSeg ((f ^ g),i)) /\ (LSeg ((f ^ g),j)) c= {(f /. 1)} by A2, A33, XBOOLE_1:27; now__::_thesis:_LSeg_((f_^_g),i)_misses_LSeg_((f_^_g),j) percases ( (LSeg ((f ^ g),i)) /\ (LSeg ((f ^ g),j)) = {} or (LSeg ((f ^ g),i)) /\ (LSeg ((f ^ g),j)) = {(f /. 1)} ) by A36, ZFMISC_1:33; suppose (LSeg ((f ^ g),i)) /\ (LSeg ((f ^ g),j)) = {} ; ::_thesis: LSeg ((f ^ g),i) misses LSeg ((f ^ g),j) hence LSeg ((f ^ g),i) misses LSeg ((f ^ g),j) by XBOOLE_0:def_7; ::_thesis: verum end; suppose (LSeg ((f ^ g),i)) /\ (LSeg ((f ^ g),j)) = {(f /. 1)} ; ::_thesis: LSeg ((f ^ g),i) misses LSeg ((f ^ g),j) then A37: f /. 1 in (LSeg ((f ^ g),i)) /\ (LSeg ((f ^ g),j)) by TARSKI:def_1; then A38: f /. 1 in LSeg ((f ^ g),i) by XBOOLE_0:def_4; j9 < len g by A22, NAT_1:13; then A39: j9 in dom g by A34, FINSEQ_3:25; j9 + 1 >= 1 by NAT_1:11; then A40: j9 + 1 in dom g by A21, FINSEQ_3:25; f /. 1 in LSeg ((f ^ g),j) by A37, XBOOLE_0:def_4; then j9 + 1 = len g by A1, A3, A35, A39, A40, GOBOARD2:2; hence LSeg ((f ^ g),i) misses LSeg ((f ^ g),j) by A7, A16, A28, A32, A38, JORDAN5B:30; ::_thesis: verum end; end; end; hence LSeg ((f ^ g),i) misses LSeg ((f ^ g),j) ; ::_thesis: verum end; end; end; hence LSeg ((f ^ g),i) misses LSeg ((f ^ g),j) ; ::_thesis: verum end; end; end; hence LSeg ((f ^ g),i) misses LSeg ((f ^ g),j) ; ::_thesis: verum end; supposeA41: i > len f ; ::_thesis: LSeg ((f ^ g),i) misses LSeg ((f ^ g),j) then j <> len f by A6, NAT_1:13; then len f < j by A20, XXREAL_0:1; then (len f) + 1 <= j by NAT_1:13; then 1 <= j9 by XREAL_1:19; then A42: LSeg ((f ^ g),((len f) + j9)) = LSeg (g,j9) by SPPOL_2:7; reconsider i9 = i - (len f) as Element of NAT by A41, INT_1:5; (len f) + 1 <= i by A41, NAT_1:13; then 1 <= i9 by XREAL_1:19; then A43: LSeg ((f ^ g),((len f) + i9)) = LSeg (g,i9) by SPPOL_2:7; (i + 1) - (len f) < j9 by A6, XREAL_1:9; then i9 + 1 < j9 ; hence LSeg ((f ^ g),i) misses LSeg ((f ^ g),j) by A1, A43, A42, TOPREAL1:def_7; ::_thesis: verum end; end; end; hence LSeg ((f ^ g),i) misses LSeg ((f ^ g),j) ; ::_thesis: verum end; end; end; hence LSeg ((f ^ g),i) misses LSeg ((f ^ g),j) ; ::_thesis: verum end; end; end; theorem :: JORDAN1J:42 for n being Element of NAT for C being non empty compact non horizontal non vertical Subset of (TOP-REAL 2) ex i being Element of NAT st ( 1 <= i & i + 1 <= len (Gauge (C,n)) & W-min C in cell ((Gauge (C,n)),1,i) & W-min C <> (Gauge (C,n)) * (2,i) ) proof let n be Element of NAT ; ::_thesis: for C being non empty compact non horizontal non vertical Subset of (TOP-REAL 2) ex i being Element of NAT st ( 1 <= i & i + 1 <= len (Gauge (C,n)) & W-min C in cell ((Gauge (C,n)),1,i) & W-min C <> (Gauge (C,n)) * (2,i) ) let C be non empty compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ex i being Element of NAT st ( 1 <= i & i + 1 <= len (Gauge (C,n)) & W-min C in cell ((Gauge (C,n)),1,i) & W-min C <> (Gauge (C,n)) * (2,i) ) set G = Gauge (C,n); defpred S1[ Nat] means ( 1 <= $1 & $1 < len (Gauge (C,n)) & ((Gauge (C,n)) * (2,$1)) `2 < (W-min C) `2 ); A1: for k being Nat st S1[k] holds k <= len (Gauge (C,n)) ; A2: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1; (SW-corner C) `2 <= (W-min C) `2 by PSCOMP_1:30; then A3: S-bound C <= (W-min C) `2 by EUCLID:52; A4: len (Gauge (C,n)) >= 4 by JORDAN8:10; then A5: 1 < len (Gauge (C,n)) by XXREAL_0:2; A6: 2 <= len (Gauge (C,n)) by A4, XXREAL_0:2; then ((Gauge (C,n)) * (2,2)) `2 = S-bound C by JORDAN8:13; then ((Gauge (C,n)) * (2,1)) `2 < S-bound C by A2, A6, GOBOARD5:4; then ((Gauge (C,n)) * (2,1)) `2 < (W-min C) `2 by A3, XXREAL_0:2; then A7: ex k being Nat st S1[k] by A5; ex i being Nat st ( S1[i] & ( for n being Nat st S1[n] holds n <= i ) ) from NAT_1:sch_6(A1, A7); then consider i being Nat such that A8: 1 <= i and A9: i < len (Gauge (C,n)) and A10: ((Gauge (C,n)) * (2,i)) `2 < (W-min C) `2 and A11: for n being Nat st S1[n] holds n <= i ; reconsider i = i as Element of NAT by ORDINAL1:def_12; A12: (W-min C) `1 = W-bound C by EUCLID:52; then A13: ((Gauge (C,n)) * (2,i)) `1 = (W-min C) `1 by A8, A9, JORDAN8:11; A14: i + 1 <= len (Gauge (C,n)) by A9, NAT_1:13; then A15: (W-min C) `1 = ((Gauge (C,n)) * (2,(i + 1))) `1 by A12, JORDAN8:11, NAT_1:12; A16: i < i + 1 by NAT_1:13; A17: 1 <= i + 1 by NAT_1:12; now__::_thesis:_not_i_+_1_=_len_(Gauge_(C,n)) assume i + 1 = len (Gauge (C,n)) ; ::_thesis: contradiction then (len (Gauge (C,n))) -' 1 = i by NAT_D:34; then A18: ((Gauge (C,n)) * (2,i)) `2 = N-bound C by A6, JORDAN8:14; (NW-corner C) `2 >= (W-min C) `2 by PSCOMP_1:30; hence contradiction by A10, A18, EUCLID:52; ::_thesis: verum end; then i + 1 < len (Gauge (C,n)) by A14, XXREAL_0:1; then (W-min C) `2 <= ((Gauge (C,n)) * (2,(i + 1))) `2 by A11, A17, A16; then A19: W-min C in LSeg (((Gauge (C,n)) * (2,i)),((Gauge (C,n)) * (2,(i + 1)))) by A10, A13, A15, GOBOARD7:7; take i ; ::_thesis: ( 1 <= i & i + 1 <= len (Gauge (C,n)) & W-min C in cell ((Gauge (C,n)),1,i) & W-min C <> (Gauge (C,n)) * (2,i) ) thus ( 1 <= i & i + 1 <= len (Gauge (C,n)) ) by A8, A9, NAT_1:13; ::_thesis: ( W-min C in cell ((Gauge (C,n)),1,i) & W-min C <> (Gauge (C,n)) * (2,i) ) len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1; then LSeg (((Gauge (C,n)) * ((1 + 1),i)),((Gauge (C,n)) * ((1 + 1),(i + 1)))) c= cell ((Gauge (C,n)),1,i) by A5, A8, A9, GOBOARD5:18; hence W-min C in cell ((Gauge (C,n)),1,i) by A19; ::_thesis: W-min C <> (Gauge (C,n)) * (2,i) thus W-min C <> (Gauge (C,n)) * (2,i) by A10; ::_thesis: verum end; theorem Th43: :: JORDAN1J:43 for f being S-Sequence_in_R2 for p being Point of (TOP-REAL 2) st p in L~ f & f . (len f) in L~ (R_Cut (f,p)) holds f . (len f) = p proof let f be S-Sequence_in_R2; ::_thesis: for p being Point of (TOP-REAL 2) st p in L~ f & f . (len f) in L~ (R_Cut (f,p)) holds f . (len f) = p let p be Point of (TOP-REAL 2); ::_thesis: ( p in L~ f & f . (len f) in L~ (R_Cut (f,p)) implies f . (len f) = p ) assume that A1: p in L~ f and A2: f . (len f) in L~ (R_Cut (f,p)) ; ::_thesis: f . (len f) = p A3: L~ f = L~ (Rev f) by SPPOL_2:22; A4: (Rev f) . 1 = f . (len f) by FINSEQ_5:62; L_Cut ((Rev f),p) = Rev (R_Cut (f,p)) by A1, JORDAN3:22; then (Rev f) . 1 in L~ (L_Cut ((Rev f),p)) by A2, A4, SPPOL_2:22; hence f . (len f) = p by A1, A3, A4, JORDAN1E:7; ::_thesis: verum end; theorem Th44: :: JORDAN1J:44 for f being FinSequence of (TOP-REAL 2) for p being Point of (TOP-REAL 2) holds R_Cut (f,p) <> {} proof let f be FinSequence of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) holds R_Cut (f,p) <> {} let p be Point of (TOP-REAL 2); ::_thesis: R_Cut (f,p) <> {} percases ( p <> f . 1 or p = f . 1 ) ; suppose p <> f . 1 ; ::_thesis: R_Cut (f,p) <> {} then R_Cut (f,p) = (mid (f,1,(Index (p,f)))) ^ <*p*> by JORDAN3:def_4; hence R_Cut (f,p) <> {} ; ::_thesis: verum end; suppose p = f . 1 ; ::_thesis: R_Cut (f,p) <> {} then R_Cut (f,p) = <*p*> by JORDAN3:def_4; hence R_Cut (f,p) <> {} ; ::_thesis: verum end; end; end; theorem Th45: :: JORDAN1J:45 for f being FinSequence of (TOP-REAL 2) for p being Point of (TOP-REAL 2) st p in L~ f holds (R_Cut (f,p)) /. (len (R_Cut (f,p))) = p proof let f be FinSequence of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st p in L~ f holds (R_Cut (f,p)) /. (len (R_Cut (f,p))) = p let p be Point of (TOP-REAL 2); ::_thesis: ( p in L~ f implies (R_Cut (f,p)) /. (len (R_Cut (f,p))) = p ) assume A1: p in L~ f ; ::_thesis: (R_Cut (f,p)) /. (len (R_Cut (f,p))) = p not R_Cut (f,p) is empty by Th44; then len (R_Cut (f,p)) in dom (R_Cut (f,p)) by FINSEQ_5:6; hence (R_Cut (f,p)) /. (len (R_Cut (f,p))) = (R_Cut (f,p)) . (len (R_Cut (f,p))) by PARTFUN1:def_6 .= p by A1, JORDAN3:24 ; ::_thesis: verum end; theorem Th46: :: JORDAN1J:46 for n being Element of NAT for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) for p being Point of (TOP-REAL 2) st p in L~ (Upper_Seq (C,n)) & p `1 = E-bound (L~ (Cage (C,n))) holds p = E-max (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) for p being Point of (TOP-REAL 2) st p in L~ (Upper_Seq (C,n)) & p `1 = E-bound (L~ (Cage (C,n))) holds p = E-max (L~ (Cage (C,n))) let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st p in L~ (Upper_Seq (C,n)) & p `1 = E-bound (L~ (Cage (C,n))) holds p = E-max (L~ (Cage (C,n))) let p be Point of (TOP-REAL 2); ::_thesis: ( p in L~ (Upper_Seq (C,n)) & p `1 = E-bound (L~ (Cage (C,n))) implies p = E-max (L~ (Cage (C,n))) ) set Ca = Cage (C,n); set US = Upper_Seq (C,n); set LS = Lower_Seq (C,n); set Wmin = W-min (L~ (Cage (C,n))); set Smax = S-max (L~ (Cage (C,n))); set Smin = S-min (L~ (Cage (C,n))); set Emin = E-min (L~ (Cage (C,n))); set Emax = E-max (L~ (Cage (C,n))); set Wbo = W-bound (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 NE = NE-corner (L~ (Cage (C,n))); assume that A1: p in L~ (Upper_Seq (C,n)) and A2: p `1 = E-bound (L~ (Cage (C,n))) and A3: p <> E-max (L~ (Cage (C,n))) ; ::_thesis: contradiction A4: (Upper_Seq (C,n)) /. 1 = W-min (L~ (Cage (C,n))) by JORDAN1F:5; 1 in dom (Upper_Seq (C,n)) by FINSEQ_5:6; then A5: (Upper_Seq (C,n)) . 1 = W-min (L~ (Cage (C,n))) by A4, PARTFUN1:def_6; W-bound (L~ (Cage (C,n))) <> E-bound (L~ (Cage (C,n))) by SPRECT_1:31; then p <> (Upper_Seq (C,n)) . 1 by A2, A5, EUCLID:52; then reconsider g = R_Cut ((Upper_Seq (C,n)),p) as being_S-Seq FinSequence of (TOP-REAL 2) by A1, JORDAN3:35; <*p*> is_in_the_area_of Cage (C,n) by A1, JORDAN1E:17, SPRECT_3:46; then A6: g is_in_the_area_of Cage (C,n) by A1, JORDAN1E:17, SPRECT_3:52; len g in dom g by FINSEQ_5:6; then A7: g /. (len g) = g . (len g) by PARTFUN1:def_6 .= p by A1, JORDAN3:24 ; (g /. 1) `1 = ((Upper_Seq (C,n)) /. 1) `1 by A1, SPRECT_3:22 .= (W-min (L~ (Cage (C,n)))) `1 by JORDAN1F:5 .= W-bound (L~ (Cage (C,n))) by EUCLID:52 ; then A8: g is_a_h.c._for Cage (C,n) by A2, A6, A7, SPRECT_2:def_2; A9: (Lower_Seq (C,n)) /. 1 = E-max (L~ (Cage (C,n))) by JORDAN1F:6; 1 in dom (Lower_Seq (C,n)) by FINSEQ_5:6; then A10: (Lower_Seq (C,n)) . 1 = E-max (L~ (Cage (C,n))) by A9, PARTFUN1:def_6; len (Cage (C,n)) > 4 by GOBOARD7:34; then A11: rng (Cage (C,n)) c= L~ (Cage (C,n)) by SPPOL_2:18, XXREAL_0:2; now__::_thesis:_contradiction percases ( E-max (L~ (Cage (C,n))) <> NE-corner (L~ (Cage (C,n))) or E-max (L~ (Cage (C,n))) = NE-corner (L~ (Cage (C,n))) ) ; supposeA12: E-max (L~ (Cage (C,n))) <> NE-corner (L~ (Cage (C,n))) ; ::_thesis: contradiction A13: not NE-corner (L~ (Cage (C,n))) in rng (Cage (C,n)) proof A14: (NE-corner (L~ (Cage (C,n)))) `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52; A15: (NE-corner (L~ (Cage (C,n)))) `2 = N-bound (L~ (Cage (C,n))) by EUCLID:52; then (NE-corner (L~ (Cage (C,n)))) `2 >= S-bound (L~ (Cage (C,n))) by SPRECT_1:22; then NE-corner (L~ (Cage (C,n))) in { p1 where p1 is Point of (TOP-REAL 2) : ( p1 `1 = E-bound (L~ (Cage (C,n))) & p1 `2 <= N-bound (L~ (Cage (C,n))) & p1 `2 >= S-bound (L~ (Cage (C,n))) ) } by A14, A15; then A16: 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 A11, A16, XBOOLE_0:def_4; then A17: (NE-corner (L~ (Cage (C,n)))) `2 <= (E-max (L~ (Cage (C,n)))) `2 by PSCOMP_1:47; A18: (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 A17, XXREAL_0:1; hence contradiction by A12, A18, TOPREAL3:6; ::_thesis: verum end; S-max (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) by Th12; then R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))) = mid ((Lower_Seq (C,n)),1,((S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)))) by JORDAN1G:49; then A19: rng (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n)))))) c= rng (Lower_Seq (C,n)) by FINSEQ_6:119; rng (Lower_Seq (C,n)) c= rng (Cage (C,n)) by JORDAN1G:39; then rng (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n)))))) c= rng (Cage (C,n)) by A19, XBOOLE_1:1; then not NE-corner (L~ (Cage (C,n))) in rng (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n)))))) by A13; then rng (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n)))))) misses {(NE-corner (L~ (Cage (C,n))))} by ZFMISC_1:50; then rng (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n)))))) misses rng <*(NE-corner (L~ (Cage (C,n))))*> by FINSEQ_1:38; then A20: rng (Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) misses rng <*(NE-corner (L~ (Cage (C,n))))*> by FINSEQ_5:57; set h = (Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>; A21: <*(NE-corner (L~ (Cage (C,n))))*> is one-to-one by FINSEQ_3:93; A22: (((Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) /. (len ((Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>))) `2 = (((Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) /. ((len (Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n)))))))) + 1)) `2 by FINSEQ_2:16 .= (NE-corner (L~ (Cage (C,n)))) `2 by FINSEQ_4:67 .= N-bound (L~ (Cage (C,n))) by EUCLID:52 ; E-min (L~ (Cage (C,n))) in L~ (Cage (C,n)) by SPRECT_1:14; then A23: S-bound (L~ (Cage (C,n))) <= (E-min (L~ (Cage (C,n)))) `2 by PSCOMP_1:24; A24: (Index ((S-max (L~ (Cage (C,n)))),(Lower_Seq (C,n)))) + 1 >= 0 + 1 by XREAL_1:7; A25: S-max (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) by Th12; then <*(S-max (L~ (Cage (C,n))))*> is_in_the_area_of Cage (C,n) by JORDAN1E:18, SPRECT_3:46; then R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))) is_in_the_area_of Cage (C,n) by A25, JORDAN1E:18, SPRECT_3:52; then A26: Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n)))))) is_in_the_area_of Cage (C,n) by SPRECT_3:51; (E-min (L~ (Cage (C,n)))) `2 < (E-max (L~ (Cage (C,n)))) `2 by SPRECT_2:53; then A27: S-max (L~ (Cage (C,n))) <> (Lower_Seq (C,n)) . 1 by A10, A23, EUCLID:52; then reconsider RCutLS = R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))) as being_S-Seq FinSequence of (TOP-REAL 2) by A25, JORDAN3:35; len ((Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) = (len (Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n)))))))) + 1 by FINSEQ_2:16 .= (len (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) + 1 by FINSEQ_5:def_3 .= ((Index ((S-max (L~ (Cage (C,n)))),(Lower_Seq (C,n)))) + 1) + 1 by A25, A27, JORDAN3:25 ; then A28: len ((Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) >= 1 + 1 by A24, XREAL_1:7; A29: 2 <= len g by TOPREAL1:def_8; 1 in dom (Rev RCutLS) by FINSEQ_5:6; then ((Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) /. 1 = (Rev RCutLS) /. 1 by FINSEQ_4:68 .= (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n)))))) /. (len (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) by FINSEQ_5:65 .= S-max (L~ (Cage (C,n))) by A25, Th45 ; then A30: (((Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) /. 1) `2 = S-bound (L~ (Cage (C,n))) by EUCLID:52; <*(NE-corner (L~ (Cage (C,n))))*> is_in_the_area_of Cage (C,n) by SPRECT_2:25; then (Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) ^ <*(NE-corner (L~ (Cage (C,n))))*> is_in_the_area_of Cage (C,n) by A26, SPRECT_2:24; then A31: (Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) ^ <*(NE-corner (L~ (Cage (C,n))))*> is_a_v.c._for Cage (C,n) by A30, A22, SPRECT_2:def_3; A32: len (Lower_Seq (C,n)) in dom (Lower_Seq (C,n)) by FINSEQ_5:6; A33: (Rev RCutLS) /. (len (Rev RCutLS)) = (Rev RCutLS) /. (len RCutLS) by FINSEQ_5:def_3 .= RCutLS /. 1 by FINSEQ_5:65 .= (Lower_Seq (C,n)) /. 1 by A25, SPRECT_3:22 .= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ; then ((Rev RCutLS) /. (len (Rev RCutLS))) `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52 .= (NE-corner (L~ (Cage (C,n)))) `1 by EUCLID:52 .= (<*(NE-corner (L~ (Cage (C,n))))*> /. 1) `1 by FINSEQ_4:16 ; then ( (Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) ^ <*(NE-corner (L~ (Cage (C,n))))*> is one-to-one & (Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) ^ <*(NE-corner (L~ (Cage (C,n))))*> is special ) by A20, A21, FINSEQ_3:91, GOBOARD2:8; then L~ g meets L~ ((Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) by A8, A29, A28, A31, SPRECT_2:29; then consider x being set such that A34: x in L~ g and A35: x in L~ ((Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) by XBOOLE_0:3; reconsider x = x as Point of (TOP-REAL 2) by A34; A36: L~ ((Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) = (L~ (Rev RCutLS)) \/ (LSeg (((Rev RCutLS) /. (len (Rev RCutLS))),(NE-corner (L~ (Cage (C,n)))))) by SPPOL_2:19; A37: L~ RCutLS c= L~ (Lower_Seq (C,n)) by Th12, JORDAN3:41; A38: len (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by FINSEQ_5:6; A39: L~ g c= L~ (Upper_Seq (C,n)) by A1, JORDAN3:41; then A40: x in L~ (Upper_Seq (C,n)) by A34; now__::_thesis:_contradiction percases ( x in L~ (Rev RCutLS) or x in LSeg (((Rev RCutLS) /. (len (Rev RCutLS))),(NE-corner (L~ (Cage (C,n))))) ) by A35, A36, XBOOLE_0:def_3; suppose x in L~ (Rev RCutLS) ; ::_thesis: contradiction then A41: x in L~ RCutLS by SPPOL_2:22; then x in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A34, A39, A37, XBOOLE_0:def_4; then A42: 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 A42, TARSKI:def_2; suppose x = W-min (L~ (Cage (C,n))) ; ::_thesis: contradiction then (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) in L~ (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n)))))) by A41, JORDAN1F:8; then (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) in L~ (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n)))))) by A32, PARTFUN1:def_6; then (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) = S-max (L~ (Cage (C,n))) by A25, Th43; then (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) = S-max (L~ (Cage (C,n))) by A32, PARTFUN1:def_6; then A43: W-min (L~ (Cage (C,n))) = S-max (L~ (Cage (C,n))) by JORDAN1F:8; S-min (L~ (Cage (C,n))) in L~ (Cage (C,n)) by SPRECT_1:12; then A44: W-bound (L~ (Cage (C,n))) <= (S-min (L~ (Cage (C,n)))) `1 by PSCOMP_1:24; (S-min (L~ (Cage (C,n)))) `1 < (S-max (L~ (Cage (C,n)))) `1 by SPRECT_2:55; hence contradiction by A43, A44, EUCLID:52; ::_thesis: verum end; suppose x = E-max (L~ (Cage (C,n))) ; ::_thesis: contradiction then (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) in L~ (R_Cut ((Upper_Seq (C,n)),p)) by A34, JORDAN1F:7; then (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) in L~ (R_Cut ((Upper_Seq (C,n)),p)) by A38, PARTFUN1:def_6; then (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) = p by A1, Th43; then (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) = p by A38, PARTFUN1:def_6; hence contradiction by A3, JORDAN1F:7; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; supposeA45: x in LSeg (((Rev RCutLS) /. (len (Rev RCutLS))),(NE-corner (L~ (Cage (C,n))))) ; ::_thesis: contradiction (E-max (L~ (Cage (C,n)))) `2 <= (NE-corner (L~ (Cage (C,n)))) `2 by PSCOMP_1:46; then A46: (E-max (L~ (Cage (C,n)))) `2 <= x `2 by A33, A45, TOPREAL1:4; A47: (E-max (L~ (Cage (C,n)))) `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52; (NE-corner (L~ (Cage (C,n)))) `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52; then A48: x `1 = E-bound (L~ (Cage (C,n))) by A33, A45, A47, GOBOARD7:5; L~ (Cage (C,n)) = (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n))) by JORDAN1E:13; then L~ (Upper_Seq (C,n)) c= L~ (Cage (C,n)) by XBOOLE_1:7; then x in E-most (L~ (Cage (C,n))) by A40, A48, SPRECT_2:13; then x `2 <= (E-max (L~ (Cage (C,n)))) `2 by PSCOMP_1:47; then x `2 = (E-max (L~ (Cage (C,n)))) `2 by A46, XXREAL_0:1; then x = E-max (L~ (Cage (C,n))) by A47, A48, TOPREAL3:6; then (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) in L~ (R_Cut ((Upper_Seq (C,n)),p)) by A34, JORDAN1F:7; then (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) in L~ (R_Cut ((Upper_Seq (C,n)),p)) by A38, PARTFUN1:def_6; then (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) = p by A1, Th43; then (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) = p by A38, PARTFUN1:def_6; hence contradiction by A3, JORDAN1F:7; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; supposeA49: E-max (L~ (Cage (C,n))) = NE-corner (L~ (Cage (C,n))) ; ::_thesis: contradiction E-min (L~ (Cage (C,n))) in L~ (Cage (C,n)) by SPRECT_1:14; then A50: S-bound (L~ (Cage (C,n))) <= (E-min (L~ (Cage (C,n)))) `2 by PSCOMP_1:24; set h = Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n)))))); A51: 2 <= len g by TOPREAL1:def_8; A52: S-max (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) by Th12; then <*(S-max (L~ (Cage (C,n))))*> is_in_the_area_of Cage (C,n) by JORDAN1E:18, SPRECT_3:46; then R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))) is_in_the_area_of Cage (C,n) by A52, JORDAN1E:18, SPRECT_3:52; then A53: Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n)))))) is_in_the_area_of Cage (C,n) by SPRECT_3:51; (E-min (L~ (Cage (C,n)))) `2 < (E-max (L~ (Cage (C,n)))) `2 by SPRECT_2:53; then S-max (L~ (Cage (C,n))) <> (Lower_Seq (C,n)) . 1 by A10, A50, EUCLID:52; then reconsider RCutLS = R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))) as being_S-Seq FinSequence of (TOP-REAL 2) by A52, JORDAN3:35; 1 in dom (Rev RCutLS) by FINSEQ_5:6; then (Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) /. 1 = (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n)))))) /. (len (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) by FINSEQ_5:65 .= S-max (L~ (Cage (C,n))) by A52, Th45 ; then A54: ((Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) /. 1) `2 = S-bound (L~ (Cage (C,n))) by EUCLID:52; A55: Rev RCutLS is special ; len RCutLS >= 2 by TOPREAL1:def_8; then A56: len (Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) >= 2 by FINSEQ_5:def_3; (Rev RCutLS) /. (len (Rev RCutLS)) = (Rev RCutLS) /. (len RCutLS) by FINSEQ_5:def_3 .= RCutLS /. 1 by FINSEQ_5:65 .= (Lower_Seq (C,n)) /. 1 by A52, SPRECT_3:22 .= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ; then ((Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) /. (len (Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))))) `2 = N-bound (L~ (Cage (C,n))) by A49, EUCLID:52; then Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n)))))) is_a_v.c._for Cage (C,n) by A53, A54, SPRECT_2:def_3; then L~ g meets L~ (Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) by A8, A55, A51, A56, SPRECT_2:29; then consider x being set such that A57: x in L~ g and A58: x in L~ (Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) by XBOOLE_0:3; reconsider x = x as Point of (TOP-REAL 2) by A57; A59: x in L~ RCutLS by A58, SPPOL_2:22; A60: L~ g c= L~ (Upper_Seq (C,n)) by A1, JORDAN3:41; A61: len (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by FINSEQ_5:6; A62: len (Lower_Seq (C,n)) in dom (Lower_Seq (C,n)) by FINSEQ_5:6; L~ RCutLS c= L~ (Lower_Seq (C,n)) by Th12, JORDAN3:41; then x in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A57, A60, A59, XBOOLE_0:def_4; then A63: 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 A63, TARSKI:def_2; suppose x = W-min (L~ (Cage (C,n))) ; ::_thesis: contradiction then (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) in L~ (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n)))))) by A59, JORDAN1F:8; then (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) in L~ (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n)))))) by A62, PARTFUN1:def_6; then (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) = S-max (L~ (Cage (C,n))) by A52, Th43; then (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) = S-max (L~ (Cage (C,n))) by A62, PARTFUN1:def_6; then A64: W-min (L~ (Cage (C,n))) = S-max (L~ (Cage (C,n))) by JORDAN1F:8; S-min (L~ (Cage (C,n))) in L~ (Cage (C,n)) by SPRECT_1:12; then A65: W-bound (L~ (Cage (C,n))) <= (S-min (L~ (Cage (C,n)))) `1 by PSCOMP_1:24; (S-min (L~ (Cage (C,n)))) `1 < (S-max (L~ (Cage (C,n)))) `1 by SPRECT_2:55; hence contradiction by A64, A65, EUCLID:52; ::_thesis: verum end; suppose x = E-max (L~ (Cage (C,n))) ; ::_thesis: contradiction then (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) in L~ (R_Cut ((Upper_Seq (C,n)),p)) by A57, JORDAN1F:7; then (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) in L~ (R_Cut ((Upper_Seq (C,n)),p)) by A61, PARTFUN1:def_6; then (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) = p by A1, Th43; then (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) = p by A61, PARTFUN1:def_6; hence contradiction by A3, JORDAN1F:7; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; theorem :: JORDAN1J:47 for n being Element of NAT for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) for p being Point of (TOP-REAL 2) st p in L~ (Lower_Seq (C,n)) & p `1 = W-bound (L~ (Cage (C,n))) holds p = 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) for p being Point of (TOP-REAL 2) st p in L~ (Lower_Seq (C,n)) & p `1 = W-bound (L~ (Cage (C,n))) holds p = W-min (L~ (Cage (C,n))) let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st p in L~ (Lower_Seq (C,n)) & p `1 = W-bound (L~ (Cage (C,n))) holds p = W-min (L~ (Cage (C,n))) let p be Point of (TOP-REAL 2); ::_thesis: ( p in L~ (Lower_Seq (C,n)) & p `1 = W-bound (L~ (Cage (C,n))) implies p = W-min (L~ (Cage (C,n))) ) set Ca = Cage (C,n); set LS = Lower_Seq (C,n); set US = Upper_Seq (C,n); set Emax = E-max (L~ (Cage (C,n))); set Nmin = N-min (L~ (Cage (C,n))); set Nmax = N-max (L~ (Cage (C,n))); set Wmax = W-max (L~ (Cage (C,n))); set Wmin = W-min (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 Nbo = N-bound (L~ (Cage (C,n))); set SW = SW-corner (L~ (Cage (C,n))); assume that A1: p in L~ (Lower_Seq (C,n)) and A2: p `1 = W-bound (L~ (Cage (C,n))) and A3: p <> W-min (L~ (Cage (C,n))) ; ::_thesis: contradiction A4: (Lower_Seq (C,n)) /. 1 = E-max (L~ (Cage (C,n))) by JORDAN1F:6; 1 in dom (Lower_Seq (C,n)) by FINSEQ_5:6; then A5: (Lower_Seq (C,n)) . 1 = E-max (L~ (Cage (C,n))) by A4, PARTFUN1:def_6; E-bound (L~ (Cage (C,n))) <> W-bound (L~ (Cage (C,n))) by SPRECT_1:31; then p <> (Lower_Seq (C,n)) . 1 by A2, A5, EUCLID:52; then reconsider g1 = R_Cut ((Lower_Seq (C,n)),p) as being_S-Seq FinSequence of (TOP-REAL 2) by A1, JORDAN3:35; len g1 in dom g1 by FINSEQ_5:6; then A6: g1 /. (len g1) = g1 . (len g1) by PARTFUN1:def_6 .= p by A1, JORDAN3:24 ; reconsider g = Rev g1 as being_S-Seq FinSequence of (TOP-REAL 2) ; <*p*> is_in_the_area_of Cage (C,n) by A1, JORDAN1E:18, SPRECT_3:46; then g1 is_in_the_area_of Cage (C,n) by A1, JORDAN1E:18, SPRECT_3:52; then A7: g is_in_the_area_of Cage (C,n) by SPRECT_3:51; A8: g /. 1 = g1 /. (len g1) by FINSEQ_5:65; A9: g /. (len g) = g /. (len g1) by FINSEQ_5:def_3 .= g1 /. 1 by FINSEQ_5:65 ; (g1 /. 1) `1 = ((Lower_Seq (C,n)) /. 1) `1 by A1, SPRECT_3:22 .= (E-max (L~ (Cage (C,n)))) `1 by JORDAN1F:6 .= E-bound (L~ (Cage (C,n))) by EUCLID:52 ; then A10: g is_a_h.c._for Cage (C,n) by A2, A7, A8, A9, A6, SPRECT_2:def_2; A11: (Upper_Seq (C,n)) /. 1 = W-min (L~ (Cage (C,n))) by JORDAN1F:5; 1 in dom (Upper_Seq (C,n)) by FINSEQ_5:6; then A12: (Upper_Seq (C,n)) . 1 = W-min (L~ (Cage (C,n))) by A11, PARTFUN1:def_6; A13: L~ g = L~ g1 by SPPOL_2:22; len (Cage (C,n)) > 4 by GOBOARD7:34; then A14: rng (Cage (C,n)) c= L~ (Cage (C,n)) by SPPOL_2:18, XXREAL_0:2; now__::_thesis:_contradiction percases ( W-min (L~ (Cage (C,n))) <> SW-corner (L~ (Cage (C,n))) or W-min (L~ (Cage (C,n))) = SW-corner (L~ (Cage (C,n))) ) ; supposeA15: W-min (L~ (Cage (C,n))) <> SW-corner (L~ (Cage (C,n))) ; ::_thesis: contradiction A16: not SW-corner (L~ (Cage (C,n))) in rng (Cage (C,n)) proof A17: (SW-corner (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52; A18: (SW-corner (L~ (Cage (C,n)))) `2 = S-bound (L~ (Cage (C,n))) by EUCLID:52; then (SW-corner (L~ (Cage (C,n)))) `2 <= N-bound (L~ (Cage (C,n))) by SPRECT_1:22; then SW-corner (L~ (Cage (C,n))) in { p1 where p1 is Point of (TOP-REAL 2) : ( p1 `1 = W-bound (L~ (Cage (C,n))) & p1 `2 <= N-bound (L~ (Cage (C,n))) & p1 `2 >= S-bound (L~ (Cage (C,n))) ) } by A17, A18; then A19: SW-corner (L~ (Cage (C,n))) in LSeg ((SW-corner (L~ (Cage (C,n)))),(NW-corner (L~ (Cage (C,n))))) by SPRECT_1:26; assume SW-corner (L~ (Cage (C,n))) in rng (Cage (C,n)) ; ::_thesis: contradiction then SW-corner (L~ (Cage (C,n))) in (LSeg ((SW-corner (L~ (Cage (C,n)))),(NW-corner (L~ (Cage (C,n)))))) /\ (L~ (Cage (C,n))) by A14, A19, XBOOLE_0:def_4; then A20: (SW-corner (L~ (Cage (C,n)))) `2 >= (W-min (L~ (Cage (C,n)))) `2 by PSCOMP_1:31; A21: (W-min (L~ (Cage (C,n)))) `1 = (SW-corner (L~ (Cage (C,n)))) `1 by PSCOMP_1:29; (W-min (L~ (Cage (C,n)))) `2 >= (SW-corner (L~ (Cage (C,n)))) `2 by PSCOMP_1:30; then (W-min (L~ (Cage (C,n)))) `2 = (SW-corner (L~ (Cage (C,n)))) `2 by A20, XXREAL_0:1; hence contradiction by A15, A21, TOPREAL3:6; ::_thesis: verum end; N-min (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) by Th7; then R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n))))) = mid ((Upper_Seq (C,n)),1,((N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)))) by JORDAN1G:49; then A22: rng (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n)))))) c= rng (Upper_Seq (C,n)) by FINSEQ_6:119; rng (Upper_Seq (C,n)) c= rng (Cage (C,n)) by JORDAN1G:39; then rng (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n)))))) c= rng (Cage (C,n)) by A22, XBOOLE_1:1; then not SW-corner (L~ (Cage (C,n))) in rng (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n)))))) by A16; then rng (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n)))))) misses {(SW-corner (L~ (Cage (C,n))))} by ZFMISC_1:50; then rng (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n)))))) misses rng <*(SW-corner (L~ (Cage (C,n))))*> by FINSEQ_1:38; then A23: rng (Rev (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n))))))) misses rng <*(SW-corner (L~ (Cage (C,n))))*> by FINSEQ_5:57; set h1 = (Rev (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n))))))) ^ <*(SW-corner (L~ (Cage (C,n))))*>; A24: <*(SW-corner (L~ (Cage (C,n))))*> is one-to-one by FINSEQ_3:93; W-max (L~ (Cage (C,n))) in L~ (Cage (C,n)) by SPRECT_1:13; then A25: N-bound (L~ (Cage (C,n))) >= (W-max (L~ (Cage (C,n)))) `2 by PSCOMP_1:24; A26: N-min (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) by Th7; then <*(N-min (L~ (Cage (C,n))))*> is_in_the_area_of Cage (C,n) by JORDAN1E:17, SPRECT_3:46; then R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n))))) is_in_the_area_of Cage (C,n) by A26, JORDAN1E:17, SPRECT_3:52; then A27: Rev (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n)))))) is_in_the_area_of Cage (C,n) by SPRECT_3:51; (W-max (L~ (Cage (C,n)))) `2 > (W-min (L~ (Cage (C,n)))) `2 by SPRECT_2:57; then A28: N-min (L~ (Cage (C,n))) <> (Upper_Seq (C,n)) . 1 by A12, A25, EUCLID:52; then reconsider RCutUS = R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n))))) as being_S-Seq FinSequence of (TOP-REAL 2) by A26, JORDAN3:35; A29: (Rev RCutUS) /. (len (Rev RCutUS)) = (Rev RCutUS) /. (len RCutUS) by FINSEQ_5:def_3 .= RCutUS /. 1 by FINSEQ_5:65 .= (Upper_Seq (C,n)) /. 1 by A26, SPRECT_3:22 .= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ; then ((Rev RCutUS) /. (len (Rev RCutUS))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52 .= (SW-corner (L~ (Cage (C,n)))) `1 by EUCLID:52 .= (<*(SW-corner (L~ (Cage (C,n))))*> /. 1) `1 by FINSEQ_4:16 ; then reconsider h1 = (Rev (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n))))))) ^ <*(SW-corner (L~ (Cage (C,n))))*> as one-to-one special FinSequence of (TOP-REAL 2) by A23, A24, FINSEQ_3:91, GOBOARD2:8; set h = Rev h1; A30: Rev h1 is special by SPPOL_2:40; <*(SW-corner (L~ (Cage (C,n))))*> is_in_the_area_of Cage (C,n) by SPRECT_2:28; then h1 is_in_the_area_of Cage (C,n) by A27, SPRECT_2:24; then A31: Rev h1 is_in_the_area_of Cage (C,n) by SPRECT_3:51; L~ (Rev h1) = L~ h1 by SPPOL_2:22; then A32: L~ (Rev h1) = (L~ (Rev RCutUS)) \/ (LSeg (((Rev RCutUS) /. (len (Rev RCutUS))),(SW-corner (L~ (Cage (C,n)))))) by SPPOL_2:19; A33: (Index ((N-min (L~ (Cage (C,n)))),(Upper_Seq (C,n)))) + 1 >= 0 + 1 by XREAL_1:7; A34: 2 <= len g by TOPREAL1:def_8; len h1 = (len (Rev (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n)))))))) + 1 by FINSEQ_2:16 .= (len (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n))))))) + 1 by FINSEQ_5:def_3 .= ((Index ((N-min (L~ (Cage (C,n)))),(Upper_Seq (C,n)))) + 1) + 1 by A26, A28, JORDAN3:25 ; then len h1 >= 1 + 1 by A33, XREAL_1:7; then A35: len (Rev h1) >= 2 by FINSEQ_5:def_3; A36: (Rev h1) /. 1 = h1 /. (len h1) by FINSEQ_5:65; A37: len (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by FINSEQ_5:6; 1 in dom (Rev RCutUS) by FINSEQ_5:6; then h1 /. 1 = (Rev RCutUS) /. 1 by FINSEQ_4:68 .= (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n)))))) /. (len (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n))))))) by FINSEQ_5:65 .= N-min (L~ (Cage (C,n))) by A26, Th45 ; then A38: (h1 /. 1) `2 = N-bound (L~ (Cage (C,n))) by EUCLID:52; A39: (h1 /. (len h1)) `2 = (h1 /. ((len (Rev (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n)))))))) + 1)) `2 by FINSEQ_2:16 .= (SW-corner (L~ (Cage (C,n)))) `2 by FINSEQ_4:67 .= S-bound (L~ (Cage (C,n))) by EUCLID:52 ; A40: len (Lower_Seq (C,n)) in dom (Lower_Seq (C,n)) by FINSEQ_5:6; (Rev h1) /. (len (Rev h1)) = (Rev h1) /. (len h1) by FINSEQ_5:def_3 .= h1 /. 1 by FINSEQ_5:65 ; then Rev h1 is_a_v.c._for Cage (C,n) by A31, A38, A36, A39, SPRECT_2:def_3; then L~ g meets L~ (Rev h1) by A10, A30, A34, A35, SPRECT_2:29; then consider x being set such that A41: x in L~ g and A42: x in L~ (Rev h1) by XBOOLE_0:3; reconsider x = x as Point of (TOP-REAL 2) by A41; A43: L~ RCutUS c= L~ (Upper_Seq (C,n)) by Th7, JORDAN3:41; A44: L~ g c= L~ (Lower_Seq (C,n)) by A1, A13, JORDAN3:41; then A45: x in L~ (Lower_Seq (C,n)) by A41; now__::_thesis:_contradiction percases ( x in L~ (Rev RCutUS) or x in LSeg (((Rev RCutUS) /. (len (Rev RCutUS))),(SW-corner (L~ (Cage (C,n))))) ) by A42, A32, XBOOLE_0:def_3; suppose x in L~ (Rev RCutUS) ; ::_thesis: contradiction then A46: x in L~ RCutUS by SPPOL_2:22; then x in (L~ (Lower_Seq (C,n))) /\ (L~ (Upper_Seq (C,n))) by A41, A44, A43, XBOOLE_0:def_4; then A47: x in {(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} by JORDAN1E:16; now__::_thesis:_contradiction percases ( x = E-max (L~ (Cage (C,n))) or x = W-min (L~ (Cage (C,n))) ) by A47, TARSKI:def_2; suppose x = E-max (L~ (Cage (C,n))) ; ::_thesis: contradiction then (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) in L~ (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n)))))) by A46, JORDAN1F:7; then (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) in L~ (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n)))))) by A37, PARTFUN1:def_6; then (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) = N-min (L~ (Cage (C,n))) by A26, Th43; then (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) = N-min (L~ (Cage (C,n))) by A37, PARTFUN1:def_6; then A48: E-max (L~ (Cage (C,n))) = N-min (L~ (Cage (C,n))) by JORDAN1F:7; N-max (L~ (Cage (C,n))) in L~ (Cage (C,n)) by SPRECT_1:11; then A49: E-bound (L~ (Cage (C,n))) >= (N-max (L~ (Cage (C,n)))) `1 by PSCOMP_1:24; (N-max (L~ (Cage (C,n)))) `1 > (N-min (L~ (Cage (C,n)))) `1 by SPRECT_2:51; hence contradiction by A48, A49, EUCLID:52; ::_thesis: verum end; suppose x = W-min (L~ (Cage (C,n))) ; ::_thesis: contradiction then (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) in L~ (R_Cut ((Lower_Seq (C,n)),p)) by A13, A41, JORDAN1F:8; then (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) in L~ (R_Cut ((Lower_Seq (C,n)),p)) by A40, PARTFUN1:def_6; then (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) = p by A1, Th43; then (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) = p by A40, PARTFUN1:def_6; hence contradiction by A3, JORDAN1F:8; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; supposeA50: x in LSeg (((Rev RCutUS) /. (len (Rev RCutUS))),(SW-corner (L~ (Cage (C,n))))) ; ::_thesis: contradiction (W-min (L~ (Cage (C,n)))) `2 >= (SW-corner (L~ (Cage (C,n)))) `2 by PSCOMP_1:30; then A51: (W-min (L~ (Cage (C,n)))) `2 >= x `2 by A29, A50, TOPREAL1:4; A52: (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52; (SW-corner (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52; then A53: x `1 = W-bound (L~ (Cage (C,n))) by A29, A50, A52, GOBOARD7:5; L~ (Cage (C,n)) = (L~ (Lower_Seq (C,n))) \/ (L~ (Upper_Seq (C,n))) by JORDAN1E:13; then L~ (Lower_Seq (C,n)) c= L~ (Cage (C,n)) by XBOOLE_1:7; then x in W-most (L~ (Cage (C,n))) by A45, A53, 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 A51, XXREAL_0:1; then x = W-min (L~ (Cage (C,n))) by A52, A53, TOPREAL3:6; then (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) in L~ (R_Cut ((Lower_Seq (C,n)),p)) by A13, A41, JORDAN1F:8; then (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) in L~ (R_Cut ((Lower_Seq (C,n)),p)) by A40, PARTFUN1:def_6; then (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) = p by A1, Th43; then (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) = p by A40, PARTFUN1:def_6; hence contradiction by A3, JORDAN1F:8; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; supposeA54: W-min (L~ (Cage (C,n))) = SW-corner (L~ (Cage (C,n))) ; ::_thesis: contradiction set h = R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n))))); A55: 2 <= len g by TOPREAL1:def_8; W-max (L~ (Cage (C,n))) in L~ (Cage (C,n)) by SPRECT_1:13; then A56: N-bound (L~ (Cage (C,n))) >= (W-max (L~ (Cage (C,n)))) `2 by PSCOMP_1:24; A57: N-min (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) by Th7; then <*(N-min (L~ (Cage (C,n))))*> is_in_the_area_of Cage (C,n) by JORDAN1E:17, SPRECT_3:46; then A58: R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n))))) is_in_the_area_of Cage (C,n) by A57, JORDAN1E:17, SPRECT_3:52; (W-max (L~ (Cage (C,n)))) `2 > (W-min (L~ (Cage (C,n)))) `2 by SPRECT_2:57; then N-min (L~ (Cage (C,n))) <> (Upper_Seq (C,n)) . 1 by A12, A56, EUCLID:52; then reconsider RCutUS = R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n))))) as being_S-Seq FinSequence of (TOP-REAL 2) by A57, JORDAN3:35; A59: len RCutUS >= 2 by TOPREAL1:def_8; (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n)))))) /. (len (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n))))))) = N-min (L~ (Cage (C,n))) by A57, Th45; then A60: ((R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n)))))) /. (len (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n)))))))) `2 = N-bound (L~ (Cage (C,n))) by EUCLID:52; RCutUS /. 1 = (Upper_Seq (C,n)) /. 1 by A57, SPRECT_3:22 .= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ; then ((R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n)))))) /. 1) `2 = S-bound (L~ (Cage (C,n))) by A54, EUCLID:52; then R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n))))) is_a_v.c._for Cage (C,n) by A58, A60, SPRECT_2:def_3; then L~ g meets L~ (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n)))))) by A10, A55, A59, SPRECT_2:29; then consider x being set such that A61: x in L~ g and A62: x in L~ (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n)))))) by XBOOLE_0:3; reconsider x = x as Point of (TOP-REAL 2) by A61; A63: len (Lower_Seq (C,n)) in dom (Lower_Seq (C,n)) by FINSEQ_5:6; A64: L~ g c= L~ (Lower_Seq (C,n)) by A1, A13, JORDAN3:41; A65: len (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by FINSEQ_5:6; L~ RCutUS c= L~ (Upper_Seq (C,n)) by Th7, JORDAN3:41; then x in (L~ (Lower_Seq (C,n))) /\ (L~ (Upper_Seq (C,n))) by A61, A62, A64, XBOOLE_0:def_4; then A66: x in {(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} by JORDAN1E:16; now__::_thesis:_contradiction percases ( x = E-max (L~ (Cage (C,n))) or x = W-min (L~ (Cage (C,n))) ) by A66, TARSKI:def_2; suppose x = E-max (L~ (Cage (C,n))) ; ::_thesis: contradiction then (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) in L~ (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n)))))) by A62, JORDAN1F:7; then (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) in L~ (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n)))))) by A65, PARTFUN1:def_6; then (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) = N-min (L~ (Cage (C,n))) by A57, Th43; then (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) = N-min (L~ (Cage (C,n))) by A65, PARTFUN1:def_6; then A67: E-max (L~ (Cage (C,n))) = N-min (L~ (Cage (C,n))) by JORDAN1F:7; N-max (L~ (Cage (C,n))) in L~ (Cage (C,n)) by SPRECT_1:11; then A68: E-bound (L~ (Cage (C,n))) >= (N-max (L~ (Cage (C,n)))) `1 by PSCOMP_1:24; (N-max (L~ (Cage (C,n)))) `1 > (N-min (L~ (Cage (C,n)))) `1 by SPRECT_2:51; hence contradiction by A67, A68, EUCLID:52; ::_thesis: verum end; suppose x = W-min (L~ (Cage (C,n))) ; ::_thesis: contradiction then (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) in L~ (R_Cut ((Lower_Seq (C,n)),p)) by A13, A61, JORDAN1F:8; then (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) in L~ (R_Cut ((Lower_Seq (C,n)),p)) by A63, PARTFUN1:def_6; then (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) = p by A1, Th43; then (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) = p by A63, PARTFUN1:def_6; hence contradiction by A3, JORDAN1F:8; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; theorem :: JORDAN1J:48 for G being Go-board for f, g being FinSequence of (TOP-REAL 2) for k being Element of NAT st 1 <= k & k < len f & f ^ g is_sequence_on G holds ( left_cell ((f ^ g),k,G) = left_cell (f,k,G) & right_cell ((f ^ g),k,G) = right_cell (f,k,G) ) proof let G be Go-board; ::_thesis: for f, g being FinSequence of (TOP-REAL 2) for k being Element of NAT st 1 <= k & k < len f & f ^ g is_sequence_on G holds ( left_cell ((f ^ g),k,G) = left_cell (f,k,G) & right_cell ((f ^ g),k,G) = right_cell (f,k,G) ) let f, g be FinSequence of (TOP-REAL 2); ::_thesis: for k being Element of NAT st 1 <= k & k < len f & f ^ g is_sequence_on G holds ( left_cell ((f ^ g),k,G) = left_cell (f,k,G) & right_cell ((f ^ g),k,G) = right_cell (f,k,G) ) let k be Element of NAT ; ::_thesis: ( 1 <= k & k < len f & f ^ g is_sequence_on G implies ( left_cell ((f ^ g),k,G) = left_cell (f,k,G) & right_cell ((f ^ g),k,G) = right_cell (f,k,G) ) ) assume that A1: 1 <= k and A2: k < len f and A3: f ^ g is_sequence_on G ; ::_thesis: ( left_cell ((f ^ g),k,G) = left_cell (f,k,G) & right_cell ((f ^ g),k,G) = right_cell (f,k,G) ) A4: k + 1 <= len f by A2, NAT_1:13; A5: (f ^ g) | (len f) = f by FINSEQ_5:23; len f <= (len f) + (len g) by NAT_1:11; then len f <= len (f ^ g) by FINSEQ_1:22; then k + 1 <= len (f ^ g) by A4, XXREAL_0:2; hence ( left_cell ((f ^ g),k,G) = left_cell (f,k,G) & right_cell ((f ^ g),k,G) = right_cell (f,k,G) ) by A1, A3, A5, A4, GOBRD13:31; ::_thesis: verum end; theorem Th49: :: JORDAN1J:49 for D being set for f, g being FinSequence of D for i being Element of NAT st i <= len f holds (f ^' g) | i = f | i proof let D be set ; ::_thesis: for f, g being FinSequence of D for i being Element of NAT st i <= len f holds (f ^' g) | i = f | i let f, g be FinSequence of D; ::_thesis: for i being Element of NAT st i <= len f holds (f ^' g) | i = f | i let i be Element of NAT ; ::_thesis: ( i <= len f implies (f ^' g) | i = f | i ) assume A1: i <= len f ; ::_thesis: (f ^' g) | i = f | i then A2: len (f | i) = i by FINSEQ_1:59; percases ( g <> {} or g = {} ) ; supposeA3: g <> {} ; ::_thesis: (f ^' g) | i = f | i then len g >= 0 + 1 by NAT_1:13; then i + 1 <= (len f) + (len g) by A1, XREAL_1:7; then i + 1 <= (len (f ^' g)) + 1 by A3, GRAPH_2:13; then i <= len (f ^' g) by XREAL_1:6; then A4: len ((f ^' g) | i) = i by FINSEQ_1:59; then A5: dom ((f ^' g) | i) = Seg i by FINSEQ_1:def_3; now__::_thesis:_for_j_being_Nat_st_j_in_dom_((f_^'_g)_|_i)_holds_ ((f_^'_g)_|_i)_._j_=_(f_|_i)_._j let j be Nat; ::_thesis: ( j in dom ((f ^' g) | i) implies ((f ^' g) | i) . j = (f | i) . j ) A6: j in NAT by ORDINAL1:def_12; assume A7: j in dom ((f ^' g) | i) ; ::_thesis: ((f ^' g) | i) . j = (f | i) . j then A8: 1 <= j by A5, FINSEQ_1:1; j <= i by A5, A7, FINSEQ_1:1; then A9: j <= len f by A1, XXREAL_0:2; thus ((f ^' g) | i) . j = ((f ^' g) | (Seg i)) . j by FINSEQ_1:def_15 .= (f ^' g) . j by A5, A7, FUNCT_1:49 .= f . j by A6, A8, A9, GRAPH_2:14 .= (f | (Seg i)) . j by A5, A7, FUNCT_1:49 .= (f | i) . j by FINSEQ_1:def_15 ; ::_thesis: verum end; hence (f ^' g) | i = f | i by A2, A4, FINSEQ_2:9; ::_thesis: verum end; suppose g = {} ; ::_thesis: (f ^' g) | i = f | i hence (f ^' g) | i = f | i by GRAPH_2:55; ::_thesis: verum end; end; end; theorem Th50: :: JORDAN1J:50 for D being set for f, g being FinSequence of D holds (f ^' g) | (len f) = f proof let D be set ; ::_thesis: for f, g being FinSequence of D holds (f ^' g) | (len f) = f let f, g be FinSequence of D; ::_thesis: (f ^' g) | (len f) = f f | (len f) = f | (Seg (len f)) by FINSEQ_1:def_15; hence (f ^' g) | (len f) = f by Th49, FINSEQ_2:20; ::_thesis: verum end; theorem Th51: :: JORDAN1J:51 for G being Go-board for f, g being FinSequence of (TOP-REAL 2) for k being Element of NAT st 1 <= k & k < len f & f ^' g is_sequence_on G holds ( left_cell ((f ^' g),k,G) = left_cell (f,k,G) & right_cell ((f ^' g),k,G) = right_cell (f,k,G) ) proof let G be Go-board; ::_thesis: for f, g being FinSequence of (TOP-REAL 2) for k being Element of NAT st 1 <= k & k < len f & f ^' g is_sequence_on G holds ( left_cell ((f ^' g),k,G) = left_cell (f,k,G) & right_cell ((f ^' g),k,G) = right_cell (f,k,G) ) let f, g be FinSequence of (TOP-REAL 2); ::_thesis: for k being Element of NAT st 1 <= k & k < len f & f ^' g is_sequence_on G holds ( left_cell ((f ^' g),k,G) = left_cell (f,k,G) & right_cell ((f ^' g),k,G) = right_cell (f,k,G) ) let k be Element of NAT ; ::_thesis: ( 1 <= k & k < len f & f ^' g is_sequence_on G implies ( left_cell ((f ^' g),k,G) = left_cell (f,k,G) & right_cell ((f ^' g),k,G) = right_cell (f,k,G) ) ) assume that A1: 1 <= k and A2: k < len f and A3: f ^' g is_sequence_on G ; ::_thesis: ( left_cell ((f ^' g),k,G) = left_cell (f,k,G) & right_cell ((f ^' g),k,G) = right_cell (f,k,G) ) A4: k + 1 <= len f by A2, NAT_1:13; A5: (f ^' g) | (len f) = f by Th50; len f <= len (f ^' g) by TOPREAL8:7; then k + 1 <= len (f ^' g) by A4, XXREAL_0:2; hence ( left_cell ((f ^' g),k,G) = left_cell (f,k,G) & right_cell ((f ^' g),k,G) = right_cell (f,k,G) ) by A1, A3, A5, A4, GOBRD13:31; ::_thesis: verum end; theorem Th52: :: JORDAN1J:52 for G being Go-board for f being S-Sequence_in_R2 for p being Point of (TOP-REAL 2) for k being Element of NAT st 1 <= k & k < p .. f & f is_sequence_on G & p in rng f holds ( left_cell ((R_Cut (f,p)),k,G) = left_cell (f,k,G) & right_cell ((R_Cut (f,p)),k,G) = right_cell (f,k,G) ) proof let G be Go-board; ::_thesis: for f being S-Sequence_in_R2 for p being Point of (TOP-REAL 2) for k being Element of NAT st 1 <= k & k < p .. f & f is_sequence_on G & p in rng f holds ( left_cell ((R_Cut (f,p)),k,G) = left_cell (f,k,G) & right_cell ((R_Cut (f,p)),k,G) = right_cell (f,k,G) ) let f be S-Sequence_in_R2; ::_thesis: for p being Point of (TOP-REAL 2) for k being Element of NAT st 1 <= k & k < p .. f & f is_sequence_on G & p in rng f holds ( left_cell ((R_Cut (f,p)),k,G) = left_cell (f,k,G) & right_cell ((R_Cut (f,p)),k,G) = right_cell (f,k,G) ) let p be Point of (TOP-REAL 2); ::_thesis: for k being Element of NAT st 1 <= k & k < p .. f & f is_sequence_on G & p in rng f holds ( left_cell ((R_Cut (f,p)),k,G) = left_cell (f,k,G) & right_cell ((R_Cut (f,p)),k,G) = right_cell (f,k,G) ) let k be Element of NAT ; ::_thesis: ( 1 <= k & k < p .. f & f is_sequence_on G & p in rng f implies ( left_cell ((R_Cut (f,p)),k,G) = left_cell (f,k,G) & right_cell ((R_Cut (f,p)),k,G) = right_cell (f,k,G) ) ) assume that A1: 1 <= k and A2: k < p .. f and A3: f is_sequence_on G and A4: p in rng f ; ::_thesis: ( left_cell ((R_Cut (f,p)),k,G) = left_cell (f,k,G) & right_cell ((R_Cut (f,p)),k,G) = right_cell (f,k,G) ) A5: f | (p .. f) = mid (f,1,(p .. f)) by A1, A2, FINSEQ_6:116, XXREAL_0:2 .= R_Cut (f,p) by A4, JORDAN1G:49 ; A6: k + 1 <= p .. f by A2, NAT_1:13; p .. f <= len f by A4, FINSEQ_4:21; then k + 1 <= len f by A6, XXREAL_0:2; hence ( left_cell ((R_Cut (f,p)),k,G) = left_cell (f,k,G) & right_cell ((R_Cut (f,p)),k,G) = right_cell (f,k,G) ) by A1, A3, A5, A6, GOBRD13:31; ::_thesis: verum end; theorem Th53: :: JORDAN1J:53 for G being Go-board for f being FinSequence of (TOP-REAL 2) for p being Point of (TOP-REAL 2) for k being Element of NAT st 1 <= k & k < p .. f & f is_sequence_on G holds ( left_cell ((f -: p),k,G) = left_cell (f,k,G) & right_cell ((f -: p),k,G) = right_cell (f,k,G) ) proof let G be Go-board; ::_thesis: for f being FinSequence of (TOP-REAL 2) for p being Point of (TOP-REAL 2) for k being Element of NAT st 1 <= k & k < p .. f & f is_sequence_on G holds ( left_cell ((f -: p),k,G) = left_cell (f,k,G) & right_cell ((f -: p),k,G) = right_cell (f,k,G) ) let f be FinSequence of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) for k being Element of NAT st 1 <= k & k < p .. f & f is_sequence_on G holds ( left_cell ((f -: p),k,G) = left_cell (f,k,G) & right_cell ((f -: p),k,G) = right_cell (f,k,G) ) let p be Point of (TOP-REAL 2); ::_thesis: for k being Element of NAT st 1 <= k & k < p .. f & f is_sequence_on G holds ( left_cell ((f -: p),k,G) = left_cell (f,k,G) & right_cell ((f -: p),k,G) = right_cell (f,k,G) ) let k be Element of NAT ; ::_thesis: ( 1 <= k & k < p .. f & f is_sequence_on G implies ( left_cell ((f -: p),k,G) = left_cell (f,k,G) & right_cell ((f -: p),k,G) = right_cell (f,k,G) ) ) assume that A1: 1 <= k and A2: k < p .. f and A3: f is_sequence_on G ; ::_thesis: ( left_cell ((f -: p),k,G) = left_cell (f,k,G) & right_cell ((f -: p),k,G) = right_cell (f,k,G) ) A4: k + 1 <= p .. f by A2, NAT_1:13; A5: f | (p .. f) = f -: p by FINSEQ_5:def_1; percases ( p in rng f or p .. f = 0 ) by TOPREAL8:4; suppose p in rng f ; ::_thesis: ( left_cell ((f -: p),k,G) = left_cell (f,k,G) & right_cell ((f -: p),k,G) = right_cell (f,k,G) ) then p .. f <= len f by FINSEQ_4:21; then k + 1 <= len f by A4, XXREAL_0:2; hence ( left_cell ((f -: p),k,G) = left_cell (f,k,G) & right_cell ((f -: p),k,G) = right_cell (f,k,G) ) by A1, A3, A5, A4, GOBRD13:31; ::_thesis: verum end; suppose p .. f = 0 ; ::_thesis: ( left_cell ((f -: p),k,G) = left_cell (f,k,G) & right_cell ((f -: p),k,G) = right_cell (f,k,G) ) hence ( left_cell ((f -: p),k,G) = left_cell (f,k,G) & right_cell ((f -: p),k,G) = right_cell (f,k,G) ) by A2; ::_thesis: verum end; end; end; theorem Th54: :: JORDAN1J:54 for f, g being FinSequence of (TOP-REAL 2) st f is unfolded & f is s.n.c. & f is one-to-one & g is unfolded & g is s.n.c. & g is one-to-one & f /. (len f) = g /. 1 & (L~ f) /\ (L~ g) = {(g /. 1)} holds f ^' g is s.n.c. proof let f, g be FinSequence of (TOP-REAL 2); ::_thesis: ( f is unfolded & f is s.n.c. & f is one-to-one & g is unfolded & g is s.n.c. & g is one-to-one & f /. (len f) = g /. 1 & (L~ f) /\ (L~ g) = {(g /. 1)} implies f ^' g is s.n.c. ) assume that A1: ( f is unfolded & f is s.n.c. & f is one-to-one ) and A2: ( g is unfolded & g is s.n.c. & g is one-to-one ) and A3: f /. (len f) = g /. 1 and A4: (L~ f) /\ (L~ g) = {(g /. 1)} ; ::_thesis: f ^' g is s.n.c. now__::_thesis:_for_i,_j_being_Nat_st_i_+_1_<_j_holds_ LSeg_((f_^'_g),i)_misses_LSeg_((f_^'_g),j) let i, j be Nat; ::_thesis: ( i + 1 < j implies LSeg ((f ^' g),i) misses LSeg ((f ^' g),j) ) assume A5: i + 1 < j ; ::_thesis: LSeg ((f ^' g),i) misses LSeg ((f ^' g),j) A6: i in NAT by ORDINAL1:def_12; A7: j in NAT by ORDINAL1:def_12; now__::_thesis:_LSeg_((f_^'_g),i)_misses_LSeg_((f_^'_g),j) percases ( j < len f or j >= len f ) ; supposeA8: j < len f ; ::_thesis: LSeg ((f ^' g),i) misses LSeg ((f ^' g),j) then i + 1 < len f by A5, XXREAL_0:2; then i < len f by NAT_1:13; then A9: LSeg ((f ^' g),i) = LSeg (f,i) by A6, TOPREAL8:28; LSeg ((f ^' g),j) = LSeg (f,j) by A7, A8, TOPREAL8:28; hence LSeg ((f ^' g),i) misses LSeg ((f ^' g),j) by A1, A5, A9, TOPREAL1:def_7; ::_thesis: verum end; suppose j >= len f ; ::_thesis: LSeg ((f ^' g),i) misses LSeg ((f ^' g),j) then consider k being Nat such that A10: j = (len f) + k by NAT_1:10; A11: now__::_thesis:_not_g_is_trivial assume g is trivial ; ::_thesis: contradiction then len g < 2 by NAT_D:60; then ( len g = 0 or len g = 1 ) by NAT_1:23; then L~ g = {} by TOPREAL1:22; hence contradiction by A4; ::_thesis: verum end; reconsider k = k as Element of NAT by ORDINAL1:def_12; A12: now__::_thesis:_not_f_is_empty assume f is empty ; ::_thesis: contradiction then len f = 0 ; then L~ f = {} by TOPREAL1:22; hence contradiction by A4; ::_thesis: verum end; now__::_thesis:_LSeg_((f_^'_g),i)_misses_LSeg_((f_^'_g),j) percases ( ( i >= 1 & j + 1 <= len (f ^' g) ) or j + 1 > len (f ^' g) or i < 1 ) ; supposeA13: ( i >= 1 & j + 1 <= len (f ^' g) ) ; ::_thesis: LSeg ((f ^' g),i) misses LSeg ((f ^' g),j) then j + 1 < (len (f ^' g)) + 1 by NAT_1:13; then (len f) + (k + 1) < (len f) + (len g) by A10, A11, GRAPH_2:13; then A14: k + 1 < len g by XREAL_1:7; then A15: LSeg ((f ^' g),((len f) + k)) = LSeg (g,(k + 1)) by A3, A12, A11, TOPREAL8:31; then A16: LSeg ((f ^' g),j) c= L~ g by A10, TOPREAL3:19; now__::_thesis:_not_LSeg_((f_^'_g),i)_meets_LSeg_((f_^'_g),j) percases ( i < len f or i >= len f ) ; supposeA17: i < len f ; ::_thesis: not LSeg ((f ^' g),i) meets LSeg ((f ^' g),j) then A18: i + 1 <= len f by NAT_1:13; i + 1 > 1 by A13, NAT_1:13; then A19: i + 1 in dom f by A18, FINSEQ_3:25; A20: len g >= 2 by A11, NAT_D:60; A21: LSeg ((f ^' g),i) = LSeg (f,i) by A6, A17, TOPREAL8:28; then LSeg ((f ^' g),i) c= L~ f by TOPREAL3:19; then A22: (LSeg ((f ^' g),i)) /\ (LSeg ((f ^' g),j)) c= {(g /. 1)} by A4, A16, XBOOLE_1:27; assume LSeg ((f ^' g),i) meets LSeg ((f ^' g),j) ; ::_thesis: contradiction then consider x being set such that A23: x in LSeg ((f ^' g),i) and A24: x in LSeg ((f ^' g),j) by XBOOLE_0:3; x in (LSeg ((f ^' g),i)) /\ (LSeg ((f ^' g),j)) by A23, A24, XBOOLE_0:def_4; then A25: x = g /. 1 by A22, TARSKI:def_1; i in dom f by A13, A17, FINSEQ_3:25; then (len f) + 0 < (len f) + k by A1, A3, A5, A10, A21, A23, A25, A19, GOBOARD2:2; then k > 0 ; then k + 1 > 0 + 1 by XREAL_1:6; hence contradiction by A2, A10, A15, A24, A25, A20, JORDAN5B:30; ::_thesis: verum end; suppose i >= len f ; ::_thesis: LSeg ((f ^' g),i) misses LSeg ((f ^' g),j) then consider l being Nat such that A26: i = (len f) + l by NAT_1:10; reconsider l = l as Element of NAT by ORDINAL1:def_12; (len f) + (l + 1) < (len f) + k by A5, A10, A26; then l + 1 < k by XREAL_1:7; then A27: (l + 1) + 1 < k + 1 by XREAL_1:6; then (l + 1) + 1 < len g by A14, XXREAL_0:2; then l + 1 < len g by NAT_1:13; then LSeg ((f ^' g),((len f) + l)) = LSeg (g,(l + 1)) by A3, A12, A11, TOPREAL8:31; hence LSeg ((f ^' g),i) misses LSeg ((f ^' g),j) by A2, A10, A15, A26, A27, TOPREAL1:def_7; ::_thesis: verum end; end; end; hence LSeg ((f ^' g),i) misses LSeg ((f ^' g),j) ; ::_thesis: verum end; suppose j + 1 > len (f ^' g) ; ::_thesis: LSeg ((f ^' g),i) misses LSeg ((f ^' g),j) then LSeg ((f ^' g),j) = {} by TOPREAL1:def_3; hence LSeg ((f ^' g),i) misses LSeg ((f ^' g),j) by XBOOLE_1:65; ::_thesis: verum end; suppose i < 1 ; ::_thesis: LSeg ((f ^' g),i) misses LSeg ((f ^' g),j) then LSeg ((f ^' g),i) = {} by TOPREAL1:def_3; hence LSeg ((f ^' g),i) misses LSeg ((f ^' g),j) by XBOOLE_1:65; ::_thesis: verum end; end; end; hence LSeg ((f ^' g),i) misses LSeg ((f ^' g),j) ; ::_thesis: verum end; end; end; hence LSeg ((f ^' g),i) misses LSeg ((f ^' g),j) ; ::_thesis: verum end; hence f ^' g is s.n.c. by TOPREAL1:def_7; ::_thesis: verum end; theorem Th55: :: JORDAN1J:55 for f, g being FinSequence of (TOP-REAL 2) st f is one-to-one & g is one-to-one & (rng f) /\ (rng g) c= {(g /. 1)} holds f ^' g is one-to-one proof let f, g be FinSequence of (TOP-REAL 2); ::_thesis: ( f is one-to-one & g is one-to-one & (rng f) /\ (rng g) c= {(g /. 1)} implies f ^' g is one-to-one ) assume that A1: f is one-to-one and A2: g is one-to-one and A3: (rng f) /\ (rng g) c= {(g /. 1)} ; ::_thesis: f ^' g is one-to-one percases ( rng g <> {} or rng g = {} ) ; supposeA4: rng g <> {} ; ::_thesis: f ^' g is one-to-one now__::_thesis:_for_i,_j_being_Element_of_NAT_st_i_in_dom_(f_^'_g)_&_j_in_dom_(f_^'_g)_&_(f_^'_g)_/._i_=_(f_^'_g)_/._j_holds_ i_=_j A5: (len (f ^' g)) + 1 = (len f) + (len g) by A4, GRAPH_2:13, RELAT_1:38; let i, j be Element of NAT ; ::_thesis: ( i in dom (f ^' g) & j in dom (f ^' g) & (f ^' g) /. i = (f ^' g) /. j implies i = j ) assume that A6: i in dom (f ^' g) and A7: j in dom (f ^' g) and A8: (f ^' g) /. i = (f ^' g) /. j ; ::_thesis: i = j A9: 1 <= i by A6, FINSEQ_3:25; j <= len (f ^' g) by A7, FINSEQ_3:25; then A10: j < (len f) + (len g) by A5, NAT_1:13; A11: len f = (len f) + 0 ; i <= len (f ^' g) by A6, FINSEQ_3:25; then A12: i < (len f) + (len g) by A5, NAT_1:13; A13: 1 <= j by A7, FINSEQ_3:25; A14: 1 in dom g by A4, FINSEQ_3:32; now__::_thesis:_i_=_j percases ( ( i <= len f & j <= len f ) or ( i > len f & j > len f ) or ( i <= len f & j > len f ) or ( j <= len f & i > len f ) ) ; supposeA15: ( i <= len f & j <= len f ) ; ::_thesis: i = j then A16: i in dom f by A9, FINSEQ_3:25; A17: (f ^' g) /. j = f /. j by A13, A15, GRAPH_2:57; A18: j in dom f by A13, A15, FINSEQ_3:25; (f ^' g) /. i = f /. i by A9, A15, GRAPH_2:57; hence i = j by A1, A8, A17, A16, A18, PARTFUN2:10; ::_thesis: verum end; supposeA19: ( i > len f & j > len f ) ; ::_thesis: i = j then consider l being Nat such that A20: j = (len f) + l by NAT_1:10; consider k being Nat such that A21: i = (len f) + k by A19, NAT_1:10; reconsider k = k, l = l as Element of NAT by ORDINAL1:def_12; l > 0 by A11, A19, A20; then A22: l >= 0 + 1 by NAT_1:13; then A23: l + 1 > 1 by NAT_1:13; k > 0 by A11, A19, A21; then A24: k >= 0 + 1 by NAT_1:13; then A25: k + 1 > 1 by NAT_1:13; A26: l < len g by A10, A20, XREAL_1:7; then A27: (f ^' g) /. j = g /. (l + 1) by A20, A22, GRAPH_2:58; A28: k < len g by A12, A21, XREAL_1:7; then k + 1 <= len g by NAT_1:13; then A29: k + 1 in dom g by A25, FINSEQ_3:25; l + 1 <= len g by A26, NAT_1:13; then A30: l + 1 in dom g by A23, FINSEQ_3:25; (f ^' g) /. i = g /. (k + 1) by A21, A28, A24, GRAPH_2:58; then k + 1 = l + 1 by A2, A8, A27, A29, A30, PARTFUN2:10; hence i = j by A21, A20; ::_thesis: verum end; supposeA31: ( i <= len f & j > len f ) ; ::_thesis: i = j then A32: i in dom f by A9, FINSEQ_3:25; (f ^' g) /. i = f /. i by A9, A31, GRAPH_2:57; then A33: (f ^' g) /. i in rng f by A32, PARTFUN2:2; consider l being Nat such that A34: j = (len f) + l by A31, NAT_1:10; reconsider l = l as Element of NAT by ORDINAL1:def_12; l > 0 by A11, A31, A34; then A35: l >= 0 + 1 by NAT_1:13; then A36: l + 1 > 1 by NAT_1:13; A37: l < len g by A10, A34, XREAL_1:7; then l + 1 <= len g by NAT_1:13; then A38: l + 1 in dom g by A36, FINSEQ_3:25; A39: (f ^' g) /. j = g /. (l + 1) by A34, A37, A35, GRAPH_2:58; then (f ^' g) /. j in rng g by A38, PARTFUN2:2; then (f ^' g) /. j in (rng f) /\ (rng g) by A8, A33, XBOOLE_0:def_4; then g /. (l + 1) = g /. 1 by A3, A39, TARSKI:def_1; hence i = j by A2, A14, A36, A38, PARTFUN2:10; ::_thesis: verum end; supposeA40: ( j <= len f & i > len f ) ; ::_thesis: i = j then A41: j in dom f by A13, FINSEQ_3:25; (f ^' g) /. j = f /. j by A13, A40, GRAPH_2:57; then A42: (f ^' g) /. j in rng f by A41, PARTFUN2:2; consider l being Nat such that A43: i = (len f) + l by A40, NAT_1:10; reconsider l = l as Element of NAT by ORDINAL1:def_12; l > 0 by A11, A40, A43; then A44: l >= 0 + 1 by NAT_1:13; then A45: l + 1 > 1 by NAT_1:13; A46: l < len g by A12, A43, XREAL_1:7; then l + 1 <= len g by NAT_1:13; then A47: l + 1 in dom g by A45, FINSEQ_3:25; A48: (f ^' g) /. i = g /. (l + 1) by A43, A46, A44, GRAPH_2:58; then (f ^' g) /. i in rng g by A47, PARTFUN2:2; then (f ^' g) /. i in (rng f) /\ (rng g) by A8, A42, XBOOLE_0:def_4; then g /. (l + 1) = g /. 1 by A3, A48, TARSKI:def_1; hence i = j by A2, A14, A45, A47, PARTFUN2:10; ::_thesis: verum end; end; end; hence i = j ; ::_thesis: verum end; hence f ^' g is one-to-one by PARTFUN2:9; ::_thesis: verum end; suppose rng g = {} ; ::_thesis: f ^' g is one-to-one then g = {} by RELAT_1:41; hence f ^' g is one-to-one by A1, GRAPH_2:55; ::_thesis: verum end; end; end; theorem Th56: :: JORDAN1J:56 for f being FinSequence of (TOP-REAL 2) for p being Point of (TOP-REAL 2) st f is being_S-Seq & p in rng f & p <> f . 1 holds (Index (p,f)) + 1 = p .. f proof let f be FinSequence of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st f is being_S-Seq & p in rng f & p <> f . 1 holds (Index (p,f)) + 1 = p .. f let p be Point of (TOP-REAL 2); ::_thesis: ( f is being_S-Seq & p in rng f & p <> f . 1 implies (Index (p,f)) + 1 = p .. f ) assume that A1: f is being_S-Seq and A2: p in rng f and A3: p <> f . 1 ; ::_thesis: (Index (p,f)) + 1 = p .. f A4: 1 <= p .. f by A2, FINSEQ_4:21; p .. f <> 1 by A2, A3, FINSEQ_4:19; then A5: 1 < p .. f by A4, XXREAL_0:1; A6: f . (p .. f) = p by A2, FINSEQ_4:19; p .. f <= len f by A2, FINSEQ_4:21; hence (Index (p,f)) + 1 = p .. f by A1, A5, A6, JORDAN3:12; ::_thesis: verum end; theorem Th57: :: JORDAN1J:57 for n being Element of NAT for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) holds j <> k 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, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) holds j <> k let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) holds j <> k let i, j, k be Element of NAT ; ::_thesis: ( 1 < i & i < len (Gauge (C,n)) & 1 <= j & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) implies j <> k ) assume that A1: 1 < i and A2: i < len (Gauge (C,n)) and A3: 1 <= j and A4: k <= width (Gauge (C,n)) and A5: (Gauge (C,n)) * (i,k) in L~ (Upper_Seq (C,n)) and A6: (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) and A7: j = k ; ::_thesis: contradiction A8: [i,j] in Indices (Gauge (C,n)) by A1, A2, A3, A4, A7, MATRIX_1:36; (Gauge (C,n)) * (i,k) in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A5, A6, A7, XBOOLE_0:def_4; then A9: (Gauge (C,n)) * (i,k) in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16; A10: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1; len (Gauge (C,n)) >= 4 by JORDAN8:10; then A11: len (Gauge (C,n)) >= 1 by XXREAL_0:2; then A12: [(len (Gauge (C,n))),j] in Indices (Gauge (C,n)) by A3, A4, A7, MATRIX_1:36; A13: [1,j] in Indices (Gauge (C,n)) by A3, A4, A7, A11, MATRIX_1:36; percases ( (Gauge (C,n)) * (i,k) = W-min (L~ (Cage (C,n))) or (Gauge (C,n)) * (i,k) = E-max (L~ (Cage (C,n))) ) by A9, TARSKI:def_2; supposeA14: (Gauge (C,n)) * (i,k) = W-min (L~ (Cage (C,n))) ; ::_thesis: contradiction ((Gauge (C,n)) * (1,j)) `1 = W-bound (L~ (Cage (C,n))) by A3, A4, A7, A10, JORDAN1A:73; then (W-min (L~ (Cage (C,n)))) `1 <> W-bound (L~ (Cage (C,n))) by A1, A7, A8, A13, A14, JORDAN1G:7; hence contradiction by EUCLID:52; ::_thesis: verum end; supposeA15: (Gauge (C,n)) * (i,k) = E-max (L~ (Cage (C,n))) ; ::_thesis: contradiction ((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 = E-bound (L~ (Cage (C,n))) by A3, A4, A7, A10, JORDAN1A:71; then (E-max (L~ (Cage (C,n)))) `1 <> E-bound (L~ (Cage (C,n))) by A2, A7, A8, A12, A15, JORDAN1G:7; hence contradiction by EUCLID:52; ::_thesis: verum end; end; end; theorem Th58: :: JORDAN1J:58 for n being Element of NAT for C being Simple_closed_curve for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j))} holds LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C proof let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j))} holds LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C let C be Simple_closed_curve; ::_thesis: for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j))} holds LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C let i, j, k be Element of NAT ; ::_thesis: ( 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j))} implies LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C ) set Ga = Gauge (C,n); set US = Upper_Seq (C,n); set LS = Lower_Seq (C,n); set LA = Lower_Arc C; 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 Gij = (Gauge (C,n)) * (i,j); set Gik = (Gauge (C,n)) * (i,k); assume that A1: 1 < i and A2: i < len (Gauge (C,n)) and A3: 1 <= j and A4: j <= k and A5: k <= width (Gauge (C,n)) and A6: (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k))} and A7: (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j))} and A8: LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) misses Lower_Arc C ; ::_thesis: contradiction (Gauge (C,n)) * (i,j) in {((Gauge (C,n)) * (i,j))} by TARSKI:def_1; then A9: (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) by A7, XBOOLE_0:def_4; (Gauge (C,n)) * (i,k) in {((Gauge (C,n)) * (i,k))} by TARSKI:def_1; then A10: (Gauge (C,n)) * (i,k) in L~ (Upper_Seq (C,n)) by A6, XBOOLE_0:def_4; then A11: j <> k by A1, A2, A3, A5, A9, Th57; A12: j <= width (Gauge (C,n)) by A4, A5, XXREAL_0:2; A13: 1 <= k by A3, A4, XXREAL_0:2; A14: [i,j] in Indices (Gauge (C,n)) by A1, A2, A3, A12, MATRIX_1:36; A15: [i,k] in Indices (Gauge (C,n)) by A1, A2, A5, A13, MATRIX_1:36; set go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,k))); set do = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))); A16: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1; A17: len (Upper_Seq (C,n)) >= 3 by JORDAN1E:15; then len (Upper_Seq (C,n)) >= 1 by XXREAL_0:2; then 1 in dom (Upper_Seq (C,n)) by FINSEQ_3:25; then A18: (Upper_Seq (C,n)) . 1 = (Upper_Seq (C,n)) /. 1 by PARTFUN1:def_6 .= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ; A19: (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52 .= ((Gauge (C,n)) * (1,k)) `1 by A5, A13, A16, JORDAN1A:73 ; len (Gauge (C,n)) >= 4 by JORDAN8:10; then A20: len (Gauge (C,n)) >= 1 by XXREAL_0:2; then A21: [1,k] in Indices (Gauge (C,n)) by A5, A13, MATRIX_1:36; then A22: (Gauge (C,n)) * (i,k) <> (Upper_Seq (C,n)) . 1 by A1, A15, A18, A19, JORDAN1G:7; then reconsider go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,k))) as being_S-Seq FinSequence of (TOP-REAL 2) by A10, JORDAN3:35; A23: len (Lower_Seq (C,n)) >= 1 + 2 by JORDAN1E:15; then A24: len (Lower_Seq (C,n)) >= 1 by XXREAL_0:2; then A25: 1 in dom (Lower_Seq (C,n)) by FINSEQ_3:25; len (Lower_Seq (C,n)) in dom (Lower_Seq (C,n)) by A24, FINSEQ_3:25; then A26: (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by PARTFUN1:def_6 .= W-min (L~ (Cage (C,n))) by JORDAN1F:8 ; A27: (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52 .= ((Gauge (C,n)) * (1,k)) `1 by A5, A13, A16, JORDAN1A:73 ; A28: [i,j] in Indices (Gauge (C,n)) by A1, A2, A3, A12, MATRIX_1:36; then A29: (Gauge (C,n)) * (i,j) <> (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) by A1, A21, A26, A27, JORDAN1G:7; then reconsider do = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))) as being_S-Seq FinSequence of (TOP-REAL 2) by A9, JORDAN3:34; A30: [(len (Gauge (C,n))),k] in Indices (Gauge (C,n)) by A5, A13, A20, MATRIX_1:36; A31: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by A25, PARTFUN1:def_6 .= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ; (E-max (L~ (Cage (C,n)))) `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52 .= ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by A5, A13, A16, JORDAN1A:71 ; then A32: (Gauge (C,n)) * (i,j) <> (Lower_Seq (C,n)) . 1 by A2, A28, A30, A31, JORDAN1G:7; A33: len go >= 1 + 1 by TOPREAL1:def_8; A34: (Gauge (C,n)) * (i,k) in rng (Upper_Seq (C,n)) by A1, A2, A5, A10, A13, Th40, JORDAN1G:4; then A35: go is_sequence_on Gauge (C,n) by Th38, JORDAN1G:4; A36: len do >= 1 + 1 by TOPREAL1:def_8; A37: (Gauge (C,n)) * (i,j) in rng (Lower_Seq (C,n)) by A1, A2, A3, A9, A12, Th40, JORDAN1G:5; then A38: do is_sequence_on Gauge (C,n) by Th39, JORDAN1G:5; reconsider go = go as non constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A33, A35, JGRAPH_1:12, JORDAN8:5; reconsider do = do as non constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A36, A38, JGRAPH_1:12, JORDAN8:5; A39: len go > 1 by A33, NAT_1:13; then A40: len go in dom go by FINSEQ_3:25; then A41: go /. (len go) = go . (len go) by PARTFUN1:def_6 .= (Gauge (C,n)) * (i,k) by A10, JORDAN3:24 ; len do >= 1 by A36, XXREAL_0:2; then 1 in dom do by FINSEQ_3:25; then A42: do /. 1 = do . 1 by PARTFUN1:def_6 .= (Gauge (C,n)) * (i,j) by A9, JORDAN3:23 ; reconsider m = (len go) - 1 as Element of NAT by A40, FINSEQ_3:26; A43: m + 1 = len go ; then A44: (len go) -' 1 = m by NAT_D:34; A45: LSeg (go,m) c= L~ go by TOPREAL3:19; A46: L~ go c= L~ (Upper_Seq (C,n)) by A10, JORDAN3:41; then LSeg (go,m) c= L~ (Upper_Seq (C,n)) by A45, XBOOLE_1:1; then A47: (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) c= {((Gauge (C,n)) * (i,k))} by A6, XBOOLE_1:26; m >= 1 by A33, XREAL_1:19; then A48: LSeg (go,m) = LSeg ((go /. m),((Gauge (C,n)) * (i,k))) by A41, A43, TOPREAL1:def_3; {((Gauge (C,n)) * (i,k))} c= (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i,k))} or x in (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) ) A49: (Gauge (C,n)) * (i,k) in LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) by RLTOPSP1:68; assume x in {((Gauge (C,n)) * (i,k))} ; ::_thesis: x in (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) then A50: x = (Gauge (C,n)) * (i,k) by TARSKI:def_1; (Gauge (C,n)) * (i,k) in LSeg (go,m) by A48, RLTOPSP1:68; hence x in (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) by A50, A49, XBOOLE_0:def_4; ::_thesis: verum end; then A51: (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) = {((Gauge (C,n)) * (i,k))} by A47, XBOOLE_0:def_10; A52: LSeg (do,1) c= L~ do by TOPREAL3:19; A53: L~ do c= L~ (Lower_Seq (C,n)) by A9, JORDAN3:42; then LSeg (do,1) c= L~ (Lower_Seq (C,n)) by A52, XBOOLE_1:1; then A54: (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) c= {((Gauge (C,n)) * (i,j))} by A7, XBOOLE_1:26; A55: LSeg (do,1) = LSeg (((Gauge (C,n)) * (i,j)),(do /. (1 + 1))) by A36, A42, TOPREAL1:def_3; {((Gauge (C,n)) * (i,j))} c= (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i,j))} or x in (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) ) A56: (Gauge (C,n)) * (i,j) in LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) by RLTOPSP1:68; assume x in {((Gauge (C,n)) * (i,j))} ; ::_thesis: x in (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) then A57: x = (Gauge (C,n)) * (i,j) by TARSKI:def_1; (Gauge (C,n)) * (i,j) in LSeg (do,1) by A55, RLTOPSP1:68; hence x in (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) by A57, A56, XBOOLE_0:def_4; ::_thesis: verum end; then A58: (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) /\ (LSeg (do,1)) = {((Gauge (C,n)) * (i,j))} by A54, XBOOLE_0:def_10; A59: go /. 1 = (Upper_Seq (C,n)) /. 1 by A10, SPRECT_3:22 .= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ; then A60: go /. 1 = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by JORDAN1F:8 .= do /. (len do) by A9, Th35 ; A61: rng go c= L~ go by A33, SPPOL_2:18; A62: rng do c= L~ do by A36, SPPOL_2:18; A63: {(go /. 1)} c= (L~ go) /\ (L~ do) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(go /. 1)} or x in (L~ go) /\ (L~ do) ) assume x in {(go /. 1)} ; ::_thesis: x in (L~ go) /\ (L~ do) then A64: x = go /. 1 by TARSKI:def_1; then A65: x in rng go by FINSEQ_6:42; x in rng do by A60, A64, REVROT_1:3; hence x in (L~ go) /\ (L~ do) by A61, A62, A65, XBOOLE_0:def_4; ::_thesis: verum end; A66: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by A25, PARTFUN1:def_6 .= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ; A67: [(len (Gauge (C,n))),j] in Indices (Gauge (C,n)) by A3, A12, A20, MATRIX_1:36; (L~ go) /\ (L~ do) c= {(go /. 1)} proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ go) /\ (L~ do) or x in {(go /. 1)} ) assume A68: x in (L~ go) /\ (L~ do) ; ::_thesis: x in {(go /. 1)} then A69: x in L~ do by XBOOLE_0:def_4; A70: now__::_thesis:_not_x_=_E-max_(L~_(Cage_(C,n))) assume x = E-max (L~ (Cage (C,n))) ; ::_thesis: contradiction then A71: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i,j) by A9, A66, A69, JORDAN1E:7; ((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 = E-bound (L~ (Cage (C,n))) by A3, A12, A16, JORDAN1A:71; then (E-max (L~ (Cage (C,n)))) `1 <> E-bound (L~ (Cage (C,n))) by A2, A14, A67, A71, JORDAN1G:7; hence contradiction by EUCLID:52; ::_thesis: verum end; x in L~ go by A68, XBOOLE_0:def_4; then x in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A46, A53, A69, XBOOLE_0:def_4; then x in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16; then ( x = W-min (L~ (Cage (C,n))) or x = E-max (L~ (Cage (C,n))) ) by TARSKI:def_2; hence x in {(go /. 1)} by A59, A70, TARSKI:def_1; ::_thesis: verum end; then A72: (L~ go) /\ (L~ do) = {(go /. 1)} by A63, XBOOLE_0:def_10; set W2 = go /. 2; A73: 2 in dom go by A33, FINSEQ_3:25; A74: now__::_thesis:_not_((Gauge_(C,n))_*_(i,k))_`1_=_W-bound_(L~_(Cage_(C,n))) assume ((Gauge (C,n)) * (i,k)) `1 = W-bound (L~ (Cage (C,n))) ; ::_thesis: contradiction then ((Gauge (C,n)) * (1,k)) `1 = ((Gauge (C,n)) * (i,k)) `1 by A5, A13, A16, JORDAN1A:73; hence contradiction by A1, A15, A21, JORDAN1G:7; ::_thesis: verum end; go = mid ((Upper_Seq (C,n)),1,(((Gauge (C,n)) * (i,k)) .. (Upper_Seq (C,n)))) by A34, JORDAN1G:49 .= (Upper_Seq (C,n)) | (((Gauge (C,n)) * (i,k)) .. (Upper_Seq (C,n))) by A34, FINSEQ_4:21, FINSEQ_6:116 ; then A75: go /. 2 = (Upper_Seq (C,n)) /. 2 by A73, FINSEQ_4:70; set pion = <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*>; A76: now__::_thesis:_for_n_being_Element_of_NAT_st_n_in_dom_<*((Gauge_(C,n))_*_(i,k)),((Gauge_(C,n))_*_(i,j))*>_holds_ ex_i,_j_being_Element_of_NAT_st_ (_[i,j]_in_Indices_(Gauge_(C,n))_&_<*((Gauge_(C,n))_*_(i,k)),((Gauge_(C,n))_*_(i,j))*>_/._n_=_(Gauge_(C,n))_*_(i,j)_) let n be Element of NAT ; ::_thesis: ( n in dom <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> implies ex i, j being Element of NAT st ( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. n = (Gauge (C,n)) * (i,j) ) ) assume n in dom <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> ; ::_thesis: ex i, j being Element of NAT st ( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. n = (Gauge (C,n)) * (i,j) ) then n in Seg 2 by FINSEQ_1:89; then ( n = 1 or n = 2 ) by FINSEQ_1:2, TARSKI:def_2; hence ex i, j being Element of NAT st ( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. n = (Gauge (C,n)) * (i,j) ) by A14, A15, FINSEQ_4:17; ::_thesis: verum end; A77: (Gauge (C,n)) * (i,k) <> (Gauge (C,n)) * (i,j) by A11, A14, A15, GOBOARD1:5; A78: ((Gauge (C,n)) * (i,k)) `1 = ((Gauge (C,n)) * (i,1)) `1 by A1, A2, A5, A13, GOBOARD5:2 .= ((Gauge (C,n)) * (i,j)) `1 by A1, A2, A3, A12, GOBOARD5:2 ; then LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) is vertical by SPPOL_1:16; then <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> is being_S-Seq by A77, JORDAN1B:7; then consider pion1 being FinSequence of (TOP-REAL 2) such that A79: pion1 is_sequence_on Gauge (C,n) and A80: pion1 is being_S-Seq and A81: L~ <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> = L~ pion1 and A82: <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. 1 = pion1 /. 1 and A83: <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. (len <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*>) = pion1 /. (len pion1) and A84: len <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> <= len pion1 by A76, GOBOARD3:2; reconsider pion1 = pion1 as being_S-Seq FinSequence of (TOP-REAL 2) by A80; set godo = (go ^' pion1) ^' do; A85: 1 + 1 <= len (Cage (C,n)) by GOBOARD7:34, XXREAL_0:2; A86: 1 + 1 <= len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by GOBOARD7:34, XXREAL_0:2; len (go ^' pion1) >= len go by TOPREAL8:7; then A87: len (go ^' pion1) >= 1 + 1 by A33, XXREAL_0:2; then A88: len (go ^' pion1) > 1 + 0 by NAT_1:13; A89: len ((go ^' pion1) ^' do) >= len (go ^' pion1) by TOPREAL8:7; then A90: 1 + 1 <= len ((go ^' pion1) ^' do) by A87, XXREAL_0:2; A91: Upper_Seq (C,n) is_sequence_on Gauge (C,n) by JORDAN1G:4; A92: go /. (len go) = pion1 /. 1 by A41, A82, FINSEQ_4:17; then A93: go ^' pion1 is_sequence_on Gauge (C,n) by A35, A79, TOPREAL8:12; A94: (go ^' pion1) /. (len (go ^' pion1)) = <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. (len <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*>) by A83, GRAPH_2:54 .= <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. 2 by FINSEQ_1:44 .= do /. 1 by A42, FINSEQ_4:17 ; then A95: (go ^' pion1) ^' do is_sequence_on Gauge (C,n) by A38, A93, TOPREAL8:12; LSeg (pion1,1) c= L~ <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> by A81, TOPREAL3:19; then LSeg (pion1,1) c= LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) by SPPOL_2:21; then A96: (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) c= {((Gauge (C,n)) * (i,k))} by A44, A51, XBOOLE_1:27; A97: len pion1 >= 1 + 1 by A84, FINSEQ_1:44; {((Gauge (C,n)) * (i,k))} c= (LSeg (go,m)) /\ (LSeg (pion1,1)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i,k))} or x in (LSeg (go,m)) /\ (LSeg (pion1,1)) ) assume x in {((Gauge (C,n)) * (i,k))} ; ::_thesis: x in (LSeg (go,m)) /\ (LSeg (pion1,1)) then A98: x = (Gauge (C,n)) * (i,k) by TARSKI:def_1; A99: (Gauge (C,n)) * (i,k) in LSeg (go,m) by A48, RLTOPSP1:68; (Gauge (C,n)) * (i,k) in LSeg (pion1,1) by A41, A92, A97, TOPREAL1:21; hence x in (LSeg (go,m)) /\ (LSeg (pion1,1)) by A98, A99, XBOOLE_0:def_4; ::_thesis: verum end; then (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) = {(go /. (len go))} by A41, A44, A96, XBOOLE_0:def_10; then A100: go ^' pion1 is unfolded by A92, TOPREAL8:34; len pion1 >= 2 + 0 by A84, FINSEQ_1:44; then A101: (len pion1) - 2 >= 0 by XREAL_1:19; ((len (go ^' pion1)) + 1) - 1 = ((len go) + (len pion1)) - 1 by GRAPH_2:13; then (len (go ^' pion1)) - 1 = (len go) + ((len pion1) - 2) .= (len go) + ((len pion1) -' 2) by A101, XREAL_0:def_2 ; then A102: (len (go ^' pion1)) -' 1 = (len go) + ((len pion1) -' 2) by XREAL_0:def_2; A103: (len pion1) - 1 >= 1 by A97, XREAL_1:19; then A104: (len pion1) -' 1 = (len pion1) - 1 by XREAL_0:def_2; A105: ((len pion1) -' 2) + 1 = ((len pion1) - 2) + 1 by A101, XREAL_0:def_2 .= (len pion1) -' 1 by A103, XREAL_0:def_2 ; ((len pion1) - 1) + 1 <= len pion1 ; then A106: (len pion1) -' 1 < len pion1 by A104, NAT_1:13; LSeg (pion1,((len pion1) -' 1)) c= L~ <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> by A81, TOPREAL3:19; then LSeg (pion1,((len pion1) -' 1)) c= LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) by SPPOL_2:21; then A107: (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) c= {((Gauge (C,n)) * (i,j))} by A58, XBOOLE_1:27; {((Gauge (C,n)) * (i,j))} c= (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i,j))} or x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) ) assume x in {((Gauge (C,n)) * (i,j))} ; ::_thesis: x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) then A108: x = (Gauge (C,n)) * (i,j) by TARSKI:def_1; pion1 /. (((len pion1) -' 1) + 1) = <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. 2 by A83, A104, FINSEQ_1:44 .= (Gauge (C,n)) * (i,j) by FINSEQ_4:17 ; then A109: (Gauge (C,n)) * (i,j) in LSeg (pion1,((len pion1) -' 1)) by A103, A104, TOPREAL1:21; (Gauge (C,n)) * (i,j) in LSeg (do,1) by A55, RLTOPSP1:68; hence x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) by A108, A109, XBOOLE_0:def_4; ::_thesis: verum end; then (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) = {((Gauge (C,n)) * (i,j))} by A107, XBOOLE_0:def_10; then A110: (LSeg ((go ^' pion1),((len go) + ((len pion1) -' 2)))) /\ (LSeg (do,1)) = {((go ^' pion1) /. (len (go ^' pion1)))} by A42, A92, A94, A105, A106, TOPREAL8:31; A111: not go ^' pion1 is trivial by A87, NAT_D:60; A112: rng pion1 c= L~ pion1 by A97, SPPOL_2:18; A113: {(pion1 /. 1)} c= (L~ go) /\ (L~ pion1) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(pion1 /. 1)} or x in (L~ go) /\ (L~ pion1) ) assume x in {(pion1 /. 1)} ; ::_thesis: x in (L~ go) /\ (L~ pion1) then A114: x = pion1 /. 1 by TARSKI:def_1; then A115: x in rng pion1 by FINSEQ_6:42; x in rng go by A92, A114, REVROT_1:3; hence x in (L~ go) /\ (L~ pion1) by A61, A112, A115, XBOOLE_0:def_4; ::_thesis: verum end; (L~ go) /\ (L~ pion1) c= {(pion1 /. 1)} proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ go) /\ (L~ pion1) or x in {(pion1 /. 1)} ) assume A116: x in (L~ go) /\ (L~ pion1) ; ::_thesis: x in {(pion1 /. 1)} then A117: x in L~ pion1 by XBOOLE_0:def_4; x in L~ go by A116, XBOOLE_0:def_4; then x in (L~ pion1) /\ (L~ (Upper_Seq (C,n))) by A46, A117, XBOOLE_0:def_4; hence x in {(pion1 /. 1)} by A6, A41, A81, A92, SPPOL_2:21; ::_thesis: verum end; then A118: (L~ go) /\ (L~ pion1) = {(pion1 /. 1)} by A113, XBOOLE_0:def_10; then A119: go ^' pion1 is s.n.c. by A92, Th54; (rng go) /\ (rng pion1) c= {(pion1 /. 1)} by A61, A112, A118, XBOOLE_1:27; then A120: go ^' pion1 is one-to-one by Th55; A121: <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. (len <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*>) = <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. 2 by FINSEQ_1:44 .= do /. 1 by A42, FINSEQ_4:17 ; A122: {(pion1 /. (len pion1))} c= (L~ do) /\ (L~ pion1) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(pion1 /. (len pion1))} or x in (L~ do) /\ (L~ pion1) ) assume x in {(pion1 /. (len pion1))} ; ::_thesis: x in (L~ do) /\ (L~ pion1) then A123: x = pion1 /. (len pion1) by TARSKI:def_1; then A124: x in rng pion1 by REVROT_1:3; x in rng do by A83, A121, A123, FINSEQ_6:42; hence x in (L~ do) /\ (L~ pion1) by A62, A112, A124, XBOOLE_0:def_4; ::_thesis: verum end; (L~ do) /\ (L~ pion1) c= {(pion1 /. (len pion1))} proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ do) /\ (L~ pion1) or x in {(pion1 /. (len pion1))} ) assume A125: x in (L~ do) /\ (L~ pion1) ; ::_thesis: x in {(pion1 /. (len pion1))} then A126: x in L~ pion1 by XBOOLE_0:def_4; x in L~ do by A125, XBOOLE_0:def_4; then x in (L~ pion1) /\ (L~ (Lower_Seq (C,n))) by A53, A126, XBOOLE_0:def_4; hence x in {(pion1 /. (len pion1))} by A7, A42, A81, A83, A121, SPPOL_2:21; ::_thesis: verum end; then A127: (L~ do) /\ (L~ pion1) = {(pion1 /. (len pion1))} by A122, XBOOLE_0:def_10; A128: (L~ (go ^' pion1)) /\ (L~ do) = ((L~ go) \/ (L~ pion1)) /\ (L~ do) by A92, TOPREAL8:35 .= {(go /. 1)} \/ {(do /. 1)} by A72, A83, A121, A127, XBOOLE_1:23 .= {((go ^' pion1) /. 1)} \/ {(do /. 1)} by GRAPH_2:53 .= {((go ^' pion1) /. 1),(do /. 1)} by ENUMSET1:1 ; do /. (len do) = (go ^' pion1) /. 1 by A60, GRAPH_2:53; then reconsider godo = (go ^' pion1) ^' do as non constant standard special_circular_sequence by A90, A94, A95, A100, A102, A110, A111, A119, A120, A128, JORDAN8:4, JORDAN8:5, TOPREAL8:11, TOPREAL8:33, TOPREAL8:34; A129: Lower_Arc C is_an_arc_of E-max C, W-min C by JORDAN6:def_9; then A130: Lower_Arc C is connected by JORDAN6:10; A131: W-min C in Lower_Arc C by A129, TOPREAL1:1; A132: E-max C in Lower_Arc C by A129, TOPREAL1:1; set ff = Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))); W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43; then A133: (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 = W-min (L~ (Cage (C,n))) by FINSEQ_6:92; A134: L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33; then (W-max (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A133, SPRECT_5:22; then (N-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A133, A134, SPRECT_5:23, XXREAL_0:2; then (N-max (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A133, A134, SPRECT_5:24, XXREAL_0:2; then A135: (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A133, A134, SPRECT_5:25, XXREAL_0:2; A136: now__::_thesis:_not_((Gauge_(C,n))_*_(i,k))_.._(Upper_Seq_(C,n))_<=_1 assume A137: ((Gauge (C,n)) * (i,k)) .. (Upper_Seq (C,n)) <= 1 ; ::_thesis: contradiction ((Gauge (C,n)) * (i,k)) .. (Upper_Seq (C,n)) >= 1 by A34, FINSEQ_4:21; then ((Gauge (C,n)) * (i,k)) .. (Upper_Seq (C,n)) = 1 by A137, XXREAL_0:1; then (Gauge (C,n)) * (i,k) = (Upper_Seq (C,n)) /. 1 by A34, FINSEQ_5:38; hence contradiction by A18, A22, JORDAN1F:5; ::_thesis: verum end; A138: Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def_1; then A139: Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))) is_sequence_on Gauge (C,n) by REVROT_1:34; A140: (right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo) c= RightComp godo by A90, A95, JORDAN9:27; A141: L~ godo = (L~ (go ^' pion1)) \/ (L~ do) by A94, TOPREAL8:35 .= ((L~ go) \/ (L~ pion1)) \/ (L~ do) by A92, TOPREAL8:35 ; A142: L~ (Cage (C,n)) = (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n))) by JORDAN1E:13; then A143: L~ (Upper_Seq (C,n)) c= L~ (Cage (C,n)) by XBOOLE_1:7; A144: L~ (Lower_Seq (C,n)) c= L~ (Cage (C,n)) by A142, XBOOLE_1:7; A145: L~ go c= L~ (Cage (C,n)) by A46, A143, XBOOLE_1:1; A146: L~ do c= L~ (Cage (C,n)) by A53, A144, XBOOLE_1:1; A147: W-min C in C by SPRECT_1:13; A148: L~ <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> = LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) by SPPOL_2:21; A149: now__::_thesis:_not_W-min_C_in_L~_godo assume W-min C in L~ godo ; ::_thesis: contradiction then A150: ( W-min C in (L~ go) \/ (L~ pion1) or W-min C in L~ do ) by A141, XBOOLE_0:def_3; percases ( W-min C in L~ go or W-min C in L~ pion1 or W-min C in L~ do ) by A150, XBOOLE_0:def_3; suppose W-min C in L~ go ; ::_thesis: contradiction then C meets L~ (Cage (C,n)) by A145, A147, XBOOLE_0:3; hence contradiction by JORDAN10:5; ::_thesis: verum end; suppose W-min C in L~ pion1 ; ::_thesis: contradiction hence contradiction by A8, A81, A131, A148, XBOOLE_0:3; ::_thesis: verum end; suppose W-min C in L~ do ; ::_thesis: contradiction then C meets L~ (Cage (C,n)) by A146, A147, XBOOLE_0:3; hence contradiction by JORDAN10:5; ::_thesis: verum end; end; end; right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1) = right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(GoB (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) by A86, JORDAN1H:23 .= right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(GoB (Cage (C,n)))) by REVROT_1:28 .= right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(Gauge (C,n))) by JORDAN1H:44 .= right_cell (((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n))))),1,(Gauge (C,n))) by A135, A139, Th53 .= right_cell ((Upper_Seq (C,n)),1,(Gauge (C,n))) by JORDAN1E:def_1 .= right_cell ((R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,k)))),1,(Gauge (C,n))) by A34, A91, A136, Th52 .= right_cell ((go ^' pion1),1,(Gauge (C,n))) by A39, A93, Th51 .= right_cell (godo,1,(Gauge (C,n))) by A88, A95, Th51 ; then W-min C in right_cell (godo,1,(Gauge (C,n))) by JORDAN1I:6; then A151: W-min C in (right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo) by A149, XBOOLE_0:def_5; A152: godo /. 1 = (go ^' pion1) /. 1 by GRAPH_2:53 .= W-min (L~ (Cage (C,n))) by A59, GRAPH_2:53 ; A153: len (Upper_Seq (C,n)) >= 2 by A17, XXREAL_0:2; A154: godo /. 2 = (go ^' pion1) /. 2 by A87, GRAPH_2:57 .= (Upper_Seq (C,n)) /. 2 by A33, A75, GRAPH_2:57 .= ((Upper_Seq (C,n)) ^' (Lower_Seq (C,n))) /. 2 by A153, GRAPH_2:57 .= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 2 by JORDAN1E:11 ; A155: (L~ go) \/ (L~ do) is compact by COMPTS_1:10; W-min (L~ (Cage (C,n))) in rng go by A59, FINSEQ_6:42; then W-min (L~ (Cage (C,n))) in (L~ go) \/ (L~ do) by A61, XBOOLE_0:def_3; then A156: W-min ((L~ go) \/ (L~ do)) = W-min (L~ (Cage (C,n))) by A145, A146, A155, Th21, XBOOLE_1:8; A157: (W-min ((L~ go) \/ (L~ do))) `1 = W-bound ((L~ go) \/ (L~ do)) by EUCLID:52; A158: (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52; W-bound (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) = ((Gauge (C,n)) * (i,k)) `1 by A78, SPRECT_1:54; then A159: W-bound (L~ pion1) = ((Gauge (C,n)) * (i,k)) `1 by A81, SPPOL_2:21; ((Gauge (C,n)) * (i,k)) `1 >= W-bound (L~ (Cage (C,n))) by A10, A143, PSCOMP_1:24; then ((Gauge (C,n)) * (i,k)) `1 > W-bound (L~ (Cage (C,n))) by A74, XXREAL_0:1; then W-min (((L~ go) \/ (L~ do)) \/ (L~ pion1)) = W-min ((L~ go) \/ (L~ do)) by A155, A156, A157, A158, A159, Th33; then A160: W-min (L~ godo) = W-min (L~ (Cage (C,n))) by A141, A156, XBOOLE_1:4; A161: rng godo c= L~ godo by A87, A89, SPPOL_2:18, XXREAL_0:2; 2 in dom godo by A90, FINSEQ_3:25; then A162: godo /. 2 in rng godo by PARTFUN2:2; godo /. 2 in W-most (L~ (Cage (C,n))) by A154, JORDAN1I:25; then (godo /. 2) `1 = (W-min (L~ godo)) `1 by A160, PSCOMP_1:31 .= W-bound (L~ godo) by EUCLID:52 ; then godo /. 2 in W-most (L~ godo) by A161, A162, SPRECT_2:12; then (Rotate (godo,(W-min (L~ godo)))) /. 2 in W-most (L~ godo) by A152, A160, FINSEQ_6:89; then reconsider godo = godo as non constant standard clockwise_oriented special_circular_sequence by JORDAN1I:25; len (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by FINSEQ_5:6; then A163: (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) = (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) by PARTFUN1:def_6 .= E-max (L~ (Cage (C,n))) by JORDAN1F:7 ; A164: east_halfline (E-max C) misses L~ go proof assume east_halfline (E-max C) meets L~ go ; ::_thesis: contradiction then consider p being set such that A165: p in east_halfline (E-max C) and A166: p in L~ go by XBOOLE_0:3; reconsider p = p as Point of (TOP-REAL 2) by A165; p in L~ (Upper_Seq (C,n)) by A46, A166; then p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A143, A165, XBOOLE_0:def_4; then A167: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50; then A168: p = E-max (L~ (Cage (C,n))) by A46, A166, Th46; then E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i,k) by A10, A163, A166, Th43; then ((Gauge (C,n)) * (i,k)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by A5, A13, A16, A167, A168, JORDAN1A:71; hence contradiction by A2, A15, A30, JORDAN1G:7; ::_thesis: verum end; now__::_thesis:_not_east_halfline_(E-max_C)_meets_L~_godo assume east_halfline (E-max C) meets L~ godo ; ::_thesis: contradiction then A169: ( east_halfline (E-max C) meets (L~ go) \/ (L~ pion1) or east_halfline (E-max C) meets L~ do ) by A141, XBOOLE_1:70; percases ( east_halfline (E-max C) meets L~ go or east_halfline (E-max C) meets L~ pion1 or east_halfline (E-max C) meets L~ do ) by A169, XBOOLE_1:70; suppose east_halfline (E-max C) meets L~ go ; ::_thesis: contradiction hence contradiction by A164; ::_thesis: verum end; suppose east_halfline (E-max C) meets L~ pion1 ; ::_thesis: contradiction then consider p being set such that A170: p in east_halfline (E-max C) and A171: p in L~ pion1 by XBOOLE_0:3; reconsider p = p as Point of (TOP-REAL 2) by A170; A172: p `2 = (E-max C) `2 by A170, TOPREAL1:def_11; i + 1 <= len (Gauge (C,n)) by A2, NAT_1:13; then (i + 1) - 1 <= (len (Gauge (C,n))) - 1 by XREAL_1:9; then A173: i <= (len (Gauge (C,n))) -' 1 by XREAL_0:def_2; A174: (len (Gauge (C,n))) -' 1 <= len (Gauge (C,n)) by NAT_D:35; p `1 = ((Gauge (C,n)) * (i,k)) `1 by A78, A81, A148, A171, GOBOARD7:5; then p `1 <= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),1)) `1 by A1, A5, A13, A16, A20, A173, A174, JORDAN1A:18; then p `1 <= E-bound C by A20, JORDAN8:12; then A175: p `1 <= (E-max C) `1 by EUCLID:52; p `1 >= (E-max C) `1 by A170, TOPREAL1:def_11; then p `1 = (E-max C) `1 by A175, XXREAL_0:1; then p = E-max C by A172, TOPREAL3:6; hence contradiction by A8, A81, A132, A148, A171, XBOOLE_0:3; ::_thesis: verum end; suppose east_halfline (E-max C) meets L~ do ; ::_thesis: contradiction then consider p being set such that A176: p in east_halfline (E-max C) and A177: p in L~ do by XBOOLE_0:3; reconsider p = p as Point of (TOP-REAL 2) by A176; A178: (E-max C) `2 = p `2 by A176, TOPREAL1:def_11; set tt = ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1; set RC = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))); A179: L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33; consider t being Nat such that A180: t in dom (Lower_Seq (C,n)) and A181: (Lower_Seq (C,n)) . t = (Gauge (C,n)) * (i,j) by A37, FINSEQ_2:10; 1 <= t by A180, FINSEQ_3:25; then A182: 1 < t by A32, A181, XXREAL_0:1; t <= len (Lower_Seq (C,n)) by A180, FINSEQ_3:25; then (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1 = t by A181, A182, JORDAN3:12; then A183: len (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) = (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) by A9, A181, JORDAN3:26; Index (p,do) < len do by A177, JORDAN3:8; then Index (p,do) < (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) by A183, XREAL_0:def_2; then (Index (p,do)) + 1 <= (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) by NAT_1:13; then A184: Index (p,do) <= ((len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n))))) - 1 by XREAL_1:19; A185: do = mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n)))) by A37, Th37; A186: len (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = len (Cage (C,n)) by REVROT_1:14; p in L~ (Lower_Seq (C,n)) by A53, A177; then p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A144, A176, XBOOLE_0:def_4; then A187: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50; A188: GoB (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = GoB (Cage (C,n)) by REVROT_1:28 .= Gauge (C,n) by JORDAN1H:44 ; A189: 1 + 1 <= len (Lower_Seq (C,n)) by A23, XXREAL_0:2; then A190: 2 in dom (Lower_Seq (C,n)) by FINSEQ_3:25; consider jj2 being Element of NAT such that A191: 1 <= jj2 and A192: jj2 <= width (Gauge (C,n)) and A193: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * ((len (Gauge (C,n))),jj2) by JORDAN1D:25; A194: len (Gauge (C,n)) >= 4 by JORDAN8:10; then len (Gauge (C,n)) >= 1 by XXREAL_0:2; then A195: [(len (Gauge (C,n))),jj2] in Indices (Gauge (C,n)) by A191, A192, MATRIX_1:36; A196: 1 <= Index (p,do) by A177, JORDAN3:8; Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) by JORDAN1G:18; then A197: LSeg ((Lower_Seq (C,n)),1) = LSeg ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by A189, SPPOL_2:9; A198: E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46; Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))) is_sequence_on Gauge (C,n) by A138, REVROT_1:34; then consider ii, jj being Element of NAT such that A199: [ii,(jj + 1)] in Indices (Gauge (C,n)) and A200: [ii,jj] in Indices (Gauge (C,n)) and A201: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = (Gauge (C,n)) * (ii,(jj + 1)) and A202: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1) = (Gauge (C,n)) * (ii,jj) by A85, A179, A186, A198, FINSEQ_6:92, JORDAN1I:23; A203: (jj + 1) + 1 <> jj ; A204: 1 <= jj by A200, MATRIX_1:38; (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = E-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))))) by A179, A198, FINSEQ_6:92; then A205: ii = len (Gauge (C,n)) by A179, A199, A201, A193, A195, GOBOARD1:5; then ii - 1 >= 4 - 1 by A194, XREAL_1:9; then A206: ii - 1 >= 1 by XXREAL_0:2; then A207: 1 <= ii -' 1 by XREAL_0:def_2; A208: jj <= width (Gauge (C,n)) by A200, MATRIX_1:38; then A209: ((Gauge (C,n)) * ((len (Gauge (C,n))),jj)) `1 = E-bound (L~ (Cage (C,n))) by A16, A204, JORDAN1A:71; A210: jj + 1 <= width (Gauge (C,n)) by A199, MATRIX_1:38; ii + 1 <> ii ; then A211: right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) = cell ((Gauge (C,n)),(ii -' 1),jj) by A85, A186, A188, A199, A200, A201, A202, A203, GOBOARD5:def_6; A212: ii <= len (Gauge (C,n)) by A200, MATRIX_1:38; A213: 1 <= ii by A200, MATRIX_1:38; A214: ii <= len (Gauge (C,n)) by A199, MATRIX_1:38; A215: 1 <= jj + 1 by A199, MATRIX_1:38; then A216: E-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * ((len (Gauge (C,n))),(jj + 1))) `1 by A16, A210, JORDAN1A:71; A217: 1 <= ii by A199, MATRIX_1:38; then A218: (ii -' 1) + 1 = ii by XREAL_1:235; then A219: ii -' 1 < len (Gauge (C,n)) by A214, NAT_1:13; then A220: ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 = ((Gauge (C,n)) * (1,(jj + 1))) `2 by A215, A210, A207, GOBOARD5:1 .= ((Gauge (C,n)) * (ii,(jj + 1))) `2 by A217, A214, A215, A210, GOBOARD5:1 ; A221: E-max C in right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by JORDAN1I:7; then A222: ((Gauge (C,n)) * ((ii -' 1),jj)) `2 <= (E-max C) `2 by A214, A210, A204, A211, A218, A206, JORDAN9:17; A223: (E-max C) `2 <= ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 by A221, A214, A210, A204, A211, A218, A206, JORDAN9:17; ((Gauge (C,n)) * ((ii -' 1),jj)) `2 = ((Gauge (C,n)) * (1,jj)) `2 by A204, A208, A207, A219, GOBOARD5:1 .= ((Gauge (C,n)) * (ii,jj)) `2 by A213, A212, A204, A208, GOBOARD5:1 ; then p in LSeg (((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1),((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1))) by A187, A178, A201, A202, A205, A222, A223, A220, A209, A216, GOBOARD7:7; then A224: p in LSeg ((Lower_Seq (C,n)),1) by A85, A197, A186, TOPREAL1:def_3; A225: ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) <= len (Lower_Seq (C,n)) by A37, FINSEQ_4:21; ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) <> len (Lower_Seq (C,n)) by A29, A37, FINSEQ_4:19; then A226: ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) < len (Lower_Seq (C,n)) by A225, XXREAL_0:1; A227: (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1 = ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) by A32, A37, Th56; 0 + (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) < len (Lower_Seq (C,n)) by A9, JORDAN3:8; then (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) > 0 by XREAL_1:20; then Index (p,do) <= ((len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n))))) - 1 by A184, XREAL_0:def_2; then Index (p,do) <= (len (Lower_Seq (C,n))) - (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))) by A227; then Index (p,do) <= (len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))) by XREAL_0:def_2; then A228: Index (p,do) < ((len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) + 1 by NAT_1:13; A229: p in LSeg (do,(Index (p,do))) by A177, JORDAN3:9; 1 <= ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) by A37, FINSEQ_4:21; then A230: LSeg ((mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n))))),(Index (p,do))) = LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1)) by A226, A196, A228, JORDAN4:19; 1 <= Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n))) by A9, JORDAN3:8; then A231: 1 + 1 <= ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) by A227, XREAL_1:7; then (Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))) >= (1 + 1) + 1 by A196, XREAL_1:7; then ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) - 1 >= ((1 + 1) + 1) - 1 by XREAL_1:9; then A232: ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1 >= 1 + 1 by XREAL_0:def_2; now__::_thesis:_contradiction percases ( ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1 > 1 + 1 or ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ) by A232, XXREAL_0:1; suppose ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1 > 1 + 1 ; ::_thesis: contradiction then LSeg ((Lower_Seq (C,n)),1) misses LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1)) by TOPREAL1:def_7; hence contradiction by A224, A229, A185, A230, XBOOLE_0:3; ::_thesis: verum end; supposeA233: ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ; ::_thesis: contradiction then 1 + 1 = ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) - 1 by XREAL_0:def_2; then (1 + 1) + 1 = (Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))) ; then A234: ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) = 2 by A196, A231, JORDAN1E:6; (LSeg ((Lower_Seq (C,n)),1)) /\ (LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1))) = {((Lower_Seq (C,n)) /. 2)} by A23, A233, TOPREAL1:def_6; then p in {((Lower_Seq (C,n)) /. 2)} by A224, A229, A185, A230, XBOOLE_0:def_4; then A235: p = (Lower_Seq (C,n)) /. 2 by TARSKI:def_1; then A236: p in rng (Lower_Seq (C,n)) by A190, PARTFUN2:2; p .. (Lower_Seq (C,n)) = 2 by A190, A235, FINSEQ_5:41; then p = (Gauge (C,n)) * (i,j) by A37, A234, A236, FINSEQ_5:9; then ((Gauge (C,n)) * (i,j)) `1 = E-bound (L~ (Cage (C,n))) by A235, JORDAN1G:32; then ((Gauge (C,n)) * (i,j)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 by A3, A12, A16, JORDAN1A:71; hence contradiction by A2, A14, A67, JORDAN1G:7; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; end; end; then east_halfline (E-max C) c= (L~ godo) ` by SUBSET_1:23; then consider W being Subset of (TOP-REAL 2) such that A237: W is_a_component_of (L~ godo) ` and A238: east_halfline (E-max C) c= W by GOBOARD9:3; not W is bounded by A238, JORDAN2C:121, RLTOPSP1:42; then W is_outside_component_of L~ godo by A237, JORDAN2C:def_3; then W c= UBD (L~ godo) by JORDAN2C:23; then A239: east_halfline (E-max C) c= UBD (L~ godo) by A238, XBOOLE_1:1; E-max C in east_halfline (E-max C) by TOPREAL1:38; then E-max C in UBD (L~ godo) by A239; then E-max C in LeftComp godo by GOBRD14:36; then Lower_Arc C meets L~ godo by A130, A131, A132, A140, A151, Th36; then A240: ( Lower_Arc C meets (L~ go) \/ (L~ pion1) or Lower_Arc C meets L~ do ) by A141, XBOOLE_1:70; A241: Lower_Arc C c= C by JORDAN6:61; percases ( Lower_Arc C meets L~ go or Lower_Arc C meets L~ pion1 or Lower_Arc C meets L~ do ) by A240, XBOOLE_1:70; suppose Lower_Arc C meets L~ go ; ::_thesis: contradiction then Lower_Arc C meets L~ (Cage (C,n)) by A46, A143, XBOOLE_1:1, XBOOLE_1:63; hence contradiction by A241, JORDAN10:5, XBOOLE_1:63; ::_thesis: verum end; suppose Lower_Arc C meets L~ pion1 ; ::_thesis: contradiction hence contradiction by A8, A81, A148; ::_thesis: verum end; suppose Lower_Arc C meets L~ do ; ::_thesis: contradiction then Lower_Arc C meets L~ (Cage (C,n)) by A53, A144, XBOOLE_1:1, XBOOLE_1:63; hence contradiction by A241, JORDAN10:5, XBOOLE_1:63; ::_thesis: verum end; end; end; theorem Th59: :: JORDAN1J:59 for n being Element of NAT for C being Simple_closed_curve for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j))} holds LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C proof let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j))} holds LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C let C be Simple_closed_curve; ::_thesis: for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j))} holds LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C let i, j, k be Element of NAT ; ::_thesis: ( 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j))} implies LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C ) set Ga = Gauge (C,n); set US = Upper_Seq (C,n); set LS = Lower_Seq (C,n); set UA = Upper_Arc C; 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 Gij = (Gauge (C,n)) * (i,j); set Gik = (Gauge (C,n)) * (i,k); assume that A1: 1 < i and A2: i < len (Gauge (C,n)) and A3: 1 <= j and A4: j <= k and A5: k <= width (Gauge (C,n)) and A6: (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k))} and A7: (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j))} and A8: LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) misses Upper_Arc C ; ::_thesis: contradiction (Gauge (C,n)) * (i,j) in {((Gauge (C,n)) * (i,j))} by TARSKI:def_1; then A9: (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) by A7, XBOOLE_0:def_4; (Gauge (C,n)) * (i,k) in {((Gauge (C,n)) * (i,k))} by TARSKI:def_1; then A10: (Gauge (C,n)) * (i,k) in L~ (Upper_Seq (C,n)) by A6, XBOOLE_0:def_4; then A11: j <> k by A1, A2, A3, A5, A9, Th57; A12: j <= width (Gauge (C,n)) by A4, A5, XXREAL_0:2; A13: 1 <= k by A3, A4, XXREAL_0:2; A14: [i,j] in Indices (Gauge (C,n)) by A1, A2, A3, A12, MATRIX_1:36; A15: [i,k] in Indices (Gauge (C,n)) by A1, A2, A5, A13, MATRIX_1:36; set go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,k))); set do = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))); A16: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1; A17: len (Upper_Seq (C,n)) >= 3 by JORDAN1E:15; then len (Upper_Seq (C,n)) >= 1 by XXREAL_0:2; then 1 in dom (Upper_Seq (C,n)) by FINSEQ_3:25; then A18: (Upper_Seq (C,n)) . 1 = (Upper_Seq (C,n)) /. 1 by PARTFUN1:def_6 .= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ; A19: (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52 .= ((Gauge (C,n)) * (1,k)) `1 by A5, A13, A16, JORDAN1A:73 ; len (Gauge (C,n)) >= 4 by JORDAN8:10; then A20: len (Gauge (C,n)) >= 1 by XXREAL_0:2; then A21: [1,k] in Indices (Gauge (C,n)) by A5, A13, MATRIX_1:36; then A22: (Gauge (C,n)) * (i,k) <> (Upper_Seq (C,n)) . 1 by A1, A15, A18, A19, JORDAN1G:7; then reconsider go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,k))) as being_S-Seq FinSequence of (TOP-REAL 2) by A10, JORDAN3:35; A23: len (Lower_Seq (C,n)) >= 1 + 2 by JORDAN1E:15; then A24: len (Lower_Seq (C,n)) >= 1 by XXREAL_0:2; then A25: 1 in dom (Lower_Seq (C,n)) by FINSEQ_3:25; len (Lower_Seq (C,n)) in dom (Lower_Seq (C,n)) by A24, FINSEQ_3:25; then A26: (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by PARTFUN1:def_6 .= W-min (L~ (Cage (C,n))) by JORDAN1F:8 ; A27: (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52 .= ((Gauge (C,n)) * (1,k)) `1 by A5, A13, A16, JORDAN1A:73 ; A28: [i,j] in Indices (Gauge (C,n)) by A1, A2, A3, A12, MATRIX_1:36; then A29: (Gauge (C,n)) * (i,j) <> (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) by A1, A21, A26, A27, JORDAN1G:7; then reconsider do = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))) as being_S-Seq FinSequence of (TOP-REAL 2) by A9, JORDAN3:34; A30: [(len (Gauge (C,n))),k] in Indices (Gauge (C,n)) by A5, A13, A20, MATRIX_1:36; A31: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by A25, PARTFUN1:def_6 .= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ; (E-max (L~ (Cage (C,n)))) `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52 .= ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by A5, A13, A16, JORDAN1A:71 ; then A32: (Gauge (C,n)) * (i,j) <> (Lower_Seq (C,n)) . 1 by A2, A28, A30, A31, JORDAN1G:7; A33: len go >= 1 + 1 by TOPREAL1:def_8; A34: (Gauge (C,n)) * (i,k) in rng (Upper_Seq (C,n)) by A1, A2, A5, A10, A13, Th40, JORDAN1G:4; then A35: go is_sequence_on Gauge (C,n) by Th38, JORDAN1G:4; A36: len do >= 1 + 1 by TOPREAL1:def_8; A37: (Gauge (C,n)) * (i,j) in rng (Lower_Seq (C,n)) by A1, A2, A3, A9, A12, Th40, JORDAN1G:5; then A38: do is_sequence_on Gauge (C,n) by Th39, JORDAN1G:5; reconsider go = go as non constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A33, A35, JGRAPH_1:12, JORDAN8:5; reconsider do = do as non constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A36, A38, JGRAPH_1:12, JORDAN8:5; A39: len go > 1 by A33, NAT_1:13; then A40: len go in dom go by FINSEQ_3:25; then A41: go /. (len go) = go . (len go) by PARTFUN1:def_6 .= (Gauge (C,n)) * (i,k) by A10, JORDAN3:24 ; len do >= 1 by A36, XXREAL_0:2; then 1 in dom do by FINSEQ_3:25; then A42: do /. 1 = do . 1 by PARTFUN1:def_6 .= (Gauge (C,n)) * (i,j) by A9, JORDAN3:23 ; reconsider m = (len go) - 1 as Element of NAT by A40, FINSEQ_3:26; A43: m + 1 = len go ; then A44: (len go) -' 1 = m by NAT_D:34; A45: LSeg (go,m) c= L~ go by TOPREAL3:19; A46: L~ go c= L~ (Upper_Seq (C,n)) by A10, JORDAN3:41; then LSeg (go,m) c= L~ (Upper_Seq (C,n)) by A45, XBOOLE_1:1; then A47: (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) c= {((Gauge (C,n)) * (i,k))} by A6, XBOOLE_1:26; m >= 1 by A33, XREAL_1:19; then A48: LSeg (go,m) = LSeg ((go /. m),((Gauge (C,n)) * (i,k))) by A41, A43, TOPREAL1:def_3; {((Gauge (C,n)) * (i,k))} c= (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i,k))} or x in (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) ) A49: (Gauge (C,n)) * (i,k) in LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) by RLTOPSP1:68; assume x in {((Gauge (C,n)) * (i,k))} ; ::_thesis: x in (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) then A50: x = (Gauge (C,n)) * (i,k) by TARSKI:def_1; (Gauge (C,n)) * (i,k) in LSeg (go,m) by A48, RLTOPSP1:68; hence x in (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) by A50, A49, XBOOLE_0:def_4; ::_thesis: verum end; then A51: (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) = {((Gauge (C,n)) * (i,k))} by A47, XBOOLE_0:def_10; A52: LSeg (do,1) c= L~ do by TOPREAL3:19; A53: L~ do c= L~ (Lower_Seq (C,n)) by A9, JORDAN3:42; then LSeg (do,1) c= L~ (Lower_Seq (C,n)) by A52, XBOOLE_1:1; then A54: (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) c= {((Gauge (C,n)) * (i,j))} by A7, XBOOLE_1:26; A55: LSeg (do,1) = LSeg (((Gauge (C,n)) * (i,j)),(do /. (1 + 1))) by A36, A42, TOPREAL1:def_3; {((Gauge (C,n)) * (i,j))} c= (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i,j))} or x in (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) ) A56: (Gauge (C,n)) * (i,j) in LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) by RLTOPSP1:68; assume x in {((Gauge (C,n)) * (i,j))} ; ::_thesis: x in (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) then A57: x = (Gauge (C,n)) * (i,j) by TARSKI:def_1; (Gauge (C,n)) * (i,j) in LSeg (do,1) by A55, RLTOPSP1:68; hence x in (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) by A57, A56, XBOOLE_0:def_4; ::_thesis: verum end; then A58: (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) /\ (LSeg (do,1)) = {((Gauge (C,n)) * (i,j))} by A54, XBOOLE_0:def_10; A59: go /. 1 = (Upper_Seq (C,n)) /. 1 by A10, SPRECT_3:22 .= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ; then A60: go /. 1 = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by JORDAN1F:8 .= do /. (len do) by A9, Th35 ; A61: rng go c= L~ go by A33, SPPOL_2:18; A62: rng do c= L~ do by A36, SPPOL_2:18; A63: {(go /. 1)} c= (L~ go) /\ (L~ do) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(go /. 1)} or x in (L~ go) /\ (L~ do) ) assume x in {(go /. 1)} ; ::_thesis: x in (L~ go) /\ (L~ do) then A64: x = go /. 1 by TARSKI:def_1; then A65: x in rng go by FINSEQ_6:42; x in rng do by A60, A64, REVROT_1:3; hence x in (L~ go) /\ (L~ do) by A61, A62, A65, XBOOLE_0:def_4; ::_thesis: verum end; A66: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by A25, PARTFUN1:def_6 .= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ; A67: [(len (Gauge (C,n))),j] in Indices (Gauge (C,n)) by A3, A12, A20, MATRIX_1:36; (L~ go) /\ (L~ do) c= {(go /. 1)} proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ go) /\ (L~ do) or x in {(go /. 1)} ) assume A68: x in (L~ go) /\ (L~ do) ; ::_thesis: x in {(go /. 1)} then A69: x in L~ do by XBOOLE_0:def_4; A70: now__::_thesis:_not_x_=_E-max_(L~_(Cage_(C,n))) assume x = E-max (L~ (Cage (C,n))) ; ::_thesis: contradiction then A71: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i,j) by A9, A66, A69, JORDAN1E:7; ((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 = E-bound (L~ (Cage (C,n))) by A3, A12, A16, JORDAN1A:71; then (E-max (L~ (Cage (C,n)))) `1 <> E-bound (L~ (Cage (C,n))) by A2, A14, A67, A71, JORDAN1G:7; hence contradiction by EUCLID:52; ::_thesis: verum end; x in L~ go by A68, XBOOLE_0:def_4; then x in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A46, A53, A69, XBOOLE_0:def_4; then x in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16; then ( x = W-min (L~ (Cage (C,n))) or x = E-max (L~ (Cage (C,n))) ) by TARSKI:def_2; hence x in {(go /. 1)} by A59, A70, TARSKI:def_1; ::_thesis: verum end; then A72: (L~ go) /\ (L~ do) = {(go /. 1)} by A63, XBOOLE_0:def_10; set W2 = go /. 2; A73: 2 in dom go by A33, FINSEQ_3:25; A74: now__::_thesis:_not_((Gauge_(C,n))_*_(i,k))_`1_=_W-bound_(L~_(Cage_(C,n))) assume ((Gauge (C,n)) * (i,k)) `1 = W-bound (L~ (Cage (C,n))) ; ::_thesis: contradiction then ((Gauge (C,n)) * (1,k)) `1 = ((Gauge (C,n)) * (i,k)) `1 by A5, A13, A16, JORDAN1A:73; hence contradiction by A1, A15, A21, JORDAN1G:7; ::_thesis: verum end; go = mid ((Upper_Seq (C,n)),1,(((Gauge (C,n)) * (i,k)) .. (Upper_Seq (C,n)))) by A34, JORDAN1G:49 .= (Upper_Seq (C,n)) | (((Gauge (C,n)) * (i,k)) .. (Upper_Seq (C,n))) by A34, FINSEQ_4:21, FINSEQ_6:116 ; then A75: go /. 2 = (Upper_Seq (C,n)) /. 2 by A73, FINSEQ_4:70; set pion = <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*>; A76: now__::_thesis:_for_n_being_Element_of_NAT_st_n_in_dom_<*((Gauge_(C,n))_*_(i,k)),((Gauge_(C,n))_*_(i,j))*>_holds_ ex_i,_j_being_Element_of_NAT_st_ (_[i,j]_in_Indices_(Gauge_(C,n))_&_<*((Gauge_(C,n))_*_(i,k)),((Gauge_(C,n))_*_(i,j))*>_/._n_=_(Gauge_(C,n))_*_(i,j)_) let n be Element of NAT ; ::_thesis: ( n in dom <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> implies ex i, j being Element of NAT st ( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. n = (Gauge (C,n)) * (i,j) ) ) assume n in dom <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> ; ::_thesis: ex i, j being Element of NAT st ( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. n = (Gauge (C,n)) * (i,j) ) then n in Seg 2 by FINSEQ_1:89; then ( n = 1 or n = 2 ) by FINSEQ_1:2, TARSKI:def_2; hence ex i, j being Element of NAT st ( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. n = (Gauge (C,n)) * (i,j) ) by A14, A15, FINSEQ_4:17; ::_thesis: verum end; A77: (Gauge (C,n)) * (i,k) <> (Gauge (C,n)) * (i,j) by A11, A14, A15, GOBOARD1:5; A78: ((Gauge (C,n)) * (i,k)) `1 = ((Gauge (C,n)) * (i,1)) `1 by A1, A2, A5, A13, GOBOARD5:2 .= ((Gauge (C,n)) * (i,j)) `1 by A1, A2, A3, A12, GOBOARD5:2 ; then LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) is vertical by SPPOL_1:16; then <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> is being_S-Seq by A77, JORDAN1B:7; then consider pion1 being FinSequence of (TOP-REAL 2) such that A79: pion1 is_sequence_on Gauge (C,n) and A80: pion1 is being_S-Seq and A81: L~ <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> = L~ pion1 and A82: <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. 1 = pion1 /. 1 and A83: <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. (len <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*>) = pion1 /. (len pion1) and A84: len <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> <= len pion1 by A76, GOBOARD3:2; reconsider pion1 = pion1 as being_S-Seq FinSequence of (TOP-REAL 2) by A80; set godo = (go ^' pion1) ^' do; A85: 1 + 1 <= len (Cage (C,n)) by GOBOARD7:34, XXREAL_0:2; A86: 1 + 1 <= len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by GOBOARD7:34, XXREAL_0:2; len (go ^' pion1) >= len go by TOPREAL8:7; then A87: len (go ^' pion1) >= 1 + 1 by A33, XXREAL_0:2; then A88: len (go ^' pion1) > 1 + 0 by NAT_1:13; A89: len ((go ^' pion1) ^' do) >= len (go ^' pion1) by TOPREAL8:7; then A90: 1 + 1 <= len ((go ^' pion1) ^' do) by A87, XXREAL_0:2; A91: Upper_Seq (C,n) is_sequence_on Gauge (C,n) by JORDAN1G:4; A92: go /. (len go) = pion1 /. 1 by A41, A82, FINSEQ_4:17; then A93: go ^' pion1 is_sequence_on Gauge (C,n) by A35, A79, TOPREAL8:12; A94: (go ^' pion1) /. (len (go ^' pion1)) = <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. (len <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*>) by A83, GRAPH_2:54 .= <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. 2 by FINSEQ_1:44 .= do /. 1 by A42, FINSEQ_4:17 ; then A95: (go ^' pion1) ^' do is_sequence_on Gauge (C,n) by A38, A93, TOPREAL8:12; LSeg (pion1,1) c= L~ <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> by A81, TOPREAL3:19; then LSeg (pion1,1) c= LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) by SPPOL_2:21; then A96: (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) c= {((Gauge (C,n)) * (i,k))} by A44, A51, XBOOLE_1:27; A97: len pion1 >= 1 + 1 by A84, FINSEQ_1:44; {((Gauge (C,n)) * (i,k))} c= (LSeg (go,m)) /\ (LSeg (pion1,1)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i,k))} or x in (LSeg (go,m)) /\ (LSeg (pion1,1)) ) assume x in {((Gauge (C,n)) * (i,k))} ; ::_thesis: x in (LSeg (go,m)) /\ (LSeg (pion1,1)) then A98: x = (Gauge (C,n)) * (i,k) by TARSKI:def_1; A99: (Gauge (C,n)) * (i,k) in LSeg (go,m) by A48, RLTOPSP1:68; (Gauge (C,n)) * (i,k) in LSeg (pion1,1) by A41, A92, A97, TOPREAL1:21; hence x in (LSeg (go,m)) /\ (LSeg (pion1,1)) by A98, A99, XBOOLE_0:def_4; ::_thesis: verum end; then (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) = {(go /. (len go))} by A41, A44, A96, XBOOLE_0:def_10; then A100: go ^' pion1 is unfolded by A92, TOPREAL8:34; len pion1 >= 2 + 0 by A84, FINSEQ_1:44; then A101: (len pion1) - 2 >= 0 by XREAL_1:19; ((len (go ^' pion1)) + 1) - 1 = ((len go) + (len pion1)) - 1 by GRAPH_2:13; then (len (go ^' pion1)) - 1 = (len go) + ((len pion1) - 2) .= (len go) + ((len pion1) -' 2) by A101, XREAL_0:def_2 ; then A102: (len (go ^' pion1)) -' 1 = (len go) + ((len pion1) -' 2) by XREAL_0:def_2; A103: (len pion1) - 1 >= 1 by A97, XREAL_1:19; then A104: (len pion1) -' 1 = (len pion1) - 1 by XREAL_0:def_2; A105: ((len pion1) -' 2) + 1 = ((len pion1) - 2) + 1 by A101, XREAL_0:def_2 .= (len pion1) -' 1 by A103, XREAL_0:def_2 ; ((len pion1) - 1) + 1 <= len pion1 ; then A106: (len pion1) -' 1 < len pion1 by A104, NAT_1:13; LSeg (pion1,((len pion1) -' 1)) c= L~ <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> by A81, TOPREAL3:19; then LSeg (pion1,((len pion1) -' 1)) c= LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) by SPPOL_2:21; then A107: (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) c= {((Gauge (C,n)) * (i,j))} by A58, XBOOLE_1:27; {((Gauge (C,n)) * (i,j))} c= (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i,j))} or x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) ) assume x in {((Gauge (C,n)) * (i,j))} ; ::_thesis: x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) then A108: x = (Gauge (C,n)) * (i,j) by TARSKI:def_1; pion1 /. (((len pion1) -' 1) + 1) = <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. 2 by A83, A104, FINSEQ_1:44 .= (Gauge (C,n)) * (i,j) by FINSEQ_4:17 ; then A109: (Gauge (C,n)) * (i,j) in LSeg (pion1,((len pion1) -' 1)) by A103, A104, TOPREAL1:21; (Gauge (C,n)) * (i,j) in LSeg (do,1) by A55, RLTOPSP1:68; hence x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) by A108, A109, XBOOLE_0:def_4; ::_thesis: verum end; then (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) = {((Gauge (C,n)) * (i,j))} by A107, XBOOLE_0:def_10; then A110: (LSeg ((go ^' pion1),((len go) + ((len pion1) -' 2)))) /\ (LSeg (do,1)) = {((go ^' pion1) /. (len (go ^' pion1)))} by A42, A92, A94, A105, A106, TOPREAL8:31; A111: not go ^' pion1 is trivial by A87, NAT_D:60; A112: rng pion1 c= L~ pion1 by A97, SPPOL_2:18; A113: {(pion1 /. 1)} c= (L~ go) /\ (L~ pion1) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(pion1 /. 1)} or x in (L~ go) /\ (L~ pion1) ) assume x in {(pion1 /. 1)} ; ::_thesis: x in (L~ go) /\ (L~ pion1) then A114: x = pion1 /. 1 by TARSKI:def_1; then A115: x in rng pion1 by FINSEQ_6:42; x in rng go by A92, A114, REVROT_1:3; hence x in (L~ go) /\ (L~ pion1) by A61, A112, A115, XBOOLE_0:def_4; ::_thesis: verum end; (L~ go) /\ (L~ pion1) c= {(pion1 /. 1)} proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ go) /\ (L~ pion1) or x in {(pion1 /. 1)} ) assume A116: x in (L~ go) /\ (L~ pion1) ; ::_thesis: x in {(pion1 /. 1)} then A117: x in L~ pion1 by XBOOLE_0:def_4; x in L~ go by A116, XBOOLE_0:def_4; then x in (L~ pion1) /\ (L~ (Upper_Seq (C,n))) by A46, A117, XBOOLE_0:def_4; hence x in {(pion1 /. 1)} by A6, A41, A81, A92, SPPOL_2:21; ::_thesis: verum end; then A118: (L~ go) /\ (L~ pion1) = {(pion1 /. 1)} by A113, XBOOLE_0:def_10; then A119: go ^' pion1 is s.n.c. by A92, Th54; (rng go) /\ (rng pion1) c= {(pion1 /. 1)} by A61, A112, A118, XBOOLE_1:27; then A120: go ^' pion1 is one-to-one by Th55; A121: <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. (len <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*>) = <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. 2 by FINSEQ_1:44 .= do /. 1 by A42, FINSEQ_4:17 ; A122: {(pion1 /. (len pion1))} c= (L~ do) /\ (L~ pion1) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(pion1 /. (len pion1))} or x in (L~ do) /\ (L~ pion1) ) assume x in {(pion1 /. (len pion1))} ; ::_thesis: x in (L~ do) /\ (L~ pion1) then A123: x = pion1 /. (len pion1) by TARSKI:def_1; then A124: x in rng pion1 by REVROT_1:3; x in rng do by A83, A121, A123, FINSEQ_6:42; hence x in (L~ do) /\ (L~ pion1) by A62, A112, A124, XBOOLE_0:def_4; ::_thesis: verum end; (L~ do) /\ (L~ pion1) c= {(pion1 /. (len pion1))} proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ do) /\ (L~ pion1) or x in {(pion1 /. (len pion1))} ) assume A125: x in (L~ do) /\ (L~ pion1) ; ::_thesis: x in {(pion1 /. (len pion1))} then A126: x in L~ pion1 by XBOOLE_0:def_4; x in L~ do by A125, XBOOLE_0:def_4; then x in (L~ pion1) /\ (L~ (Lower_Seq (C,n))) by A53, A126, XBOOLE_0:def_4; hence x in {(pion1 /. (len pion1))} by A7, A42, A81, A83, A121, SPPOL_2:21; ::_thesis: verum end; then A127: (L~ do) /\ (L~ pion1) = {(pion1 /. (len pion1))} by A122, XBOOLE_0:def_10; A128: (L~ (go ^' pion1)) /\ (L~ do) = ((L~ go) \/ (L~ pion1)) /\ (L~ do) by A92, TOPREAL8:35 .= {(go /. 1)} \/ {(do /. 1)} by A72, A83, A121, A127, XBOOLE_1:23 .= {((go ^' pion1) /. 1)} \/ {(do /. 1)} by GRAPH_2:53 .= {((go ^' pion1) /. 1),(do /. 1)} by ENUMSET1:1 ; do /. (len do) = (go ^' pion1) /. 1 by A60, GRAPH_2:53; then reconsider godo = (go ^' pion1) ^' do as non constant standard special_circular_sequence by A90, A94, A95, A100, A102, A110, A111, A119, A120, A128, JORDAN8:4, JORDAN8:5, TOPREAL8:11, TOPREAL8:33, TOPREAL8:34; A129: Upper_Arc C is_an_arc_of W-min C, E-max C by JORDAN6:def_8; then A130: Upper_Arc C is connected by JORDAN6:10; A131: W-min C in Upper_Arc C by A129, TOPREAL1:1; A132: E-max C in Upper_Arc C by A129, TOPREAL1:1; set ff = Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))); W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43; then A133: (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 = W-min (L~ (Cage (C,n))) by FINSEQ_6:92; A134: L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33; then (W-max (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A133, SPRECT_5:22; then (N-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A133, A134, SPRECT_5:23, XXREAL_0:2; then (N-max (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A133, A134, SPRECT_5:24, XXREAL_0:2; then A135: (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A133, A134, SPRECT_5:25, XXREAL_0:2; A136: now__::_thesis:_not_((Gauge_(C,n))_*_(i,k))_.._(Upper_Seq_(C,n))_<=_1 assume A137: ((Gauge (C,n)) * (i,k)) .. (Upper_Seq (C,n)) <= 1 ; ::_thesis: contradiction ((Gauge (C,n)) * (i,k)) .. (Upper_Seq (C,n)) >= 1 by A34, FINSEQ_4:21; then ((Gauge (C,n)) * (i,k)) .. (Upper_Seq (C,n)) = 1 by A137, XXREAL_0:1; then (Gauge (C,n)) * (i,k) = (Upper_Seq (C,n)) /. 1 by A34, FINSEQ_5:38; hence contradiction by A18, A22, JORDAN1F:5; ::_thesis: verum end; A138: Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def_1; then A139: Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))) is_sequence_on Gauge (C,n) by REVROT_1:34; A140: (right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo) c= RightComp godo by A90, A95, JORDAN9:27; A141: L~ godo = (L~ (go ^' pion1)) \/ (L~ do) by A94, TOPREAL8:35 .= ((L~ go) \/ (L~ pion1)) \/ (L~ do) by A92, TOPREAL8:35 ; A142: L~ (Cage (C,n)) = (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n))) by JORDAN1E:13; then A143: L~ (Upper_Seq (C,n)) c= L~ (Cage (C,n)) by XBOOLE_1:7; A144: L~ (Lower_Seq (C,n)) c= L~ (Cage (C,n)) by A142, XBOOLE_1:7; A145: L~ go c= L~ (Cage (C,n)) by A46, A143, XBOOLE_1:1; A146: L~ do c= L~ (Cage (C,n)) by A53, A144, XBOOLE_1:1; A147: W-min C in C by SPRECT_1:13; A148: L~ <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> = LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) by SPPOL_2:21; A149: now__::_thesis:_not_W-min_C_in_L~_godo assume W-min C in L~ godo ; ::_thesis: contradiction then A150: ( W-min C in (L~ go) \/ (L~ pion1) or W-min C in L~ do ) by A141, XBOOLE_0:def_3; percases ( W-min C in L~ go or W-min C in L~ pion1 or W-min C in L~ do ) by A150, XBOOLE_0:def_3; suppose W-min C in L~ go ; ::_thesis: contradiction then C meets L~ (Cage (C,n)) by A145, A147, XBOOLE_0:3; hence contradiction by JORDAN10:5; ::_thesis: verum end; suppose W-min C in L~ pion1 ; ::_thesis: contradiction hence contradiction by A8, A81, A131, A148, XBOOLE_0:3; ::_thesis: verum end; suppose W-min C in L~ do ; ::_thesis: contradiction then C meets L~ (Cage (C,n)) by A146, A147, XBOOLE_0:3; hence contradiction by JORDAN10:5; ::_thesis: verum end; end; end; right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1) = right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(GoB (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) by A86, JORDAN1H:23 .= right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(GoB (Cage (C,n)))) by REVROT_1:28 .= right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(Gauge (C,n))) by JORDAN1H:44 .= right_cell (((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n))))),1,(Gauge (C,n))) by A135, A139, Th53 .= right_cell ((Upper_Seq (C,n)),1,(Gauge (C,n))) by JORDAN1E:def_1 .= right_cell ((R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,k)))),1,(Gauge (C,n))) by A34, A91, A136, Th52 .= right_cell ((go ^' pion1),1,(Gauge (C,n))) by A39, A93, Th51 .= right_cell (godo,1,(Gauge (C,n))) by A88, A95, Th51 ; then W-min C in right_cell (godo,1,(Gauge (C,n))) by JORDAN1I:6; then A151: W-min C in (right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo) by A149, XBOOLE_0:def_5; A152: godo /. 1 = (go ^' pion1) /. 1 by GRAPH_2:53 .= W-min (L~ (Cage (C,n))) by A59, GRAPH_2:53 ; A153: len (Upper_Seq (C,n)) >= 2 by A17, XXREAL_0:2; A154: godo /. 2 = (go ^' pion1) /. 2 by A87, GRAPH_2:57 .= (Upper_Seq (C,n)) /. 2 by A33, A75, GRAPH_2:57 .= ((Upper_Seq (C,n)) ^' (Lower_Seq (C,n))) /. 2 by A153, GRAPH_2:57 .= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 2 by JORDAN1E:11 ; A155: (L~ go) \/ (L~ do) is compact by COMPTS_1:10; W-min (L~ (Cage (C,n))) in rng go by A59, FINSEQ_6:42; then W-min (L~ (Cage (C,n))) in (L~ go) \/ (L~ do) by A61, XBOOLE_0:def_3; then A156: W-min ((L~ go) \/ (L~ do)) = W-min (L~ (Cage (C,n))) by A145, A146, A155, Th21, XBOOLE_1:8; A157: (W-min ((L~ go) \/ (L~ do))) `1 = W-bound ((L~ go) \/ (L~ do)) by EUCLID:52; A158: (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52; W-bound (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) = ((Gauge (C,n)) * (i,k)) `1 by A78, SPRECT_1:54; then A159: W-bound (L~ pion1) = ((Gauge (C,n)) * (i,k)) `1 by A81, SPPOL_2:21; ((Gauge (C,n)) * (i,k)) `1 >= W-bound (L~ (Cage (C,n))) by A10, A143, PSCOMP_1:24; then ((Gauge (C,n)) * (i,k)) `1 > W-bound (L~ (Cage (C,n))) by A74, XXREAL_0:1; then W-min (((L~ go) \/ (L~ do)) \/ (L~ pion1)) = W-min ((L~ go) \/ (L~ do)) by A155, A156, A157, A158, A159, Th33; then A160: W-min (L~ godo) = W-min (L~ (Cage (C,n))) by A141, A156, XBOOLE_1:4; A161: rng godo c= L~ godo by A87, A89, SPPOL_2:18, XXREAL_0:2; 2 in dom godo by A90, FINSEQ_3:25; then A162: godo /. 2 in rng godo by PARTFUN2:2; godo /. 2 in W-most (L~ (Cage (C,n))) by A154, JORDAN1I:25; then (godo /. 2) `1 = (W-min (L~ godo)) `1 by A160, PSCOMP_1:31 .= W-bound (L~ godo) by EUCLID:52 ; then godo /. 2 in W-most (L~ godo) by A161, A162, SPRECT_2:12; then (Rotate (godo,(W-min (L~ godo)))) /. 2 in W-most (L~ godo) by A152, A160, FINSEQ_6:89; then reconsider godo = godo as non constant standard clockwise_oriented special_circular_sequence by JORDAN1I:25; len (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by FINSEQ_5:6; then A163: (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) = (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) by PARTFUN1:def_6 .= E-max (L~ (Cage (C,n))) by JORDAN1F:7 ; A164: east_halfline (E-max C) misses L~ go proof assume east_halfline (E-max C) meets L~ go ; ::_thesis: contradiction then consider p being set such that A165: p in east_halfline (E-max C) and A166: p in L~ go by XBOOLE_0:3; reconsider p = p as Point of (TOP-REAL 2) by A165; p in L~ (Upper_Seq (C,n)) by A46, A166; then p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A143, A165, XBOOLE_0:def_4; then A167: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50; then A168: p = E-max (L~ (Cage (C,n))) by A46, A166, Th46; then E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i,k) by A10, A163, A166, Th43; then ((Gauge (C,n)) * (i,k)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by A5, A13, A16, A167, A168, JORDAN1A:71; hence contradiction by A2, A15, A30, JORDAN1G:7; ::_thesis: verum end; now__::_thesis:_not_east_halfline_(E-max_C)_meets_L~_godo assume east_halfline (E-max C) meets L~ godo ; ::_thesis: contradiction then A169: ( east_halfline (E-max C) meets (L~ go) \/ (L~ pion1) or east_halfline (E-max C) meets L~ do ) by A141, XBOOLE_1:70; percases ( east_halfline (E-max C) meets L~ go or east_halfline (E-max C) meets L~ pion1 or east_halfline (E-max C) meets L~ do ) by A169, XBOOLE_1:70; suppose east_halfline (E-max C) meets L~ go ; ::_thesis: contradiction hence contradiction by A164; ::_thesis: verum end; suppose east_halfline (E-max C) meets L~ pion1 ; ::_thesis: contradiction then consider p being set such that A170: p in east_halfline (E-max C) and A171: p in L~ pion1 by XBOOLE_0:3; reconsider p = p as Point of (TOP-REAL 2) by A170; A172: p `2 = (E-max C) `2 by A170, TOPREAL1:def_11; i + 1 <= len (Gauge (C,n)) by A2, NAT_1:13; then (i + 1) - 1 <= (len (Gauge (C,n))) - 1 by XREAL_1:9; then A173: i <= (len (Gauge (C,n))) -' 1 by XREAL_0:def_2; A174: (len (Gauge (C,n))) -' 1 <= len (Gauge (C,n)) by NAT_D:35; p `1 = ((Gauge (C,n)) * (i,k)) `1 by A78, A81, A148, A171, GOBOARD7:5; then p `1 <= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),1)) `1 by A1, A5, A13, A16, A20, A173, A174, JORDAN1A:18; then p `1 <= E-bound C by A20, JORDAN8:12; then A175: p `1 <= (E-max C) `1 by EUCLID:52; p `1 >= (E-max C) `1 by A170, TOPREAL1:def_11; then p `1 = (E-max C) `1 by A175, XXREAL_0:1; then p = E-max C by A172, TOPREAL3:6; hence contradiction by A8, A81, A132, A148, A171, XBOOLE_0:3; ::_thesis: verum end; suppose east_halfline (E-max C) meets L~ do ; ::_thesis: contradiction then consider p being set such that A176: p in east_halfline (E-max C) and A177: p in L~ do by XBOOLE_0:3; reconsider p = p as Point of (TOP-REAL 2) by A176; A178: (E-max C) `2 = p `2 by A176, TOPREAL1:def_11; set tt = ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1; set RC = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))); A179: L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33; consider t being Nat such that A180: t in dom (Lower_Seq (C,n)) and A181: (Lower_Seq (C,n)) . t = (Gauge (C,n)) * (i,j) by A37, FINSEQ_2:10; 1 <= t by A180, FINSEQ_3:25; then A182: 1 < t by A32, A181, XXREAL_0:1; t <= len (Lower_Seq (C,n)) by A180, FINSEQ_3:25; then (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1 = t by A181, A182, JORDAN3:12; then A183: len (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) = (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) by A9, A181, JORDAN3:26; Index (p,do) < len do by A177, JORDAN3:8; then Index (p,do) < (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) by A183, XREAL_0:def_2; then (Index (p,do)) + 1 <= (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) by NAT_1:13; then A184: Index (p,do) <= ((len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n))))) - 1 by XREAL_1:19; A185: do = mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n)))) by A37, Th37; A186: len (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = len (Cage (C,n)) by REVROT_1:14; p in L~ (Lower_Seq (C,n)) by A53, A177; then p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A144, A176, XBOOLE_0:def_4; then A187: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50; A188: GoB (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = GoB (Cage (C,n)) by REVROT_1:28 .= Gauge (C,n) by JORDAN1H:44 ; A189: 1 + 1 <= len (Lower_Seq (C,n)) by A23, XXREAL_0:2; then A190: 2 in dom (Lower_Seq (C,n)) by FINSEQ_3:25; consider jj2 being Element of NAT such that A191: 1 <= jj2 and A192: jj2 <= width (Gauge (C,n)) and A193: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * ((len (Gauge (C,n))),jj2) by JORDAN1D:25; A194: len (Gauge (C,n)) >= 4 by JORDAN8:10; then len (Gauge (C,n)) >= 1 by XXREAL_0:2; then A195: [(len (Gauge (C,n))),jj2] in Indices (Gauge (C,n)) by A191, A192, MATRIX_1:36; A196: 1 <= Index (p,do) by A177, JORDAN3:8; Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) by JORDAN1G:18; then A197: LSeg ((Lower_Seq (C,n)),1) = LSeg ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by A189, SPPOL_2:9; A198: E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46; Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))) is_sequence_on Gauge (C,n) by A138, REVROT_1:34; then consider ii, jj being Element of NAT such that A199: [ii,(jj + 1)] in Indices (Gauge (C,n)) and A200: [ii,jj] in Indices (Gauge (C,n)) and A201: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = (Gauge (C,n)) * (ii,(jj + 1)) and A202: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1) = (Gauge (C,n)) * (ii,jj) by A85, A179, A186, A198, FINSEQ_6:92, JORDAN1I:23; A203: (jj + 1) + 1 <> jj ; A204: 1 <= jj by A200, MATRIX_1:38; (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = E-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))))) by A179, A198, FINSEQ_6:92; then A205: ii = len (Gauge (C,n)) by A179, A199, A201, A193, A195, GOBOARD1:5; then ii - 1 >= 4 - 1 by A194, XREAL_1:9; then A206: ii - 1 >= 1 by XXREAL_0:2; then A207: 1 <= ii -' 1 by XREAL_0:def_2; A208: jj <= width (Gauge (C,n)) by A200, MATRIX_1:38; then A209: ((Gauge (C,n)) * ((len (Gauge (C,n))),jj)) `1 = E-bound (L~ (Cage (C,n))) by A16, A204, JORDAN1A:71; A210: jj + 1 <= width (Gauge (C,n)) by A199, MATRIX_1:38; ii + 1 <> ii ; then A211: right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) = cell ((Gauge (C,n)),(ii -' 1),jj) by A85, A186, A188, A199, A200, A201, A202, A203, GOBOARD5:def_6; A212: ii <= len (Gauge (C,n)) by A200, MATRIX_1:38; A213: 1 <= ii by A200, MATRIX_1:38; A214: ii <= len (Gauge (C,n)) by A199, MATRIX_1:38; A215: 1 <= jj + 1 by A199, MATRIX_1:38; then A216: E-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * ((len (Gauge (C,n))),(jj + 1))) `1 by A16, A210, JORDAN1A:71; A217: 1 <= ii by A199, MATRIX_1:38; then A218: (ii -' 1) + 1 = ii by XREAL_1:235; then A219: ii -' 1 < len (Gauge (C,n)) by A214, NAT_1:13; then A220: ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 = ((Gauge (C,n)) * (1,(jj + 1))) `2 by A215, A210, A207, GOBOARD5:1 .= ((Gauge (C,n)) * (ii,(jj + 1))) `2 by A217, A214, A215, A210, GOBOARD5:1 ; A221: E-max C in right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by JORDAN1I:7; then A222: ((Gauge (C,n)) * ((ii -' 1),jj)) `2 <= (E-max C) `2 by A214, A210, A204, A211, A218, A206, JORDAN9:17; A223: (E-max C) `2 <= ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 by A221, A214, A210, A204, A211, A218, A206, JORDAN9:17; ((Gauge (C,n)) * ((ii -' 1),jj)) `2 = ((Gauge (C,n)) * (1,jj)) `2 by A204, A208, A207, A219, GOBOARD5:1 .= ((Gauge (C,n)) * (ii,jj)) `2 by A213, A212, A204, A208, GOBOARD5:1 ; then p in LSeg (((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1),((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1))) by A187, A178, A201, A202, A205, A222, A223, A220, A209, A216, GOBOARD7:7; then A224: p in LSeg ((Lower_Seq (C,n)),1) by A85, A197, A186, TOPREAL1:def_3; A225: ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) <= len (Lower_Seq (C,n)) by A37, FINSEQ_4:21; ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) <> len (Lower_Seq (C,n)) by A29, A37, FINSEQ_4:19; then A226: ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) < len (Lower_Seq (C,n)) by A225, XXREAL_0:1; A227: (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1 = ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) by A32, A37, Th56; 0 + (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) < len (Lower_Seq (C,n)) by A9, JORDAN3:8; then (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) > 0 by XREAL_1:20; then Index (p,do) <= ((len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n))))) - 1 by A184, XREAL_0:def_2; then Index (p,do) <= (len (Lower_Seq (C,n))) - (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))) by A227; then Index (p,do) <= (len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))) by XREAL_0:def_2; then A228: Index (p,do) < ((len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) + 1 by NAT_1:13; A229: p in LSeg (do,(Index (p,do))) by A177, JORDAN3:9; 1 <= ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) by A37, FINSEQ_4:21; then A230: LSeg ((mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n))))),(Index (p,do))) = LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1)) by A226, A196, A228, JORDAN4:19; 1 <= Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n))) by A9, JORDAN3:8; then A231: 1 + 1 <= ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) by A227, XREAL_1:7; then (Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))) >= (1 + 1) + 1 by A196, XREAL_1:7; then ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) - 1 >= ((1 + 1) + 1) - 1 by XREAL_1:9; then A232: ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1 >= 1 + 1 by XREAL_0:def_2; now__::_thesis:_contradiction percases ( ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1 > 1 + 1 or ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ) by A232, XXREAL_0:1; suppose ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1 > 1 + 1 ; ::_thesis: contradiction then LSeg ((Lower_Seq (C,n)),1) misses LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1)) by TOPREAL1:def_7; hence contradiction by A224, A229, A185, A230, XBOOLE_0:3; ::_thesis: verum end; supposeA233: ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ; ::_thesis: contradiction then 1 + 1 = ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) - 1 by XREAL_0:def_2; then (1 + 1) + 1 = (Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))) ; then A234: ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) = 2 by A196, A231, JORDAN1E:6; (LSeg ((Lower_Seq (C,n)),1)) /\ (LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1))) = {((Lower_Seq (C,n)) /. 2)} by A23, A233, TOPREAL1:def_6; then p in {((Lower_Seq (C,n)) /. 2)} by A224, A229, A185, A230, XBOOLE_0:def_4; then A235: p = (Lower_Seq (C,n)) /. 2 by TARSKI:def_1; then A236: p in rng (Lower_Seq (C,n)) by A190, PARTFUN2:2; p .. (Lower_Seq (C,n)) = 2 by A190, A235, FINSEQ_5:41; then p = (Gauge (C,n)) * (i,j) by A37, A234, A236, FINSEQ_5:9; then ((Gauge (C,n)) * (i,j)) `1 = E-bound (L~ (Cage (C,n))) by A235, JORDAN1G:32; then ((Gauge (C,n)) * (i,j)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 by A3, A12, A16, JORDAN1A:71; hence contradiction by A2, A14, A67, JORDAN1G:7; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; end; end; then east_halfline (E-max C) c= (L~ godo) ` by SUBSET_1:23; then consider W being Subset of (TOP-REAL 2) such that A237: W is_a_component_of (L~ godo) ` and A238: east_halfline (E-max C) c= W by GOBOARD9:3; not W is bounded by A238, JORDAN2C:121, RLTOPSP1:42; then W is_outside_component_of L~ godo by A237, JORDAN2C:def_3; then W c= UBD (L~ godo) by JORDAN2C:23; then A239: east_halfline (E-max C) c= UBD (L~ godo) by A238, XBOOLE_1:1; E-max C in east_halfline (E-max C) by TOPREAL1:38; then E-max C in UBD (L~ godo) by A239; then E-max C in LeftComp godo by GOBRD14:36; then Upper_Arc C meets L~ godo by A130, A131, A132, A140, A151, Th36; then A240: ( Upper_Arc C meets (L~ go) \/ (L~ pion1) or Upper_Arc C meets L~ do ) by A141, XBOOLE_1:70; A241: Upper_Arc C c= C by JORDAN6:61; percases ( Upper_Arc C meets L~ go or Upper_Arc C meets L~ pion1 or Upper_Arc C meets L~ do ) by A240, XBOOLE_1:70; suppose Upper_Arc C meets L~ go ; ::_thesis: contradiction then Upper_Arc C meets L~ (Cage (C,n)) by A46, A143, XBOOLE_1:1, XBOOLE_1:63; hence contradiction by A241, JORDAN10:5, XBOOLE_1:63; ::_thesis: verum end; suppose Upper_Arc C meets L~ pion1 ; ::_thesis: contradiction hence contradiction by A8, A81, A148; ::_thesis: verum end; suppose Upper_Arc C meets L~ do ; ::_thesis: contradiction then Upper_Arc C meets L~ (Cage (C,n)) by A53, A144, XBOOLE_1:1, XBOOLE_1:63; hence contradiction by A241, JORDAN10:5, XBOOLE_1:63; ::_thesis: verum end; end; end; theorem Th60: :: JORDAN1J:60 for n being Element of NAT for C being Simple_closed_curve for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Upper_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Lower_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,j))} holds LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C proof let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Upper_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Lower_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,j))} holds LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C let C be Simple_closed_curve; ::_thesis: for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Upper_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Lower_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,j))} holds LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C let i, j, k be Element of NAT ; ::_thesis: ( 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Upper_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Lower_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,j))} implies LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C ) assume that A1: 1 < i and A2: i < len (Gauge (C,n)) and A3: 1 <= j and A4: j <= k and A5: k <= width (Gauge (C,n)) and A6: n > 0 and A7: (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Upper_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,k))} and A8: (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Lower_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,j))} ; ::_thesis: LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C A9: L~ (Lower_Seq (C,n)) = Lower_Arc (L~ (Cage (C,n))) by A6, JORDAN1G:56; L~ (Upper_Seq (C,n)) = Upper_Arc (L~ (Cage (C,n))) by A6, JORDAN1G:55; hence LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C by A1, A2, A3, A4, A5, A7, A8, A9, Th58; ::_thesis: verum end; theorem Th61: :: JORDAN1J:61 for n being Element of NAT for C being Simple_closed_curve for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Upper_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Lower_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,j))} holds LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C proof let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Upper_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Lower_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,j))} holds LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C let C be Simple_closed_curve; ::_thesis: for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Upper_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Lower_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,j))} holds LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C let i, j, k be Element of NAT ; ::_thesis: ( 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Upper_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Lower_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,j))} implies LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C ) assume that A1: 1 < i and A2: i < len (Gauge (C,n)) and A3: 1 <= j and A4: j <= k and A5: k <= width (Gauge (C,n)) and A6: n > 0 and A7: (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Upper_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,k))} and A8: (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Lower_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,j))} ; ::_thesis: LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C A9: L~ (Lower_Seq (C,n)) = Lower_Arc (L~ (Cage (C,n))) by A6, JORDAN1G:56; L~ (Upper_Seq (C,n)) = Upper_Arc (L~ (Cage (C,n))) by A6, JORDAN1G:55; hence LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C by A1, A2, A3, A4, A5, A7, A8, A9, Th59; ::_thesis: verum end; theorem :: JORDAN1J:62 for n being Element of NAT for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) for j being Element of NAT st (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j) in Upper_Arc (L~ (Cage (C,(n + 1)))) & 1 <= j & j <= width (Gauge (C,(n + 1))) holds LSeg (((Gauge (C,1)) * ((Center (Gauge (C,1))),1)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))) meets Lower_Arc (L~ (Cage (C,(n + 1)))) proof let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) for j being Element of NAT st (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j) in Upper_Arc (L~ (Cage (C,(n + 1)))) & 1 <= j & j <= width (Gauge (C,(n + 1))) holds LSeg (((Gauge (C,1)) * ((Center (Gauge (C,1))),1)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))) meets Lower_Arc (L~ (Cage (C,(n + 1)))) let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for j being Element of NAT st (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j) in Upper_Arc (L~ (Cage (C,(n + 1)))) & 1 <= j & j <= width (Gauge (C,(n + 1))) holds LSeg (((Gauge (C,1)) * ((Center (Gauge (C,1))),1)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))) meets Lower_Arc (L~ (Cage (C,(n + 1)))) let j be Element of NAT ; ::_thesis: ( (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j) in Upper_Arc (L~ (Cage (C,(n + 1)))) & 1 <= j & j <= width (Gauge (C,(n + 1))) implies LSeg (((Gauge (C,1)) * ((Center (Gauge (C,1))),1)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))) meets Lower_Arc (L~ (Cage (C,(n + 1)))) ) assume that A1: (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j) in Upper_Arc (L~ (Cage (C,(n + 1)))) and A2: 1 <= j and A3: j <= width (Gauge (C,(n + 1))) ; ::_thesis: LSeg (((Gauge (C,1)) * ((Center (Gauge (C,1))),1)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))) meets Lower_Arc (L~ (Cage (C,(n + 1)))) set in1 = Center (Gauge (C,(n + 1))); A4: 1 <= Center (Gauge (C,(n + 1))) by JORDAN1B:11; A5: Upper_Arc (L~ (Cage (C,(n + 1)))) c= L~ (Cage (C,(n + 1))) by JORDAN6:61; A6: Center (Gauge (C,(n + 1))) <= len (Gauge (C,(n + 1))) by JORDAN1B:13; n + 1 >= 0 + 1 by NAT_1:11; then LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),1)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))) c= LSeg (((Gauge (C,1)) * ((Center (Gauge (C,1))),1)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))) by A2, A3, JORDAN1A:45; hence LSeg (((Gauge (C,1)) * ((Center (Gauge (C,1))),1)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))) meets Lower_Arc (L~ (Cage (C,(n + 1)))) by A1, A2, A3, A4, A6, A5, JORDAN1G:57, XBOOLE_1:63; ::_thesis: verum end; theorem :: JORDAN1J:63 for n being Element of NAT for C being Simple_closed_curve for j, k being Element of NAT st 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Upper_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))} & (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Lower_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))} holds LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))) meets Lower_Arc C proof let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve for j, k being Element of NAT st 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Upper_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))} & (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Lower_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))} holds LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))) meets Lower_Arc C let C be Simple_closed_curve; ::_thesis: for j, k being Element of NAT st 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Upper_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))} & (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Lower_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))} holds LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))) meets Lower_Arc C let j, k be Element of NAT ; ::_thesis: ( 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Upper_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))} & (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Lower_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))} implies LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))) meets Lower_Arc C ) assume that A1: 1 <= j and A2: j <= k and A3: k <= width (Gauge (C,(n + 1))) and A4: (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Upper_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))} and A5: (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Lower_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))} ; ::_thesis: LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))) meets Lower_Arc C A6: len (Gauge (C,(n + 1))) >= 4 by JORDAN8:10; then len (Gauge (C,(n + 1))) >= 2 by XXREAL_0:2; then A7: 1 < Center (Gauge (C,(n + 1))) by JORDAN1B:14; len (Gauge (C,(n + 1))) >= 3 by A6, XXREAL_0:2; hence LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))) meets Lower_Arc C by A1, A2, A3, A4, A5, A7, Th60, JORDAN1B:15; ::_thesis: verum end; theorem :: JORDAN1J:64 for n being Element of NAT for C being Simple_closed_curve for j, k being Element of NAT st 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Upper_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))} & (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Lower_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))} holds LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))) meets Upper_Arc C proof let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve for j, k being Element of NAT st 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Upper_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))} & (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Lower_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))} holds LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))) meets Upper_Arc C let C be Simple_closed_curve; ::_thesis: for j, k being Element of NAT st 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Upper_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))} & (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Lower_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))} holds LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))) meets Upper_Arc C let j, k be Element of NAT ; ::_thesis: ( 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Upper_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))} & (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Lower_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))} implies LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))) meets Upper_Arc C ) assume that A1: 1 <= j and A2: j <= k and A3: k <= width (Gauge (C,(n + 1))) and A4: (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Upper_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))} and A5: (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Lower_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))} ; ::_thesis: LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))) meets Upper_Arc C A6: len (Gauge (C,(n + 1))) >= 4 by JORDAN8:10; then len (Gauge (C,(n + 1))) >= 2 by XXREAL_0:2; then A7: 1 < Center (Gauge (C,(n + 1))) by JORDAN1B:14; len (Gauge (C,(n + 1))) >= 3 by A6, XXREAL_0:2; hence LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))) meets Upper_Arc C by A1, A2, A3, A4, A5, A7, Th61, JORDAN1B:15; ::_thesis: verum end; theorem :: JORDAN1J:65 for X, Y being non empty compact Subset of (TOP-REAL 2) st X c= Y & ( W-min Y in X or W-max Y in X ) holds W-bound X = W-bound Y proof let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( X c= Y & ( W-min Y in X or W-max Y in X ) implies W-bound X = W-bound Y ) assume that A1: X c= Y and A2: ( W-min Y in X or W-max Y in X ) ; ::_thesis: W-bound X = W-bound Y A3: (W-max X) `1 = W-bound X by EUCLID:52; A4: (W-max Y) `1 = W-bound Y by EUCLID:52; A5: (W-min Y) `1 = W-bound Y by EUCLID:52; (W-min X) `1 = W-bound X by EUCLID:52; hence W-bound X = W-bound Y by A1, A2, A3, A5, A4, Th21, Th22; ::_thesis: verum end; theorem :: JORDAN1J:66 for X, Y being non empty compact Subset of (TOP-REAL 2) st X c= Y & ( E-min Y in X or E-max Y in X ) holds E-bound X = E-bound Y proof let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( X c= Y & ( E-min Y in X or E-max Y in X ) implies E-bound X = E-bound Y ) assume that A1: X c= Y and A2: ( E-min Y in X or E-max Y in X ) ; ::_thesis: E-bound X = E-bound Y A3: (E-max X) `1 = E-bound X by EUCLID:52; A4: (E-max Y) `1 = E-bound Y by EUCLID:52; A5: (E-min Y) `1 = E-bound Y by EUCLID:52; (E-min X) `1 = E-bound X by EUCLID:52; hence E-bound X = E-bound Y by A1, A2, A3, A5, A4, Th17, Th18; ::_thesis: verum end; theorem :: JORDAN1J:67 for X, Y being non empty compact Subset of (TOP-REAL 2) st X c= Y & ( N-min Y in X or N-max Y in X ) holds N-bound X = N-bound Y proof let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( X c= Y & ( N-min Y in X or N-max Y in X ) implies N-bound X = N-bound Y ) assume that A1: X c= Y and A2: ( N-min Y in X or N-max Y in X ) ; ::_thesis: N-bound X = N-bound Y A3: (N-max X) `2 = N-bound X by EUCLID:52; A4: (N-max Y) `2 = N-bound Y by EUCLID:52; A5: (N-min Y) `2 = N-bound Y by EUCLID:52; (N-min X) `2 = N-bound X by EUCLID:52; hence N-bound X = N-bound Y by A1, A2, A3, A5, A4, Th15, Th16; ::_thesis: verum end; theorem :: JORDAN1J:68 for X, Y being non empty compact Subset of (TOP-REAL 2) st X c= Y & ( S-min Y in X or S-max Y in X ) holds S-bound X = S-bound Y proof let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( X c= Y & ( S-min Y in X or S-max Y in X ) implies S-bound X = S-bound Y ) assume that A1: X c= Y and A2: ( S-min Y in X or S-max Y in X ) ; ::_thesis: S-bound X = S-bound Y A3: (S-max X) `2 = S-bound X by EUCLID:52; A4: (S-max Y) `2 = S-bound Y by EUCLID:52; A5: (S-min Y) `2 = S-bound Y by EUCLID:52; (S-min X) `2 = S-bound X by EUCLID:52; hence S-bound X = S-bound Y by A1, A2, A3, A5, A4, Th19, Th20; ::_thesis: verum end;