let n be Element of NAT ; :: thesis:
let C be Simple_closed_curve; :: thesis:
let i, j, k be Element of NAT ; :: thesis:
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) * j,i;
set Gik = (Gauge C,n) * k,i;
assume that
A1: ( 1 < j & j <= k & k < len (Gauge C,n) ) and
A2: ( 1 <= i & i <= width (Gauge C,n) ) and
A3: (LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Upper_Seq C,n)) = {((Gauge C,n) * j,i)} and
A4: (LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n)) = {((Gauge C,n) * k,i)} and
A5: LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i) misses Lower_Arc C ; :: thesis:
(Gauge C,n) * k,i in {((Gauge C,n) * k,i)} by TARSKI:def 1;
then A6: (Gauge C,n) * k,i in L~ (Lower_Seq C,n) by A4, XBOOLE_0:def 4;
(Gauge C,n) * j,i in {((Gauge C,n) * j,i)} by TARSKI:def 1;
then A7: (Gauge C,n) * j,i in L~ (Upper_Seq C,n) by A3, XBOOLE_0:def 4;
A8: len (Gauge C,n) = width (Gauge C,n) by JORDAN8:def 1;
A9: j <> k by A1, A2, A6, A7, Th29;
A10: ( 1 < j & j < width (Gauge C,n) ) by A1, A8, XXREAL_0:2;
A11: ( 1 < k & k < width (Gauge C,n) ) by A1, JORDAN8:def 1, XXREAL_0:2;
A12: [j,i] in Indices (Gauge C,n) by A2, A8, A10, MATRIX_1:37;
A13: [k,i] in Indices (Gauge C,n) by A2, A8, A11, MATRIX_1:37;
set go = R_Cut (Upper_Seq C,n),((Gauge C,n) * j,i);
set do = L_Cut (Lower_Seq C,n),((Gauge C,n) * k,i);
A14: len (Upper_Seq C,n) >= 3 by JORDAN1E:19;
then len (Upper_Seq C,n) >= 1 by XXREAL_0:2;
then 1 in dom (Upper_Seq C,n) by FINSEQ_3:27;
then A15: (Upper_Seq C,n) . 1 = (Upper_Seq C,n) /. 1 by PARTFUN1:def 8
.= W-min (L~ (Cage C,n)) by JORDAN1F:5 ;
A16: (W-min (L~ (Cage C,n))) `1 = W-bound (L~ (Cage C,n)) by EUCLID:56
.= ((Gauge C,n) * 1,i) `1 by A2, A8, JORDAN1A:94 ;
len (Gauge C,n) >= 4 by JORDAN8:13;
then A17: len (Gauge C,n) >= 1 by XXREAL_0:2;
then A18: [1,k] in Indices (Gauge C,n) by A11, MATRIX_1:37;
A19: [1,i] in Indices (Gauge C,n) by A2, A17, MATRIX_1:37;
then A20: (Gauge C,n) * j,i <> (Upper_Seq C,n) . 1 by A1, A12, A15, A16, JORDAN1G:7;
then reconsider go = R_Cut (Upper_Seq C,n),((Gauge C,n) * j,i) as being_S-Seq FinSequence of (TOP-REAL 2) by A7, JORDAN3:70;
A21: len (Lower_Seq C,n) >= 1 + 2 by JORDAN1E:19;
then len (Lower_Seq C,n) >= 1 by XXREAL_0:2;
then A22: ( 1 in dom (Lower_Seq C,n) & len (Lower_Seq C,n) in dom (Lower_Seq C,n) ) by FINSEQ_3:27;
then A23: (Lower_Seq C,n) . (len (Lower_Seq C,n)) = (Lower_Seq C,n) /. (len (Lower_Seq C,n)) by PARTFUN1:def 8
.= W-min (L~ (Cage C,n)) by JORDAN1F:8 ;
(W-min (L~ (Cage C,n))) `1 = W-bound (L~ (Cage C,n)) by EUCLID:56
.= ((Gauge C,n) * 1,i) `1 by A2, A8, JORDAN1A:94 ;
then A24: (Gauge C,n) * k,i <> (Lower_Seq C,n) . (len (Lower_Seq C,n)) by A1, A13, A19, A23, JORDAN1G:7;
then reconsider do = L_Cut (Lower_Seq C,n),((Gauge C,n) * k,i) as being_S-Seq FinSequence of (TOP-REAL 2) by A6, JORDAN3:69;
A25: [(len (Gauge C,n)),k] in Indices (Gauge C,n) by A11, A17, MATRIX_1:37;
A26: (Lower_Seq C,n) . 1 = (Lower_Seq C,n) /. 1 by A22, PARTFUN1:def 8
.= E-max (L~ (Cage C,n)) by JORDAN1F:6 ;
(E-max (L~ (Cage C,n))) `1 = E-bound (L~ (Cage C,n)) by EUCLID:56
.= ((Gauge C,n) * (len (Gauge C,n)),k) `1 by A8, A11, JORDAN1A:92 ;
then A27: (Gauge C,n) * k,i <> (Lower_Seq C,n) . 1 by A1, A13, A25, A26, JORDAN1G:7;
A28: len go >= 1 + 1 by TOPREAL1:def 10;
A29: (Gauge C,n) * j,i in rng (Upper_Seq C,n) by A2, A7, A8, A10, JORDAN1G:4, JORDAN1J:40;
then A30: go is_sequence_on Gauge C,n by JORDAN1G:4, JORDAN1J:38;
A31: len do >= 1 + 1 by TOPREAL1:def 10;
A32: (Gauge C,n) * k,i in rng (Lower_Seq C,n) by A2, A6, A8, A11, JORDAN1G:5, JORDAN1J:40;
then A33: do is_sequence_on Gauge C,n by JORDAN1G:5, JORDAN1J:39;
reconsider go = go as non constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A28, A30, JGRAPH_1:16, JORDAN8:8;
reconsider do = do as non constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A31, A33, JGRAPH_1:16, JORDAN8:8;
A34: len go > 1 by A28, NAT_1:13;
then A35: len go in dom go by FINSEQ_3:27;
then A36: go /. (len go) = go . (len go) by PARTFUN1:def 8
.= (Gauge C,n) * j,i by A7, JORDAN3:59 ;
len do >= 1 by A31, XXREAL_0:2;
then 1 in dom do by FINSEQ_3:27;
then A37: do /. 1 = do . 1 by PARTFUN1:def 8
.= (Gauge C,n) * k,i by A6, JORDAN3:58 ;
reconsider m = (len go) - 1 as Element of NAT by A35, FINSEQ_3:28;
A38: m + 1 = len go ;
then A39: (len go) -' 1 = m by NAT_D:34;
A40: LSeg go,m c= L~ go by TOPREAL3:26;
A41: L~ go c= L~ (Upper_Seq C,n) by A7, JORDAN3:76;
then LSeg go,m c= L~ (Upper_Seq C,n) by A40, XBOOLE_1:1;
then A42: (LSeg go,m) /\ (LSeg ((Gauge C,n) * k,i),((Gauge C,n) * j,i)) c= {((Gauge C,n) * j,i)} by A3, XBOOLE_1:26;
m >= 1 by A28, XREAL_1:21;
then A43: LSeg go,m = LSeg (go /. m),((Gauge C,n) * j,i) by A36, A38, TOPREAL1:def 5;
{((Gauge C,n) * j,i)} c= (LSeg go,m) /\ (LSeg ((Gauge C,n) * k,i),((Gauge C,n) * j,i))

proof

let x be set ; :: according to TARSKI:def 3 :: thesis:
assume x in {((Gauge C,n) * j,i)} ; :: thesis:
then A44: x = (Gauge C,n) * j,i by TARSKI:def 1;
A45: (Gauge C,n) * j,i in LSeg go,m by A43, RLTOPSP1:69;
(Gauge C,n) * j,i in LSeg ((Gauge C,n) * k,i),((Gauge C,n) * j,i) by RLTOPSP1:69;
hence x in (LSeg go,m) /\ (LSeg ((Gauge C,n) * k,i),((Gauge C,n) * j,i)) by A44, A45, XBOOLE_0:def 4; :: thesis:

end;

then A46: (LSeg go,m) /\ (LSeg ((Gauge C,n) * k,i),((Gauge C,n) * j,i)) = {((Gauge C,n) * j,i)} by A42, XBOOLE_0:def 10;
A47: LSeg do,1 c= L~ do by TOPREAL3:26;
A48: L~ do c= L~ (Lower_Seq C,n) by A6, JORDAN3:77;
then LSeg do,1 c= L~ (Lower_Seq C,n) by A47, XBOOLE_1:1;
then A49: (LSeg do,1) /\ (LSeg ((Gauge C,n) * k,i),((Gauge C,n) * j,i)) c= {((Gauge C,n) * k,i)} by A4, XBOOLE_1:26;
A50: LSeg do,1 = LSeg ((Gauge C,n) * k,i),(do /. (1 + 1)) by A31, A37, TOPREAL1:def 5;
{((Gauge C,n) * k,i)} c= (LSeg do,1) /\ (LSeg ((Gauge C,n) * k,i),((Gauge C,n) * j,i))

proof

let x be set ; :: according to TARSKI:def 3 :: thesis:
assume x in {((Gauge C,n) * k,i)} ; :: thesis:
then A51: x = (Gauge C,n) * k,i by TARSKI:def 1;
A52: (Gauge C,n) * k,i in LSeg do,1 by A50, RLTOPSP1:69;
(Gauge C,n) * k,i in LSeg ((Gauge C,n) * k,i),((Gauge C,n) * j,i) by RLTOPSP1:69;
hence x in (LSeg do,1) /\ (LSeg ((Gauge C,n) * k,i),((Gauge C,n) * j,i)) by A51, A52, XBOOLE_0:def 4; :: thesis:

end;

then A53: (LSeg ((Gauge C,n) * k,i),((Gauge C,n) * j,i)) /\ (LSeg do,1) = {((Gauge C,n) * k,i)} by A49, XBOOLE_0:def 10;
A54: go /. 1 = (Upper_Seq C,n) /. 1 by A7, SPRECT_3:39
.= W-min (L~ (Cage C,n)) by JORDAN1F:5 ;
then A55: go /. 1 = (Lower_Seq C,n) /. (len (Lower_Seq C,n)) by JORDAN1F:8
.= do /. (len do) by A6, JORDAN1J:35 ;
A56: ( rng go c= L~ go & rng do c= L~ do ) by A28, A31, SPPOL_2:18;
A57: {(go /. 1)} c= (L~ go) /\ (L~ do)

proof

let x be set ; :: according to TARSKI:def 3 :: thesis:
assume x in {(go /. 1)} ; :: thesis:
then x = go /. 1 by TARSKI:def 1;
then ( x in rng go & x in rng do ) by A55, FINSEQ_6:46, REVROT_1:3;
hence x in (L~ go) /\ (L~ do) by A56, XBOOLE_0:def 4; :: thesis:

end;

A58: (Lower_Seq C,n) . 1 = (Lower_Seq C,n) /. 1 by A22, PARTFUN1:def 8
.= E-max (L~ (Cage C,n)) by JORDAN1F:6 ;
A59: [(len (Gauge C,n)),j] in Indices (Gauge C,n) by A10, A17, MATRIX_1:37;
(L~ go) /\ (L~ do) c= {(go /. 1)}

proof

let x be set ; :: according to TARSKI:def 3 :: thesis:
assume x in (L~ go) /\ (L~ do) ; :: thesis:
then A60: ( x in L~ go & x in L~ do ) by XBOOLE_0:def 4;
then x in (L~ (Upper_Seq C,n)) /\ (L~ (Lower_Seq C,n)) by A41, A48, XBOOLE_0:def 4;
then x in {(W-min (L~ (Cage C,n))),(E-max (L~ (Cage C,n)))} by JORDAN1E:20;
then A61: ( x = W-min (L~ (Cage C,n)) or x = E-max (L~ (Cage C,n)) ) by TARSKI:def 2;

now

assume x = E-max (L~ (Cage C,n)) ; :: thesis:
then A62: E-max (L~ (Cage C,n)) = (Gauge C,n) * k,i by A6, A58, A60, JORDAN1E:11;
((Gauge C,n) * (len (Gauge C,n)),j) `1 = E-bound (L~ (Cage C,n)) by A8, A10, JORDAN1A:92;
then (E-max (L~ (Cage C,n))) `1 <> E-bound (L~ (Cage C,n)) by A1, A13, A59, A62, JORDAN1G:7;
hence contradiction by EUCLID:56; :: thesis:

end;

hence x in {(go /. 1)} by A54, A61, TARSKI:def 1; :: thesis:

end;

then A63: (L~ go) /\ (L~ do) = {(go /. 1)} by A57, XBOOLE_0:def 10;
set W2 = go /. 2;
A64: 2 in dom go by A28, FINSEQ_3:27;
A65: ((Gauge C,n) * j,i) .. (Upper_Seq C,n) >= 1 by A29, FINSEQ_4:31;

A66: now

assume ((Gauge C,n) * j,i) `1 = W-bound (L~ (Cage C,n)) ; :: thesis:
then ((Gauge C,n) * 1,k) `1 = ((Gauge C,n) * j,i) `1 by A8, A11, JORDAN1A:94;
hence contradiction by A1, A12, A18, JORDAN1G:7; :: thesis:

end;

go = mid (Upper_Seq C,n),1,(((Gauge C,n) * j,i) .. (Upper_Seq C,n)) by A29, JORDAN1G:57
.= (Upper_Seq C,n) | (((Gauge C,n) * j,i) .. (Upper_Seq C,n)) by A65, JORDAN3:25 ;
then A67: go /. 2 = (Upper_Seq C,n) /. 2 by A64, FINSEQ_4:85;
A68: W-min (L~ (Cage C,n)) in rng go by A54, FINSEQ_6:46;
set pion = <*((Gauge C,n) * j,i),((Gauge C,n) * k,i)*>;

A69: now

let n be Element of NAT ; :: thesis:
assume n in dom <*((Gauge C,n) * j,i),((Gauge C,n) * k,i)*> ; :: thesis:
then n in {1,2} by FINSEQ_1:4, FINSEQ_3:29;
then ( n = 1 or n = 2 ) by TARSKI:def 2;
then ( <*((Gauge C,n) * j,i),((Gauge C,n) * k,i)*> /. n = (Gauge C,n) * k,i or <*((Gauge C,n) * j,i),((Gauge C,n) * k,i)*> /. n = (Gauge C,n) * j,i ) by FINSEQ_4:26;
hence ex j, i being Element of NAT st
( [j,i] in Indices (Gauge C,n) & <*((Gauge C,n) * j,i),((Gauge C,n) * k,i)*> /. n = (Gauge C,n) * j,i ) by A12, A13; :: thesis:

end;

A70: (Gauge C,n) * k,i <> (Gauge C,n) * j,i by A9, A12, A13, GOBOARD1:21;
((Gauge C,n) * k,i) `2 = ((Gauge C,n) * 1,i) `2 by A2, A8, A11, GOBOARD5:2
.= ((Gauge C,n) * j,i) `2 by A2, A8, A10, GOBOARD5:2 ;
then LSeg ((Gauge C,n) * k,i),((Gauge C,n) * j,i) is horizontal by SPPOL_1:36;
then <*((Gauge C,n) * j,i),((Gauge C,n) * k,i)*> is being_S-Seq by A70, JORDAN1B:9;
then consider pion1 being FinSequence of (TOP-REAL 2) such that
A71: pion1 is_sequence_on Gauge C,n and
A72: pion1 is being_S-Seq and
A73: L~ <*((Gauge C,n) * j,i),((Gauge C,n) * k,i)*> = L~ pion1 and
A74: <*((Gauge C,n) * j,i),((Gauge C,n) * k,i)*> /. 1 = pion1 /. 1 and
A75: <*((Gauge C,n) * j,i),((Gauge C,n) * k,i)*> /. (len <*((Gauge C,n) * j,i),((Gauge C,n) * k,i)*>) = pion1 /. (len pion1) and
A76: len <*((Gauge C,n) * j,i),((Gauge C,n) * k,i)*> <= len pion1 by A69, GOBOARD3:2;
reconsider pion1 = pion1 as being_S-Seq FinSequence of (TOP-REAL 2) by A72;
set godo = (go ^' pion1) ^' do;
A77: 1 + 1 <= len (Cage C,n) by GOBOARD7:36, XXREAL_0:2;
then A78: 1 + 1 <= len (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) by REVROT_1:14;
len (go ^' pion1) >= len go by TOPREAL8:7;
then A79: len (go ^' pion1) >= 1 + 1 by A28, XXREAL_0:2;
then A80: len (go ^' pion1) > 1 + 0 by NAT_1:13;
A81: len ((go ^' pion1) ^' do) >= len (go ^' pion1) by TOPREAL8:7;
then A82: 1 + 1 <= len ((go ^' pion1) ^' do) by A79, XXREAL_0:2;
A83: Upper_Seq C,n is_sequence_on Gauge C,n by JORDAN1G:4;
A84: go /. (len go) = pion1 /. 1 by A36, A74, FINSEQ_4:26;
then A85: go ^' pion1 is_sequence_on Gauge C,n by A30, A71, TOPREAL8:12;
A86: (go ^' pion1) /. (len (go ^' pion1)) = <*((Gauge C,n) * j,i),((Gauge C,n) * k,i)*> /. (len <*((Gauge C,n) * j,i),((Gauge C,n) * k,i)*>) by A75, GRAPH_2:58
.= <*((Gauge C,n) * j,i),((Gauge C,n) * k,i)*> /. 2 by FINSEQ_1:61
.= do /. 1 by A37, FINSEQ_4:26 ;
then A87: (go ^' pion1) ^' do is_sequence_on Gauge C,n by A33, A85, TOPREAL8:12;
LSeg pion1,1 c= L~ <*((Gauge C,n) * j,i),((Gauge C,n) * k,i)*> by A73, TOPREAL3:26;
then LSeg pion1,1 c= LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i) by SPPOL_2:21;
then A88: (LSeg go,((len go) -' 1)) /\ (LSeg pion1,1) c= {((Gauge C,n) * j,i)} by A39, A46, XBOOLE_1:27;
A89: len pion1 >= 1 + 1 by A76, FINSEQ_1:61;
{((Gauge C,n) * j,i)} c= (LSeg go,m) /\ (LSeg pion1,1)

proof

let x be set ; :: according to TARSKI:def 3 :: thesis:
assume x in {((Gauge C,n) * j,i)} ; :: thesis:
then A90: x = (Gauge C,n) * j,i by TARSKI:def 1;
A91: (Gauge C,n) * j,i in LSeg go,m by A43, RLTOPSP1:69;
(Gauge C,n) * j,i in LSeg pion1,1 by A36, A84, A89, TOPREAL1:27;
hence x in (LSeg go,m) /\ (LSeg pion1,1) by A90, A91, XBOOLE_0:def 4; :: thesis:

end;

then (LSeg go,((len go) -' 1)) /\ (LSeg pion1,1) = {(go /. (len go))} by A36, A39, A88, XBOOLE_0:def 10;
then A92: go ^' pion1 is unfolded by A84, TOPREAL8:34;
len pion1 >= 2 + 0 by A76, FINSEQ_1:61;
then A93: (len pion1) - 2 >= 0 by XREAL_1:21;
((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 A93, XREAL_0:def 2 ;
then A94: (len (go ^' pion1)) -' 1 = (len go) + ((len pion1) -' 2) by XREAL_0:def 2;
A95: (len pion1) - 1 >= 1 by A89, XREAL_1:21;
then A96: (len pion1) -' 1 = (len pion1) - 1 by XREAL_0:def 2;
A97: ((len pion1) -' 2) + 1 = ((len pion1) - 2) + 1 by A93, XREAL_0:def 2
.= (len pion1) -' 1 by A95, XREAL_0:def 2 ;
((len pion1) - 1) + 1 <= len pion1 ;
then A98: (len pion1) -' 1 < len pion1 by A96, NAT_1:13;
LSeg pion1,((len pion1) -' 1) c= L~ <*((Gauge C,n) * j,i),((Gauge C,n) * k,i)*> by A73, TOPREAL3:26;
then LSeg pion1,((len pion1) -' 1) c= LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i) by SPPOL_2:21;
then A99: (LSeg pion1,((len pion1) -' 1)) /\ (LSeg do,1) c= {((Gauge C,n) * k,i)} by A53, XBOOLE_1:27;
{((Gauge C,n) * k,i)} c= (LSeg pion1,((len pion1) -' 1)) /\ (LSeg do,1)

proof

let x be set ; :: according to TARSKI:def 3 :: thesis:
assume x in {((Gauge C,n) * k,i)} ; :: thesis:
then A100: x = (Gauge C,n) * k,i by TARSKI:def 1;
A101: (Gauge C,n) * k,i in LSeg do,1 by A50, RLTOPSP1:69;
pion1 /. (((len pion1) -' 1) + 1) = <*((Gauge C,n) * j,i),((Gauge C,n) * k,i)*> /. 2 by A75, A96, FINSEQ_1:61
.= (Gauge C,n) * k,i by FINSEQ_4:26 ;
then (Gauge C,n) * k,i in LSeg pion1,((len pion1) -' 1) by A95, A96, TOPREAL1:27;
hence x in (LSeg pion1,((len pion1) -' 1)) /\ (LSeg do,1) by A100, A101, XBOOLE_0:def 4; :: thesis:

end;

then (LSeg pion1,((len pion1) -' 1)) /\ (LSeg do,1) = {((Gauge C,n) * k,i)} by A99, XBOOLE_0:def 10;
then A102: (LSeg (go ^' pion1),((len go) + ((len pion1) -' 2))) /\ (LSeg do,1) = {((go ^' pion1) /. (len (go ^' pion1)))} by A37, A84, A86, A97, A98, TOPREAL8:31;
A103: not go ^' pion1 is trivial by A79, REALSET1:13;
A104: rng pion1 c= L~ pion1 by A89, SPPOL_2:18;
A105: {(pion1 /. 1)} c= (L~ go) /\ (L~ pion1)

proof

let x be set ; :: according to TARSKI:def 3 :: thesis:
assume x in {(pion1 /. 1)} ; :: thesis:
then x = pion1 /. 1 by TARSKI:def 1;
then ( x in rng go & x in rng pion1 ) by A84, FINSEQ_6:46, REVROT_1:3;
hence x in (L~ go) /\ (L~ pion1) by A56, A104, XBOOLE_0:def 4; :: thesis:

end;

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

proof

let x be set ; :: according to TARSKI:def 3 :: thesis:
assume x in (L~ go) /\ (L~ pion1) ; :: thesis:
then ( x in L~ go & x in L~ pion1 ) by XBOOLE_0:def 4;
then x in (L~ pion1) /\ (L~ (Upper_Seq C,n)) by A41, XBOOLE_0:def 4;
hence x in {(pion1 /. 1)} by A3, A36, A73, A84, SPPOL_2:21; :: thesis:

end;

then A106: (L~ go) /\ (L~ pion1) = {(pion1 /. 1)} by A105, XBOOLE_0:def 10;
then A107: go ^' pion1 is s.n.c. by A84, JORDAN1J:54;
(rng go) /\ (rng pion1) c= {(pion1 /. 1)} by A56, A104, A106, XBOOLE_1:27;
then A108: go ^' pion1 is one-to-one by JORDAN1J:55;
A109: <*((Gauge C,n) * j,i),((Gauge C,n) * k,i)*> /. (len <*((Gauge C,n) * j,i),((Gauge C,n) * k,i)*>) = <*((Gauge C,n) * j,i),((Gauge C,n) * k,i)*> /. 2 by FINSEQ_1:61
.= do /. 1 by A37, FINSEQ_4:26 ;
A110: {(pion1 /. (len pion1))} c= (L~ do) /\ (L~ pion1)

proof

let x be set ; :: according to TARSKI:def 3 :: thesis:
assume x in {(pion1 /. (len pion1))} ; :: thesis:
then x = pion1 /. (len pion1) by TARSKI:def 1;
then ( x in rng do & x in rng pion1 ) by A75, A109, FINSEQ_6:46, REVROT_1:3;
hence x in (L~ do) /\ (L~ pion1) by A56, A104, XBOOLE_0:def 4; :: thesis:

end;

(L~ do) /\ (L~ pion1) c= {(pion1 /. (len pion1))}

proof

let x be set ; :: according to TARSKI:def 3 :: thesis:
assume x in (L~ do) /\ (L~ pion1) ; :: thesis:
then ( x in L~ do & x in L~ pion1 ) by XBOOLE_0:def 4;
then x in (L~ pion1) /\ (L~ (Lower_Seq C,n)) by A48, XBOOLE_0:def 4;
hence x in {(pion1 /. (len pion1))} by A4, A37, A73, A75, A109, SPPOL_2:21; :: thesis:

end;

then A111: (L~ do) /\ (L~ pion1) = {(pion1 /. (len pion1))} by A110, XBOOLE_0:def 10;
A112: (L~ (go ^' pion1)) /\ (L~ do) = ((L~ go) \/ (L~ pion1)) /\ (L~ do) by A84, TOPREAL8:35
.= {(go /. 1)} \/ {(do /. 1)} by A63, A75, A109, A111, XBOOLE_1:23
.= {((go ^' pion1) /. 1)} \/ {(do /. 1)} by GRAPH_2:57
.= {((go ^' pion1) /. 1),(do /. 1)} by ENUMSET1:41 ;
do /. (len do) = (go ^' pion1) /. 1 by A55, GRAPH_2:57;
then reconsider godo = (go ^' pion1) ^' do as non constant standard special_circular_sequence by A82, A86, A87, A92, A94, A102, A103, A107, A108, A112, JORDAN8:7, JORDAN8:8, TOPREAL8:11, TOPREAL8:33, TOPREAL8:34;
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 JORDAN6:11;
A115: ( W-min C in Lower_Arc C & E-max C in Lower_Arc C ) by A113, TOPREAL1:4;
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_2:47;
then A116: (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) /. 1 = W-min (L~ (Cage C,n)) by FINSEQ_6:98;
A117: 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 A116, SPRECT_5:23;
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 A116, A117, SPRECT_5:24, 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 A116, A117, SPRECT_5:25, XXREAL_0:2;
then A118: (E-max (L~ (Cage C,n))) .. (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) > 1 by A116, A117, SPRECT_5:26, XXREAL_0:2;

A119: now

assume A120: ((Gauge C,n) * j,i) .. (Upper_Seq C,n) <= 1 ; :: thesis:
((Gauge C,n) * j,i) .. (Upper_Seq C,n) >= 1 by A29, FINSEQ_4:31;
then ((Gauge C,n) * j,i) .. (Upper_Seq C,n) = 1 by A120, XXREAL_0:1;
then (Gauge C,n) * j,i = (Upper_Seq C,n) /. 1 by A29, FINSEQ_5:41;
hence contradiction by A15, A20, JORDAN1F:5; :: thesis:

end;

A121: Cage C,n is_sequence_on Gauge C,n by JORDAN9:def 1;
then A122: Rotate (Cage C,n),(W-min (L~ (Cage C,n))) is_sequence_on Gauge C,n by REVROT_1:34;
A123: (right_cell godo,1,(Gauge C,n)) \ (L~ godo) c= RightComp godo by A82, A87, JORDAN9:29;
A124: L~ godo = (L~ (go ^' pion1)) \/ (L~ do) by A86, TOPREAL8:35
.= ((L~ go) \/ (L~ pion1)) \/ (L~ do) by A84, TOPREAL8:35 ;
L~ (Cage C,n) = (L~ (Upper_Seq C,n)) \/ (L~ (Lower_Seq C,n)) by JORDAN1E:17;
then A125: ( L~ (Upper_Seq C,n) c= L~ (Cage C,n) & L~ (Lower_Seq C,n) c= L~ (Cage C,n) ) by XBOOLE_1:7;
then A126: ( L~ go c= L~ (Cage C,n) & L~ do c= L~ (Cage C,n) ) by A41, A48, XBOOLE_1:1;
A127: W-min C in C by SPRECT_1:15;
A128: L~ <*((Gauge C,n) * j,i),((Gauge C,n) * k,i)*> = LSeg ((Gauge C,n) * k,i),((Gauge C,n) * j,i) by SPPOL_2:21;

A129: now

assume W-min C in L~ godo ; :: thesis:
then A130: ( W-min C in (L~ go) \/ (L~ pion1) or W-min C in L~ do ) by A124, XBOOLE_0:def 3;

per cases by A130, XBOOLE_0:def 3;

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

then C meets L~ (Cage C,n) by A126, A127, XBOOLE_0:3;
hence contradiction by JORDAN10:5; :: thesis:

end;

suppose W-min C in L~ pion1 ; :: thesis:

hence contradiction by A5, A73, A115, A128, XBOOLE_0:3; :: thesis:

end;

suppose W-min C in L~ do ; :: thesis:

then C meets L~ (Cage C,n) by A126, A127, XBOOLE_0:3;
hence contradiction by JORDAN10:5; :: thesis:

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 A78, JORDAN1H:29
.= 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:52
.= right_cell ((Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) -: (E-max (L~ (Cage C,n)))),1,(Gauge C,n) by A118, A122, 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) * j,i)),1,(Gauge C,n) by A29, A83, A119, JORDAN1J:52
.= right_cell (go ^' pion1),1,(Gauge C,n) by A34, A85, JORDAN1J:51
.= right_cell godo,1,(Gauge C,n) by A80, A87, JORDAN1J:51 ;
then W-min C in right_cell godo,1,(Gauge C,n) by JORDAN1I:8;
then A131: W-min C in (right_cell godo,1,(Gauge C,n)) \ (L~ godo) by A129, XBOOLE_0:def 5;
A132: godo /. 1 = (go ^' pion1) /. 1 by GRAPH_2:57
.= W-min (L~ (Cage C,n)) by A54, GRAPH_2:57 ;
A133: len (Upper_Seq C,n) >= 2 by A14, XXREAL_0:2;
A134: godo /. 2 = (go ^' pion1) /. 2 by A79, GRAPH_2:61
.= (Upper_Seq C,n) /. 2 by A28, A67, GRAPH_2:61
.= ((Upper_Seq C,n) ^' (Lower_Seq C,n)) /. 2 by A133, GRAPH_2:61
.= (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) /. 2 by JORDAN1E:15 ;
A135: (L~ go) \/ (L~ do) is compact by COMPTS_1:19;
W-min (L~ (Cage C,n)) in (L~ go) \/ (L~ do) by A56, A68, XBOOLE_0:def 3;
then A136: W-min ((L~ go) \/ (L~ do)) = W-min (L~ (Cage C,n)) by A126, A135, JORDAN1J:21, XBOOLE_1:8;
A137: ( (W-min ((L~ go) \/ (L~ do))) `1 = W-bound ((L~ go) \/ (L~ do)) & (W-min (L~ (Cage C,n))) `1 = W-bound (L~ (Cage C,n)) ) by EUCLID:56;
A138: ((Gauge C,n) * j,i) `1 <= ((Gauge C,n) * k,i) `1 by A1, A2, SPRECT_3:25;
then W-bound (LSeg ((Gauge C,n) * k,i),((Gauge C,n) * j,i)) = ((Gauge C,n) * j,i) `1 by SPRECT_1:62;
then A139: W-bound (L~ pion1) = ((Gauge C,n) * j,i) `1 by A73, SPPOL_2:21;
((Gauge C,n) * j,i) `1 >= W-bound (L~ (Cage C,n)) by A7, A125, PSCOMP_1:71;
then ((Gauge C,n) * j,i) `1 > W-bound (L~ (Cage C,n)) by A66, XXREAL_0:1;
then W-min (((L~ go) \/ (L~ do)) \/ (L~ pion1)) = W-min ((L~ go) \/ (L~ do)) by A135, A136, A137, A139, JORDAN1J:33;
then A140: W-min (L~ godo) = W-min (L~ (Cage C,n)) by A124, A136, XBOOLE_1:4;
A141: rng godo c= L~ godo by A79, A81, SPPOL_2:18, XXREAL_0:2;
2 in dom godo by A82, FINSEQ_3:27;
then A142: godo /. 2 in rng godo by PARTFUN2:4;
godo /. 2 in W-most (L~ (Cage C,n)) by A134, JORDAN1I:27;
then (godo /. 2) `1 = (W-min (L~ godo)) `1 by A140, PSCOMP_1:88
.= W-bound (L~ godo) by EUCLID:56 ;
then godo /. 2 in W-most (L~ godo) by A141, A142, SPRECT_2:16;
then (Rotate godo,(W-min (L~ godo))) /. 2 in W-most (L~ godo) by A132, A140, FINSEQ_6:95;
then reconsider godo = godo as non constant standard clockwise_oriented special_circular_sequence by JORDAN1I:27;
len (Upper_Seq C,n) in dom (Upper_Seq C,n) by FINSEQ_5:6;
then A143: (Upper_Seq C,n) . (len (Upper_Seq C,n)) = (Upper_Seq C,n) /. (len (Upper_Seq C,n)) by PARTFUN1:def 8
.= E-max (L~ (Cage C,n)) by JORDAN1F:7 ;
A144: E-max C in E-most C by PSCOMP_1:111;
A145: east_halfline (E-max C) misses L~ go

proof

assume east_halfline (E-max C) meets L~ go ; :: thesis:
then consider p being set such that
A146: p in east_halfline (E-max C) and
A147: p in L~ go by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A146;
p in L~ (Upper_Seq C,n) by A41, A147;
then p in (east_halfline (E-max C)) /\ (L~ (Cage C,n)) by A125, A146, XBOOLE_0:def 4;
then A148: p `1 = E-bound (L~ (Cage C,n)) by A144, JORDAN1A:104;
then A149: p = E-max (L~ (Cage C,n)) by A41, A147, JORDAN1J:46;
then E-max (L~ (Cage C,n)) = (Gauge C,n) * j,i by A7, A143, A147, JORDAN1J:43;
then ((Gauge C,n) * j,i) `1 = ((Gauge C,n) * (len (Gauge C,n)),k) `1 by A8, A11, A148, A149, JORDAN1A:92;
hence contradiction by A1, A12, A25, JORDAN1G:7; :: thesis:

end;

now

assume east_halfline (E-max C) meets L~ godo ; :: thesis:
then A150: ( east_halfline (E-max C) meets (L~ go) \/ (L~ pion1) or east_halfline (E-max C) meets L~ do ) by A124, XBOOLE_1:70;

per cases by A150, XBOOLE_1:70;

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

hence contradiction by A145; :: thesis:

end;

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

then consider p being set such that
A151: p in east_halfline (E-max C) and
A152: p in L~ pion1 by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A151;
A153: p `1 <= ((Gauge C,n) * k,i) `1 by A73, A128, A138, A152, TOPREAL1:9;
k + 1 <= len (Gauge C,n) by A1, NAT_1:13;
then (k + 1) - 1 <= (len (Gauge C,n)) - 1 by XREAL_1:11;
then A154: k <= (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) * k,i) `1 <= ((Gauge C,n) * ((len (Gauge C,n)) -' 1),1) `1 by A2, A8, A11, A17, A154, JORDAN1A:39;
then p `1 <= ((Gauge C,n) * ((len (Gauge C,n)) -' 1),1) `1 by A153, XXREAL_0:2;
then p `1 <= E-bound C by A17, JORDAN8:15;
then A155: p `1 <= (E-max C) `1 by EUCLID:56;
p `1 >= (E-max C) `1 by A151, TOPREAL1:def 13;
then A156: p `1 = (E-max C) `1 by A155, XXREAL_0:1;
p `2 = (E-max C) `2 by A151, TOPREAL1:def 13;
then p = E-max C by A156, TOPREAL3:11;
hence contradiction by A5, A73, A115, A128, A152, XBOOLE_0:3; :: thesis:

end;

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

then consider p being set such that
A157: p in east_halfline (E-max C) and
A158: p in L~ do by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A157;
p in L~ (Lower_Seq C,n) by A48, A158;
then p in (east_halfline (E-max C)) /\ (L~ (Cage C,n)) by A125, A157, XBOOLE_0:def 4;
then A159: p `1 = E-bound (L~ (Cage C,n)) by A144, JORDAN1A:104;
A160: (E-max C) `2 = p `2 by A157, TOPREAL1:def 13;
set RC = Rotate (Cage C,n),(E-max (L~ (Cage C,n)));
A161: E-max C in right_cell (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))),1 by JORDAN1I:9;
A162: 1 + 1 <= len (Lower_Seq C,n) by A21, XXREAL_0:2;
Lower_Seq C,n = (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) -: (W-min (L~ (Cage C,n))) by JORDAN1G:26;
then A163: LSeg (Lower_Seq C,n),1 = LSeg (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))),1 by A162, SPPOL_2:9;
A164: L~ (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) = L~ (Cage C,n) by REVROT_1:33;
A165: len (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) = len (Cage C,n) by REVROT_1:14;
A166: GoB (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) = GoB (Cage C,n) by REVROT_1:28
.= Gauge C,n by JORDAN1H:52 ;
A167: E-max (L~ (Cage C,n)) in rng (Cage C,n) by SPRECT_2:50;
A168: Rotate (Cage C,n),(E-max (L~ (Cage C,n))) is_sequence_on Gauge C,n by A121, REVROT_1:34;
A169: (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 A164, A167, FINSEQ_6:98;
then consider ii, g being Element of NAT such that
A170: [ii,(g + 1)] in Indices (Gauge C,n) and
A171: [ii,g] in Indices (Gauge C,n) and
A172: (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) /. 1 = (Gauge C,n) * ii,(g + 1) and
A173: (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) /. (1 + 1) = (Gauge C,n) * ii,g by A77, A165, A168, JORDAN1I:25;
consider g2 being Element of NAT such that
A174: ( 1 <= g2 & g2 <= width (Gauge C,n) ) and
A175: E-max (L~ (Cage C,n)) = (Gauge C,n) * (len (Gauge C,n)),g2 by JORDAN1D:29;
A176: len (Gauge C,n) >= 4 by JORDAN8:13;
then len (Gauge C,n) >= 1 by XXREAL_0:2;
then [(len (Gauge C,n)),g2] in Indices (Gauge C,n) by A174, MATRIX_1:37;
then A177: ii = len (Gauge C,n) by A164, A169, A170, A172, A175, GOBOARD1:21;
A178: ( 1 <= ii & ii <= len (Gauge C,n) & 1 <= g + 1 & g + 1 <= width (Gauge C,n) ) by A170, MATRIX_1:39;
A179: ( 1 <= ii & ii <= len (Gauge C,n) & 1 <= g & g <= width (Gauge C,n) ) by A171, MATRIX_1:39;
A180: ii + 1 <> ii ;
(g + 1) + 1 <> g ;
then A181: right_cell (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))),1 = cell (Gauge C,n),(ii -' 1),g by A77, A165, A166, A170, A171, A172, A173, A180, GOBOARD5:def 6;
A182: (ii -' 1) + 1 = ii by A178, XREAL_1:237;
ii - 1 >= 4 - 1 by A176, A177, XREAL_1:11;
then A183: ii - 1 >= 1 by XXREAL_0:2;
then A184: 1 <= ii -' 1 by XREAL_0:def 2;
A185: ( ((Gauge C,n) * (ii -' 1),g) `2 <= p `2 & p `2 <= ((Gauge C,n) * (ii -' 1),(g + 1)) `2 ) by A160, A161, A178, A179, A181, A182, A183, JORDAN9:19;
A186: ii -' 1 < len (Gauge C,n) by A178, A182, NAT_1:13;
then A187: ((Gauge C,n) * (ii -' 1),g) `2 = ((Gauge C,n) * 1,g) `2 by A179, A184, GOBOARD5:2
.= ((Gauge C,n) * ii,g) `2 by A179, GOBOARD5:2 ;
A188: ((Gauge C,n) * (ii -' 1),(g + 1)) `2 = ((Gauge C,n) * 1,(g + 1)) `2 by A178, A184, A186, GOBOARD5:2
.= ((Gauge C,n) * ii,(g + 1)) `2 by A178, GOBOARD5:2 ;
( ((Gauge C,n) * (len (Gauge C,n)),g) `1 = E-bound (L~ (Cage C,n)) & E-bound (L~ (Cage C,n)) = ((Gauge C,n) * (len (Gauge C,n)),(g + 1)) `1 ) by A8, A178, A179, JORDAN1A:92;
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 A159, A172, A173, A177, A185, A187, A188, GOBOARD7:8;
then A189: p in LSeg (Lower_Seq C,n),1 by A77, A163, A165, TOPREAL1:def 5;
A190: p in LSeg do,(Index p,do) by A158, JORDAN3:42;
A191: do = mid (Lower_Seq C,n),(((Gauge C,n) * k,i) .. (Lower_Seq C,n)),(len (Lower_Seq C,n)) by A32, JORDAN1J:37;
A192: ( 1 <= ((Gauge C,n) * k,i) .. (Lower_Seq C,n) & ((Gauge C,n) * k,i) .. (Lower_Seq C,n) <= len (Lower_Seq C,n) ) by A32, FINSEQ_4:31;
((Gauge C,n) * k,i) .. (Lower_Seq C,n) <> len (Lower_Seq C,n) by A24, A32, FINSEQ_4:29;
then A193: ((Gauge C,n) * k,i) .. (Lower_Seq C,n) < len (Lower_Seq C,n) by A192, XXREAL_0:1;
A194: ( 1 <= Index p,do & Index p,do < len do ) by A158, JORDAN3:41;
A195: (Index ((Gauge C,n) * k,i),(Lower_Seq C,n)) + 1 = ((Gauge C,n) * k,i) .. (Lower_Seq C,n) by A27, A32, JORDAN1J:56;
consider t being Nat such that
A196: t in dom (Lower_Seq C,n) and
A197: (Lower_Seq C,n) . t = (Gauge C,n) * k,i by A32, FINSEQ_2:11;
A198: ( 1 <= t & t <= len (Lower_Seq C,n) ) by A196, FINSEQ_3:27;
then 1 < t by A27, A197, XXREAL_0:1;
then (Index ((Gauge C,n) * k,i),(Lower_Seq C,n)) + 1 = t by A197, A198, JORDAN3:45;
then A199: len (L_Cut (Lower_Seq C,n),((Gauge C,n) * k,i)) = (len (Lower_Seq C,n)) - (Index ((Gauge C,n) * k,i),(Lower_Seq C,n)) by A6, A197, JORDAN3:61;
set tt = ((Index p,do) + (((Gauge C,n) * k,i) .. (Lower_Seq C,n))) -' 1;
A200: ( 1 <= Index ((Gauge C,n) * k,i),(Lower_Seq C,n) & 0 + (Index ((Gauge C,n) * k,i),(Lower_Seq C,n)) < len (Lower_Seq C,n) ) by A6, JORDAN3:41;
then A201: (len (Lower_Seq C,n)) - (Index ((Gauge C,n) * k,i),(Lower_Seq C,n)) > 0 by XREAL_1:22;
Index p,do < (len (Lower_Seq C,n)) -' (Index ((Gauge C,n) * k,i),(Lower_Seq C,n)) by A194, A199, XREAL_0:def 2;
then (Index p,do) + 1 <= (len (Lower_Seq C,n)) -' (Index ((Gauge C,n) * k,i),(Lower_Seq C,n)) by NAT_1:13;
then Index p,do <= ((len (Lower_Seq C,n)) -' (Index ((Gauge C,n) * k,i),(Lower_Seq C,n))) - 1 by XREAL_1:21;
then Index p,do <= ((len (Lower_Seq C,n)) - (Index ((Gauge C,n) * k,i),(Lower_Seq C,n))) - 1 by A201, XREAL_0:def 2;
then Index p,do <= (len (Lower_Seq C,n)) - (((Gauge C,n) * k,i) .. (Lower_Seq C,n)) by A195;
then Index p,do <= (len (Lower_Seq C,n)) -' (((Gauge C,n) * k,i) .. (Lower_Seq C,n)) by XREAL_0:def 2;
then Index p,do < ((len (Lower_Seq C,n)) -' (((Gauge C,n) * k,i) .. (Lower_Seq C,n))) + 1 by NAT_1:13;
then A202: LSeg (mid (Lower_Seq C,n),(((Gauge C,n) * k,i) .. (Lower_Seq C,n)),(len (Lower_Seq C,n))),(Index p,do) = LSeg (Lower_Seq C,n),(((Index p,do) + (((Gauge C,n) * k,i) .. (Lower_Seq C,n))) -' 1) by A192, A193, A194, JORDAN4:31;
A203: 1 + 1 <= ((Gauge C,n) * k,i) .. (Lower_Seq C,n) by A195, A200, XREAL_1:9;
then (Index p,do) + (((Gauge C,n) * k,i) .. (Lower_Seq C,n)) >= (1 + 1) + 1 by A194, XREAL_1:9;
then ((Index p,do) + (((Gauge C,n) * k,i) .. (Lower_Seq C,n))) - 1 >= ((1 + 1) + 1) - 1 by XREAL_1:11;
then A204: ((Index p,do) + (((Gauge C,n) * k,i) .. (Lower_Seq C,n))) -' 1 >= 1 + 1 by XREAL_0:def 2;
A205: 2 in dom (Lower_Seq C,n) by A162, FINSEQ_3:27;

now

per cases ) + (((Gauge C,n) * k,i) .. (Lower_Seq C,n))) -' 1 > 1 + 1 or ((Index p,do) + (((Gauge C,n) * k,i) .. (Lower_Seq C,n))) -' 1 = 1 + 1 ) by A204, XXREAL_0:1;

suppose ((Index p,do) + (((Gauge C,n) * k,i) .. (Lower_Seq C,n))) -' 1 > 1 + 1 ; :: thesis:

then LSeg (Lower_Seq C,n),1 misses LSeg (Lower_Seq C,n),(((Index p,do) + (((Gauge C,n) * k,i) .. (Lower_Seq C,n))) -' 1) by TOPREAL1:def 9;
hence contradiction by A189, A190, A191, A202, XBOOLE_0:3; :: thesis:

end;

suppose A206: ((Index p,do) + (((Gauge C,n) * k,i) .. (Lower_Seq C,n))) -' 1 = 1 + 1 ; :: thesis:

then (LSeg (Lower_Seq C,n),1) /\ (LSeg (Lower_Seq C,n),(((Index p,do) + (((Gauge C,n) * k,i) .. (Lower_Seq C,n))) -' 1)) = {((Lower_Seq C,n) /. 2)} by A21, TOPREAL1:def 8;
then p in {((Lower_Seq C,n) /. 2)} by A189, A190, A191, A202, XBOOLE_0:def 4;
then A207: p = (Lower_Seq C,n) /. 2 by TARSKI:def 1;
then A208: p .. (Lower_Seq C,n) = 2 by A205, FINSEQ_5:44;
1 + 1 = ((Index p,do) + (((Gauge C,n) * k,i) .. (Lower_Seq C,n))) - 1 by A206, XREAL_0:def 2;
then (1 + 1) + 1 = (Index p,do) + (((Gauge C,n) * k,i) .. (Lower_Seq C,n)) ;
then A209: ((Gauge C,n) * k,i) .. (Lower_Seq C,n) = 2 by A194, A203, JORDAN1E:10;
p in rng (Lower_Seq C,n) by A205, A207, PARTFUN2:4;
then p = (Gauge C,n) * k,i by A32, A208, A209, FINSEQ_5:10;
then ((Gauge C,n) * k,i) `1 = E-bound (L~ (Cage C,n)) by A207, JORDAN1G:40;
then ((Gauge C,n) * k,i) `1 = ((Gauge C,n) * (len (Gauge C,n)),j) `1 by A8, A10, JORDAN1A:92;
hence contradiction by A1, A13, A59, JORDAN1G:7; :: thesis:

end;

hence contradiction ; :: thesis:

end;

then east_halfline (E-max C) c= (L~ godo) ` by SUBSET_1:43;
then consider W being Subset of (TOP-REAL 2) such that
A210: W is_a_component_of (L~ godo) ` and
A211: east_halfline (E-max C) c= W by GOBOARD9:5;
not W is Bounded by A211, JORDAN2C:16, JORDAN2C:129;
then W is_outside_component_of L~ godo by A210, JORDAN2C:def 4;
then W c= UBD (L~ godo) by JORDAN2C:27;
then A212: east_halfline (E-max C) c= UBD (L~ godo) by A211, XBOOLE_1:1;
E-max C in east_halfline (E-max C) by TOPREAL1:45;
then E-max C in UBD (L~ godo) by A212;
then E-max C in LeftComp godo by GOBRD14:46;
then Lower_Arc C meets L~ godo by A114, A115, A123, A131, JORDAN1J:36;
then A213: ( Lower_Arc C meets (L~ go) \/ (L~ pion1) or Lower_Arc C meets L~ do ) by A124, XBOOLE_1:70;
A214: Lower_Arc C c= C by JORDAN6:76;

per cases by A213, XBOOLE_1:70;

suppose Lower_Arc C meets L~ go ; :: thesis:

then Lower_Arc C meets L~ (Cage C,n) by A41, A125, XBOOLE_1:1, XBOOLE_1:63;
hence contradiction by A214, JORDAN10:5, XBOOLE_1:63; :: thesis:

end;

suppose Lower_Arc C meets L~ pion1 ; :: thesis:

hence contradiction by A5, A73, A128; :: thesis:

end;

suppose Lower_Arc C meets L~ do ; :: thesis:

then Lower_Arc C meets L~ (Cage C,n) by A48, A125, XBOOLE_1:1, XBOOLE_1:63;
hence contradiction by A214, JORDAN10:5, XBOOLE_1:63; :: thesis:

end;