let n be Nat; :: thesis: for C being Simple_closed_curve

for i1, i2, j, k being Nat st 1 < i2 & i2 <= i1 & i1 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} holds

(LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Lower_Arc C

let C be Simple_closed_curve; :: thesis: for i1, i2, j, k being Nat st 1 < i2 & i2 <= i1 & i1 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} holds

(LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Lower_Arc C

let i1, i2, j, k be Nat; :: thesis: ( 1 < i2 & i2 <= i1 & i1 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} implies (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Lower_Arc C )

set G = Gauge (C,n);

set pio = LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)));

set poz = LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)));

set US = Upper_Seq (C,n);

set LS = Lower_Seq (C,n);

assume that

A1: 1 < i2 and

A2: i2 <= i1 and

A3: i1 < len (Gauge (C,n)) and

A4: 1 <= j and

A5: j <= k and

A6: k <= width (Gauge (C,n)) and

A7: ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} and

A8: ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} and

A9: (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) misses Lower_Arc C ; :: thesis: contradiction

set Gi1k = (Gauge (C,n)) * (i1,k);

set Gik = (Gauge (C,n)) * (i2,k);

A10: 1 <= k by A4, A5, XXREAL_0:2;

A11: i2 < len (Gauge (C,n)) by A2, A3, XXREAL_0:2;

then A12: [i2,k] in Indices (Gauge (C,n)) by A1, A6, A10, MATRIX_0:30;

set Wmin = W-min (L~ (Cage (C,n)));

set Wbo = W-bound (L~ (Cage (C,n)));

A13: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def 1;

set go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i2,k)));

A14: 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 A15: (Upper_Seq (C,n)) . 1 = (Upper_Seq (C,n)) /. 1 by PARTFUN1:def 6

.= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ;

set Gij = (Gauge (C,n)) * (i1,j);

set co = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i1,j)));

(Gauge (C,n)) * (i1,j) in {((Gauge (C,n)) * (i1,j))} by TARSKI:def 1;

then A16: (Gauge (C,n)) * (i1,j) in L~ (Lower_Seq (C,n)) by A8, XBOOLE_0:def 4;

A17: 1 < i1 by A1, A2, XXREAL_0:2;

then A18: ((Gauge (C,n)) * (i1,k)) `2 = ((Gauge (C,n)) * (1,k)) `2 by A3, A6, A10, GOBOARD5:1

.= ((Gauge (C,n)) * (i2,k)) `2 by A1, A6, A11, A10, GOBOARD5:1 ;

A19: j <= width (Gauge (C,n)) by A5, A6, XXREAL_0:2;

then A20: [i1,j] in Indices (Gauge (C,n)) by A3, A4, A17, MATRIX_0:30;

len (Gauge (C,n)) >= 4 by JORDAN8:10;

then A21: len (Gauge (C,n)) >= 1 by XXREAL_0:2;

then A22: [(len (Gauge (C,n))),j] in Indices (Gauge (C,n)) by A4, A19, MATRIX_0:30;

A23: [1,k] in Indices (Gauge (C,n)) by A6, A10, A21, MATRIX_0:30;

set pion = <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>;

A26: (Gauge (C,n)) * (i1,k) in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) by RLTOPSP1:68;

set LA = Lower_Arc C;

A27: (Gauge (C,n)) * (i1,k) in LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) by RLTOPSP1:68;

A28: [i1,k] in Indices (Gauge (C,n)) by A3, A6, A17, A10, MATRIX_0:30;

then A30: (Gauge (C,n)) * (i2,k) in L~ (Upper_Seq (C,n)) by A7, XBOOLE_0:def 4;

set Emax = E-max (L~ (Cage (C,n)));

A31: len (Lower_Seq (C,n)) >= 1 + 2 by JORDAN1E:15;

then A32: len (Lower_Seq (C,n)) >= 1 by XXREAL_0:2;

then A33: 1 in dom (Lower_Seq (C,n)) by FINSEQ_3:25;

then A34: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by PARTFUN1:def 6

.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;

len (Lower_Seq (C,n)) in dom (Lower_Seq (C,n)) by A32, FINSEQ_3:25;

then A35: (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 ;

set Ebo = E-bound (L~ (Cage (C,n)));

A36: L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> = (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) by TOPREAL3:16;

(W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52

.= ((Gauge (C,n)) * (1,k)) `1 by A6, A10, A13, JORDAN1A:73 ;

then A37: (Gauge (C,n)) * (i2,k) <> (Upper_Seq (C,n)) . 1 by A1, A12, A15, A23, JORDAN1G:7;

then reconsider go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i2,k))) as being_S-Seq FinSequence of (TOP-REAL 2) by A30, JORDAN3:35;

A38: (Gauge (C,n)) * (i2,k) in rng (Upper_Seq (C,n)) by A1, A6, A11, A30, A10, JORDAN1G:4, JORDAN1J:40;

then A39: go is_sequence_on Gauge (C,n) by JORDAN1G:4, JORDAN1J:38;

((Gauge (C,n)) * (i1,k)) `1 = ((Gauge (C,n)) * (i1,1)) `1 by A3, A6, A17, A10, GOBOARD5:2

.= ((Gauge (C,n)) * (i1,j)) `1 by A3, A4, A17, A19, GOBOARD5:2 ;

then A40: (Gauge (C,n)) * (i1,k) = |[(((Gauge (C,n)) * (i1,j)) `1),(((Gauge (C,n)) * (i2,k)) `2)]| by A18, EUCLID:53;

A41: [(len (Gauge (C,n))),k] in Indices (Gauge (C,n)) by A6, A10, A21, MATRIX_0:30;

A42: len go >= 1 + 1 by TOPREAL1:def 8;

(W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52

.= ((Gauge (C,n)) * (1,k)) `1 by A6, A10, A13, JORDAN1A:73 ;

then A43: (Gauge (C,n)) * (i1,j) <> (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) by A1, A2, A23, A35, A20, JORDAN1G:7;

then reconsider co = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i1,j))) as being_S-Seq FinSequence of (TOP-REAL 2) by A16, JORDAN3:34;

A44: (Gauge (C,n)) * (i1,j) in rng (Lower_Seq (C,n)) by A3, A4, A17, A16, A19, JORDAN1G:5, JORDAN1J:40;

then A45: co is_sequence_on Gauge (C,n) by JORDAN1G:5, JORDAN1J:39;

(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 A6, A10, A13, JORDAN1A:71 ;

then A46: (Gauge (C,n)) * (i1,j) <> (Lower_Seq (C,n)) . 1 by A3, A20, A41, A34, JORDAN1G:7;

A47: len co >= 1 + 1 by TOPREAL1:def 8;

then reconsider co = co as V29() being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A45, JGRAPH_1:12, JORDAN8:5;

A48: L~ co c= L~ (Lower_Seq (C,n)) by A16, JORDAN3:42;

len co >= 1 by A47, XXREAL_0:2;

then 1 in dom co by FINSEQ_3:25;

then A49: co /. 1 = co . 1 by PARTFUN1:def 6

.= (Gauge (C,n)) * (i1,j) by A16, JORDAN3:23 ;

then A50: LSeg (co,1) = LSeg (((Gauge (C,n)) * (i1,j)),(co /. (1 + 1))) by A47, TOPREAL1:def 3;

A51: {((Gauge (C,n)) * (i1,j))} c= (LSeg (co,1)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>)

then LSeg (co,1) c= L~ (Lower_Seq (C,n)) by A48;

then (LSeg (co,1)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) c= {((Gauge (C,n)) * (i1,j))} by A8, A36, XBOOLE_1:26;

then A54: (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) /\ (LSeg (co,1)) = {((Gauge (C,n)) * (i1,j))} by A51;

A55: rng co c= L~ co by A47, SPPOL_2:18;

reconsider go = go as V29() being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A42, A39, JGRAPH_1:12, JORDAN8:5;

A56: L~ go c= L~ (Upper_Seq (C,n)) by A30, JORDAN3:41;

A57: len go > 1 by A42, NAT_1:13;

then A58: len go in dom go by FINSEQ_3:25;

then A59: go /. (len go) = go . (len go) by PARTFUN1:def 6

.= (Gauge (C,n)) * (i2,k) by A30, JORDAN3:24 ;

reconsider m = (len go) - 1 as Nat by A58, FINSEQ_3:26;

A60: m + 1 = len go ;

then A61: (len go) -' 1 = m by NAT_D:34;

m >= 1 by A42, XREAL_1:19;

then A62: LSeg (go,m) = LSeg ((go /. m),((Gauge (C,n)) * (i2,k))) by A59, A60, TOPREAL1:def 3;

A63: {((Gauge (C,n)) * (i2,k))} c= (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>)

then LSeg (go,m) c= L~ (Upper_Seq (C,n)) by A56;

then (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) c= {((Gauge (C,n)) * (i2,k))} by A7, A36, XBOOLE_1:26;

then A66: (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) = {((Gauge (C,n)) * (i2,k))} by A63;

A67: go /. 1 = (Upper_Seq (C,n)) /. 1 by A30, SPRECT_3:22

.= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ;

then A68: W-min (L~ (Cage (C,n))) in rng go by FINSEQ_6:42;

A69: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by A33, PARTFUN1:def 6

.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;

A70: (L~ go) /\ (L~ co) c= {(go /. 1)}

A75: 2 in dom go by A42, FINSEQ_3:25;

go = mid ((Upper_Seq (C,n)),1,(((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n)))) by A38, JORDAN1G:49

.= (Upper_Seq (C,n)) | (((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n))) by A38, FINSEQ_4:21, FINSEQ_6:116 ;

then A76: go /. 2 = (Upper_Seq (C,n)) /. 2 by A75, FINSEQ_4:70;

A77: rng go c= L~ go by A42, SPPOL_2:18;

A78: go /. 1 = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by A67, JORDAN1F:8

.= co /. (len co) by A16, JORDAN1J:35 ;

{(go /. 1)} c= (L~ go) /\ (L~ co)

for i1, i2, j, k being Nat st 1 < i2 & i2 <= i1 & i1 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} holds

(LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Lower_Arc C

let C be Simple_closed_curve; :: thesis: for i1, i2, j, k being Nat st 1 < i2 & i2 <= i1 & i1 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} holds

(LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Lower_Arc C

let i1, i2, j, k be Nat; :: thesis: ( 1 < i2 & i2 <= i1 & i1 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} implies (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Lower_Arc C )

set G = Gauge (C,n);

set pio = LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)));

set poz = LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)));

set US = Upper_Seq (C,n);

set LS = Lower_Seq (C,n);

assume that

A1: 1 < i2 and

A2: i2 <= i1 and

A3: i1 < len (Gauge (C,n)) and

A4: 1 <= j and

A5: j <= k and

A6: k <= width (Gauge (C,n)) and

A7: ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} and

A8: ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} and

A9: (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) misses Lower_Arc C ; :: thesis: contradiction

set Gi1k = (Gauge (C,n)) * (i1,k);

set Gik = (Gauge (C,n)) * (i2,k);

A10: 1 <= k by A4, A5, XXREAL_0:2;

A11: i2 < len (Gauge (C,n)) by A2, A3, XXREAL_0:2;

then A12: [i2,k] in Indices (Gauge (C,n)) by A1, A6, A10, MATRIX_0:30;

set Wmin = W-min (L~ (Cage (C,n)));

set Wbo = W-bound (L~ (Cage (C,n)));

A13: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def 1;

set go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i2,k)));

A14: 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 A15: (Upper_Seq (C,n)) . 1 = (Upper_Seq (C,n)) /. 1 by PARTFUN1:def 6

.= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ;

set Gij = (Gauge (C,n)) * (i1,j);

set co = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i1,j)));

(Gauge (C,n)) * (i1,j) in {((Gauge (C,n)) * (i1,j))} by TARSKI:def 1;

then A16: (Gauge (C,n)) * (i1,j) in L~ (Lower_Seq (C,n)) by A8, XBOOLE_0:def 4;

A17: 1 < i1 by A1, A2, XXREAL_0:2;

then A18: ((Gauge (C,n)) * (i1,k)) `2 = ((Gauge (C,n)) * (1,k)) `2 by A3, A6, A10, GOBOARD5:1

.= ((Gauge (C,n)) * (i2,k)) `2 by A1, A6, A11, A10, GOBOARD5:1 ;

A19: j <= width (Gauge (C,n)) by A5, A6, XXREAL_0:2;

then A20: [i1,j] in Indices (Gauge (C,n)) by A3, A4, A17, MATRIX_0:30;

len (Gauge (C,n)) >= 4 by JORDAN8:10;

then A21: len (Gauge (C,n)) >= 1 by XXREAL_0:2;

then A22: [(len (Gauge (C,n))),j] in Indices (Gauge (C,n)) by A4, A19, MATRIX_0:30;

A23: [1,k] in Indices (Gauge (C,n)) by A6, A10, A21, MATRIX_0:30;

A24: now :: thesis: not ((Gauge (C,n)) * (i2,k)) `1 = W-bound (L~ (Cage (C,n)))

A25:
[i1,j] in Indices (Gauge (C,n))
by A3, A4, A17, A19, MATRIX_0:30;assume
((Gauge (C,n)) * (i2,k)) `1 = W-bound (L~ (Cage (C,n)))
; :: thesis: contradiction

then ((Gauge (C,n)) * (1,k)) `1 = ((Gauge (C,n)) * (i2,k)) `1 by A6, A10, A13, JORDAN1A:73;

hence contradiction by A1, A12, A23, JORDAN1G:7; :: thesis: verum

end;then ((Gauge (C,n)) * (1,k)) `1 = ((Gauge (C,n)) * (i2,k)) `1 by A6, A10, A13, JORDAN1A:73;

hence contradiction by A1, A12, A23, JORDAN1G:7; :: thesis: verum

set pion = <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>;

A26: (Gauge (C,n)) * (i1,k) in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) by RLTOPSP1:68;

set LA = Lower_Arc C;

A27: (Gauge (C,n)) * (i1,k) in LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) by RLTOPSP1:68;

A28: [i1,k] in Indices (Gauge (C,n)) by A3, A6, A17, A10, MATRIX_0:30;

A29: now :: thesis: for n being Nat st n in dom <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> holds

ex i, j being Nat st

( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. n = (Gauge (C,n)) * (i,j) )

(Gauge (C,n)) * (i2,k) in {((Gauge (C,n)) * (i2,k))}
by TARSKI:def 1;ex i, j being Nat st

( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. n = (Gauge (C,n)) * (i,j) )

let n be Nat; :: thesis: ( n in dom <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> implies ex i, j being Nat st

( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. n = (Gauge (C,n)) * (i,j) ) )

assume n in dom <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> ; :: thesis: ex i, j being Nat st

( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. n = (Gauge (C,n)) * (i,j) )

then n in {1,2,3} by FINSEQ_1:89, FINSEQ_3:1;

then ( n = 1 or n = 2 or n = 3 ) by ENUMSET1:def 1;

hence ex i, j being Nat st

( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. n = (Gauge (C,n)) * (i,j) ) by A25, A12, A28, FINSEQ_4:18; :: thesis: verum

end;( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. n = (Gauge (C,n)) * (i,j) ) )

assume n in dom <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> ; :: thesis: ex i, j being Nat st

( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. n = (Gauge (C,n)) * (i,j) )

then n in {1,2,3} by FINSEQ_1:89, FINSEQ_3:1;

then ( n = 1 or n = 2 or n = 3 ) by ENUMSET1:def 1;

hence ex i, j being Nat st

( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. n = (Gauge (C,n)) * (i,j) ) by A25, A12, A28, FINSEQ_4:18; :: thesis: verum

then A30: (Gauge (C,n)) * (i2,k) in L~ (Upper_Seq (C,n)) by A7, XBOOLE_0:def 4;

set Emax = E-max (L~ (Cage (C,n)));

A31: len (Lower_Seq (C,n)) >= 1 + 2 by JORDAN1E:15;

then A32: len (Lower_Seq (C,n)) >= 1 by XXREAL_0:2;

then A33: 1 in dom (Lower_Seq (C,n)) by FINSEQ_3:25;

then A34: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by PARTFUN1:def 6

.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;

len (Lower_Seq (C,n)) in dom (Lower_Seq (C,n)) by A32, FINSEQ_3:25;

then A35: (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 ;

set Ebo = E-bound (L~ (Cage (C,n)));

A36: L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> = (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) by TOPREAL3:16;

(W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52

.= ((Gauge (C,n)) * (1,k)) `1 by A6, A10, A13, JORDAN1A:73 ;

then A37: (Gauge (C,n)) * (i2,k) <> (Upper_Seq (C,n)) . 1 by A1, A12, A15, A23, JORDAN1G:7;

then reconsider go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i2,k))) as being_S-Seq FinSequence of (TOP-REAL 2) by A30, JORDAN3:35;

A38: (Gauge (C,n)) * (i2,k) in rng (Upper_Seq (C,n)) by A1, A6, A11, A30, A10, JORDAN1G:4, JORDAN1J:40;

then A39: go is_sequence_on Gauge (C,n) by JORDAN1G:4, JORDAN1J:38;

((Gauge (C,n)) * (i1,k)) `1 = ((Gauge (C,n)) * (i1,1)) `1 by A3, A6, A17, A10, GOBOARD5:2

.= ((Gauge (C,n)) * (i1,j)) `1 by A3, A4, A17, A19, GOBOARD5:2 ;

then A40: (Gauge (C,n)) * (i1,k) = |[(((Gauge (C,n)) * (i1,j)) `1),(((Gauge (C,n)) * (i2,k)) `2)]| by A18, EUCLID:53;

A41: [(len (Gauge (C,n))),k] in Indices (Gauge (C,n)) by A6, A10, A21, MATRIX_0:30;

A42: len go >= 1 + 1 by TOPREAL1:def 8;

(W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52

.= ((Gauge (C,n)) * (1,k)) `1 by A6, A10, A13, JORDAN1A:73 ;

then A43: (Gauge (C,n)) * (i1,j) <> (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) by A1, A2, A23, A35, A20, JORDAN1G:7;

then reconsider co = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i1,j))) as being_S-Seq FinSequence of (TOP-REAL 2) by A16, JORDAN3:34;

A44: (Gauge (C,n)) * (i1,j) in rng (Lower_Seq (C,n)) by A3, A4, A17, A16, A19, JORDAN1G:5, JORDAN1J:40;

then A45: co is_sequence_on Gauge (C,n) by JORDAN1G:5, JORDAN1J:39;

(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 A6, A10, A13, JORDAN1A:71 ;

then A46: (Gauge (C,n)) * (i1,j) <> (Lower_Seq (C,n)) . 1 by A3, A20, A41, A34, JORDAN1G:7;

A47: len co >= 1 + 1 by TOPREAL1:def 8;

then reconsider co = co as V29() being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A45, JGRAPH_1:12, JORDAN8:5;

A48: L~ co c= L~ (Lower_Seq (C,n)) by A16, JORDAN3:42;

len co >= 1 by A47, XXREAL_0:2;

then 1 in dom co by FINSEQ_3:25;

then A49: co /. 1 = co . 1 by PARTFUN1:def 6

.= (Gauge (C,n)) * (i1,j) by A16, JORDAN3:23 ;

then A50: LSeg (co,1) = LSeg (((Gauge (C,n)) * (i1,j)),(co /. (1 + 1))) by A47, TOPREAL1:def 3;

A51: {((Gauge (C,n)) * (i1,j))} c= (LSeg (co,1)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>)

proof

LSeg (co,1) c= L~ co
by TOPREAL3:19;
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in {((Gauge (C,n)) * (i1,j))} or x in (LSeg (co,1)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) )

assume x in {((Gauge (C,n)) * (i1,j))} ; :: thesis: x in (LSeg (co,1)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>)

then A52: x = (Gauge (C,n)) * (i1,j) by TARSKI:def 1;

(Gauge (C,n)) * (i1,j) in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))) by RLTOPSP1:68;

then (Gauge (C,n)) * (i1,j) in (LSeg (((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j)))) by XBOOLE_0:def 3;

then A53: (Gauge (C,n)) * (i1,j) in L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> by SPRECT_1:8;

(Gauge (C,n)) * (i1,j) in LSeg (co,1) by A50, RLTOPSP1:68;

hence x in (LSeg (co,1)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) by A52, A53, XBOOLE_0:def 4; :: thesis: verum

end;assume x in {((Gauge (C,n)) * (i1,j))} ; :: thesis: x in (LSeg (co,1)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>)

then A52: x = (Gauge (C,n)) * (i1,j) by TARSKI:def 1;

(Gauge (C,n)) * (i1,j) in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))) by RLTOPSP1:68;

then (Gauge (C,n)) * (i1,j) in (LSeg (((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j)))) by XBOOLE_0:def 3;

then A53: (Gauge (C,n)) * (i1,j) in L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> by SPRECT_1:8;

(Gauge (C,n)) * (i1,j) in LSeg (co,1) by A50, RLTOPSP1:68;

hence x in (LSeg (co,1)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) by A52, A53, XBOOLE_0:def 4; :: thesis: verum

then LSeg (co,1) c= L~ (Lower_Seq (C,n)) by A48;

then (LSeg (co,1)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) c= {((Gauge (C,n)) * (i1,j))} by A8, A36, XBOOLE_1:26;

then A54: (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) /\ (LSeg (co,1)) = {((Gauge (C,n)) * (i1,j))} by A51;

A55: rng co c= L~ co by A47, SPPOL_2:18;

reconsider go = go as V29() being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A42, A39, JGRAPH_1:12, JORDAN8:5;

A56: L~ go c= L~ (Upper_Seq (C,n)) by A30, JORDAN3:41;

A57: len go > 1 by A42, NAT_1:13;

then A58: len go in dom go by FINSEQ_3:25;

then A59: go /. (len go) = go . (len go) by PARTFUN1:def 6

.= (Gauge (C,n)) * (i2,k) by A30, JORDAN3:24 ;

reconsider m = (len go) - 1 as Nat by A58, FINSEQ_3:26;

A60: m + 1 = len go ;

then A61: (len go) -' 1 = m by NAT_D:34;

m >= 1 by A42, XREAL_1:19;

then A62: LSeg (go,m) = LSeg ((go /. m),((Gauge (C,n)) * (i2,k))) by A59, A60, TOPREAL1:def 3;

A63: {((Gauge (C,n)) * (i2,k))} c= (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>)

proof

LSeg (go,m) c= L~ go
by TOPREAL3:19;
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in {((Gauge (C,n)) * (i2,k))} or x in (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) )

assume x in {((Gauge (C,n)) * (i2,k))} ; :: thesis: x in (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>)

then A64: x = (Gauge (C,n)) * (i2,k) by TARSKI:def 1;

(Gauge (C,n)) * (i2,k) in LSeg (((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k))) by RLTOPSP1:68;

then (Gauge (C,n)) * (i2,k) in (LSeg (((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j)))) by XBOOLE_0:def 3;

then A65: (Gauge (C,n)) * (i2,k) in L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> by SPRECT_1:8;

(Gauge (C,n)) * (i2,k) in LSeg (go,m) by A62, RLTOPSP1:68;

hence x in (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) by A64, A65, XBOOLE_0:def 4; :: thesis: verum

end;assume x in {((Gauge (C,n)) * (i2,k))} ; :: thesis: x in (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>)

then A64: x = (Gauge (C,n)) * (i2,k) by TARSKI:def 1;

(Gauge (C,n)) * (i2,k) in LSeg (((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k))) by RLTOPSP1:68;

then (Gauge (C,n)) * (i2,k) in (LSeg (((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j)))) by XBOOLE_0:def 3;

then A65: (Gauge (C,n)) * (i2,k) in L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> by SPRECT_1:8;

(Gauge (C,n)) * (i2,k) in LSeg (go,m) by A62, RLTOPSP1:68;

hence x in (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) by A64, A65, XBOOLE_0:def 4; :: thesis: verum

then LSeg (go,m) c= L~ (Upper_Seq (C,n)) by A56;

then (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) c= {((Gauge (C,n)) * (i2,k))} by A7, A36, XBOOLE_1:26;

then A66: (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) = {((Gauge (C,n)) * (i2,k))} by A63;

A67: go /. 1 = (Upper_Seq (C,n)) /. 1 by A30, SPRECT_3:22

.= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ;

then A68: W-min (L~ (Cage (C,n))) in rng go by FINSEQ_6:42;

A69: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by A33, PARTFUN1:def 6

.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;

A70: (L~ go) /\ (L~ co) c= {(go /. 1)}

proof

set W2 = go /. 2;
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in (L~ go) /\ (L~ co) or x in {(go /. 1)} )

assume A71: x in (L~ go) /\ (L~ co) ; :: thesis: x in {(go /. 1)}

then A72: x in L~ co by XBOOLE_0:def 4;

then x in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A56, A48, A72, 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 A67, A73, TARSKI:def 1; :: thesis: verum

end;assume A71: x in (L~ go) /\ (L~ co) ; :: thesis: x in {(go /. 1)}

then A72: x in L~ co by XBOOLE_0:def 4;

A73: now :: thesis: not x = E-max (L~ (Cage (C,n)))

x in L~ go
by A71, XBOOLE_0:def 4;assume
x = E-max (L~ (Cage (C,n)))
; :: thesis: contradiction

then A74: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i1,j) by A16, A69, A72, JORDAN1E:7;

((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 = E-bound (L~ (Cage (C,n))) by A4, A19, A13, JORDAN1A:71;

then (E-max (L~ (Cage (C,n)))) `1 <> E-bound (L~ (Cage (C,n))) by A3, A25, A22, A74, JORDAN1G:7;

hence contradiction by EUCLID:52; :: thesis: verum

end;then A74: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i1,j) by A16, A69, A72, JORDAN1E:7;

((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 = E-bound (L~ (Cage (C,n))) by A4, A19, A13, JORDAN1A:71;

then (E-max (L~ (Cage (C,n)))) `1 <> E-bound (L~ (Cage (C,n))) by A3, A25, A22, A74, JORDAN1G:7;

hence contradiction by EUCLID:52; :: thesis: verum

then x in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A56, A48, A72, 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 A67, A73, TARSKI:def 1; :: thesis: verum

A75: 2 in dom go by A42, FINSEQ_3:25;

go = mid ((Upper_Seq (C,n)),1,(((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n)))) by A38, JORDAN1G:49

.= (Upper_Seq (C,n)) | (((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n))) by A38, FINSEQ_4:21, FINSEQ_6:116 ;

then A76: go /. 2 = (Upper_Seq (C,n)) /. 2 by A75, FINSEQ_4:70;

A77: rng go c= L~ go by A42, SPPOL_2:18;

A78: go /. 1 = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by A67, JORDAN1F:8

.= co /. (len co) by A16, JORDAN1J:35 ;

{(go /. 1)} c= (L~ go) /\ (L~ co)

proof

then A81:
(L~ go) /\ (L~ co) = {(go /. 1)}
by A70;
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in {(go /. 1)} or x in (L~ go) /\ (L~ co) )

assume x in {(go /. 1)} ; :: thesis: x in (L~ go) /\ (L~ co)

then A79: x = go /. 1 by TARSKI:def 1;

then A80: x in rng go by FINSEQ_6:42;

x in rng co by A78, A79, FINSEQ_6:168;

hence x in (L~ go) /\ (L~ co) by A77, A55, A80, XBOOLE_0:def 4; :: thesis: verum

end;assume x in {(go /. 1)} ; :: thesis: x in (L~ go) /\ (L~ co)

then A79: x = go /. 1 by TARSKI:def 1;

then A80: x in rng go by FINSEQ_6:42;

x in rng co by A78, A79, FINSEQ_6:168;

hence x in (L~ go) /\ (L~ co) by A77, A55, A80, XBOOLE_0:def 4; :: thesis: verum

now :: thesis: contradictionend;

hence
contradiction
; :: thesis: verumper cases
( ( ((Gauge (C,n)) * (i1,j)) `1 <> ((Gauge (C,n)) * (i2,k)) `1 & ((Gauge (C,n)) * (i1,j)) `2 <> ((Gauge (C,n)) * (i2,k)) `2 ) or ((Gauge (C,n)) * (i1,j)) `1 = ((Gauge (C,n)) * (i2,k)) `1 or ((Gauge (C,n)) * (i1,j)) `2 = ((Gauge (C,n)) * (i2,k)) `2 )
;

end;

suppose
( ((Gauge (C,n)) * (i1,j)) `1 <> ((Gauge (C,n)) * (i2,k)) `1 & ((Gauge (C,n)) * (i1,j)) `2 <> ((Gauge (C,n)) * (i2,k)) `2 )
; :: thesis: contradiction

then
<*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> is being_S-Seq
by A40, TOPREAL3:35;

then consider pion1 being FinSequence of (TOP-REAL 2) such that

A82: pion1 is_sequence_on Gauge (C,n) and

A83: pion1 is being_S-Seq and

A84: L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> = L~ pion1 and

A85: <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. 1 = pion1 /. 1 and

A86: <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. (len <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) = pion1 /. (len pion1) and

A87: len <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> <= len pion1 by A29, GOBOARD3:2;

reconsider pion1 = pion1 as being_S-Seq FinSequence of (TOP-REAL 2) by A83;

A88: (go ^' pion1) /. (len (go ^' pion1)) = <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. (len <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) by A86, FINSEQ_6:156

.= <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. 3 by FINSEQ_1:45

.= co /. 1 by A49, FINSEQ_4:18 ;

A89: go /. (len go) = pion1 /. 1 by A59, A85, FINSEQ_4:18;

A90: (L~ go) /\ (L~ pion1) c= {(pion1 /. 1)}

then A93: len pion1 > 1 + 1 by NAT_1:13;

then A94: rng pion1 c= L~ pion1 by SPPOL_2:18;

{(pion1 /. 1)} c= (L~ go) /\ (L~ pion1)

then A98: go ^' pion1 is s.n.c. by A89, JORDAN1J:54;

A99: {((Gauge (C,n)) * (i2,k))} c= (LSeg (go,m)) /\ (LSeg (pion1,1))

then (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) c= {((Gauge (C,n)) * (i2,k))} by A61, A66, XBOOLE_1:27;

then (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) = {(go /. (len go))} by A59, A61, A99;

then A102: go ^' pion1 is unfolded by A89, TOPREAL8:34;

len (go ^' pion1) >= len go by TOPREAL8:7;

then A103: len (go ^' pion1) >= 1 + 1 by A42, XXREAL_0:2;

then A104: len (go ^' pion1) > 1 + 0 by NAT_1:13;

A105: <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. (len <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) = <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. 3 by FINSEQ_1:45

.= co /. 1 by A49, FINSEQ_4:18 ;

A106: {(pion1 /. (len pion1))} c= (L~ co) /\ (L~ pion1)

A112: (L~ (go ^' pion1)) /\ (L~ co) = ((L~ go) \/ (L~ pion1)) /\ (L~ co) by A89, TOPREAL8:35

.= {(go /. 1)} \/ {(co /. 1)} by A81, A86, A105, A111, XBOOLE_1:23

.= {((go ^' pion1) /. 1)} \/ {(co /. 1)} by FINSEQ_6:155

.= {((go ^' pion1) /. 1),(co /. 1)} by ENUMSET1:1 ;

A113: Lower_Arc C is_an_arc_of E-max C, W-min C by JORDAN6:def 9;

then A114: Lower_Arc C is connected by JORDAN6:10;

set godo = (go ^' pion1) ^' co;

A115: co /. (len co) = (go ^' pion1) /. 1 by A78, FINSEQ_6:155;

A116: go ^' pion1 is_sequence_on Gauge (C,n) by A39, A82, A89, TOPREAL8:12;

then A117: (go ^' pion1) ^' co is_sequence_on Gauge (C,n) by A45, A88, TOPREAL8:12;

A118: (len pion1) - 1 >= 1 by A93, XREAL_1:19;

then A119: (len pion1) -' 1 = (len pion1) - 1 by XREAL_0:def 2;

A120: {((Gauge (C,n)) * (i1,j))} c= (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (co,1))

then (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (co,1)) c= {((Gauge (C,n)) * (i1,j))} by A54, XBOOLE_1:27;

then A123: (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (co,1)) = {((Gauge (C,n)) * (i1,j))} by A120;

((len pion1) - 1) + 1 <= len pion1 ;

then A124: (len pion1) -' 1 < len pion1 by A119, NAT_1:13;

len pion1 >= 2 + 1 by A87, FINSEQ_1:45;

then A125: (len pion1) - 2 >= 0 by XREAL_1:19;

then ((len pion1) -' 2) + 1 = ((len pion1) - 2) + 1 by XREAL_0:def 2

.= (len pion1) -' 1 by A118, XREAL_0:def 2 ;

then A126: (LSeg ((go ^' pion1),((len go) + ((len pion1) -' 2)))) /\ (LSeg (co,1)) = {((go ^' pion1) /. (len (go ^' pion1)))} by A49, A89, A88, A124, A123, TOPREAL8:31;

(rng go) /\ (rng pion1) c= {(pion1 /. 1)} by A77, A94, A97, XBOOLE_1:27;

then A127: go ^' pion1 is one-to-one by JORDAN1J:55;

((len (go ^' pion1)) + 1) - 1 = ((len go) + (len pion1)) - 1 by FINSEQ_6:139;

then (len (go ^' pion1)) - 1 = (len go) + ((len pion1) - 2)

.= (len go) + ((len pion1) -' 2) by A125, XREAL_0:def 2 ;

then A128: (len (go ^' pion1)) -' 1 = (len go) + ((len pion1) -' 2) by XREAL_0:def 2;

A129: L~ (Cage (C,n)) = (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n))) by JORDAN1E:13;

then A130: L~ (Upper_Seq (C,n)) c= L~ (Cage (C,n)) by XBOOLE_1:7;

then A131: L~ go c= L~ (Cage (C,n)) by A56;

A132: len ((go ^' pion1) ^' co) >= len (go ^' pion1) by TOPREAL8:7;

then A133: 1 + 1 <= len ((go ^' pion1) ^' co) by A103, XXREAL_0:2;

not go ^' pion1 is trivial by A103, NAT_D:60;

then reconsider godo = (go ^' pion1) ^' co as V29() standard special_circular_sequence by A133, A88, A117, A102, A128, A126, A98, A127, A112, A115, JORDAN8:4, JORDAN8:5, TOPREAL8:11, TOPREAL8:33, TOPREAL8:34;

A134: L~ godo = (L~ (go ^' pion1)) \/ (L~ co) by A88, TOPREAL8:35

.= ((L~ go) \/ (L~ pion1)) \/ (L~ co) by A89, TOPREAL8:35 ;

A138: ((Gauge (C,n)) * (i2,k)) `1 <= ((Gauge (C,n)) * (i1,k)) `1 by A1, A2, A3, A6, A10, JORDAN1A:18;

then A139: W-bound (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) = ((Gauge (C,n)) * (i2,k)) `1 by SPRECT_1:54;

A140: ((Gauge (C,n)) * (i1,k)) `1 = ((Gauge (C,n)) * (i1,1)) `1 by A3, A6, A17, A10, GOBOARD5:2

.= ((Gauge (C,n)) * (i1,j)) `1 by A3, A4, A17, A19, GOBOARD5:2 ;

then A141: W-bound (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) = ((Gauge (C,n)) * (i1,j)) `1 by SPRECT_1:54;

W-bound ((LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))))) = min ((W-bound (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))),(W-bound (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))))) by SPRECT_1:47

.= ((Gauge (C,n)) * (i2,k)) `1 by A140, A138, A139, A141, XXREAL_0:def 9 ;

then A142: W-bound (L~ pion1) = ((Gauge (C,n)) * (i2,k)) `1 by A84, TOPREAL3:16;

A143: Lower_Arc C c= C by JORDAN6:61;

((Gauge (C,n)) * (i2,k)) `1 >= W-bound (L~ (Cage (C,n))) by A30, A130, PSCOMP_1:24;

then A144: ((Gauge (C,n)) * (i2,k)) `1 > W-bound (L~ (Cage (C,n))) by A24, XXREAL_0:1;

A145: len (Upper_Seq (C,n)) >= 2 by A14, XXREAL_0:2;

A146: (L~ go) \/ (L~ co) is compact by COMPTS_1:10;

A147: L~ (Lower_Seq (C,n)) c= L~ (Cage (C,n)) by A129, XBOOLE_1:7;

then A148: L~ co c= L~ (Cage (C,n)) by A48;

A149: (right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo) c= RightComp godo by A133, A117, JORDAN9:27;

2 in dom godo by A133, FINSEQ_3:25;

then A150: godo /. 2 in rng godo by PARTFUN2:2;

A151: rng godo c= L~ godo by A103, A132, SPPOL_2:18, XXREAL_0:2;

A152: godo /. 1 = (go ^' pion1) /. 1 by FINSEQ_6:155

.= W-min (L~ (Cage (C,n))) by A67, FINSEQ_6:155 ;

A153: W-min C in Lower_Arc C by A113, TOPREAL1:1;

A154: W-min C in C by SPRECT_1:13;

set ff = Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))));

A158: 1 + 1 <= len (Cage (C,n)) by GOBOARD7:34, XXREAL_0:2;

W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;

then A159: (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 = W-min (L~ (Cage (C,n))) by FINSEQ_6:92;

A160: 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 A159, 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 A159, A160, 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 A159, A160, SPRECT_5:24, XXREAL_0:2;

then A161: (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A159, A160, SPRECT_5:25, XXREAL_0:2;

A162: Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def 1;

then A163: Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))) is_sequence_on Gauge (C,n) by REVROT_1:34;

1 + 1 <= len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by GOBOARD7:34, XXREAL_0:2;

then 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 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 A161, A163, JORDAN1J:53

.= 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)) * (i2,k)))),1,(Gauge (C,n))) by A38, A137, A135, JORDAN1J:52

.= right_cell ((go ^' pion1),1,(Gauge (C,n))) by A57, A116, JORDAN1J:51

.= right_cell (godo,1,(Gauge (C,n))) by A104, A117, JORDAN1J:51 ;

then W-min C in right_cell (godo,1,(Gauge (C,n))) by JORDAN1I:6;

then A164: W-min C in (right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo) by A155, XBOOLE_0:def 5;

A165: E-max C in Lower_Arc C by A113, TOPREAL1:1;

W-min (L~ (Cage (C,n))) in (L~ go) \/ (L~ co) by A77, A68, XBOOLE_0:def 3;

then A166: W-min ((L~ go) \/ (L~ co)) = W-min (L~ (Cage (C,n))) by A131, A148, A146, JORDAN1J:21, XBOOLE_1:8;

(W-min ((L~ go) \/ (L~ co))) `1 = W-bound ((L~ go) \/ (L~ co)) by EUCLID:52;

then W-min (((L~ go) \/ (L~ co)) \/ (L~ pion1)) = W-min ((L~ go) \/ (L~ co)) by A142, A146, A166, A157, A144, JORDAN1J:33;

then A167: W-min (L~ godo) = W-min (L~ (Cage (C,n))) by A134, A166, XBOOLE_1:4;

godo /. 2 = (go ^' pion1) /. 2 by A103, FINSEQ_6:159

.= (Upper_Seq (C,n)) /. 2 by A42, A76, FINSEQ_6:159

.= ((Upper_Seq (C,n)) ^' (Lower_Seq (C,n))) /. 2 by A145, FINSEQ_6:159

.= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 2 by JORDAN1E:11 ;

then godo /. 2 in W-most (L~ (Cage (C,n))) by JORDAN1I:25;

then (godo /. 2) `1 = (W-min (L~ godo)) `1 by A167, PSCOMP_1:31

.= W-bound (L~ godo) by EUCLID:52 ;

then godo /. 2 in W-most (L~ godo) by A151, A150, SPRECT_2:12;

then (Rotate (godo,(W-min (L~ godo)))) /. 2 in W-most (L~ godo) by A152, A167, FINSEQ_6:89;

then reconsider godo = godo as V29() 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 A168: (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 ;

A169: east_halfline (E-max C) misses L~ go

then consider W being Subset of (TOP-REAL 2) such that

A242: W is_a_component_of (L~ godo) ` and

A243: east_halfline (E-max C) c= W by GOBOARD9:3;

not W is bounded by A243, JORDAN2C:121, RLTOPSP1:42;

then W is_outside_component_of L~ godo by A242, JORDAN2C:def 3;

then W c= UBD (L~ godo) by JORDAN2C:23;

then A244: east_halfline (E-max C) c= UBD (L~ godo) by A243;

E-max C in east_halfline (E-max C) by TOPREAL1:38;

then E-max C in UBD (L~ godo) by A244;

then E-max C in LeftComp godo by GOBRD14:36;

then Lower_Arc C meets L~ godo by A114, A153, A165, A149, A164, JORDAN1J:36;

then A245: ( Lower_Arc C meets (L~ go) \/ (L~ pion1) or Lower_Arc C meets L~ co ) by A134, XBOOLE_1:70;

end;then consider pion1 being FinSequence of (TOP-REAL 2) such that

A82: pion1 is_sequence_on Gauge (C,n) and

A83: pion1 is being_S-Seq and

A84: L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> = L~ pion1 and

A85: <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. 1 = pion1 /. 1 and

A86: <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. (len <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) = pion1 /. (len pion1) and

A87: len <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> <= len pion1 by A29, GOBOARD3:2;

reconsider pion1 = pion1 as being_S-Seq FinSequence of (TOP-REAL 2) by A83;

A88: (go ^' pion1) /. (len (go ^' pion1)) = <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. (len <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) by A86, FINSEQ_6:156

.= <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. 3 by FINSEQ_1:45

.= co /. 1 by A49, FINSEQ_4:18 ;

A89: go /. (len go) = pion1 /. 1 by A59, A85, FINSEQ_4:18;

A90: (L~ go) /\ (L~ pion1) c= {(pion1 /. 1)}

proof

len pion1 >= 2 + 1
by A87, FINSEQ_1:45;
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in (L~ go) /\ (L~ pion1) or x in {(pion1 /. 1)} )

assume A91: x in (L~ go) /\ (L~ pion1) ; :: thesis: x in {(pion1 /. 1)}

then A92: x in L~ pion1 by XBOOLE_0:def 4;

x in L~ go by A91, XBOOLE_0:def 4;

hence x in {(pion1 /. 1)} by A7, A36, A59, A56, A84, A89, A92, XBOOLE_0:def 4; :: thesis: verum

end;assume A91: x in (L~ go) /\ (L~ pion1) ; :: thesis: x in {(pion1 /. 1)}

then A92: x in L~ pion1 by XBOOLE_0:def 4;

x in L~ go by A91, XBOOLE_0:def 4;

hence x in {(pion1 /. 1)} by A7, A36, A59, A56, A84, A89, A92, XBOOLE_0:def 4; :: thesis: verum

then A93: len pion1 > 1 + 1 by NAT_1:13;

then A94: rng pion1 c= L~ pion1 by SPPOL_2:18;

{(pion1 /. 1)} c= (L~ go) /\ (L~ pion1)

proof

then A97:
(L~ go) /\ (L~ pion1) = {(pion1 /. 1)}
by A90;
let x be object ; :: 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 A95: x = pion1 /. 1 by TARSKI:def 1;

then A96: x in rng pion1 by FINSEQ_6:42;

x in rng go by A89, A95, FINSEQ_6:168;

hence x in (L~ go) /\ (L~ pion1) by A77, A94, A96, XBOOLE_0:def 4; :: thesis: verum

end;assume x in {(pion1 /. 1)} ; :: thesis: x in (L~ go) /\ (L~ pion1)

then A95: x = pion1 /. 1 by TARSKI:def 1;

then A96: x in rng pion1 by FINSEQ_6:42;

x in rng go by A89, A95, FINSEQ_6:168;

hence x in (L~ go) /\ (L~ pion1) by A77, A94, A96, XBOOLE_0:def 4; :: thesis: verum

then A98: go ^' pion1 is s.n.c. by A89, JORDAN1J:54;

A99: {((Gauge (C,n)) * (i2,k))} c= (LSeg (go,m)) /\ (LSeg (pion1,1))

proof

LSeg (pion1,1) c= L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>
by A84, TOPREAL3:19;
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in {((Gauge (C,n)) * (i2,k))} or x in (LSeg (go,m)) /\ (LSeg (pion1,1)) )

assume x in {((Gauge (C,n)) * (i2,k))} ; :: thesis: x in (LSeg (go,m)) /\ (LSeg (pion1,1))

then A100: x = (Gauge (C,n)) * (i2,k) by TARSKI:def 1;

A101: (Gauge (C,n)) * (i2,k) in LSeg (go,m) by A62, RLTOPSP1:68;

(Gauge (C,n)) * (i2,k) in LSeg (pion1,1) by A59, A89, A93, TOPREAL1:21;

hence x in (LSeg (go,m)) /\ (LSeg (pion1,1)) by A100, A101, XBOOLE_0:def 4; :: thesis: verum

end;assume x in {((Gauge (C,n)) * (i2,k))} ; :: thesis: x in (LSeg (go,m)) /\ (LSeg (pion1,1))

then A100: x = (Gauge (C,n)) * (i2,k) by TARSKI:def 1;

A101: (Gauge (C,n)) * (i2,k) in LSeg (go,m) by A62, RLTOPSP1:68;

(Gauge (C,n)) * (i2,k) in LSeg (pion1,1) by A59, A89, A93, TOPREAL1:21;

hence x in (LSeg (go,m)) /\ (LSeg (pion1,1)) by A100, A101, XBOOLE_0:def 4; :: thesis: verum

then (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) c= {((Gauge (C,n)) * (i2,k))} by A61, A66, XBOOLE_1:27;

then (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) = {(go /. (len go))} by A59, A61, A99;

then A102: go ^' pion1 is unfolded by A89, TOPREAL8:34;

len (go ^' pion1) >= len go by TOPREAL8:7;

then A103: len (go ^' pion1) >= 1 + 1 by A42, XXREAL_0:2;

then A104: len (go ^' pion1) > 1 + 0 by NAT_1:13;

A105: <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. (len <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) = <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. 3 by FINSEQ_1:45

.= co /. 1 by A49, FINSEQ_4:18 ;

A106: {(pion1 /. (len pion1))} c= (L~ co) /\ (L~ pion1)

proof

(L~ co) /\ (L~ pion1) c= {(pion1 /. (len pion1))}
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in {(pion1 /. (len pion1))} or x in (L~ co) /\ (L~ pion1) )

assume x in {(pion1 /. (len pion1))} ; :: thesis: x in (L~ co) /\ (L~ pion1)

then A107: x = pion1 /. (len pion1) by TARSKI:def 1;

then A108: x in rng pion1 by FINSEQ_6:168;

x in rng co by A86, A105, A107, FINSEQ_6:42;

hence x in (L~ co) /\ (L~ pion1) by A55, A94, A108, XBOOLE_0:def 4; :: thesis: verum

end;assume x in {(pion1 /. (len pion1))} ; :: thesis: x in (L~ co) /\ (L~ pion1)

then A107: x = pion1 /. (len pion1) by TARSKI:def 1;

then A108: x in rng pion1 by FINSEQ_6:168;

x in rng co by A86, A105, A107, FINSEQ_6:42;

hence x in (L~ co) /\ (L~ pion1) by A55, A94, A108, XBOOLE_0:def 4; :: thesis: verum

proof

then A111:
(L~ co) /\ (L~ pion1) = {(pion1 /. (len pion1))}
by A106;
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in (L~ co) /\ (L~ pion1) or x in {(pion1 /. (len pion1))} )

assume A109: x in (L~ co) /\ (L~ pion1) ; :: thesis: x in {(pion1 /. (len pion1))}

then A110: x in L~ pion1 by XBOOLE_0:def 4;

x in L~ co by A109, XBOOLE_0:def 4;

hence x in {(pion1 /. (len pion1))} by A8, A36, A49, A48, A84, A86, A105, A110, XBOOLE_0:def 4; :: thesis: verum

end;assume A109: x in (L~ co) /\ (L~ pion1) ; :: thesis: x in {(pion1 /. (len pion1))}

then A110: x in L~ pion1 by XBOOLE_0:def 4;

x in L~ co by A109, XBOOLE_0:def 4;

hence x in {(pion1 /. (len pion1))} by A8, A36, A49, A48, A84, A86, A105, A110, XBOOLE_0:def 4; :: thesis: verum

A112: (L~ (go ^' pion1)) /\ (L~ co) = ((L~ go) \/ (L~ pion1)) /\ (L~ co) by A89, TOPREAL8:35

.= {(go /. 1)} \/ {(co /. 1)} by A81, A86, A105, A111, XBOOLE_1:23

.= {((go ^' pion1) /. 1)} \/ {(co /. 1)} by FINSEQ_6:155

.= {((go ^' pion1) /. 1),(co /. 1)} by ENUMSET1:1 ;

A113: Lower_Arc C is_an_arc_of E-max C, W-min C by JORDAN6:def 9;

then A114: Lower_Arc C is connected by JORDAN6:10;

set godo = (go ^' pion1) ^' co;

A115: co /. (len co) = (go ^' pion1) /. 1 by A78, FINSEQ_6:155;

A116: go ^' pion1 is_sequence_on Gauge (C,n) by A39, A82, A89, TOPREAL8:12;

then A117: (go ^' pion1) ^' co is_sequence_on Gauge (C,n) by A45, A88, TOPREAL8:12;

A118: (len pion1) - 1 >= 1 by A93, XREAL_1:19;

then A119: (len pion1) -' 1 = (len pion1) - 1 by XREAL_0:def 2;

A120: {((Gauge (C,n)) * (i1,j))} c= (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (co,1))

proof

LSeg (pion1,((len pion1) -' 1)) c= L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>
by A84, TOPREAL3:19;
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in {((Gauge (C,n)) * (i1,j))} or x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (co,1)) )

assume x in {((Gauge (C,n)) * (i1,j))} ; :: thesis: x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (co,1))

then A121: x = (Gauge (C,n)) * (i1,j) by TARSKI:def 1;

pion1 /. (((len pion1) -' 1) + 1) = <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. 3 by A86, A119, FINSEQ_1:45

.= (Gauge (C,n)) * (i1,j) by FINSEQ_4:18 ;

then A122: (Gauge (C,n)) * (i1,j) in LSeg (pion1,((len pion1) -' 1)) by A118, A119, TOPREAL1:21;

(Gauge (C,n)) * (i1,j) in LSeg (co,1) by A50, RLTOPSP1:68;

hence x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (co,1)) by A121, A122, XBOOLE_0:def 4; :: thesis: verum

end;assume x in {((Gauge (C,n)) * (i1,j))} ; :: thesis: x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (co,1))

then A121: x = (Gauge (C,n)) * (i1,j) by TARSKI:def 1;

pion1 /. (((len pion1) -' 1) + 1) = <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. 3 by A86, A119, FINSEQ_1:45

.= (Gauge (C,n)) * (i1,j) by FINSEQ_4:18 ;

then A122: (Gauge (C,n)) * (i1,j) in LSeg (pion1,((len pion1) -' 1)) by A118, A119, TOPREAL1:21;

(Gauge (C,n)) * (i1,j) in LSeg (co,1) by A50, RLTOPSP1:68;

hence x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (co,1)) by A121, A122, XBOOLE_0:def 4; :: thesis: verum

then (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (co,1)) c= {((Gauge (C,n)) * (i1,j))} by A54, XBOOLE_1:27;

then A123: (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (co,1)) = {((Gauge (C,n)) * (i1,j))} by A120;

((len pion1) - 1) + 1 <= len pion1 ;

then A124: (len pion1) -' 1 < len pion1 by A119, NAT_1:13;

len pion1 >= 2 + 1 by A87, FINSEQ_1:45;

then A125: (len pion1) - 2 >= 0 by XREAL_1:19;

then ((len pion1) -' 2) + 1 = ((len pion1) - 2) + 1 by XREAL_0:def 2

.= (len pion1) -' 1 by A118, XREAL_0:def 2 ;

then A126: (LSeg ((go ^' pion1),((len go) + ((len pion1) -' 2)))) /\ (LSeg (co,1)) = {((go ^' pion1) /. (len (go ^' pion1)))} by A49, A89, A88, A124, A123, TOPREAL8:31;

(rng go) /\ (rng pion1) c= {(pion1 /. 1)} by A77, A94, A97, XBOOLE_1:27;

then A127: go ^' pion1 is one-to-one by JORDAN1J:55;

((len (go ^' pion1)) + 1) - 1 = ((len go) + (len pion1)) - 1 by FINSEQ_6:139;

then (len (go ^' pion1)) - 1 = (len go) + ((len pion1) - 2)

.= (len go) + ((len pion1) -' 2) by A125, XREAL_0:def 2 ;

then A128: (len (go ^' pion1)) -' 1 = (len go) + ((len pion1) -' 2) by XREAL_0:def 2;

A129: L~ (Cage (C,n)) = (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n))) by JORDAN1E:13;

then A130: L~ (Upper_Seq (C,n)) c= L~ (Cage (C,n)) by XBOOLE_1:7;

then A131: L~ go c= L~ (Cage (C,n)) by A56;

A132: len ((go ^' pion1) ^' co) >= len (go ^' pion1) by TOPREAL8:7;

then A133: 1 + 1 <= len ((go ^' pion1) ^' co) by A103, XXREAL_0:2;

not go ^' pion1 is trivial by A103, NAT_D:60;

then reconsider godo = (go ^' pion1) ^' co as V29() standard special_circular_sequence by A133, A88, A117, A102, A128, A126, A98, A127, A112, A115, JORDAN8:4, JORDAN8:5, TOPREAL8:11, TOPREAL8:33, TOPREAL8:34;

A134: L~ godo = (L~ (go ^' pion1)) \/ (L~ co) by A88, TOPREAL8:35

.= ((L~ go) \/ (L~ pion1)) \/ (L~ co) by A89, TOPREAL8:35 ;

A135: now :: thesis: not ((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n)) <= 1

A137:
Upper_Seq (C,n) is_sequence_on Gauge (C,n)
by JORDAN1G:4;assume A136:
((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n)) <= 1
; :: thesis: contradiction

((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n)) >= 1 by A38, FINSEQ_4:21;

then ((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n)) = 1 by A136, XXREAL_0:1;

then (Gauge (C,n)) * (i2,k) = (Upper_Seq (C,n)) /. 1 by A38, FINSEQ_5:38;

hence contradiction by A15, A37, JORDAN1F:5; :: thesis: verum

end;((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n)) >= 1 by A38, FINSEQ_4:21;

then ((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n)) = 1 by A136, XXREAL_0:1;

then (Gauge (C,n)) * (i2,k) = (Upper_Seq (C,n)) /. 1 by A38, FINSEQ_5:38;

hence contradiction by A15, A37, JORDAN1F:5; :: thesis: verum

A138: ((Gauge (C,n)) * (i2,k)) `1 <= ((Gauge (C,n)) * (i1,k)) `1 by A1, A2, A3, A6, A10, JORDAN1A:18;

then A139: W-bound (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) = ((Gauge (C,n)) * (i2,k)) `1 by SPRECT_1:54;

A140: ((Gauge (C,n)) * (i1,k)) `1 = ((Gauge (C,n)) * (i1,1)) `1 by A3, A6, A17, A10, GOBOARD5:2

.= ((Gauge (C,n)) * (i1,j)) `1 by A3, A4, A17, A19, GOBOARD5:2 ;

then A141: W-bound (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) = ((Gauge (C,n)) * (i1,j)) `1 by SPRECT_1:54;

W-bound ((LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))))) = min ((W-bound (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))),(W-bound (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))))) by SPRECT_1:47

.= ((Gauge (C,n)) * (i2,k)) `1 by A140, A138, A139, A141, XXREAL_0:def 9 ;

then A142: W-bound (L~ pion1) = ((Gauge (C,n)) * (i2,k)) `1 by A84, TOPREAL3:16;

A143: Lower_Arc C c= C by JORDAN6:61;

((Gauge (C,n)) * (i2,k)) `1 >= W-bound (L~ (Cage (C,n))) by A30, A130, PSCOMP_1:24;

then A144: ((Gauge (C,n)) * (i2,k)) `1 > W-bound (L~ (Cage (C,n))) by A24, XXREAL_0:1;

A145: len (Upper_Seq (C,n)) >= 2 by A14, XXREAL_0:2;

A146: (L~ go) \/ (L~ co) is compact by COMPTS_1:10;

A147: L~ (Lower_Seq (C,n)) c= L~ (Cage (C,n)) by A129, XBOOLE_1:7;

then A148: L~ co c= L~ (Cage (C,n)) by A48;

A149: (right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo) c= RightComp godo by A133, A117, JORDAN9:27;

2 in dom godo by A133, FINSEQ_3:25;

then A150: godo /. 2 in rng godo by PARTFUN2:2;

A151: rng godo c= L~ godo by A103, A132, SPPOL_2:18, XXREAL_0:2;

A152: godo /. 1 = (go ^' pion1) /. 1 by FINSEQ_6:155

.= W-min (L~ (Cage (C,n))) by A67, FINSEQ_6:155 ;

A153: W-min C in Lower_Arc C by A113, TOPREAL1:1;

A154: W-min C in C by SPRECT_1:13;

A155: now :: thesis: not W-min C in L~ godo

A157:
(W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n)))
by EUCLID:52;assume
W-min C in L~ godo
; :: thesis: contradiction

then A156: ( W-min C in (L~ go) \/ (L~ pion1) or W-min C in L~ co ) by A134, XBOOLE_0:def 3;

end;then A156: ( W-min C in (L~ go) \/ (L~ pion1) or W-min C in L~ co ) by A134, XBOOLE_0:def 3;

per cases
( W-min C in L~ go or W-min C in L~ pion1 or W-min C in L~ co )
by A156, XBOOLE_0:def 3;

end;

suppose
W-min C in L~ go
; :: thesis: contradiction

then
C meets L~ (Cage (C,n))
by A131, A154, XBOOLE_0:3;

hence contradiction by JORDAN10:5; :: thesis: verum

end;hence contradiction by JORDAN10:5; :: thesis: verum

suppose
W-min C in L~ co
; :: thesis: contradiction

then
C meets L~ (Cage (C,n))
by A148, A154, XBOOLE_0:3;

hence contradiction by JORDAN10:5; :: thesis: verum

end;hence contradiction by JORDAN10:5; :: thesis: verum

set ff = Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))));

A158: 1 + 1 <= len (Cage (C,n)) by GOBOARD7:34, XXREAL_0:2;

W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;

then A159: (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 = W-min (L~ (Cage (C,n))) by FINSEQ_6:92;

A160: 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 A159, 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 A159, A160, 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 A159, A160, SPRECT_5:24, XXREAL_0:2;

then A161: (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A159, A160, SPRECT_5:25, XXREAL_0:2;

A162: Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def 1;

then A163: Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))) is_sequence_on Gauge (C,n) by REVROT_1:34;

1 + 1 <= len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by GOBOARD7:34, XXREAL_0:2;

then 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 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 A161, A163, JORDAN1J:53

.= 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)) * (i2,k)))),1,(Gauge (C,n))) by A38, A137, A135, JORDAN1J:52

.= right_cell ((go ^' pion1),1,(Gauge (C,n))) by A57, A116, JORDAN1J:51

.= right_cell (godo,1,(Gauge (C,n))) by A104, A117, JORDAN1J:51 ;

then W-min C in right_cell (godo,1,(Gauge (C,n))) by JORDAN1I:6;

then A164: W-min C in (right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo) by A155, XBOOLE_0:def 5;

A165: E-max C in Lower_Arc C by A113, TOPREAL1:1;

W-min (L~ (Cage (C,n))) in (L~ go) \/ (L~ co) by A77, A68, XBOOLE_0:def 3;

then A166: W-min ((L~ go) \/ (L~ co)) = W-min (L~ (Cage (C,n))) by A131, A148, A146, JORDAN1J:21, XBOOLE_1:8;

(W-min ((L~ go) \/ (L~ co))) `1 = W-bound ((L~ go) \/ (L~ co)) by EUCLID:52;

then W-min (((L~ go) \/ (L~ co)) \/ (L~ pion1)) = W-min ((L~ go) \/ (L~ co)) by A142, A146, A166, A157, A144, JORDAN1J:33;

then A167: W-min (L~ godo) = W-min (L~ (Cage (C,n))) by A134, A166, XBOOLE_1:4;

godo /. 2 = (go ^' pion1) /. 2 by A103, FINSEQ_6:159

.= (Upper_Seq (C,n)) /. 2 by A42, A76, FINSEQ_6:159

.= ((Upper_Seq (C,n)) ^' (Lower_Seq (C,n))) /. 2 by A145, FINSEQ_6:159

.= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 2 by JORDAN1E:11 ;

then godo /. 2 in W-most (L~ (Cage (C,n))) by JORDAN1I:25;

then (godo /. 2) `1 = (W-min (L~ godo)) `1 by A167, PSCOMP_1:31

.= W-bound (L~ godo) by EUCLID:52 ;

then godo /. 2 in W-most (L~ godo) by A151, A150, SPRECT_2:12;

then (Rotate (godo,(W-min (L~ godo)))) /. 2 in W-most (L~ godo) by A152, A167, FINSEQ_6:89;

then reconsider godo = godo as V29() 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 A168: (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 ;

A169: east_halfline (E-max C) misses L~ go

proof

assume
east_halfline (E-max C) meets L~ go
; :: thesis: contradiction

then consider p being object such that

A170: p in east_halfline (E-max C) and

A171: p in L~ go by XBOOLE_0:3;

reconsider p = p as Point of (TOP-REAL 2) by A170;

p in L~ (Upper_Seq (C,n)) by A56, A171;

then p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A130, A170, XBOOLE_0:def 4;

then A172: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;

then A173: p = E-max (L~ (Cage (C,n))) by A56, A171, JORDAN1J:46;

then E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i2,k) by A30, A168, A171, JORDAN1J:43;

then ((Gauge (C,n)) * (i2,k)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by A6, A10, A13, A172, A173, JORDAN1A:71;

hence contradiction by A2, A3, A12, A41, JORDAN1G:7; :: thesis: verum

end;then consider p being object such that

A170: p in east_halfline (E-max C) and

A171: p in L~ go by XBOOLE_0:3;

reconsider p = p as Point of (TOP-REAL 2) by A170;

p in L~ (Upper_Seq (C,n)) by A56, A171;

then p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A130, A170, XBOOLE_0:def 4;

then A172: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;

then A173: p = E-max (L~ (Cage (C,n))) by A56, A171, JORDAN1J:46;

then E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i2,k) by A30, A168, A171, JORDAN1J:43;

then ((Gauge (C,n)) * (i2,k)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by A6, A10, A13, A172, A173, JORDAN1A:71;

hence contradiction by A2, A3, A12, A41, JORDAN1G:7; :: thesis: verum

now :: thesis: not east_halfline (E-max C) meets L~ godo

then
east_halfline (E-max C) c= (L~ godo) `
by SUBSET_1:23;assume
east_halfline (E-max C) meets L~ godo
; :: thesis: contradiction

then A174: ( east_halfline (E-max C) meets (L~ go) \/ (L~ pion1) or east_halfline (E-max C) meets L~ co ) by A134, XBOOLE_1:70;

end;then A174: ( east_halfline (E-max C) meets (L~ go) \/ (L~ pion1) or east_halfline (E-max C) meets L~ co ) by A134, XBOOLE_1:70;

per cases
( 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~ co )
by A174, XBOOLE_1:70;

end;

suppose
east_halfline (E-max C) meets L~ pion1
; :: thesis: contradiction

then consider p being object such that

A175: p in east_halfline (E-max C) and

A176: p in L~ pion1 by XBOOLE_0:3;

reconsider p = p as Point of (TOP-REAL 2) by A175;

A177: p `2 = (E-max C) `2 by A175, TOPREAL1:def 11;

then (i1 + 1) - 1 <= (len (Gauge (C,n))) - 1 by XREAL_1:9;

then A179: i1 <= (len (Gauge (C,n))) -' 1 by XREAL_0:def 2;

(len (Gauge (C,n))) -' 1 <= len (Gauge (C,n)) by NAT_D:35;

then ((Gauge (C,n)) * (i1,k)) `1 <= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),1)) `1 by A6, A17, A10, A13, A21, A179, JORDAN1A:18;

then p `1 <= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),1)) `1 by A178, XXREAL_0:2;

then p `1 <= E-bound C by A21, JORDAN8:12;

then A180: p `1 <= (E-max C) `1 by EUCLID:52;

p `1 >= (E-max C) `1 by A175, TOPREAL1:def 11;

then p `1 = (E-max C) `1 by A180, XXREAL_0:1;

then p = E-max C by A177, TOPREAL3:6;

hence contradiction by A9, A36, A84, A165, A176, XBOOLE_0:3; :: thesis: verum

end;A175: p in east_halfline (E-max C) and

A176: p in L~ pion1 by XBOOLE_0:3;

reconsider p = p as Point of (TOP-REAL 2) by A175;

A177: p `2 = (E-max C) `2 by A175, TOPREAL1:def 11;

A178: now :: thesis: p `1 <= ((Gauge (C,n)) * (i1,k)) `1 end;

i1 + 1 <= len (Gauge (C,n))
by A3, NAT_1:13;per cases
( p in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) or p in LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) )
by A36, A84, A176, XBOOLE_0:def 3;

end;

then (i1 + 1) - 1 <= (len (Gauge (C,n))) - 1 by XREAL_1:9;

then A179: i1 <= (len (Gauge (C,n))) -' 1 by XREAL_0:def 2;

(len (Gauge (C,n))) -' 1 <= len (Gauge (C,n)) by NAT_D:35;

then ((Gauge (C,n)) * (i1,k)) `1 <= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),1)) `1 by A6, A17, A10, A13, A21, A179, JORDAN1A:18;

then p `1 <= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),1)) `1 by A178, XXREAL_0:2;

then p `1 <= E-bound C by A21, JORDAN8:12;

then A180: p `1 <= (E-max C) `1 by EUCLID:52;

p `1 >= (E-max C) `1 by A175, TOPREAL1:def 11;

then p `1 = (E-max C) `1 by A180, XXREAL_0:1;

then p = E-max C by A177, TOPREAL3:6;

hence contradiction by A9, A36, A84, A165, A176, XBOOLE_0:3; :: thesis: verum

suppose
east_halfline (E-max C) meets L~ co
; :: thesis: contradiction

then consider p being object such that

A181: p in east_halfline (E-max C) and

A182: p in L~ co by XBOOLE_0:3;

reconsider p = p as Point of (TOP-REAL 2) by A181;

A183: p in LSeg (co,(Index (p,co))) by A182, JORDAN3:9;

consider t being Nat such that

A184: t in dom (Lower_Seq (C,n)) and

A185: (Lower_Seq (C,n)) . t = (Gauge (C,n)) * (i1,j) by A44, FINSEQ_2:10;

1 <= t by A184, FINSEQ_3:25;

then A186: 1 < t by A46, A185, XXREAL_0:1;

t <= len (Lower_Seq (C,n)) by A184, FINSEQ_3:25;

then (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) + 1 = t by A185, A186, JORDAN3:12;

then A187: len (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i1,j)))) = (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) by A16, A185, JORDAN3:26;

Index (p,co) < len co by A182, JORDAN3:8;

then Index (p,co) < (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) by A187, XREAL_0:def 2;

then (Index (p,co)) + 1 <= (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) by NAT_1:13;

then A188: Index (p,co) <= ((len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n))))) - 1 by XREAL_1:19;

A189: co = mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n)))) by A44, JORDAN1J:37;

p in L~ (Lower_Seq (C,n)) by A48, A182;

then p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A147, A181, XBOOLE_0:def 4;

then A190: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;

A191: (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) + 1 = ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) by A46, A44, JORDAN1J:56;

0 + (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) < len (Lower_Seq (C,n)) by A16, JORDAN3:8;

then (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) > 0 by XREAL_1:20;

then Index (p,co) <= ((len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n))))) - 1 by A188, XREAL_0:def 2;

then Index (p,co) <= (len (Lower_Seq (C,n))) - (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))) by A191;

then Index (p,co) <= (len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))) by XREAL_0:def 2;

then A192: Index (p,co) < ((len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) + 1 by NAT_1:13;

A193: 1 <= Index (p,co) by A182, JORDAN3:8;

A194: ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) <= len (Lower_Seq (C,n)) by A44, FINSEQ_4:21;

((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) <> len (Lower_Seq (C,n)) by A43, A44, FINSEQ_4:19;

then A195: ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) < len (Lower_Seq (C,n)) by A194, XXREAL_0:1;

A196: 1 + 1 <= len (Lower_Seq (C,n)) by A31, XXREAL_0:2;

then A197: 2 in dom (Lower_Seq (C,n)) by FINSEQ_3:25;

set tt = ((Index (p,co)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1;

set RC = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))));

A198: E-max C in right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by JORDAN1I:7;

A199: 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 ;

A200: L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33;

consider jj2 being Nat such that

A201: 1 <= jj2 and

A202: jj2 <= width (Gauge (C,n)) and

A203: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * ((len (Gauge (C,n))),jj2) by JORDAN1D:25;

A204: len (Gauge (C,n)) >= 4 by JORDAN8:10;

then len (Gauge (C,n)) >= 1 by XXREAL_0:2;

then A205: [(len (Gauge (C,n))),jj2] in Indices (Gauge (C,n)) by A201, A202, MATRIX_0:30;

A206: len (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = len (Cage (C,n)) by FINSEQ_6:179;

Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) by JORDAN1G:18;

then A207: LSeg ((Lower_Seq (C,n)),1) = LSeg ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by A196, SPPOL_2:9;

A208: 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 A162, REVROT_1:34;

then consider ii, jj being Nat such that

A209: [ii,(jj + 1)] in Indices (Gauge (C,n)) and

A210: [ii,jj] in Indices (Gauge (C,n)) and

A211: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = (Gauge (C,n)) * (ii,(jj + 1)) and

A212: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1) = (Gauge (C,n)) * (ii,jj) by A158, A200, A206, A208, FINSEQ_6:92, JORDAN1I:23;

A213: (jj + 1) + 1 <> jj ;

A214: 1 <= jj by A210, MATRIX_0:32;

(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 A200, A208, FINSEQ_6:92;

then A215: ii = len (Gauge (C,n)) by A200, A209, A211, A203, A205, GOBOARD1:5;

then ii - 1 >= 4 - 1 by A204, XREAL_1:9;

then A216: ii - 1 >= 1 by XXREAL_0:2;

then A217: 1 <= ii -' 1 by XREAL_0:def 2;

A218: jj <= width (Gauge (C,n)) by A210, MATRIX_0:32;

then A219: ((Gauge (C,n)) * ((len (Gauge (C,n))),jj)) `1 = E-bound (L~ (Cage (C,n))) by A13, A214, JORDAN1A:71;

A220: jj + 1 <= width (Gauge (C,n)) by A209, MATRIX_0:32;

ii + 1 <> ii ;

then A221: right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) = cell ((Gauge (C,n)),(ii -' 1),jj) by A158, A206, A199, A209, A210, A211, A212, A213, GOBOARD5:def 6;

A222: ii <= len (Gauge (C,n)) by A210, MATRIX_0:32;

A223: 1 <= ii by A210, MATRIX_0:32;

A224: ii <= len (Gauge (C,n)) by A209, MATRIX_0:32;

A225: 1 <= jj + 1 by A209, MATRIX_0:32;

then A226: E-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * ((len (Gauge (C,n))),(jj + 1))) `1 by A13, A220, JORDAN1A:71;

A227: 1 <= ii by A209, MATRIX_0:32;

then A228: (ii -' 1) + 1 = ii by XREAL_1:235;

then A229: ii -' 1 < len (Gauge (C,n)) by A224, NAT_1:13;

then A230: ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 = ((Gauge (C,n)) * (1,(jj + 1))) `2 by A225, A220, A217, GOBOARD5:1

.= ((Gauge (C,n)) * (ii,(jj + 1))) `2 by A227, A224, A225, A220, GOBOARD5:1 ;

A231: (E-max C) `2 = p `2 by A181, TOPREAL1:def 11;

then A232: p `2 <= ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 by A198, A224, A220, A214, A221, A228, A216, JORDAN9:17;

A233: ((Gauge (C,n)) * ((ii -' 1),jj)) `2 = ((Gauge (C,n)) * (1,jj)) `2 by A214, A218, A217, A229, GOBOARD5:1

.= ((Gauge (C,n)) * (ii,jj)) `2 by A223, A222, A214, A218, GOBOARD5:1 ;

((Gauge (C,n)) * ((ii -' 1),jj)) `2 <= p `2 by A231, A198, A224, A220, A214, A221, A228, A216, JORDAN9:17;

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 A190, A211, A212, A215, A232, A233, A230, A219, A226, GOBOARD7:7;

then A234: p in LSeg ((Lower_Seq (C,n)),1) by A158, A207, A206, TOPREAL1:def 3;

1 <= ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) by A44, FINSEQ_4:21;

then A235: LSeg ((mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n))))),(Index (p,co))) = LSeg ((Lower_Seq (C,n)),(((Index (p,co)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1)) by A195, A193, A192, JORDAN4:19;

1 <= Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n))) by A16, JORDAN3:8;

then A236: 1 + 1 <= ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) by A191, XREAL_1:7;

then (Index (p,co)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))) >= (1 + 1) + 1 by A193, XREAL_1:7;

then ((Index (p,co)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) - 1 >= ((1 + 1) + 1) - 1 by XREAL_1:9;

then A237: ((Index (p,co)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1 >= 1 + 1 by XREAL_0:def 2;

end;A181: p in east_halfline (E-max C) and

A182: p in L~ co by XBOOLE_0:3;

reconsider p = p as Point of (TOP-REAL 2) by A181;

A183: p in LSeg (co,(Index (p,co))) by A182, JORDAN3:9;

consider t being Nat such that

A184: t in dom (Lower_Seq (C,n)) and

A185: (Lower_Seq (C,n)) . t = (Gauge (C,n)) * (i1,j) by A44, FINSEQ_2:10;

1 <= t by A184, FINSEQ_3:25;

then A186: 1 < t by A46, A185, XXREAL_0:1;

t <= len (Lower_Seq (C,n)) by A184, FINSEQ_3:25;

then (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) + 1 = t by A185, A186, JORDAN3:12;

then A187: len (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i1,j)))) = (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) by A16, A185, JORDAN3:26;

Index (p,co) < len co by A182, JORDAN3:8;

then Index (p,co) < (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) by A187, XREAL_0:def 2;

then (Index (p,co)) + 1 <= (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) by NAT_1:13;

then A188: Index (p,co) <= ((len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n))))) - 1 by XREAL_1:19;

A189: co = mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n)))) by A44, JORDAN1J:37;

p in L~ (Lower_Seq (C,n)) by A48, A182;

then p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A147, A181, XBOOLE_0:def 4;

then A190: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;

A191: (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) + 1 = ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) by A46, A44, JORDAN1J:56;

0 + (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) < len (Lower_Seq (C,n)) by A16, JORDAN3:8;

then (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) > 0 by XREAL_1:20;

then Index (p,co) <= ((len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n))))) - 1 by A188, XREAL_0:def 2;

then Index (p,co) <= (len (Lower_Seq (C,n))) - (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))) by A191;

then Index (p,co) <= (len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))) by XREAL_0:def 2;

then A192: Index (p,co) < ((len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) + 1 by NAT_1:13;

A193: 1 <= Index (p,co) by A182, JORDAN3:8;

A194: ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) <= len (Lower_Seq (C,n)) by A44, FINSEQ_4:21;

((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) <> len (Lower_Seq (C,n)) by A43, A44, FINSEQ_4:19;

then A195: ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) < len (Lower_Seq (C,n)) by A194, XXREAL_0:1;

A196: 1 + 1 <= len (Lower_Seq (C,n)) by A31, XXREAL_0:2;

then A197: 2 in dom (Lower_Seq (C,n)) by FINSEQ_3:25;

set tt = ((Index (p,co)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1;

set RC = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))));

A198: E-max C in right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by JORDAN1I:7;

A199: 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 ;

A200: L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33;

consider jj2 being Nat such that

A201: 1 <= jj2 and

A202: jj2 <= width (Gauge (C,n)) and

A203: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * ((len (Gauge (C,n))),jj2) by JORDAN1D:25;

A204: len (Gauge (C,n)) >= 4 by JORDAN8:10;

then len (Gauge (C,n)) >= 1 by XXREAL_0:2;

then A205: [(len (Gauge (C,n))),jj2] in Indices (Gauge (C,n)) by A201, A202, MATRIX_0:30;

A206: len (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = len (Cage (C,n)) by FINSEQ_6:179;

Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) by JORDAN1G:18;

then A207: LSeg ((Lower_Seq (C,n)),1) = LSeg ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by A196, SPPOL_2:9;

A208: 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 A162, REVROT_1:34;

then consider ii, jj being Nat such that

A209: [ii,(jj + 1)] in Indices (Gauge (C,n)) and

A210: [ii,jj] in Indices (Gauge (C,n)) and

A211: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = (Gauge (C,n)) * (ii,(jj + 1)) and

A212: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1) = (Gauge (C,n)) * (ii,jj) by A158, A200, A206, A208, FINSEQ_6:92, JORDAN1I:23;

A213: (jj + 1) + 1 <> jj ;

A214: 1 <= jj by A210, MATRIX_0:32;

(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 A200, A208, FINSEQ_6:92;

then A215: ii = len (Gauge (C,n)) by A200, A209, A211, A203, A205, GOBOARD1:5;

then ii - 1 >= 4 - 1 by A204, XREAL_1:9;

then A216: ii - 1 >= 1 by XXREAL_0:2;

then A217: 1 <= ii -' 1 by XREAL_0:def 2;

A218: jj <= width (Gauge (C,n)) by A210, MATRIX_0:32;

then A219: ((Gauge (C,n)) * ((len (Gauge (C,n))),jj)) `1 = E-bound (L~ (Cage (C,n))) by A13, A214, JORDAN1A:71;

A220: jj + 1 <= width (Gauge (C,n)) by A209, MATRIX_0:32;

ii + 1 <> ii ;

then A221: right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) = cell ((Gauge (C,n)),(ii -' 1),jj) by A158, A206, A199, A209, A210, A211, A212, A213, GOBOARD5:def 6;

A222: ii <= len (Gauge (C,n)) by A210, MATRIX_0:32;

A223: 1 <= ii by A210, MATRIX_0:32;

A224: ii <= len (Gauge (C,n)) by A209, MATRIX_0:32;

A225: 1 <= jj + 1 by A209, MATRIX_0:32;

then A226: E-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * ((len (Gauge (C,n))),(jj + 1))) `1 by A13, A220, JORDAN1A:71;

A227: 1 <= ii by A209, MATRIX_0:32;

then A228: (ii -' 1) + 1 = ii by XREAL_1:235;

then A229: ii -' 1 < len (Gauge (C,n)) by A224, NAT_1:13;

then A230: ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 = ((Gauge (C,n)) * (1,(jj + 1))) `2 by A225, A220, A217, GOBOARD5:1

.= ((Gauge (C,n)) * (ii,(jj + 1))) `2 by A227, A224, A225, A220, GOBOARD5:1 ;

A231: (E-max C) `2 = p `2 by A181, TOPREAL1:def 11;

then A232: p `2 <= ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 by A198, A224, A220, A214, A221, A228, A216, JORDAN9:17;

A233: ((Gauge (C,n)) * ((ii -' 1),jj)) `2 = ((Gauge (C,n)) * (1,jj)) `2 by A214, A218, A217, A229, GOBOARD5:1

.= ((Gauge (C,n)) * (ii,jj)) `2 by A223, A222, A214, A218, GOBOARD5:1 ;

((Gauge (C,n)) * ((ii -' 1),jj)) `2 <= p `2 by A231, A198, A224, A220, A214, A221, A228, A216, JORDAN9:17;

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 A190, A211, A212, A215, A232, A233, A230, A219, A226, GOBOARD7:7;

then A234: p in LSeg ((Lower_Seq (C,n)),1) by A158, A207, A206, TOPREAL1:def 3;

1 <= ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) by A44, FINSEQ_4:21;

then A235: LSeg ((mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n))))),(Index (p,co))) = LSeg ((Lower_Seq (C,n)),(((Index (p,co)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1)) by A195, A193, A192, JORDAN4:19;

1 <= Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n))) by A16, JORDAN3:8;

then A236: 1 + 1 <= ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) by A191, XREAL_1:7;

then (Index (p,co)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))) >= (1 + 1) + 1 by A193, XREAL_1:7;

then ((Index (p,co)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) - 1 >= ((1 + 1) + 1) - 1 by XREAL_1:9;

then A237: ((Index (p,co)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1 >= 1 + 1 by XREAL_0:def 2;

now :: thesis: contradictionend;

hence
contradiction
; :: thesis: verumper cases
( ((Index (p,co)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1 > 1 + 1 or ((Index (p,co)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 )
by A237, XXREAL_0:1;

end;

suppose
((Index (p,co)) + (((Gauge (C,n)) * (i1,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,co)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1))
by TOPREAL1:def 7;

hence contradiction by A234, A183, A189, A235, XBOOLE_0:3; :: thesis: verum

end;hence contradiction by A234, A183, A189, A235, XBOOLE_0:3; :: thesis: verum

suppose A238:
((Index (p,co)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1
; :: thesis: contradiction

then
1 + 1 = ((Index (p,co)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) - 1
by XREAL_0:def 2;

then (1 + 1) + 1 = (Index (p,co)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))) ;

then A239: ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) = 2 by A193, A236, JORDAN1E:6;

(LSeg ((Lower_Seq (C,n)),1)) /\ (LSeg ((Lower_Seq (C,n)),(((Index (p,co)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1))) = {((Lower_Seq (C,n)) /. 2)} by A31, A238, TOPREAL1:def 6;

then p in {((Lower_Seq (C,n)) /. 2)} by A234, A183, A189, A235, XBOOLE_0:def 4;

then A240: p = (Lower_Seq (C,n)) /. 2 by TARSKI:def 1;

then A241: p in rng (Lower_Seq (C,n)) by A197, PARTFUN2:2;

p .. (Lower_Seq (C,n)) = 2 by A197, A240, FINSEQ_5:41;

then p = (Gauge (C,n)) * (i1,j) by A44, A239, A241, FINSEQ_5:9;

then ((Gauge (C,n)) * (i1,j)) `1 = E-bound (L~ (Cage (C,n))) by A240, JORDAN1G:32;

then ((Gauge (C,n)) * (i1,j)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 by A4, A19, A13, JORDAN1A:71;

hence contradiction by A3, A25, A22, JORDAN1G:7; :: thesis: verum

end;then (1 + 1) + 1 = (Index (p,co)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))) ;

then A239: ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) = 2 by A193, A236, JORDAN1E:6;

(LSeg ((Lower_Seq (C,n)),1)) /\ (LSeg ((Lower_Seq (C,n)),(((Index (p,co)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1))) = {((Lower_Seq (C,n)) /. 2)} by A31, A238, TOPREAL1:def 6;

then p in {((Lower_Seq (C,n)) /. 2)} by A234, A183, A189, A235, XBOOLE_0:def 4;

then A240: p = (Lower_Seq (C,n)) /. 2 by TARSKI:def 1;

then A241: p in rng (Lower_Seq (C,n)) by A197, PARTFUN2:2;

p .. (Lower_Seq (C,n)) = 2 by A197, A240, FINSEQ_5:41;

then p = (Gauge (C,n)) * (i1,j) by A44, A239, A241, FINSEQ_5:9;

then ((Gauge (C,n)) * (i1,j)) `1 = E-bound (L~ (Cage (C,n))) by A240, JORDAN1G:32;

then ((Gauge (C,n)) * (i1,j)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 by A4, A19, A13, JORDAN1A:71;

hence contradiction by A3, A25, A22, JORDAN1G:7; :: thesis: verum

then consider W being Subset of (TOP-REAL 2) such that

A242: W is_a_component_of (L~ godo) ` and

A243: east_halfline (E-max C) c= W by GOBOARD9:3;

not W is bounded by A243, JORDAN2C:121, RLTOPSP1:42;

then W is_outside_component_of L~ godo by A242, JORDAN2C:def 3;

then W c= UBD (L~ godo) by JORDAN2C:23;

then A244: east_halfline (E-max C) c= UBD (L~ godo) by A243;

E-max C in east_halfline (E-max C) by TOPREAL1:38;

then E-max C in UBD (L~ godo) by A244;

then E-max C in LeftComp godo by GOBRD14:36;

then Lower_Arc C meets L~ godo by A114, A153, A165, A149, A164, JORDAN1J:36;

then A245: ( Lower_Arc C meets (L~ go) \/ (L~ pion1) or Lower_Arc C meets L~ co ) by A134, XBOOLE_1:70;

now :: thesis: contradictionend;

hence
contradiction
; :: thesis: verumper cases
( Lower_Arc C meets L~ go or Lower_Arc C meets L~ pion1 or Lower_Arc C meets L~ co )
by A245, XBOOLE_1:70;

end;

suppose
Lower_Arc C meets L~ go
; :: thesis: contradiction

then
Lower_Arc C meets L~ (Cage (C,n))
by A56, A130, XBOOLE_1:1, XBOOLE_1:63;

hence contradiction by A143, JORDAN10:5, XBOOLE_1:63; :: thesis: verum

end;hence contradiction by A143, JORDAN10:5, XBOOLE_1:63; :: thesis: verum

suppose
Lower_Arc C meets L~ co
; :: thesis: contradiction

then
Lower_Arc C meets L~ (Cage (C,n))
by A48, A147, XBOOLE_1:1, XBOOLE_1:63;

hence contradiction by A143, JORDAN10:5, XBOOLE_1:63; :: thesis: verum

end;hence contradiction by A143, JORDAN10:5, XBOOLE_1:63; :: thesis: verum

suppose
((Gauge (C,n)) * (i1,j)) `1 = ((Gauge (C,n)) * (i2,k)) `1
; :: thesis: contradiction

then A246:
i1 = i2
by A25, A12, JORDAN1G:7;

then LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) = {((Gauge (C,n)) * (i1,k))} by RLTOPSP1:70;

then LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) c= LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) by A27, ZFMISC_1:31;

then (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) = LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) by XBOOLE_1:12;

hence contradiction by A1, A3, A4, A5, A6, A7, A8, A9, A246, JORDAN1J:58; :: thesis: verum

end;then LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) = {((Gauge (C,n)) * (i1,k))} by RLTOPSP1:70;

then LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) c= LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) by A27, ZFMISC_1:31;

then (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) = LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) by XBOOLE_1:12;

hence contradiction by A1, A3, A4, A5, A6, A7, A8, A9, A246, JORDAN1J:58; :: thesis: verum

suppose
((Gauge (C,n)) * (i1,j)) `2 = ((Gauge (C,n)) * (i2,k)) `2
; :: thesis: contradiction

then A247:
j = k
by A25, A12, JORDAN1G:6;

then LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) = {((Gauge (C,n)) * (i1,k))} by RLTOPSP1:70;

then LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) c= LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) by A26, ZFMISC_1:31;

then (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) = LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) by XBOOLE_1:12;

hence contradiction by A1, A2, A3, A4, A6, A7, A8, A9, A247, Th36; :: thesis: verum

end;then LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) = {((Gauge (C,n)) * (i1,k))} by RLTOPSP1:70;

then LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) c= LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) by A26, ZFMISC_1:31;

then (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) = LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) by XBOOLE_1:12;

hence contradiction by A1, A2, A3, A4, A6, A7, A8, A9, A247, Th36; :: thesis: verum