let n be Element of NAT ; :: thesis: for C being Simple_closed_curve
for i1, i2, j, k being Element of NAT st 1 < i1 & i1 <= i2 & i2 < 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 Element of NAT st 1 < i1 & i1 <= i2 & i2 < 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 Element of NAT ; :: thesis: ( 1 < i1 & i1 <= i2 & i2 < 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 < i1 and
A2: i1 <= i2 and
A3: i2 < 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 Gij = (Gauge C,n) * i1,j;
A10: j <= width (Gauge C,n) by A5, A6, XXREAL_0:2;
A11: i1 < len (Gauge C,n) by A2, A3, XXREAL_0:2;
then A12: [i1,j] in Indices (Gauge C,n) by A1, A4, A10, MATRIX_1:37;
set Gi1k = (Gauge C,n) * i1,k;
set Gik = (Gauge C,n) * i2,k;
A13: 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 TOPREAL3:23;
len (Gauge C,n) >= 4 by JORDAN8:13;
then A14: len (Gauge C,n) >= 1 by XXREAL_0:2;
then A15: [(len (Gauge C,n)),j] in Indices (Gauge C,n) by A4, A10, MATRIX_1:37;
A16: 1 <= k by A4, A5, XXREAL_0:2;
then A17: [1,k] in Indices (Gauge C,n) by A6, A14, MATRIX_1:37;
A18: 1 < i2 by A1, A2, XXREAL_0:2;
then A19: [i2,k] in Indices (Gauge C,n) by A3, A6, A16, MATRIX_1:37;
A20: ((Gauge C,n) * i1,k) `2 = ((Gauge C,n) * 1,k) `2 by A1, A6, A11, A16, GOBOARD5:2
.= ((Gauge C,n) * i2,k) `2 by A3, A6, A18, A16, GOBOARD5:2 ;
((Gauge C,n) * i1,k) `1 = ((Gauge C,n) * i1,1) `1 by A1, A6, A11, A16, GOBOARD5:3
.= ((Gauge C,n) * i1,j) `1 by A1, A4, A11, A10, GOBOARD5:3 ;
then A21: (Gauge C,n) * i1,k = |[(((Gauge C,n) * i1,j) `1 ),(((Gauge C,n) * i2,k) `2 )]| by A20, EUCLID:57;
A22: [(len (Gauge C,n)),k] in Indices (Gauge C,n) by A6, A16, A14, MATRIX_1:37;
A23: [i1,j] in Indices (Gauge C,n) by A1, A4, A11, A10, MATRIX_1:37;
set Wbo = W-bound (L~ (Cage C,n));
set Wmin = W-min (L~ (Cage C,n));
A24: len (Gauge C,n) = width (Gauge C,n) by JORDAN8:def 1;
set Ebo = E-bound (L~ (Cage C,n));
set Emax = E-max (L~ (Cage C,n));
A25: len (Lower_Seq C,n) >= 1 + 2 by JORDAN1E:19;
then A26: len (Lower_Seq C,n) >= 1 by XXREAL_0:2;
then A27: 1 in dom (Lower_Seq C,n) by FINSEQ_3:27;
then A28: (Lower_Seq C,n) . 1 = (Lower_Seq C,n) /. 1 by PARTFUN1:def 8
.= E-max (L~ (Cage C,n)) by JORDAN1F:6 ;
len (Lower_Seq C,n) in dom (Lower_Seq C,n) by A26, FINSEQ_3:27;
then A29: (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 ;
set do = 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 A30: (Gauge C,n) * i1,j in L~ (Lower_Seq C,n) by A8, XBOOLE_0:def 4;
(W-min (L~ (Cage C,n))) `1 = W-bound (L~ (Cage C,n)) by EUCLID:56
.= ((Gauge C,n) * 1,k) `1 by A6, A16, A24, JORDAN1A:94 ;
then A31: (Gauge C,n) * i1,j <> (Lower_Seq C,n) . (len (Lower_Seq C,n)) by A1, A17, A29, A12, JORDAN1G:7;
then reconsider do = L_Cut (Lower_Seq C,n),((Gauge C,n) * i1,j) as being_S-Seq FinSequence of (TOP-REAL 2) by A30, JORDAN3:69;
A32: (Gauge C,n) * i1,j in rng (Lower_Seq C,n) by A1, A4, A11, A30, A10, JORDAN1G:5, JORDAN1J:40;
then A33: do 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:56
.= ((Gauge C,n) * (len (Gauge C,n)),k) `1 by A6, A16, A24, JORDAN1A:92 ;
then A34: (Gauge C,n) * i1,j <> (Lower_Seq C,n) . 1 by A2, A3, A12, A22, A28, JORDAN1G:7;
A35: len do >= 1 + 1 by TOPREAL1:def 10;
then reconsider do = do as non constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A33, JGRAPH_1:16, JORDAN8:8;
A36: L~ do c= L~ (Lower_Seq C,n) by A30, JORDAN3:77;
A37: [1,j] in Indices (Gauge C,n) by A4, A10, A14, MATRIX_1:37;
A38: now
assume ((Gauge C,n) * i1,j) `1 = W-bound (L~ (Cage C,n)) ; :: thesis: contradiction
then ((Gauge C,n) * 1,j) `1 = ((Gauge C,n) * i1,j) `1 by A4, A10, A24, JORDAN1A:94;
hence contradiction by A1, A23, A37, JORDAN1G:7; :: thesis: verum
end;
set pion = <*((Gauge C,n) * i2,k),((Gauge C,n) * i1,k),((Gauge C,n) * i1,j)*>;
A39: (Gauge C,n) * i1,k in LSeg ((Gauge C,n) * i1,k),((Gauge C,n) * i2,k) by RLTOPSP1:69;
set LA = Lower_Arc C;
A40: (Gauge C,n) * i1,k in LSeg ((Gauge C,n) * i1,j),((Gauge C,n) * i1,k) by RLTOPSP1:69;
set go = R_Cut (Upper_Seq C,n),((Gauge C,n) * i2,k);
A41: 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 A42: (Upper_Seq C,n) . 1 = (Upper_Seq C,n) /. 1 by PARTFUN1:def 8
.= W-min (L~ (Cage C,n)) by JORDAN1F:5 ;
A43: [i1,k] in Indices (Gauge C,n) by A1, A6, A11, A16, MATRIX_1:37;
A44: now
let n be Element of 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 Element of 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 Element of 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_3:1, FINSEQ_3:30;
then ( n = 1 or n = 2 or n = 3 ) by ENUMSET1:def 1;
hence ex i, j being Element of 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 A23, A19, A43, FINSEQ_4:27; :: thesis: verum
end;
(Gauge C,n) * i2,k in {((Gauge C,n) * i2,k)} by TARSKI:def 1;
then A45: (Gauge C,n) * i2,k in L~ (Upper_Seq C,n) by A7, XBOOLE_0:def 4;
(W-min (L~ (Cage C,n))) `1 = W-bound (L~ (Cage C,n)) by EUCLID:56
.= ((Gauge C,n) * 1,k) `1 by A6, A16, A24, JORDAN1A:94 ;
then A46: (Gauge C,n) * i2,k <> (Upper_Seq C,n) . 1 by A1, A2, A19, A42, A17, 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 A45, JORDAN3:70;
A47: (Gauge C,n) * i2,k in rng (Upper_Seq C,n) by A3, A6, A18, A45, A16, JORDAN1G:4, JORDAN1J:40;
then A48: go is_sequence_on Gauge C,n by JORDAN1G:4, JORDAN1J:38;
len do >= 1 by A35, XXREAL_0:2;
then 1 in dom do by FINSEQ_3:27;
then A49: do /. 1 = do . 1 by PARTFUN1:def 8
.= (Gauge C,n) * i1,j by A30, JORDAN3:58 ;
then A50: LSeg do,1 = LSeg ((Gauge C,n) * i1,j),(do /. (1 + 1)) by A35, TOPREAL1:def 5;
A51: {((Gauge C,n) * i1,j)} c= (LSeg do,1) /\ (L~ <*((Gauge C,n) * i2,k),((Gauge C,n) * i1,k),((Gauge C,n) * i1,j)*>)
proof
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in {((Gauge C,n) * i1,j)} or x in (LSeg do,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 do,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 RLTOPSP1:69;
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_1:10;
(Gauge C,n) * i1,j in LSeg do,1 by A50, RLTOPSP1:69;
hence x in (LSeg do,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;
LSeg do,1 c= L~ do by TOPREAL3:26;
then LSeg do,1 c= L~ (Lower_Seq C,n) by A36, XBOOLE_1:1;
then (LSeg do,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, A13, XBOOLE_1:26;
then A54: (L~ <*((Gauge C,n) * i2,k),((Gauge C,n) * i1,k),((Gauge C,n) * i1,j)*>) /\ (LSeg do,1) = {((Gauge C,n) * i1,j)} by A51, XBOOLE_0:def 10;
A55: rng do c= L~ do by A35, SPPOL_2:18;
A56: len go >= 1 + 1 by TOPREAL1:def 10;
then reconsider go = go as non constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A48, JGRAPH_1:16, JORDAN8:8;
A57: L~ go c= L~ (Upper_Seq C,n) by A45, JORDAN3:76;
A58: len go > 1 by A56, NAT_1:13;
then A59: len go in dom go by FINSEQ_3:27;
then A60: go /. (len go) = go . (len go) by PARTFUN1:def 8
.= (Gauge C,n) * i2,k by A45, JORDAN3:59 ;
reconsider m = (len go) - 1 as Element of NAT by A59, FINSEQ_3:28;
A61: m + 1 = len go ;
then A62: (len go) -' 1 = m by NAT_D:34;
m >= 1 by A56, XREAL_1:21;
then A63: LSeg go,m = LSeg (go /. m),((Gauge C,n) * i2,k) by A60, A61, TOPREAL1:def 5;
A64: {((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
let x be set ; :: 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 A65: 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 RLTOPSP1:69;
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 A66: (Gauge C,n) * i2,k in L~ <*((Gauge C,n) * i2,k),((Gauge C,n) * i1,k),((Gauge C,n) * i1,j)*> by SPRECT_1:10;
(Gauge C,n) * i2,k in LSeg go,m by A63, RLTOPSP1:69;
hence x in (LSeg go,m) /\ (L~ <*((Gauge C,n) * i2,k),((Gauge C,n) * i1,k),((Gauge C,n) * i1,j)*>) by A65, A66, XBOOLE_0:def 4; :: thesis: verum
end;
LSeg go,m c= L~ go by TOPREAL3:26;
then LSeg go,m c= L~ (Upper_Seq C,n) by A57, XBOOLE_1:1;
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, A13, XBOOLE_1:26;
then A67: (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 A64, XBOOLE_0:def 10;
A68: go /. 1 = (Upper_Seq C,n) /. 1 by A45, SPRECT_3:39
.= W-min (L~ (Cage C,n)) by JORDAN1F:5 ;
then A69: W-min (L~ (Cage C,n)) in rng go by FINSEQ_6:46;
A70: (Lower_Seq C,n) . 1 = (Lower_Seq C,n) /. 1 by A27, PARTFUN1:def 8
.= E-max (L~ (Cage C,n)) by JORDAN1F:6 ;
A71: (L~ go) /\ (L~ do) c= {(go /. 1)}
proof
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in (L~ go) /\ (L~ do) or x in {(go /. 1)} )
assume A72: x in (L~ go) /\ (L~ do) ; :: thesis: x in {(go /. 1)}
then A73: x in L~ do by XBOOLE_0:def 4;
A74: now
assume x = E-max (L~ (Cage C,n)) ; :: thesis: contradiction
then A75: E-max (L~ (Cage C,n)) = (Gauge C,n) * i1,j by A30, A70, A73, JORDAN1E:11;
((Gauge C,n) * (len (Gauge C,n)),j) `1 = E-bound (L~ (Cage C,n)) by A4, A10, A24, JORDAN1A:92;
then (E-max (L~ (Cage C,n))) `1 <> E-bound (L~ (Cage C,n)) by A2, A3, A23, A15, A75, JORDAN1G:7;
hence contradiction by EUCLID:56; :: thesis: verum
end;
x in L~ go by A72, XBOOLE_0:def 4;
then x in (L~ (Upper_Seq C,n)) /\ (L~ (Lower_Seq C,n)) by A57, A36, A73, XBOOLE_0:def 4;
then x in {(W-min (L~ (Cage C,n))),(E-max (L~ (Cage C,n)))} by JORDAN1E:20;
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 A68, A74, TARSKI:def 1; :: thesis: verum
end;
set W2 = go /. 2;
A76: 2 in dom go by A56, FINSEQ_3:27;
go = mid (Upper_Seq C,n),1,(((Gauge C,n) * i2,k) .. (Upper_Seq C,n)) by A47, JORDAN1G:57
.= (Upper_Seq C,n) | (((Gauge C,n) * i2,k) .. (Upper_Seq C,n)) by A47, FINSEQ_4:31, JORDAN3:25 ;
then A77: go /. 2 = (Upper_Seq C,n) /. 2 by A76, FINSEQ_4:85;
A78: rng go c= L~ go by A56, SPPOL_2:18;
A79: go /. 1 = (Lower_Seq C,n) /. (len (Lower_Seq C,n)) by A68, JORDAN1F:8
.= do /. (len do) by A30, JORDAN1J:35 ;
{(go /. 1)} c= (L~ go) /\ (L~ do)
proof
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in {(go /. 1)} or x in (L~ go) /\ (L~ do) )
assume x in {(go /. 1)} ; :: thesis: x in (L~ go) /\ (L~ do)
then A80: x = go /. 1 by TARSKI:def 1;
then A81: x in rng go by FINSEQ_6:46;
x in rng do by A79, A80, REVROT_1:3;
hence x in (L~ go) /\ (L~ do) by A78, A55, A81, XBOOLE_0:def 4; :: thesis: verum
end;
then A82: (L~ go) /\ (L~ do) = {(go /. 1)} by A71, XBOOLE_0:def 10;
now
per 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 ) ;
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 A21, TOPREAL3:42;
then consider pion1 being FinSequence of (TOP-REAL 2) such that
A83: pion1 is_sequence_on Gauge C,n and
A84: pion1 is being_S-Seq and
A85: L~ <*((Gauge C,n) * i2,k),((Gauge C,n) * i1,k),((Gauge C,n) * i1,j)*> = L~ pion1 and
A86: <*((Gauge C,n) * i2,k),((Gauge C,n) * i1,k),((Gauge C,n) * i1,j)*> /. 1 = pion1 /. 1 and
A87: <*((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
A88: len <*((Gauge C,n) * i2,k),((Gauge C,n) * i1,k),((Gauge C,n) * i1,j)*> <= len pion1 by A44, GOBOARD3:2;
reconsider pion1 = pion1 as being_S-Seq FinSequence of (TOP-REAL 2) by A84;
A89: (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 A87, GRAPH_2:58
.= <*((Gauge C,n) * i2,k),((Gauge C,n) * i1,k),((Gauge C,n) * i1,j)*> /. 3 by FINSEQ_1:62
.= do /. 1 by A49, FINSEQ_4:27 ;
A90: go /. (len go) = pion1 /. 1 by A60, A86, FINSEQ_4:27;
A91: (L~ go) /\ (L~ pion1) c= {(pion1 /. 1)}
proof
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in (L~ go) /\ (L~ pion1) or x in {(pion1 /. 1)} )
assume A92: x in (L~ go) /\ (L~ pion1) ; :: thesis: x in {(pion1 /. 1)}
then A93: x in L~ pion1 by XBOOLE_0:def 4;
x in L~ go by A92, XBOOLE_0:def 4;
hence x in {(pion1 /. 1)} by A7, A13, A60, A57, A85, A90, A93, XBOOLE_0:def 4; :: thesis: verum
end;
len pion1 >= 2 + 1 by A88, FINSEQ_1:62;
then A94: len pion1 > 1 + 1 by NAT_1:13;
then A95: rng pion1 c= L~ pion1 by SPPOL_2:18;
{(pion1 /. 1)} c= (L~ go) /\ (L~ pion1)
proof
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in {(pion1 /. 1)} or x in (L~ go) /\ (L~ pion1) )
assume x in {(pion1 /. 1)} ; :: thesis: x in (L~ go) /\ (L~ pion1)
then A96: x = pion1 /. 1 by TARSKI:def 1;
then A97: x in rng pion1 by FINSEQ_6:46;
x in rng go by A90, A96, REVROT_1:3;
hence x in (L~ go) /\ (L~ pion1) by A78, A95, A97, XBOOLE_0:def 4; :: thesis: verum
end;
then A98: (L~ go) /\ (L~ pion1) = {(pion1 /. 1)} by A91, XBOOLE_0:def 10;
then A99: go ^' pion1 is s.n.c. by A90, JORDAN1J:54;
A100: <*((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_1:62
.= do /. 1 by A49, FINSEQ_4:27 ;
A101: {(pion1 /. (len pion1))} c= (L~ do) /\ (L~ pion1)
proof
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in {(pion1 /. (len pion1))} or x in (L~ do) /\ (L~ pion1) )
assume x in {(pion1 /. (len pion1))} ; :: thesis: x in (L~ do) /\ (L~ pion1)
then A102: x = pion1 /. (len pion1) by TARSKI:def 1;
then A103: x in rng pion1 by REVROT_1:3;
x in rng do by A87, A100, A102, FINSEQ_6:46;
hence x in (L~ do) /\ (L~ pion1) by A55, A95, A103, XBOOLE_0:def 4; :: thesis: verum
end;
(L~ do) /\ (L~ pion1) c= {(pion1 /. (len pion1))}
proof
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in (L~ do) /\ (L~ pion1) or x in {(pion1 /. (len pion1))} )
assume A104: x in (L~ do) /\ (L~ pion1) ; :: thesis: x in {(pion1 /. (len pion1))}
then A105: x in L~ pion1 by XBOOLE_0:def 4;
x in L~ do by A104, XBOOLE_0:def 4;
hence x in {(pion1 /. (len pion1))} by A8, A13, A49, A36, A85, A87, A100, A105, XBOOLE_0:def 4; :: thesis: verum
end;
then A106: (L~ do) /\ (L~ pion1) = {(pion1 /. (len pion1))} by A101, XBOOLE_0:def 10;
A107: (L~ (go ^' pion1)) /\ (L~ do) = ((L~ go) \/ (L~ pion1)) /\ (L~ do) by A90, TOPREAL8:35
.= {(go /. 1)} \/ {(do /. 1)} by A82, A87, A100, A106, XBOOLE_1:23
.= {((go ^' pion1) /. 1)} \/ {(do /. 1)} by GRAPH_2:57
.= {((go ^' pion1) /. 1),(do /. 1)} by ENUMSET1:41 ;
A108: Lower_Arc C is_an_arc_of E-max C, W-min C by JORDAN6:def 9;
then A109: Lower_Arc C is connected by JORDAN6:11;
set godo = (go ^' pion1) ^' do;
A110: do /. (len do) = (go ^' pion1) /. 1 by A79, GRAPH_2:57;
A111: go ^' pion1 is_sequence_on Gauge C,n by A48, A83, A90, TOPREAL8:12;
then A112: (go ^' pion1) ^' do is_sequence_on Gauge C,n by A33, A89, TOPREAL8:12;
A113: (len pion1) - 1 >= 1 by A94, XREAL_1:21;
then A114: (len pion1) -' 1 = (len pion1) - 1 by XREAL_0:def 2;
A115: {((Gauge C,n) * i1,j)} c= (LSeg pion1,((len pion1) -' 1)) /\ (LSeg do,1)
proof
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in {((Gauge C,n) * i1,j)} or x in (LSeg pion1,((len pion1) -' 1)) /\ (LSeg do,1) )
assume x in {((Gauge C,n) * i1,j)} ; :: thesis: x in (LSeg pion1,((len pion1) -' 1)) /\ (LSeg do,1)
then A116: 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 A87, A114, FINSEQ_1:62
.= (Gauge C,n) * i1,j by FINSEQ_4:27 ;
then A117: (Gauge C,n) * i1,j in LSeg pion1,((len pion1) -' 1) by A113, A114, TOPREAL1:27;
(Gauge C,n) * i1,j in LSeg do,1 by A50, RLTOPSP1:69;
hence x in (LSeg pion1,((len pion1) -' 1)) /\ (LSeg do,1) by A116, A117, XBOOLE_0:def 4; :: thesis: verum
end;
LSeg pion1,((len pion1) -' 1) c= L~ <*((Gauge C,n) * i2,k),((Gauge C,n) * i1,k),((Gauge C,n) * i1,j)*> by A85, TOPREAL3:26;
then (LSeg pion1,((len pion1) -' 1)) /\ (LSeg do,1) c= {((Gauge C,n) * i1,j)} by A54, XBOOLE_1:27;
then A118: (LSeg pion1,((len pion1) -' 1)) /\ (LSeg do,1) = {((Gauge C,n) * i1,j)} by A115, XBOOLE_0:def 10;
((len pion1) - 1) + 1 <= len pion1 ;
then A119: (len pion1) -' 1 < len pion1 by A114, NAT_1:13;
len pion1 >= 2 + 1 by A88, FINSEQ_1:62;
then A120: (len pion1) - 2 >= 0 by XREAL_1:21;
then ((len pion1) -' 2) + 1 = ((len pion1) - 2) + 1 by XREAL_0:def 2
.= (len pion1) -' 1 by A113, XREAL_0:def 2 ;
then A121: (LSeg (go ^' pion1),((len go) + ((len pion1) -' 2))) /\ (LSeg do,1) = {((go ^' pion1) /. (len (go ^' pion1)))} by A49, A90, A89, A119, A118, TOPREAL8:31;
(rng go) /\ (rng pion1) c= {(pion1 /. 1)} by A78, A95, A98, XBOOLE_1:27;
then A122: go ^' pion1 is one-to-one by JORDAN1J:55;
((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 A120, XREAL_0:def 2 ;
then A123: (len (go ^' pion1)) -' 1 = (len go) + ((len pion1) -' 2) by XREAL_0:def 2;
A124: 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) by XBOOLE_1:7;
then A126: L~ go c= L~ (Cage C,n) by A57, XBOOLE_1:1;
A127: {((Gauge C,n) * i2,k)} c= (LSeg go,m) /\ (LSeg pion1,1)
proof
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in {((Gauge C,n) * 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 A128: x = (Gauge C,n) * i2,k by TARSKI:def 1;
A129: (Gauge C,n) * i2,k in LSeg go,m by A63, RLTOPSP1:69;
(Gauge C,n) * i2,k in LSeg pion1,1 by A60, A90, A94, TOPREAL1:27;
hence x in (LSeg go,m) /\ (LSeg pion1,1) by A128, A129, XBOOLE_0:def 4; :: thesis: verum
end;
LSeg pion1,1 c= L~ <*((Gauge C,n) * i2,k),((Gauge C,n) * i1,k),((Gauge C,n) * i1,j)*> by A85, TOPREAL3:26;
then (LSeg go,((len go) -' 1)) /\ (LSeg pion1,1) c= {((Gauge C,n) * i2,k)} by A62, A67, XBOOLE_1:27;
then (LSeg go,((len go) -' 1)) /\ (LSeg pion1,1) = {(go /. (len go))} by A60, A62, A127, XBOOLE_0:def 10;
then A130: go ^' pion1 is unfolded by A90, TOPREAL8:34;
len (go ^' pion1) >= len go by TOPREAL8:7;
then A131: len (go ^' pion1) >= 1 + 1 by A56, XXREAL_0:2;
then A132: len (go ^' pion1) > 1 + 0 by NAT_1:13;
A133: now
assume A134: ((Gauge C,n) * i2,k) .. (Upper_Seq C,n) <= 1 ; :: thesis: contradiction
((Gauge C,n) * i2,k) .. (Upper_Seq C,n) >= 1 by A47, FINSEQ_4:31;
then ((Gauge C,n) * i2,k) .. (Upper_Seq C,n) = 1 by A134, XXREAL_0:1;
then (Gauge C,n) * i2,k = (Upper_Seq C,n) /. 1 by A47, FINSEQ_5:41;
hence contradiction by A42, A46, JORDAN1F:5; :: thesis: verum
end;
A135: Upper_Seq C,n is_sequence_on Gauge C,n by JORDAN1G:4;
A136: (W-min (L~ (Cage C,n))) `1 = W-bound (L~ (Cage C,n)) by EUCLID:56;
set ff = Rotate (Cage C,n),(W-min (L~ (Cage C,n)));
A137: 1 + 1 <= len (Cage C,n) by GOBOARD7:36, XXREAL_0:2;
A138: len ((go ^' pion1) ^' do) >= len (go ^' pion1) by TOPREAL8:7;
then A139: 1 + 1 <= len ((go ^' pion1) ^' do) by A131, XXREAL_0:2;
not go ^' pion1 is trivial by A131, REALSET1:13;
then reconsider godo = (go ^' pion1) ^' do as non constant standard special_circular_sequence by A139, A89, A112, A130, A123, A121, A99, A122, A107, A110, JORDAN8:7, JORDAN8:8, TOPREAL8:11, TOPREAL8:33, TOPREAL8:34;
A140: L~ godo = (L~ (go ^' pion1)) \/ (L~ do) by A89, TOPREAL8:35
.= ((L~ go) \/ (L~ pion1)) \/ (L~ do) by A90, TOPREAL8:35 ;
A141: (right_cell godo,1,(Gauge C,n)) \ (L~ godo) c= RightComp godo by A139, A112, JORDAN9:29;
2 in dom godo by A139, FINSEQ_3:27;
then A142: godo /. 2 in rng godo by PARTFUN2:4;
A143: W-min C in Lower_Arc C by A108, TOPREAL1:4;
W-min (L~ (Cage C,n)) in rng (Cage C,n) by SPRECT_2:47;
then A144: (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) /. 1 = W-min (L~ (Cage C,n)) by FINSEQ_6:98;
A145: 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 A144, 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 A144, A145, 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 A144, A145, SPRECT_5:25, XXREAL_0:2;
then A146: (E-max (L~ (Cage C,n))) .. (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) > 1 by A144, A145, SPRECT_5:26, XXREAL_0:2;
A147: Cage C,n is_sequence_on Gauge C,n by JORDAN9:def 1;
then A148: Rotate (Cage C,n),(W-min (L~ (Cage C,n))) is_sequence_on Gauge C,n by REVROT_1:34;
A149: ((Gauge C,n) * i1,k) `1 = ((Gauge C,n) * i1,1) `1 by A1, A6, A11, A16, GOBOARD5:3
.= ((Gauge C,n) * i1,j) `1 by A1, A4, A11, A10, GOBOARD5:3 ;
then A150: W-bound (LSeg ((Gauge C,n) * i1,j),((Gauge C,n) * i1,k)) = ((Gauge C,n) * i1,j) `1 by SPRECT_1:62;
A151: L~ (Lower_Seq C,n) c= L~ (Cage C,n) by A124, XBOOLE_1:7;
then A152: L~ do c= L~ (Cage C,n) by A36, XBOOLE_1:1;
A153: W-min C in C by SPRECT_1:15;
A154: now
assume W-min C in L~ godo ; :: thesis: contradiction
then A155: ( W-min C in (L~ go) \/ (L~ pion1) or W-min C in L~ do ) by A140, XBOOLE_0:def 3;
end;
A156: len (Upper_Seq C,n) >= 2 by A41, XXREAL_0:2;
A157: (L~ go) \/ (L~ do) is compact by COMPTS_1:19;
1 + 1 <= len (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) by GOBOARD7:36, 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: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 A146, A148, 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 A47, A135, A133, JORDAN1J:52
.= right_cell (go ^' pion1),1,(Gauge C,n) by A58, A111, JORDAN1J:51
.= right_cell godo,1,(Gauge C,n) by A132, A112, JORDAN1J:51 ;
then W-min C in right_cell godo,1,(Gauge C,n) by JORDAN1I:8;
then A158: W-min C in (right_cell godo,1,(Gauge C,n)) \ (L~ godo) by A154, XBOOLE_0:def 5;
A159: rng godo c= L~ godo by A131, A138, SPPOL_2:18, XXREAL_0:2;
A160: godo /. 1 = (go ^' pion1) /. 1 by GRAPH_2:57
.= W-min (L~ (Cage C,n)) by A68, GRAPH_2:57 ;
A161: ((Gauge C,n) * i1,k) `1 <= ((Gauge C,n) * i2,k) `1 by A1, A2, A3, A6, A16, JORDAN1A:39;
then A162: W-bound (LSeg ((Gauge C,n) * i1,k),((Gauge C,n) * i2,k)) = ((Gauge C,n) * i1,k) `1 by SPRECT_1:62;
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_1:54
.= ((Gauge C,n) * i1,j) `1 by A149, A162, A150 ;
then A163: W-bound (L~ pion1) = ((Gauge C,n) * i1,j) `1 by A85, TOPREAL3:23;
A164: Lower_Arc C c= C by JORDAN6:76;
((Gauge C,n) * i1,j) `1 >= W-bound (L~ (Cage C,n)) by A30, A151, PSCOMP_1:71;
then A165: ((Gauge C,n) * i1,j) `1 > W-bound (L~ (Cage C,n)) by A38, XXREAL_0:1;
A166: E-max C in Lower_Arc C by A108, TOPREAL1:4;
W-min (L~ (Cage C,n)) in (L~ go) \/ (L~ do) by A78, A69, XBOOLE_0:def 3;
then A167: W-min ((L~ go) \/ (L~ do)) = W-min (L~ (Cage C,n)) by A126, A152, A157, JORDAN1J:21, XBOOLE_1:8;
(W-min ((L~ go) \/ (L~ do))) `1 = W-bound ((L~ go) \/ (L~ do)) by EUCLID:56;
then W-min (((L~ go) \/ (L~ do)) \/ (L~ pion1)) = W-min ((L~ go) \/ (L~ do)) by A163, A157, A167, A136, A165, JORDAN1J:33;
then A168: W-min (L~ godo) = W-min (L~ (Cage C,n)) by A140, A167, XBOOLE_1:4;
godo /. 2 = (go ^' pion1) /. 2 by A131, GRAPH_2:61
.= (Upper_Seq C,n) /. 2 by A56, A77, GRAPH_2:61
.= ((Upper_Seq C,n) ^' (Lower_Seq C,n)) /. 2 by A156, GRAPH_2:61
.= (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) /. 2 by JORDAN1E:15 ;
then godo /. 2 in W-most (L~ (Cage C,n)) by JORDAN1I:27;
then (godo /. 2) `1 = (W-min (L~ godo)) `1 by A168, PSCOMP_1:88
.= W-bound (L~ godo) by EUCLID:56 ;
then godo /. 2 in W-most (L~ godo) by A159, A142, SPRECT_2:16;
then (Rotate godo,(W-min (L~ godo))) /. 2 in W-most (L~ godo) by A160, A168, 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 A169: (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 ;
A170: east_halfline (E-max C) misses L~ go
proof
assume east_halfline (E-max C) meets L~ go ; :: thesis: contradiction
then consider p being set such that
A171: p in east_halfline (E-max C) and
A172: p in L~ go by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A171;
p in L~ (Upper_Seq C,n) by A57, A172;
then p in (east_halfline (E-max C)) /\ (L~ (Cage C,n)) by A125, A171, XBOOLE_0:def 4;
then A173: p `1 = E-bound (L~ (Cage C,n)) by JORDAN1A:104, PSCOMP_1:111;
then A174: p = E-max (L~ (Cage C,n)) by A57, A172, JORDAN1J:46;
then E-max (L~ (Cage C,n)) = (Gauge C,n) * i2,k by A45, A169, A172, JORDAN1J:43;
then ((Gauge C,n) * i2,k) `1 = ((Gauge C,n) * (len (Gauge C,n)),k) `1 by A6, A16, A24, A173, A174, JORDAN1A:92;
hence contradiction by A3, A19, A22, JORDAN1G:7; :: thesis: verum
end;
now
assume east_halfline (E-max C) meets L~ godo ; :: thesis: contradiction
then A175: ( east_halfline (E-max C) meets (L~ go) \/ (L~ pion1) or east_halfline (E-max C) meets L~ do ) by A140, 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~ do ) by A175, XBOOLE_1:70;
suppose east_halfline (E-max C) meets L~ pion1 ; :: thesis: contradiction
then consider p being set such that
A176: p in east_halfline (E-max C) and
A177: p in L~ pion1 by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A176;
A178: p `2 = (E-max C) `2 by A176, TOPREAL1:def 13;
A179: now
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 A13, A85, A177, XBOOLE_0:def 3;
suppose p in LSeg ((Gauge C,n) * i1,k),((Gauge C,n) * i2,k) ; :: thesis: p `1 <= ((Gauge C,n) * i2,k) `1
hence p `1 <= ((Gauge C,n) * i2,k) `1 by A161, TOPREAL1:9; :: thesis: verum
end;
suppose p in LSeg ((Gauge C,n) * i1,j),((Gauge C,n) * i1,k) ; :: thesis: p `1 <= ((Gauge C,n) * i2,k) `1
hence p `1 <= ((Gauge C,n) * i2,k) `1 by A149, A161, GOBOARD7:5; :: thesis: verum
end;
end;
end;
i2 + 1 <= len (Gauge C,n) by A3, NAT_1:13;
then (i2 + 1) - 1 <= (len (Gauge C,n)) - 1 by XREAL_1:11;
then A180: i2 <= (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) * i2,k) `1 <= ((Gauge C,n) * ((len (Gauge C,n)) -' 1),1) `1 by A6, A18, A16, A24, A14, A180, JORDAN1A:39;
then p `1 <= ((Gauge C,n) * ((len (Gauge C,n)) -' 1),1) `1 by A179, XXREAL_0:2;
then p `1 <= E-bound C by A14, JORDAN8:15;
then A181: p `1 <= (E-max C) `1 by EUCLID:56;
p `1 >= (E-max C) `1 by A176, TOPREAL1:def 13;
then p `1 = (E-max C) `1 by A181, XXREAL_0:1;
then p = E-max C by A178, TOPREAL3:11;
hence contradiction by A9, A13, A85, A166, A177, XBOOLE_0:3; :: thesis: verum
end;
suppose east_halfline (E-max C) meets L~ do ; :: thesis: contradiction
then consider p being set such that
A182: p in east_halfline (E-max C) and
A183: p in L~ do by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A182;
A184: p in LSeg do,(Index p,do) by A183, JORDAN3:42;
consider t being Nat such that
A185: t in dom (Lower_Seq C,n) and
A186: (Lower_Seq C,n) . t = (Gauge C,n) * i1,j by A32, FINSEQ_2:11;
1 <= t by A185, FINSEQ_3:27;
then A187: 1 < t by A34, A186, XXREAL_0:1;
t <= len (Lower_Seq C,n) by A185, FINSEQ_3:27;
then (Index ((Gauge C,n) * i1,j),(Lower_Seq C,n)) + 1 = t by A186, A187, JORDAN3:45;
then A188: 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 A30, A186, JORDAN3:61;
Index p,do < len do by A183, JORDAN3:41;
then Index p,do < (len (Lower_Seq C,n)) -' (Index ((Gauge C,n) * i1,j),(Lower_Seq C,n)) by A188, XREAL_0:def 2;
then (Index p,do) + 1 <= (len (Lower_Seq C,n)) -' (Index ((Gauge C,n) * i1,j),(Lower_Seq C,n)) by NAT_1:13;
then A189: Index p,do <= ((len (Lower_Seq C,n)) -' (Index ((Gauge C,n) * i1,j),(Lower_Seq C,n))) - 1 by XREAL_1:21;
A190: do = mid (Lower_Seq C,n),(((Gauge C,n) * i1,j) .. (Lower_Seq C,n)),(len (Lower_Seq C,n)) by A32, JORDAN1J:37;
p in L~ (Lower_Seq C,n) by A36, A183;
then p in (east_halfline (E-max C)) /\ (L~ (Cage C,n)) by A151, A182, XBOOLE_0:def 4;
then A191: p `1 = E-bound (L~ (Cage C,n)) by JORDAN1A:104, PSCOMP_1:111;
A192: (Index ((Gauge C,n) * i1,j),(Lower_Seq C,n)) + 1 = ((Gauge C,n) * i1,j) .. (Lower_Seq C,n) by A34, A32, JORDAN1J:56;
0 + (Index ((Gauge C,n) * i1,j),(Lower_Seq C,n)) < len (Lower_Seq C,n) by A30, JORDAN3:41;
then (len (Lower_Seq C,n)) - (Index ((Gauge C,n) * i1,j),(Lower_Seq C,n)) > 0 by XREAL_1:22;
then Index p,do <= ((len (Lower_Seq C,n)) - (Index ((Gauge C,n) * i1,j),(Lower_Seq C,n))) - 1 by A189, XREAL_0:def 2;
then Index p,do <= (len (Lower_Seq C,n)) - (((Gauge C,n) * i1,j) .. (Lower_Seq C,n)) by A192;
then Index p,do <= (len (Lower_Seq C,n)) -' (((Gauge C,n) * i1,j) .. (Lower_Seq C,n)) by XREAL_0:def 2;
then A193: Index p,do < ((len (Lower_Seq C,n)) -' (((Gauge C,n) * i1,j) .. (Lower_Seq C,n))) + 1 by NAT_1:13;
A194: 1 <= Index p,do by A183, JORDAN3:41;
A195: ((Gauge C,n) * i1,j) .. (Lower_Seq C,n) <= len (Lower_Seq C,n) by A32, FINSEQ_4:31;
((Gauge C,n) * i1,j) .. (Lower_Seq C,n) <> len (Lower_Seq C,n) by A31, A32, FINSEQ_4:29;
then A196: ((Gauge C,n) * i1,j) .. (Lower_Seq C,n) < len (Lower_Seq C,n) by A195, XXREAL_0:1;
A197: 1 + 1 <= len (Lower_Seq C,n) by A25, XXREAL_0:2;
then A198: 2 in dom (Lower_Seq C,n) by FINSEQ_3:27;
set tt = ((Index p,do) + (((Gauge C,n) * i1,j) .. (Lower_Seq C,n))) -' 1;
set RC = Rotate (Cage C,n),(E-max (L~ (Cage C,n)));
A199: E-max C in right_cell (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))),1 by JORDAN1I:9;
A200: 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 ;
A201: L~ (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) = L~ (Cage C,n) by REVROT_1:33;
consider jj2 being Element of NAT such that
A202: 1 <= jj2 and
A203: jj2 <= width (Gauge C,n) and
A204: E-max (L~ (Cage C,n)) = (Gauge C,n) * (len (Gauge C,n)),jj2 by JORDAN1D:29;
A205: len (Gauge C,n) >= 4 by JORDAN8:13;
then len (Gauge C,n) >= 1 by XXREAL_0:2;
then A206: [(len (Gauge C,n)),jj2] in Indices (Gauge C,n) by A202, A203, MATRIX_1:37;
A207: len (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) = len (Cage C,n) by REVROT_1:14;
Lower_Seq C,n = (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) -: (W-min (L~ (Cage C,n))) by JORDAN1G:26;
then A208: LSeg (Lower_Seq C,n),1 = LSeg (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))),1 by A197, SPPOL_2:9;
A209: E-max (L~ (Cage C,n)) in rng (Cage C,n) by SPRECT_2:50;
Rotate (Cage C,n),(E-max (L~ (Cage C,n))) is_sequence_on Gauge C,n by A147, REVROT_1:34;
then consider ii, jj being Element of NAT such that
A210: [ii,(jj + 1)] in Indices (Gauge C,n) and
A211: [ii,jj] in Indices (Gauge C,n) and
A212: (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) /. 1 = (Gauge C,n) * ii,(jj + 1) and
A213: (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) /. (1 + 1) = (Gauge C,n) * ii,jj by A137, A201, A207, A209, FINSEQ_6:98, JORDAN1I:25;
A214: (jj + 1) + 1 <> jj ;
A215: 1 <= jj by A211, MATRIX_1:39;
(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 A201, A209, FINSEQ_6:98;
then A216: ii = len (Gauge C,n) by A201, A210, A212, A204, A206, GOBOARD1:21;
then ii - 1 >= 4 - 1 by A205, XREAL_1:11;
then A217: ii - 1 >= 1 by XXREAL_0:2;
then A218: 1 <= ii -' 1 by XREAL_0:def 2;
A219: jj <= width (Gauge C,n) by A211, MATRIX_1:39;
then A220: ((Gauge C,n) * (len (Gauge C,n)),jj) `1 = E-bound (L~ (Cage C,n)) by A24, A215, JORDAN1A:92;
A221: jj + 1 <= width (Gauge C,n) by A210, MATRIX_1:39;
ii + 1 <> ii ;
then A222: right_cell (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))),1 = cell (Gauge C,n),(ii -' 1),jj by A137, A207, A200, A210, A211, A212, A213, A214, GOBOARD5:def 6;
A223: ii <= len (Gauge C,n) by A211, MATRIX_1:39;
A224: 1 <= ii by A211, MATRIX_1:39;
A225: ii <= len (Gauge C,n) by A210, MATRIX_1:39;
A226: 1 <= jj + 1 by A210, MATRIX_1:39;
then A227: E-bound (L~ (Cage C,n)) = ((Gauge C,n) * (len (Gauge C,n)),(jj + 1)) `1 by A24, A221, JORDAN1A:92;
A228: 1 <= ii by A210, MATRIX_1:39;
then A229: (ii -' 1) + 1 = ii by XREAL_1:237;
then A230: ii -' 1 < len (Gauge C,n) by A225, NAT_1:13;
then A231: ((Gauge C,n) * (ii -' 1),(jj + 1)) `2 = ((Gauge C,n) * 1,(jj + 1)) `2 by A226, A221, A218, GOBOARD5:2
.= ((Gauge C,n) * ii,(jj + 1)) `2 by A228, A225, A226, A221, GOBOARD5:2 ;
A232: (E-max C) `2 = p `2 by A182, TOPREAL1:def 13;
then A233: p `2 <= ((Gauge C,n) * (ii -' 1),(jj + 1)) `2 by A199, A225, A221, A215, A222, A229, A217, JORDAN9:19;
A234: ((Gauge C,n) * (ii -' 1),jj) `2 = ((Gauge C,n) * 1,jj) `2 by A215, A219, A218, A230, GOBOARD5:2
.= ((Gauge C,n) * ii,jj) `2 by A224, A223, A215, A219, GOBOARD5:2 ;
((Gauge C,n) * (ii -' 1),jj) `2 <= p `2 by A232, A199, A225, A221, A215, A222, A229, A217, JORDAN9:19;
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 A191, A212, A213, A216, A233, A234, A231, A220, A227, GOBOARD7:8;
then A235: p in LSeg (Lower_Seq C,n),1 by A137, A208, A207, TOPREAL1:def 5;
1 <= ((Gauge C,n) * i1,j) .. (Lower_Seq C,n) by A32, FINSEQ_4:31;
then A236: LSeg (mid (Lower_Seq C,n),(((Gauge C,n) * i1,j) .. (Lower_Seq C,n)),(len (Lower_Seq C,n))),(Index p,do) = LSeg (Lower_Seq C,n),(((Index p,do) + (((Gauge C,n) * i1,j) .. (Lower_Seq C,n))) -' 1) by A196, A194, A193, JORDAN4:31;
1 <= Index ((Gauge C,n) * i1,j),(Lower_Seq C,n) by A30, JORDAN3:41;
then A237: 1 + 1 <= ((Gauge C,n) * i1,j) .. (Lower_Seq C,n) by A192, XREAL_1:9;
then (Index p,do) + (((Gauge C,n) * i1,j) .. (Lower_Seq C,n)) >= (1 + 1) + 1 by A194, XREAL_1:9;
then ((Index p,do) + (((Gauge C,n) * i1,j) .. (Lower_Seq C,n))) - 1 >= ((1 + 1) + 1) - 1 by XREAL_1:11;
then A238: ((Index p,do) + (((Gauge C,n) * i1,j) .. (Lower_Seq C,n))) -' 1 >= 1 + 1 by XREAL_0:def 2;
now
per cases ( ((Index p,do) + (((Gauge C,n) * i1,j) .. (Lower_Seq C,n))) -' 1 > 1 + 1 or ((Index p,do) + (((Gauge C,n) * i1,j) .. (Lower_Seq C,n))) -' 1 = 1 + 1 ) by A238, XXREAL_0:1;
suppose ((Index p,do) + (((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,do) + (((Gauge C,n) * i1,j) .. (Lower_Seq C,n))) -' 1) by TOPREAL1:def 9;
hence contradiction by A235, A184, A190, A236, XBOOLE_0:3; :: thesis: verum
end;
suppose A239: ((Index p,do) + (((Gauge C,n) * i1,j) .. (Lower_Seq C,n))) -' 1 = 1 + 1 ; :: thesis: contradiction
then 1 + 1 = ((Index p,do) + (((Gauge C,n) * i1,j) .. (Lower_Seq C,n))) - 1 by XREAL_0:def 2;
then (1 + 1) + 1 = (Index p,do) + (((Gauge C,n) * i1,j) .. (Lower_Seq C,n)) ;
then A240: ((Gauge C,n) * i1,j) .. (Lower_Seq C,n) = 2 by A194, A237, JORDAN1E:10;
(LSeg (Lower_Seq C,n),1) /\ (LSeg (Lower_Seq C,n),(((Index p,do) + (((Gauge C,n) * i1,j) .. (Lower_Seq C,n))) -' 1)) = {((Lower_Seq C,n) /. 2)} by A25, A239, TOPREAL1:def 8;
then p in {((Lower_Seq C,n) /. 2)} by A235, A184, A190, A236, XBOOLE_0:def 4;
then A241: p = (Lower_Seq C,n) /. 2 by TARSKI:def 1;
then A242: p in rng (Lower_Seq C,n) by A198, PARTFUN2:4;
p .. (Lower_Seq C,n) = 2 by A198, A241, FINSEQ_5:44;
then p = (Gauge C,n) * i1,j by A32, A240, A242, FINSEQ_5:10;
then ((Gauge C,n) * i1,j) `1 = E-bound (L~ (Cage C,n)) by A241, JORDAN1G:40;
then ((Gauge C,n) * i1,j) `1 = ((Gauge C,n) * (len (Gauge C,n)),j) `1 by A4, A10, A24, JORDAN1A:92;
hence contradiction by A2, A3, A23, A15, JORDAN1G:7; :: thesis: verum
end;
end;
end;
hence contradiction ; :: thesis: verum
end;
end;
end;
then east_halfline (E-max C) c= (L~ godo) ` by SUBSET_1:43;
then consider W being Subset of (TOP-REAL 2) such that
A243: W is_a_component_of (L~ godo) ` and
A244: east_halfline (E-max C) c= W by GOBOARD9:5;
not W is Bounded by A244, JORDAN2C:16, JORDAN2C:129;
then W is_outside_component_of L~ godo by A243, JORDAN2C:def 4;
then W c= UBD (L~ godo) by JORDAN2C:27;
then A245: east_halfline (E-max C) c= UBD (L~ godo) by A244, XBOOLE_1:1;
E-max C in east_halfline (E-max C) by TOPREAL1:45;
then E-max C in UBD (L~ godo) by A245;
then E-max C in LeftComp godo by GOBRD14:46;
then Lower_Arc C meets L~ godo by A109, A143, A166, A141, A158, JORDAN1J:36;
then A246: ( Lower_Arc C meets (L~ go) \/ (L~ pion1) or Lower_Arc C meets L~ do ) by A140, XBOOLE_1:70;
hence contradiction ; :: thesis: verum
end;
suppose ((Gauge C,n) * i1,j) `1 = ((Gauge C,n) * i2,k) `1 ; :: thesis: contradiction
then A247: i1 = i2 by A23, A19, JORDAN1G:7;
then LSeg ((Gauge C,n) * i1,k),((Gauge C,n) * i2,k) = {((Gauge C,n) * i1,k)} by RLTOPSP1:71;
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 A40, ZFMISC_1:37;
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, A247, JORDAN1J:58; :: thesis: verum
end;
suppose ((Gauge C,n) * i1,j) `2 = ((Gauge C,n) * i2,k) `2 ; :: thesis: contradiction
then A248: j = k by A23, A19, JORDAN1G:6;
then LSeg ((Gauge C,n) * i1,j),((Gauge C,n) * i1,k) = {((Gauge C,n) * i1,k)} by RLTOPSP1:71;
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 A39, ZFMISC_1:37;
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, A248, Th30; :: thesis: verum
end;
end;
end;
hence contradiction ; :: thesis: verum