let n be Element of NAT ; :: thesis: for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < i & i < len (Gauge C,n) & 1 <= k & k <= j & j <= width (Gauge C,n) & (LSeg ((Gauge C,n) * i,k),((Gauge C,n) * i,j)) /\ (L~ (Upper_Seq C,n)) = {((Gauge C,n) * i,k)} & (LSeg ((Gauge C,n) * i,k),((Gauge C,n) * i,j)) /\ (L~ (Lower_Seq C,n)) = {((Gauge C,n) * i,j)} holds
LSeg ((Gauge C,n) * i,k),((Gauge C,n) * i,j) meets Lower_Arc C
let C be Simple_closed_curve; :: thesis: for i, j, k being Element of NAT st 1 < i & i < len (Gauge C,n) & 1 <= k & k <= j & j <= width (Gauge C,n) & (LSeg ((Gauge C,n) * i,k),((Gauge C,n) * i,j)) /\ (L~ (Upper_Seq C,n)) = {((Gauge C,n) * i,k)} & (LSeg ((Gauge C,n) * i,k),((Gauge C,n) * i,j)) /\ (L~ (Lower_Seq C,n)) = {((Gauge C,n) * i,j)} holds
LSeg ((Gauge C,n) * i,k),((Gauge C,n) * i,j) meets Lower_Arc C
let i, j, k be Element of NAT ; :: thesis: ( 1 < i & i < len (Gauge C,n) & 1 <= k & k <= j & j <= width (Gauge C,n) & (LSeg ((Gauge C,n) * i,k),((Gauge C,n) * i,j)) /\ (L~ (Upper_Seq C,n)) = {((Gauge C,n) * i,k)} & (LSeg ((Gauge C,n) * i,k),((Gauge C,n) * i,j)) /\ (L~ (Lower_Seq C,n)) = {((Gauge C,n) * i,j)} implies LSeg ((Gauge C,n) * i,k),((Gauge C,n) * i,j) meets Lower_Arc C )
set Ga = Gauge C,n;
set US = Upper_Seq C,n;
set LS = Lower_Seq C,n;
set LA = Lower_Arc C;
set Wmin = W-min (L~ (Cage C,n));
set Emax = E-max (L~ (Cage C,n));
set Wbo = W-bound (L~ (Cage C,n));
set Ebo = E-bound (L~ (Cage C,n));
set Gik = (Gauge C,n) * i,k;
set Gij = (Gauge C,n) * i,j;
assume that
A1: 1 < i and
A2: i < len (Gauge C,n) and
A3: 1 <= k and
A4: k <= j and
A5: j <= width (Gauge C,n) and
A6: (LSeg ((Gauge C,n) * i,k),((Gauge C,n) * i,j)) /\ (L~ (Upper_Seq C,n)) = {((Gauge C,n) * i,k)} and
A7: (LSeg ((Gauge C,n) * i,k),((Gauge C,n) * i,j)) /\ (L~ (Lower_Seq C,n)) = {((Gauge C,n) * i,j)} and
A8: LSeg ((Gauge C,n) * i,k),((Gauge C,n) * i,j) misses Lower_Arc C ; :: thesis: contradiction
(Gauge C,n) * i,j in {((Gauge C,n) * i,j)} by TARSKI:def 1;
then A9: (Gauge C,n) * i,j in L~ (Lower_Seq C,n) by A7, XBOOLE_0:def 4;
(Gauge C,n) * i,k in {((Gauge C,n) * i,k)} by TARSKI:def 1;
then A10: (Gauge C,n) * i,k in L~ (Upper_Seq C,n) by A6, XBOOLE_0:def 4;
then A11: j <> k by A1, A2, A3, A5, A9, JORDAN1J:57;
A12: 1 <= j by A3, A4, XXREAL_0:2;
A13: k <= width (Gauge C,n) by A4, A5, XXREAL_0:2;
A14: [i,j] in Indices (Gauge C,n) by A1, A2, A5, A12, MATRIX_1:37;
A15: [i,k] in Indices (Gauge C,n) by A1, A2, A3, A13, MATRIX_1:37;
set do = L_Cut (Lower_Seq C,n),((Gauge C,n) * i,j);
set go = R_Cut (Upper_Seq C,n),((Gauge C,n) * i,k);
A16: len (Gauge C,n) = width (Gauge C,n) by JORDAN8:def 1;
A17: 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 A18: (Upper_Seq C,n) . 1 = (Upper_Seq C,n) /. 1 by PARTFUN1:def 8
.= W-min (L~ (Cage C,n)) by JORDAN1F:5 ;
A19: (W-min (L~ (Cage C,n))) `1 = W-bound (L~ (Cage C,n)) by EUCLID:56
.= ((Gauge C,n) * 1,k) `1 by A3, A13, A16, JORDAN1A:94 ;
len (Gauge C,n) >= 4 by JORDAN8:13;
then A20: len (Gauge C,n) >= 1 by XXREAL_0:2;
then A21: [1,k] in Indices (Gauge C,n) by A3, A13, MATRIX_1:37;
then A22: (Gauge C,n) * i,k <> (Upper_Seq C,n) . 1 by A1, A15, A18, A19, JORDAN1G:7;
then reconsider go = R_Cut (Upper_Seq C,n),((Gauge C,n) * i,k) as being_S-Seq FinSequence of (TOP-REAL 2) by A10, JORDAN3:70;
A23: len (Lower_Seq C,n) >= 1 + 2 by JORDAN1E:19;
then A24: len (Lower_Seq C,n) >= 1 by XXREAL_0:2;
then A25: 1 in dom (Lower_Seq C,n) by FINSEQ_3:27;
len (Lower_Seq C,n) in dom (Lower_Seq C,n) by A24, FINSEQ_3:27;
then A26: (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 ;
A27: (W-min (L~ (Cage C,n))) `1 = W-bound (L~ (Cage C,n)) by EUCLID:56
.= ((Gauge C,n) * 1,k) `1 by A3, A13, A16, JORDAN1A:94 ;
A28: [i,j] in Indices (Gauge C,n) by A1, A2, A5, A12, MATRIX_1:37;
then A29: (Gauge C,n) * i,j <> (Lower_Seq C,n) . (len (Lower_Seq C,n)) by A1, A21, A26, A27, JORDAN1G:7;
then reconsider do = L_Cut (Lower_Seq C,n),((Gauge C,n) * i,j) as being_S-Seq FinSequence of (TOP-REAL 2) by A9, JORDAN3:69;
A30: [(len (Gauge C,n)),k] in Indices (Gauge C,n) by A3, A13, A20, MATRIX_1:37;
A31: (Lower_Seq C,n) . 1 = (Lower_Seq C,n) /. 1 by A25, 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 A3, A13, A16, JORDAN1A:92 ;
then A32: (Gauge C,n) * i,j <> (Lower_Seq C,n) . 1 by A2, A28, A30, A31, JORDAN1G:7;
A33: len go >= 1 + 1 by TOPREAL1:def 10;
A34: (Gauge C,n) * i,k in rng (Upper_Seq C,n) by A1, A2, A3, A10, A13, JORDAN1G:4, JORDAN1J:40;
then A35: go is_sequence_on Gauge C,n by JORDAN1G:4, JORDAN1J:38;
A36: len do >= 1 + 1 by TOPREAL1:def 10;
A37: (Gauge C,n) * i,j in rng (Lower_Seq C,n) by A1, A2, A5, A9, A12, JORDAN1G:5, JORDAN1J:40;
then A38: 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 A33, A35, JGRAPH_1:16, JORDAN8:8;
reconsider do = do as non constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A36, A38, JGRAPH_1:16, JORDAN8:8;
A39: len go > 1 by A33, NAT_1:13;
then A40: len go in dom go by FINSEQ_3:27;
then A41: go /. (len go) = go . (len go) by PARTFUN1:def 8
.= (Gauge C,n) * i,k by A10, JORDAN3:59 ;
len do >= 1 by A36, XXREAL_0:2;
then 1 in dom do by FINSEQ_3:27;
then A42: do /. 1 = do . 1 by PARTFUN1:def 8
.= (Gauge C,n) * i,j by A9, JORDAN3:58 ;
reconsider m = (len go) - 1 as Element of NAT by A40, FINSEQ_3:28;
A43: m + 1 = len go ;
then A44: (len go) -' 1 = m by NAT_D:34;
A45: LSeg go,m c= L~ go by TOPREAL3:26;
A46: L~ go c= L~ (Upper_Seq C,n) by A10, JORDAN3:76;
then LSeg go,m c= L~ (Upper_Seq C,n) by A45, XBOOLE_1:1;
then A47: (LSeg go,m) /\ (LSeg ((Gauge C,n) * i,k),((Gauge C,n) * i,j)) c= {((Gauge C,n) * i,k)} by A6, XBOOLE_1:26;
m >= 1 by A33, XREAL_1:21;
then A48: LSeg go,m = LSeg (go /. m),((Gauge C,n) * i,k) by A41, A43, TOPREAL1:def 5;
{((Gauge C,n) * i,k)} c= (LSeg go,m) /\ (LSeg ((Gauge C,n) * i,k),((Gauge C,n) * i,j))
proof
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in {((Gauge C,n) * i,k)} or x in (LSeg go,m) /\ (LSeg ((Gauge C,n) * i,k),((Gauge C,n) * i,j)) )
assume x in {((Gauge C,n) * i,k)} ; :: thesis: x in (LSeg go,m) /\ (LSeg ((Gauge C,n) * i,k),((Gauge C,n) * i,j))
then A49: x = (Gauge C,n) * i,k by TARSKI:def 1;
A50: (Gauge C,n) * i,k in LSeg go,m by A48, RLTOPSP1:69;
(Gauge C,n) * i,k in LSeg ((Gauge C,n) * i,k),((Gauge C,n) * i,j) by RLTOPSP1:69;
hence x in (LSeg go,m) /\ (LSeg ((Gauge C,n) * i,k),((Gauge C,n) * i,j)) by A49, A50, XBOOLE_0:def 4; :: thesis: verum
end;
then A51: (LSeg go,m) /\ (LSeg ((Gauge C,n) * i,k),((Gauge C,n) * i,j)) = {((Gauge C,n) * i,k)} by A47, XBOOLE_0:def 10;
A52: LSeg do,1 c= L~ do by TOPREAL3:26;
A53: L~ do c= L~ (Lower_Seq C,n) by A9, JORDAN3:77;
then LSeg do,1 c= L~ (Lower_Seq C,n) by A52, XBOOLE_1:1;
then A54: (LSeg do,1) /\ (LSeg ((Gauge C,n) * i,k),((Gauge C,n) * i,j)) c= {((Gauge C,n) * i,j)} by A7, XBOOLE_1:26;
A55: LSeg do,1 = LSeg ((Gauge C,n) * i,j),(do /. (1 + 1)) by A36, A42, TOPREAL1:def 5;
{((Gauge C,n) * i,j)} c= (LSeg do,1) /\ (LSeg ((Gauge C,n) * i,k),((Gauge C,n) * i,j))
proof
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in {((Gauge C,n) * i,j)} or x in (LSeg do,1) /\ (LSeg ((Gauge C,n) * i,k),((Gauge C,n) * i,j)) )
assume x in {((Gauge C,n) * i,j)} ; :: thesis: x in (LSeg do,1) /\ (LSeg ((Gauge C,n) * i,k),((Gauge C,n) * i,j))
then A56: x = (Gauge C,n) * i,j by TARSKI:def 1;
A57: (Gauge C,n) * i,j in LSeg do,1 by A55, RLTOPSP1:69;
(Gauge C,n) * i,j in LSeg ((Gauge C,n) * i,k),((Gauge C,n) * i,j) by RLTOPSP1:69;
hence x in (LSeg do,1) /\ (LSeg ((Gauge C,n) * i,k),((Gauge C,n) * i,j)) by A56, A57, XBOOLE_0:def 4; :: thesis: verum
end;
then A58: (LSeg ((Gauge C,n) * i,k),((Gauge C,n) * i,j)) /\ (LSeg do,1) = {((Gauge C,n) * i,j)} by A54, XBOOLE_0:def 10;
A59: go /. 1 = (Upper_Seq C,n) /. 1 by A10, SPRECT_3:39
.= W-min (L~ (Cage C,n)) by JORDAN1F:5 ;
then A60: go /. 1 = (Lower_Seq C,n) /. (len (Lower_Seq C,n)) by JORDAN1F:8
.= do /. (len do) by A9, JORDAN1J:35 ;
A61: rng go c= L~ go by A33, SPPOL_2:18;
A62: rng do c= L~ do by A36, SPPOL_2:18;
A63: {(go /. 1)} c= (L~ go) /\ (L~ do)
proof
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in {(go /. 1)} or x in (L~ go) /\ (L~ do) )
assume x in {(go /. 1)} ; :: thesis: x in (L~ go) /\ (L~ do)
then A64: x = go /. 1 by TARSKI:def 1;
then A65: x in rng go by FINSEQ_6:46;
x in rng do by A60, A64, REVROT_1:3;
hence x in (L~ go) /\ (L~ do) by A61, A62, A65, XBOOLE_0:def 4; :: thesis: verum
end;
A66: (Lower_Seq C,n) . 1 = (Lower_Seq C,n) /. 1 by A25, PARTFUN1:def 8
.= E-max (L~ (Cage C,n)) by JORDAN1F:6 ;
A67: [(len (Gauge C,n)),j] in Indices (Gauge C,n) by A5, A12, A20, MATRIX_1:37;
(L~ go) /\ (L~ do) c= {(go /. 1)}
proof
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in (L~ go) /\ (L~ do) or x in {(go /. 1)} )
assume A68: x in (L~ go) /\ (L~ do) ; :: thesis: x in {(go /. 1)}
then A69: x in L~ go by XBOOLE_0:def 4;
A70: x in L~ do by A68, XBOOLE_0:def 4;
then x in (L~ (Upper_Seq C,n)) /\ (L~ (Lower_Seq C,n)) by A46, A53, A69, XBOOLE_0:def 4;
then x in {(W-min (L~ (Cage C,n))),(E-max (L~ (Cage C,n)))} by JORDAN1E:20;
then A71: ( 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: contradiction
then A72: E-max (L~ (Cage C,n)) = (Gauge C,n) * i,j by A9, A66, A70, JORDAN1E:11;
((Gauge C,n) * (len (Gauge C,n)),j) `1 = E-bound (L~ (Cage C,n)) by A5, A12, A16, JORDAN1A:92;
then (E-max (L~ (Cage C,n))) `1 <> E-bound (L~ (Cage C,n)) by A2, A14, A67, A72, JORDAN1G:7;
hence contradiction by EUCLID:56; :: thesis: verum
end;
hence x in {(go /. 1)} by A59, A71, TARSKI:def 1; :: thesis: verum
end;
then A73: (L~ go) /\ (L~ do) = {(go /. 1)} by A63, XBOOLE_0:def 10;
set W2 = go /. 2;
A74: 2 in dom go by A33, FINSEQ_3:27;
A75: now
assume ((Gauge C,n) * i,k) `1 = W-bound (L~ (Cage C,n)) ; :: thesis: contradiction
then ((Gauge C,n) * 1,k) `1 = ((Gauge C,n) * i,k) `1 by A3, A13, A16, JORDAN1A:94;
hence contradiction by A1, A15, A21, JORDAN1G:7; :: thesis: verum
end;
go = mid (Upper_Seq C,n),1,(((Gauge C,n) * i,k) .. (Upper_Seq C,n)) by A34, JORDAN1G:57
.= (Upper_Seq C,n) | (((Gauge C,n) * i,k) .. (Upper_Seq C,n)) by A34, FINSEQ_4:31, FINSEQ_6:122 ;
then A76: go /. 2 = (Upper_Seq C,n) /. 2 by A74, FINSEQ_4:85;
A77: W-min (L~ (Cage C,n)) in rng go by A59, FINSEQ_6:46;
set pion = <*((Gauge C,n) * i,k),((Gauge C,n) * i,j)*>;
A78: now
let n be Element of NAT ; :: thesis: ( n in dom <*((Gauge C,n) * i,k),((Gauge C,n) * i,j)*> implies ex i, j being Element of NAT st
( [i,j] in Indices (Gauge C,n) & <*((Gauge C,n) * i,k),((Gauge C,n) * i,j)*> /. n = (Gauge C,n) * i,j ) )
assume n in dom <*((Gauge C,n) * i,k),((Gauge C,n) * i,j)*> ; :: thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge C,n) & <*((Gauge C,n) * i,k),((Gauge C,n) * i,j)*> /. n = (Gauge C,n) * i,j )
then n in Seg 2 by FINSEQ_3:29;
then ( n = 1 or n = 2 ) by FINSEQ_1:4, TARSKI:def 2;
hence ex i, j being Element of NAT st
( [i,j] in Indices (Gauge C,n) & <*((Gauge C,n) * i,k),((Gauge C,n) * i,j)*> /. n = (Gauge C,n) * i,j ) by A14, A15, FINSEQ_4:26; :: thesis: verum
end;
A79: (Gauge C,n) * i,k <> (Gauge C,n) * i,j by A11, A14, A15, GOBOARD1:21;
A80: ((Gauge C,n) * i,k) `1 = ((Gauge C,n) * i,1) `1 by A1, A2, A3, A13, GOBOARD5:3
.= ((Gauge C,n) * i,j) `1 by A1, A2, A5, A12, GOBOARD5:3 ;
then LSeg ((Gauge C,n) * i,k),((Gauge C,n) * i,j) is vertical by SPPOL_1:37;
then <*((Gauge C,n) * i,k),((Gauge C,n) * i,j)*> is being_S-Seq by A79, JORDAN1B:8;
then consider pion1 being FinSequence of (TOP-REAL 2) such that
A81: pion1 is_sequence_on Gauge C,n and
A82: pion1 is being_S-Seq and
A83: L~ <*((Gauge C,n) * i,k),((Gauge C,n) * i,j)*> = L~ pion1 and
A84: <*((Gauge C,n) * i,k),((Gauge C,n) * i,j)*> /. 1 = pion1 /. 1 and
A85: <*((Gauge C,n) * i,k),((Gauge C,n) * i,j)*> /. (len <*((Gauge C,n) * i,k),((Gauge C,n) * i,j)*>) = pion1 /. (len pion1) and
A86: len <*((Gauge C,n) * i,k),((Gauge C,n) * i,j)*> <= len pion1 by A78, GOBOARD3:2;
reconsider pion1 = pion1 as being_S-Seq FinSequence of (TOP-REAL 2) by A82;
set godo = (go ^' pion1) ^' do;
A87: 1 + 1 <= len (Cage C,n) by GOBOARD7:36, XXREAL_0:2;
A88: 1 + 1 <= len (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) by GOBOARD7:36, XXREAL_0:2;
len (go ^' pion1) >= len go by TOPREAL8:7;
then A89: len (go ^' pion1) >= 1 + 1 by A33, XXREAL_0:2;
then A90: len (go ^' pion1) > 1 + 0 by NAT_1:13;
A91: len ((go ^' pion1) ^' do) >= len (go ^' pion1) by TOPREAL8:7;
then A92: 1 + 1 <= len ((go ^' pion1) ^' do) by A89, XXREAL_0:2;
A93: Upper_Seq C,n is_sequence_on Gauge C,n by JORDAN1G:4;
A94: go /. (len go) = pion1 /. 1 by A41, A84, FINSEQ_4:26;
then A95: go ^' pion1 is_sequence_on Gauge C,n by A35, A81, TOPREAL8:12;
A96: (go ^' pion1) /. (len (go ^' pion1)) = <*((Gauge C,n) * i,k),((Gauge C,n) * i,j)*> /. (len <*((Gauge C,n) * i,k),((Gauge C,n) * i,j)*>) by A85, GRAPH_2:58
.= <*((Gauge C,n) * i,k),((Gauge C,n) * i,j)*> /. 2 by FINSEQ_1:61
.= do /. 1 by A42, FINSEQ_4:26 ;
then A97: (go ^' pion1) ^' do is_sequence_on Gauge C,n by A38, A95, TOPREAL8:12;
LSeg pion1,1 c= L~ <*((Gauge C,n) * i,k),((Gauge C,n) * i,j)*> by A83, TOPREAL3:26;
then LSeg pion1,1 c= LSeg ((Gauge C,n) * i,k),((Gauge C,n) * i,j) by SPPOL_2:21;
then A98: (LSeg go,((len go) -' 1)) /\ (LSeg pion1,1) c= {((Gauge C,n) * i,k)} by A44, A51, XBOOLE_1:27;
A99: len pion1 >= 1 + 1 by A86, FINSEQ_1:61;
{((Gauge C,n) * i,k)} c= (LSeg go,m) /\ (LSeg pion1,1)
proof
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in {((Gauge C,n) * i,k)} or x in (LSeg go,m) /\ (LSeg pion1,1) )
assume x in {((Gauge C,n) * i,k)} ; :: thesis: x in (LSeg go,m) /\ (LSeg pion1,1)
then A100: x = (Gauge C,n) * i,k by TARSKI:def 1;
A101: (Gauge C,n) * i,k in LSeg go,m by A48, RLTOPSP1:69;
(Gauge C,n) * i,k in LSeg pion1,1 by A41, A94, A99, TOPREAL1:27;
hence x in (LSeg go,m) /\ (LSeg pion1,1) by A100, A101, XBOOLE_0:def 4; :: thesis: verum
end;
then (LSeg go,((len go) -' 1)) /\ (LSeg pion1,1) = {(go /. (len go))} by A41, A44, A98, XBOOLE_0:def 10;
then A102: go ^' pion1 is unfolded by A94, TOPREAL8:34;
len pion1 >= 2 + 0 by A86, FINSEQ_1:61;
then A103: (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 A103, XREAL_0:def 2 ;
then A104: (len (go ^' pion1)) -' 1 = (len go) + ((len pion1) -' 2) by XREAL_0:def 2;
A105: (len pion1) - 1 >= 1 by A99, XREAL_1:21;
then A106: (len pion1) -' 1 = (len pion1) - 1 by XREAL_0:def 2;
A107: ((len pion1) -' 2) + 1 = ((len pion1) - 2) + 1 by A103, XREAL_0:def 2
.= (len pion1) -' 1 by A105, XREAL_0:def 2 ;
((len pion1) - 1) + 1 <= len pion1 ;
then A108: (len pion1) -' 1 < len pion1 by A106, NAT_1:13;
LSeg pion1,((len pion1) -' 1) c= L~ <*((Gauge C,n) * i,k),((Gauge C,n) * i,j)*> by A83, TOPREAL3:26;
then LSeg pion1,((len pion1) -' 1) c= LSeg ((Gauge C,n) * i,k),((Gauge C,n) * i,j) by SPPOL_2:21;
then A109: (LSeg pion1,((len pion1) -' 1)) /\ (LSeg do,1) c= {((Gauge C,n) * i,j)} by A58, XBOOLE_1:27;
{((Gauge C,n) * i,j)} c= (LSeg pion1,((len pion1) -' 1)) /\ (LSeg do,1)
proof
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in {((Gauge C,n) * i,j)} or x in (LSeg pion1,((len pion1) -' 1)) /\ (LSeg do,1) )
assume x in {((Gauge C,n) * i,j)} ; :: thesis: x in (LSeg pion1,((len pion1) -' 1)) /\ (LSeg do,1)
then A110: x = (Gauge C,n) * i,j by TARSKI:def 1;
A111: (Gauge C,n) * i,j in LSeg do,1 by A55, RLTOPSP1:69;
pion1 /. (((len pion1) -' 1) + 1) = <*((Gauge C,n) * i,k),((Gauge C,n) * i,j)*> /. 2 by A85, A106, FINSEQ_1:61
.= (Gauge C,n) * i,j by FINSEQ_4:26 ;
then (Gauge C,n) * i,j in LSeg pion1,((len pion1) -' 1) by A105, A106, TOPREAL1:27;
hence x in (LSeg pion1,((len pion1) -' 1)) /\ (LSeg do,1) by A110, A111, XBOOLE_0:def 4; :: thesis: verum
end;
then (LSeg pion1,((len pion1) -' 1)) /\ (LSeg do,1) = {((Gauge C,n) * i,j)} by A109, XBOOLE_0:def 10;
then A112: (LSeg (go ^' pion1),((len go) + ((len pion1) -' 2))) /\ (LSeg do,1) = {((go ^' pion1) /. (len (go ^' pion1)))} by A42, A94, A96, A107, A108, TOPREAL8:31;
A113: not go ^' pion1 is trivial by A89, REALSET1:13;
A114: rng pion1 c= L~ pion1 by A99, SPPOL_2:18;
A115: {(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 A116: x = pion1 /. 1 by TARSKI:def 1;
then A117: x in rng go by A94, REVROT_1:3;
x in rng pion1 by A116, FINSEQ_6:46;
hence x in (L~ go) /\ (L~ pion1) by A61, A114, A117, XBOOLE_0:def 4; :: thesis: verum
end;
(L~ go) /\ (L~ pion1) c= {(pion1 /. 1)}
proof
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in (L~ go) /\ (L~ pion1) or x in {(pion1 /. 1)} )
assume A118: x in (L~ go) /\ (L~ pion1) ; :: thesis: x in {(pion1 /. 1)}
then A119: x in L~ go by XBOOLE_0:def 4;
x in L~ pion1 by A118, XBOOLE_0:def 4;
then x in (L~ pion1) /\ (L~ (Upper_Seq C,n)) by A46, A119, XBOOLE_0:def 4;
hence x in {(pion1 /. 1)} by A6, A41, A83, A94, SPPOL_2:21; :: thesis: verum
end;
then A120: (L~ go) /\ (L~ pion1) = {(pion1 /. 1)} by A115, XBOOLE_0:def 10;
then A121: go ^' pion1 is s.n.c. by A94, JORDAN1J:54;
(rng go) /\ (rng pion1) c= {(pion1 /. 1)} by A61, A114, A120, XBOOLE_1:27;
then A122: go ^' pion1 is one-to-one by JORDAN1J:55;
A123: <*((Gauge C,n) * i,k),((Gauge C,n) * i,j)*> /. (len <*((Gauge C,n) * i,k),((Gauge C,n) * i,j)*>) = <*((Gauge C,n) * i,k),((Gauge C,n) * i,j)*> /. 2 by FINSEQ_1:61
.= do /. 1 by A42, FINSEQ_4:26 ;
A124: {(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 A125: x = pion1 /. (len pion1) by TARSKI:def 1;
then A126: x in rng do by A85, A123, FINSEQ_6:46;
x in rng pion1 by A125, REVROT_1:3;
hence x in (L~ do) /\ (L~ pion1) by A62, A114, A126, 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 A127: x in (L~ do) /\ (L~ pion1) ; :: thesis: x in {(pion1 /. (len pion1))}
then A128: x in L~ do by XBOOLE_0:def 4;
x in L~ pion1 by A127, XBOOLE_0:def 4;
then x in (L~ pion1) /\ (L~ (Lower_Seq C,n)) by A53, A128, XBOOLE_0:def 4;
hence x in {(pion1 /. (len pion1))} by A7, A42, A83, A85, A123, SPPOL_2:21; :: thesis: verum
end;
then A129: (L~ do) /\ (L~ pion1) = {(pion1 /. (len pion1))} by A124, XBOOLE_0:def 10;
A130: (L~ (go ^' pion1)) /\ (L~ do) = ((L~ go) \/ (L~ pion1)) /\ (L~ do) by A94, TOPREAL8:35
.= {(go /. 1)} \/ {(do /. 1)} by A73, A85, A123, A129, 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 A60, GRAPH_2:57;
then reconsider godo = (go ^' pion1) ^' do as non constant standard special_circular_sequence by A92, A96, A97, A102, A104, A112, A113, A121, A122, A130, JORDAN8:7, JORDAN8:8, TOPREAL8:11, TOPREAL8:33, TOPREAL8:34;
A131: Lower_Arc C is_an_arc_of E-max C, W-min C by JORDAN6:def 9;
then A132: Lower_Arc C is connected by JORDAN6:11;
A133: W-min C in Lower_Arc C by A131, TOPREAL1:4;
A134: E-max C in Lower_Arc C by A131, 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 A135: (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) /. 1 = W-min (L~ (Cage C,n)) by FINSEQ_6:98;
A136: 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 A135, 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 A135, A136, 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 A135, A136, SPRECT_5:25, XXREAL_0:2;
then A137: (E-max (L~ (Cage C,n))) .. (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) > 1 by A135, A136, SPRECT_5:26, XXREAL_0:2;
A138: now
assume A139: ((Gauge C,n) * i,k) .. (Upper_Seq C,n) <= 1 ; :: thesis: contradiction
((Gauge C,n) * i,k) .. (Upper_Seq C,n) >= 1 by A34, FINSEQ_4:31;
then ((Gauge C,n) * i,k) .. (Upper_Seq C,n) = 1 by A139, XXREAL_0:1;
then (Gauge C,n) * i,k = (Upper_Seq C,n) /. 1 by A34, FINSEQ_5:41;
hence contradiction by A18, A22, JORDAN1F:5; :: thesis: verum
end;
A140: Cage C,n is_sequence_on Gauge C,n by JORDAN9:def 1;
then A141: Rotate (Cage C,n),(W-min (L~ (Cage C,n))) is_sequence_on Gauge C,n by REVROT_1:34;
A142: (right_cell godo,1,(Gauge C,n)) \ (L~ godo) c= RightComp godo by A92, A97, JORDAN9:29;
A143: L~ godo = (L~ (go ^' pion1)) \/ (L~ do) by A96, TOPREAL8:35
.= ((L~ go) \/ (L~ pion1)) \/ (L~ do) by A94, TOPREAL8:35 ;
A144: L~ (Cage C,n) = (L~ (Upper_Seq C,n)) \/ (L~ (Lower_Seq C,n)) by JORDAN1E:17;
then A145: L~ (Upper_Seq C,n) c= L~ (Cage C,n) by XBOOLE_1:7;
A146: L~ (Lower_Seq C,n) c= L~ (Cage C,n) by A144, XBOOLE_1:7;
A147: L~ go c= L~ (Cage C,n) by A46, A145, XBOOLE_1:1;
A148: L~ do c= L~ (Cage C,n) by A53, A146, XBOOLE_1:1;
A149: W-min C in C by SPRECT_1:15;
A150: L~ <*((Gauge C,n) * i,k),((Gauge C,n) * i,j)*> = LSeg ((Gauge C,n) * i,k),((Gauge C,n) * i,j) by SPPOL_2:21;
A151: now
assume W-min C in L~ godo ; :: thesis: contradiction
then A152: ( W-min C in (L~ go) \/ (L~ pion1) or W-min C in L~ do ) by A143, 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~ do ) by A152, XBOOLE_0:def 3;
end;
end;
right_cell (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))),1 = right_cell (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))),1,(GoB (Rotate (Cage C,n),(W-min (L~ (Cage C,n))))) by A88, 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 A137, A141, 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) * i,k)),1,(Gauge C,n) by A34, A93, A138, JORDAN1J:52
.= right_cell (go ^' pion1),1,(Gauge C,n) by A39, A95, JORDAN1J:51
.= right_cell godo,1,(Gauge C,n) by A90, A97, JORDAN1J:51 ;
then W-min C in right_cell godo,1,(Gauge C,n) by JORDAN1I:8;
then A153: W-min C in (right_cell godo,1,(Gauge C,n)) \ (L~ godo) by A151, XBOOLE_0:def 5;
A154: godo /. 1 = (go ^' pion1) /. 1 by GRAPH_2:57
.= W-min (L~ (Cage C,n)) by A59, GRAPH_2:57 ;
A155: len (Upper_Seq C,n) >= 2 by A17, XXREAL_0:2;
A156: godo /. 2 = (go ^' pion1) /. 2 by A89, GRAPH_2:61
.= (Upper_Seq C,n) /. 2 by A33, A76, GRAPH_2:61
.= ((Upper_Seq C,n) ^' (Lower_Seq C,n)) /. 2 by A155, GRAPH_2:61
.= (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) /. 2 by JORDAN1E:15 ;
A157: (L~ go) \/ (L~ do) is compact by COMPTS_1:19;
W-min (L~ (Cage C,n)) in (L~ go) \/ (L~ do) by A61, A77, XBOOLE_0:def 3;
then A158: W-min ((L~ go) \/ (L~ do)) = W-min (L~ (Cage C,n)) by A147, A148, A157, JORDAN1J:21, XBOOLE_1:8;
A159: (W-min ((L~ go) \/ (L~ do))) `1 = W-bound ((L~ go) \/ (L~ do)) by EUCLID:56;
A160: (W-min (L~ (Cage C,n))) `1 = W-bound (L~ (Cage C,n)) by EUCLID:56;
W-bound (LSeg ((Gauge C,n) * i,k),((Gauge C,n) * i,j)) = ((Gauge C,n) * i,k) `1 by A80, SPRECT_1:62;
then A161: W-bound (L~ pion1) = ((Gauge C,n) * i,k) `1 by A83, SPPOL_2:21;
((Gauge C,n) * i,k) `1 >= W-bound (L~ (Cage C,n)) by A10, A145, PSCOMP_1:71;
then ((Gauge C,n) * i,k) `1 > W-bound (L~ (Cage C,n)) by A75, XXREAL_0:1;
then W-min (((L~ go) \/ (L~ do)) \/ (L~ pion1)) = W-min ((L~ go) \/ (L~ do)) by A157, A158, A159, A160, A161, JORDAN1J:33;
then A162: W-min (L~ godo) = W-min (L~ (Cage C,n)) by A143, A158, XBOOLE_1:4;
A163: rng godo c= L~ godo by A89, A91, SPPOL_2:18, XXREAL_0:2;
2 in dom godo by A92, FINSEQ_3:27;
then A164: godo /. 2 in rng godo by PARTFUN2:4;
godo /. 2 in W-most (L~ (Cage C,n)) by A156, JORDAN1I:27;
then (godo /. 2) `1 = (W-min (L~ godo)) `1 by A162, PSCOMP_1:88
.= W-bound (L~ godo) by EUCLID:56 ;
then godo /. 2 in W-most (L~ godo) by A163, A164, SPRECT_2:16;
then (Rotate godo,(W-min (L~ godo))) /. 2 in W-most (L~ godo) by A154, A162, 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 A165: (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 ;
A166: 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
A167: p in east_halfline (E-max C) and
A168: p in L~ go by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A167;
p in L~ (Upper_Seq C,n) by A46, A168;
then p in (east_halfline (E-max C)) /\ (L~ (Cage C,n)) by A145, A167, XBOOLE_0:def 4;
then A169: p `1 = E-bound (L~ (Cage C,n)) by JORDAN1A:104, PSCOMP_1:111;
then A170: p = E-max (L~ (Cage C,n)) by A46, A168, JORDAN1J:46;
then E-max (L~ (Cage C,n)) = (Gauge C,n) * i,k by A10, A165, A168, JORDAN1J:43;
then ((Gauge C,n) * i,k) `1 = ((Gauge C,n) * (len (Gauge C,n)),k) `1 by A3, A13, A16, A169, A170, JORDAN1A:92;
hence contradiction by A2, A15, A30, JORDAN1G:7; :: thesis: verum
end;
now
assume east_halfline (E-max C) meets L~ godo ; :: thesis: contradiction
then A171: ( east_halfline (E-max C) meets (L~ go) \/ (L~ pion1) or east_halfline (E-max C) meets L~ do ) by A143, 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 A171, XBOOLE_1:70;
suppose east_halfline (E-max C) meets L~ pion1 ; :: thesis: contradiction
then consider p being set such that
A172: p in east_halfline (E-max C) and
A173: p in L~ pion1 by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A172;
A174: p `1 = ((Gauge C,n) * i,k) `1 by A80, A83, A150, A173, GOBOARD7:5;
i + 1 <= len (Gauge C,n) by A2, NAT_1:13;
then (i + 1) - 1 <= (len (Gauge C,n)) - 1 by XREAL_1:11;
then A175: i <= (len (Gauge C,n)) -' 1 by XREAL_0:def 2;
(len (Gauge C,n)) -' 1 <= len (Gauge C,n) by NAT_D:35;
then p `1 <= ((Gauge C,n) * ((len (Gauge C,n)) -' 1),1) `1 by A1, A3, A13, A16, A20, A174, A175, JORDAN1A:39;
then p `1 <= E-bound C by A20, JORDAN8:15;
then A176: p `1 <= (E-max C) `1 by EUCLID:56;
p `1 >= (E-max C) `1 by A172, TOPREAL1:def 13;
then A177: p `1 = (E-max C) `1 by A176, XXREAL_0:1;
p `2 = (E-max C) `2 by A172, TOPREAL1:def 13;
then p = E-max C by A177, TOPREAL3:11;
hence contradiction by A8, A83, A134, A150, A173, XBOOLE_0:3; :: thesis: verum
end;
suppose east_halfline (E-max C) meets L~ do ; :: thesis: contradiction
then consider p being set such that
A178: p in east_halfline (E-max C) and
A179: p in L~ do by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A178;
p in L~ (Lower_Seq C,n) by A53, A179;
then p in (east_halfline (E-max C)) /\ (L~ (Cage C,n)) by A146, A178, XBOOLE_0:def 4;
then A180: p `1 = E-bound (L~ (Cage C,n)) by JORDAN1A:104, PSCOMP_1:111;
A181: (E-max C) `2 = p `2 by A178, TOPREAL1:def 13;
set RC = Rotate (Cage C,n),(E-max (L~ (Cage C,n)));
A182: E-max C in right_cell (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))),1 by JORDAN1I:9;
A183: 1 + 1 <= len (Lower_Seq C,n) by A23, 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 A184: LSeg (Lower_Seq C,n),1 = LSeg (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))),1 by A183, SPPOL_2:9;
A185: L~ (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) = L~ (Cage C,n) by REVROT_1:33;
A186: len (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) = len (Cage C,n) by REVROT_1:14;
A187: 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 ;
A188: E-max (L~ (Cage C,n)) in rng (Cage C,n) by SPRECT_2:50;
A189: Rotate (Cage C,n),(E-max (L~ (Cage C,n))) is_sequence_on Gauge C,n by A140, REVROT_1:34;
A190: (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 A185, A188, FINSEQ_6:98;
consider ii, jj being Element of NAT such that
A191: [ii,(jj + 1)] in Indices (Gauge C,n) and
A192: [ii,jj] in Indices (Gauge C,n) and
A193: (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) /. 1 = (Gauge C,n) * ii,(jj + 1) and
A194: (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) /. (1 + 1) = (Gauge C,n) * ii,jj by A87, A185, A186, A188, A189, FINSEQ_6:98, JORDAN1I:25;
consider jj2 being Element of NAT such that
A195: 1 <= jj2 and
A196: jj2 <= width (Gauge C,n) and
A197: E-max (L~ (Cage C,n)) = (Gauge C,n) * (len (Gauge C,n)),jj2 by JORDAN1D:29;
A198: len (Gauge C,n) >= 4 by JORDAN8:13;
then len (Gauge C,n) >= 1 by XXREAL_0:2;
then [(len (Gauge C,n)),jj2] in Indices (Gauge C,n) by A195, A196, MATRIX_1:37;
then A199: ii = len (Gauge C,n) by A185, A190, A191, A193, A197, GOBOARD1:21;
A200: 1 <= ii by A191, MATRIX_1:39;
A201: ii <= len (Gauge C,n) by A191, MATRIX_1:39;
A202: 1 <= jj + 1 by A191, MATRIX_1:39;
A203: jj + 1 <= width (Gauge C,n) by A191, MATRIX_1:39;
A204: 1 <= ii by A192, MATRIX_1:39;
A205: ii <= len (Gauge C,n) by A192, MATRIX_1:39;
A206: 1 <= jj by A192, MATRIX_1:39;
A207: jj <= width (Gauge C,n) by A192, MATRIX_1:39;
A208: ii + 1 <> ii ;
(jj + 1) + 1 <> jj ;
then A209: right_cell (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))),1 = cell (Gauge C,n),(ii -' 1),jj by A87, A186, A187, A191, A192, A193, A194, A208, GOBOARD5:def 6;
A210: (ii -' 1) + 1 = ii by A200, XREAL_1:237;
ii - 1 >= 4 - 1 by A198, A199, XREAL_1:11;
then A211: ii - 1 >= 1 by XXREAL_0:2;
then A212: 1 <= ii -' 1 by XREAL_0:def 2;
A213: ((Gauge C,n) * (ii -' 1),jj) `2 <= p `2 by A181, A182, A201, A203, A206, A209, A210, A211, JORDAN9:19;
A214: p `2 <= ((Gauge C,n) * (ii -' 1),(jj + 1)) `2 by A181, A182, A201, A203, A206, A209, A210, A211, JORDAN9:19;
A215: ii -' 1 < len (Gauge C,n) by A201, A210, NAT_1:13;
then A216: ((Gauge C,n) * (ii -' 1),jj) `2 = ((Gauge C,n) * 1,jj) `2 by A206, A207, A212, GOBOARD5:2
.= ((Gauge C,n) * ii,jj) `2 by A204, A205, A206, A207, GOBOARD5:2 ;
A217: ((Gauge C,n) * (ii -' 1),(jj + 1)) `2 = ((Gauge C,n) * 1,(jj + 1)) `2 by A202, A203, A212, A215, GOBOARD5:2
.= ((Gauge C,n) * ii,(jj + 1)) `2 by A200, A201, A202, A203, GOBOARD5:2 ;
A218: ((Gauge C,n) * (len (Gauge C,n)),jj) `1 = E-bound (L~ (Cage C,n)) by A16, A206, A207, JORDAN1A:92;
E-bound (L~ (Cage C,n)) = ((Gauge C,n) * (len (Gauge C,n)),(jj + 1)) `1 by A16, A202, A203, 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 A180, A193, A194, A199, A213, A214, A216, A217, A218, GOBOARD7:8;
then A219: p in LSeg (Lower_Seq C,n),1 by A87, A184, A186, TOPREAL1:def 5;
A220: p in LSeg do,(Index p,do) by A179, JORDAN3:42;
A221: do = mid (Lower_Seq C,n),(((Gauge C,n) * i,j) .. (Lower_Seq C,n)),(len (Lower_Seq C,n)) by A37, JORDAN1J:37;
A222: 1 <= ((Gauge C,n) * i,j) .. (Lower_Seq C,n) by A37, FINSEQ_4:31;
A223: ((Gauge C,n) * i,j) .. (Lower_Seq C,n) <= len (Lower_Seq C,n) by A37, FINSEQ_4:31;
((Gauge C,n) * i,j) .. (Lower_Seq C,n) <> len (Lower_Seq C,n) by A29, A37, FINSEQ_4:29;
then A224: ((Gauge C,n) * i,j) .. (Lower_Seq C,n) < len (Lower_Seq C,n) by A223, XXREAL_0:1;
A225: 1 <= Index p,do by A179, JORDAN3:41;
A226: Index p,do < len do by A179, JORDAN3:41;
A227: (Index ((Gauge C,n) * i,j),(Lower_Seq C,n)) + 1 = ((Gauge C,n) * i,j) .. (Lower_Seq C,n) by A32, A37, JORDAN1J:56;
consider t being Nat such that
A228: t in dom (Lower_Seq C,n) and
A229: (Lower_Seq C,n) . t = (Gauge C,n) * i,j by A37, FINSEQ_2:11;
A230: 1 <= t by A228, FINSEQ_3:27;
A231: t <= len (Lower_Seq C,n) by A228, FINSEQ_3:27;
1 < t by A32, A229, A230, XXREAL_0:1;
then (Index ((Gauge C,n) * i,j),(Lower_Seq C,n)) + 1 = t by A229, A231, JORDAN3:45;
then A232: len (L_Cut (Lower_Seq C,n),((Gauge C,n) * i,j)) = (len (Lower_Seq C,n)) - (Index ((Gauge C,n) * i,j),(Lower_Seq C,n)) by A9, A229, JORDAN3:61;
set tt = ((Index p,do) + (((Gauge C,n) * i,j) .. (Lower_Seq C,n))) -' 1;
A233: 1 <= Index ((Gauge C,n) * i,j),(Lower_Seq C,n) by A9, JORDAN3:41;
0 + (Index ((Gauge C,n) * i,j),(Lower_Seq C,n)) < len (Lower_Seq C,n) by A9, JORDAN3:41;
then A234: (len (Lower_Seq C,n)) - (Index ((Gauge C,n) * i,j),(Lower_Seq C,n)) > 0 by XREAL_1:22;
Index p,do < (len (Lower_Seq C,n)) -' (Index ((Gauge C,n) * i,j),(Lower_Seq C,n)) by A226, A232, XREAL_0:def 2;
then (Index p,do) + 1 <= (len (Lower_Seq C,n)) -' (Index ((Gauge C,n) * i,j),(Lower_Seq C,n)) by NAT_1:13;
then Index p,do <= ((len (Lower_Seq C,n)) -' (Index ((Gauge C,n) * i,j),(Lower_Seq C,n))) - 1 by XREAL_1:21;
then Index p,do <= ((len (Lower_Seq C,n)) - (Index ((Gauge C,n) * i,j),(Lower_Seq C,n))) - 1 by A234, XREAL_0:def 2;
then Index p,do <= (len (Lower_Seq C,n)) - (((Gauge C,n) * i,j) .. (Lower_Seq C,n)) by A227;
then Index p,do <= (len (Lower_Seq C,n)) -' (((Gauge C,n) * i,j) .. (Lower_Seq C,n)) by XREAL_0:def 2;
then Index p,do < ((len (Lower_Seq C,n)) -' (((Gauge C,n) * i,j) .. (Lower_Seq C,n))) + 1 by NAT_1:13;
then A235: LSeg (mid (Lower_Seq C,n),(((Gauge C,n) * i,j) .. (Lower_Seq C,n)),(len (Lower_Seq C,n))),(Index p,do) = LSeg (Lower_Seq C,n),(((Index p,do) + (((Gauge C,n) * i,j) .. (Lower_Seq C,n))) -' 1) by A222, A224, A225, JORDAN4:31;
A236: 1 + 1 <= ((Gauge C,n) * i,j) .. (Lower_Seq C,n) by A227, A233, XREAL_1:9;
then (Index p,do) + (((Gauge C,n) * i,j) .. (Lower_Seq C,n)) >= (1 + 1) + 1 by A225, XREAL_1:9;
then ((Index p,do) + (((Gauge C,n) * i,j) .. (Lower_Seq C,n))) - 1 >= ((1 + 1) + 1) - 1 by XREAL_1:11;
then A237: ((Index p,do) + (((Gauge C,n) * i,j) .. (Lower_Seq C,n))) -' 1 >= 1 + 1 by XREAL_0:def 2;
A238: 2 in dom (Lower_Seq C,n) by A183, FINSEQ_3:27;
now
per cases ( ((Index p,do) + (((Gauge C,n) * i,j) .. (Lower_Seq C,n))) -' 1 > 1 + 1 or ((Index p,do) + (((Gauge C,n) * i,j) .. (Lower_Seq C,n))) -' 1 = 1 + 1 ) by A237, XXREAL_0:1;
suppose ((Index p,do) + (((Gauge C,n) * i,j) .. (Lower_Seq C,n))) -' 1 > 1 + 1 ; :: thesis: contradiction
then LSeg (Lower_Seq C,n),1 misses LSeg (Lower_Seq C,n),(((Index p,do) + (((Gauge C,n) * i,j) .. (Lower_Seq C,n))) -' 1) by TOPREAL1:def 9;
hence contradiction by A219, A220, A221, A235, XBOOLE_0:3; :: thesis: verum
end;
suppose A239: ((Index p,do) + (((Gauge C,n) * i,j) .. (Lower_Seq C,n))) -' 1 = 1 + 1 ; :: thesis: contradiction
then (LSeg (Lower_Seq C,n),1) /\ (LSeg (Lower_Seq C,n),(((Index p,do) + (((Gauge C,n) * i,j) .. (Lower_Seq C,n))) -' 1)) = {((Lower_Seq C,n) /. 2)} by A23, TOPREAL1:def 8;
then p in {((Lower_Seq C,n) /. 2)} by A219, A220, A221, A235, XBOOLE_0:def 4;
then A240: p = (Lower_Seq C,n) /. 2 by TARSKI:def 1;
then A241: p .. (Lower_Seq C,n) = 2 by A238, FINSEQ_5:44;
1 + 1 = ((Index p,do) + (((Gauge C,n) * i,j) .. (Lower_Seq C,n))) - 1 by A239, XREAL_0:def 2;
then (1 + 1) + 1 = (Index p,do) + (((Gauge C,n) * i,j) .. (Lower_Seq C,n)) ;
then A242: ((Gauge C,n) * i,j) .. (Lower_Seq C,n) = 2 by A225, A236, JORDAN1E:10;
p in rng (Lower_Seq C,n) by A238, A240, PARTFUN2:4;
then p = (Gauge C,n) * i,j by A37, A241, A242, FINSEQ_5:10;
then ((Gauge C,n) * i,j) `1 = E-bound (L~ (Cage C,n)) by A240, JORDAN1G:40;
then ((Gauge C,n) * i,j) `1 = ((Gauge C,n) * (len (Gauge C,n)),j) `1 by A5, A12, A16, JORDAN1A:92;
hence contradiction by A2, A14, A67, JORDAN1G:7; :: thesis: verum
end;
end;
end;
hence contradiction ; :: thesis: verum
end;
end;
end;
then east_halfline (E-max C) c= (L~ godo) ` by SUBSET_1:43;
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 A132, A133, A134, A142, A153, JORDAN1J:36;
then A246: ( Lower_Arc C meets (L~ go) \/ (L~ pion1) or Lower_Arc C meets L~ do ) by A143, XBOOLE_1:70;
A247: Lower_Arc C c= C by JORDAN6:76;
per cases ( Lower_Arc C meets L~ go or Lower_Arc C meets L~ pion1 or Lower_Arc C meets L~ do ) by A246, XBOOLE_1:70;
suppose Lower_Arc C meets L~ go ; :: thesis: contradiction
then Lower_Arc C meets L~ (Cage C,n) by A46, A145, XBOOLE_1:1, XBOOLE_1:63;
hence contradiction by A247, JORDAN10:5, XBOOLE_1:63; :: thesis: verum
end;
suppose Lower_Arc C meets L~ pion1 ; :: thesis: contradiction
hence contradiction by A8, A83, A150; :: thesis: verum
end;
suppose Lower_Arc C meets L~ do ; :: thesis: contradiction
then Lower_Arc C meets L~ (Cage C,n) by A53, A146, XBOOLE_1:1, XBOOLE_1:63;
hence contradiction by A247, JORDAN10:5, XBOOLE_1:63; :: thesis: verum
end;
end;