let n be Element of NAT ; :: thesis: for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < j & j <= k & k < len (Gauge C,n) & 1 <= i & i <= width (Gauge C,n) & (LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Upper_Seq C,n)) = {((Gauge C,n) * k,i)} & (LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n)) = {((Gauge C,n) * j,i)} holds
LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i) meets Upper_Arc C
let C be Simple_closed_curve; :: thesis: for i, j, k being Element of NAT st 1 < j & j <= k & k < len (Gauge C,n) & 1 <= i & i <= width (Gauge C,n) & (LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Upper_Seq C,n)) = {((Gauge C,n) * k,i)} & (LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n)) = {((Gauge C,n) * j,i)} holds
LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i) meets Upper_Arc C
let i, j, k be Element of NAT ; :: thesis: ( 1 < j & j <= k & k < len (Gauge C,n) & 1 <= i & i <= width (Gauge C,n) & (LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Upper_Seq C,n)) = {((Gauge C,n) * k,i)} & (LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n)) = {((Gauge C,n) * j,i)} implies LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i) meets Upper_Arc C )
set Ga = Gauge C,n;
set US = Upper_Seq C,n;
set LS = Lower_Seq C,n;
set UA = Upper_Arc C;
set Wmin = W-min (L~ (Cage C,n));
set Emax = E-max (L~ (Cage C,n));
set Wbo = W-bound (L~ (Cage C,n));
set Ebo = E-bound (L~ (Cage C,n));
set Gij = (Gauge C,n) * j,i;
set Gik = (Gauge C,n) * k,i;
assume that
A1: 1 < j and
A2: j <= k and
A3: k < len (Gauge C,n) and
A4: 1 <= i and
A5: i <= width (Gauge C,n) and
A6: (LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Upper_Seq C,n)) = {((Gauge C,n) * k,i)} and
A7: (LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n)) = {((Gauge C,n) * j,i)} and
A8: LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i) misses Upper_Arc C ; :: thesis: contradiction
(Gauge C,n) * j,i in {((Gauge C,n) * j,i)} by TARSKI:def 1;
then A9: (Gauge C,n) * j,i in L~ (Lower_Seq C,n) by A7, XBOOLE_0:def 4;
(Gauge C,n) * k,i in {((Gauge C,n) * k,i)} by TARSKI:def 1;
then A10: (Gauge C,n) * k,i in L~ (Upper_Seq C,n) by A6, XBOOLE_0:def 4;
A11: len (Gauge C,n) = width (Gauge C,n) by JORDAN8:def 1;
A12: j <> k by A1, A3, A4, A5, A9, A10, Th29;
A13: j <= width (Gauge C,n) by A2, A3, A11, XXREAL_0:2;
A14: 1 <= k by A1, A2, XXREAL_0:2;
A15: k <= width (Gauge C,n) by A3, JORDAN8:def 1;
A16: [j,i] in Indices (Gauge C,n) by A1, A4, A5, A11, A13, MATRIX_1:37;
A17: [k,i] in Indices (Gauge C,n) by A3, A4, A5, A14, MATRIX_1:37;
set go = R_Cut (Upper_Seq C,n),((Gauge C,n) * k,i);
set do = L_Cut (Lower_Seq C,n),((Gauge C,n) * j,i);
A18: 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 A19: (Upper_Seq C,n) . 1 = (Upper_Seq C,n) /. 1 by PARTFUN1:def 8
.= W-min (L~ (Cage C,n)) by JORDAN1F:5 ;
A20: (W-min (L~ (Cage C,n))) `1 = W-bound (L~ (Cage C,n)) by EUCLID:56
.= ((Gauge C,n) * 1,k) `1 by A3, A14, JORDAN1A:94 ;
len (Gauge C,n) >= 4 by JORDAN8:13;
then A21: len (Gauge C,n) >= 1 by XXREAL_0:2;
then A22: [1,k] in Indices (Gauge C,n) by A14, A15, MATRIX_1:37;
then A23: (Gauge C,n) * k,i <> (Upper_Seq C,n) . 1 by A1, A2, A17, A19, A20, JORDAN1G:7;
then reconsider go = R_Cut (Upper_Seq C,n),((Gauge C,n) * k,i) as being_S-Seq FinSequence of (TOP-REAL 2) by A10, JORDAN3:70;
A24: [1,j] in Indices (Gauge C,n) by A1, A13, A21, MATRIX_1:37;
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;
len (Lower_Seq C,n) in dom (Lower_Seq C,n) by A26, FINSEQ_3:27;
then A28: (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 ;
A29: (W-min (L~ (Cage C,n))) `1 = W-bound (L~ (Cage C,n)) by EUCLID:56
.= ((Gauge C,n) * 1,k) `1 by A3, A14, JORDAN1A:94 ;
A30: [j,i] in Indices (Gauge C,n) by A1, A4, A5, A11, A13, MATRIX_1:37;
then A31: (Gauge C,n) * j,i <> (Lower_Seq C,n) . (len (Lower_Seq C,n)) by A1, A22, A28, A29, JORDAN1G:7;
then reconsider do = L_Cut (Lower_Seq C,n),((Gauge C,n) * j,i) as being_S-Seq FinSequence of (TOP-REAL 2) by A9, JORDAN3:69;
A32: [(len (Gauge C,n)),k] in Indices (Gauge C,n) by A14, A15, A21, MATRIX_1:37;
A33: (Lower_Seq C,n) . 1 = (Lower_Seq C,n) /. 1 by A27, 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, A14, JORDAN1A:92 ;
then A34: (Gauge C,n) * j,i <> (Lower_Seq C,n) . 1 by A2, A3, A30, A32, A33, JORDAN1G:7;
A35: len go >= 1 + 1 by TOPREAL1:def 10;
A36: (Gauge C,n) * k,i in rng (Upper_Seq C,n) by A4, A5, A10, A11, A14, A15, JORDAN1G:4, JORDAN1J:40;
then A37: go is_sequence_on Gauge C,n by JORDAN1G:4, JORDAN1J:38;
A38: len do >= 1 + 1 by TOPREAL1:def 10;
A39: (Gauge C,n) * j,i in rng (Lower_Seq C,n) by A1, A4, A5, A9, A11, A13, JORDAN1G:5, JORDAN1J:40;
then A40: 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 A35, A37, JGRAPH_1:16, JORDAN8:8;
reconsider do = do as non constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A38, A40, JGRAPH_1:16, JORDAN8:8;
A41: len go > 1 by A35, NAT_1:13;
then A42: len go in dom go by FINSEQ_3:27;
then A43: go /. (len go) = go . (len go) by PARTFUN1:def 8
.= (Gauge C,n) * k,i by A10, JORDAN3:59 ;
len do >= 1 by A38, XXREAL_0:2;
then 1 in dom do by FINSEQ_3:27;
then A44: do /. 1 = do . 1 by PARTFUN1:def 8
.= (Gauge C,n) * j,i by A9, JORDAN3:58 ;
reconsider m = (len go) - 1 as Element of NAT by A42, FINSEQ_3:28;
A45: m + 1 = len go ;
then A46: (len go) -' 1 = m by NAT_D:34;
A47: LSeg go,m c= L~ go by TOPREAL3:26;
A48: L~ go c= L~ (Upper_Seq C,n) by A10, JORDAN3:76;
then LSeg go,m c= L~ (Upper_Seq C,n) by A47, XBOOLE_1:1;
then A49: (LSeg go,m) /\ (LSeg ((Gauge C,n) * k,i),((Gauge C,n) * j,i)) c= {((Gauge C,n) * k,i)} by A6, XBOOLE_1:26;
m >= 1 by A35, XREAL_1:21;
then A50: LSeg go,m = LSeg (go /. m),((Gauge C,n) * k,i) by A43, A45, TOPREAL1:def 5;
{((Gauge C,n) * k,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: ( not x in {((Gauge C,n) * k,i)} or x in (LSeg go,m) /\ (LSeg ((Gauge C,n) * k,i),((Gauge C,n) * j,i)) )
A51: (Gauge C,n) * k,i in LSeg ((Gauge C,n) * k,i),((Gauge C,n) * j,i) by RLTOPSP1:69;
assume x in {((Gauge C,n) * k,i)} ; :: thesis: x in (LSeg go,m) /\ (LSeg ((Gauge C,n) * k,i),((Gauge C,n) * j,i))
then A52: x = (Gauge C,n) * k,i by TARSKI:def 1;
(Gauge C,n) * k,i in LSeg go,m by A50, RLTOPSP1:69;
hence x in (LSeg go,m) /\ (LSeg ((Gauge C,n) * k,i),((Gauge C,n) * j,i)) by A52, A51, XBOOLE_0:def 4; :: thesis: verum
end;
then A53: (LSeg go,m) /\ (LSeg ((Gauge C,n) * k,i),((Gauge C,n) * j,i)) = {((Gauge C,n) * k,i)} by A49, XBOOLE_0:def 10;
A54: LSeg do,1 c= L~ do by TOPREAL3:26;
A55: L~ do c= L~ (Lower_Seq C,n) by A9, JORDAN3:77;
then LSeg do,1 c= L~ (Lower_Seq C,n) by A54, XBOOLE_1:1;
then A56: (LSeg do,1) /\ (LSeg ((Gauge C,n) * k,i),((Gauge C,n) * j,i)) c= {((Gauge C,n) * j,i)} by A7, XBOOLE_1:26;
A57: LSeg do,1 = LSeg ((Gauge C,n) * j,i),(do /. (1 + 1)) by A38, A44, TOPREAL1:def 5;
{((Gauge C,n) * j,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: ( not x in {((Gauge C,n) * j,i)} or x in (LSeg do,1) /\ (LSeg ((Gauge C,n) * k,i),((Gauge C,n) * j,i)) )
A58: (Gauge C,n) * j,i in LSeg ((Gauge C,n) * k,i),((Gauge C,n) * j,i) by RLTOPSP1:69;
assume x in {((Gauge C,n) * j,i)} ; :: thesis: x in (LSeg do,1) /\ (LSeg ((Gauge C,n) * k,i),((Gauge C,n) * j,i))
then A59: x = (Gauge C,n) * j,i by TARSKI:def 1;
(Gauge C,n) * j,i in LSeg do,1 by A57, RLTOPSP1:69;
hence x in (LSeg do,1) /\ (LSeg ((Gauge C,n) * k,i),((Gauge C,n) * j,i)) by A59, A58, XBOOLE_0:def 4; :: thesis: verum
end;
then A60: (LSeg ((Gauge C,n) * k,i),((Gauge C,n) * j,i)) /\ (LSeg do,1) = {((Gauge C,n) * j,i)} by A56, XBOOLE_0:def 10;
A61: go /. 1 = (Upper_Seq C,n) /. 1 by A10, SPRECT_3:39
.= W-min (L~ (Cage C,n)) by JORDAN1F:5 ;
then A62: go /. 1 = (Lower_Seq C,n) /. (len (Lower_Seq C,n)) by JORDAN1F:8
.= do /. (len do) by A9, JORDAN1J:35 ;
A63: rng go c= L~ go by A35, SPPOL_2:18;
A64: rng do c= L~ do by A38, SPPOL_2:18;
A65: {(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 A66: x = go /. 1 by TARSKI:def 1;
then A67: x in rng go by FINSEQ_6:46;
x in rng do by A62, A66, REVROT_1:3;
hence x in (L~ go) /\ (L~ do) by A63, A64, A67, XBOOLE_0:def 4; :: thesis: verum
end;
A68: (Lower_Seq C,n) . 1 = (Lower_Seq C,n) /. 1 by A27, PARTFUN1:def 8
.= E-max (L~ (Cage C,n)) by JORDAN1F:6 ;
A69: [(len (Gauge C,n)),j] in Indices (Gauge C,n) by A1, A13, A21, 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 A70: x in (L~ go) /\ (L~ do) ; :: thesis: x in {(go /. 1)}
then A71: x in L~ do by XBOOLE_0:def 4;
A72: now
assume x = E-max (L~ (Cage C,n)) ; :: thesis: contradiction
then A73: E-max (L~ (Cage C,n)) = (Gauge C,n) * j,i by A9, A68, A71, JORDAN1E:11;
((Gauge C,n) * (len (Gauge C,n)),j) `1 = E-bound (L~ (Cage C,n)) by A1, A11, A13, JORDAN1A:92;
then (E-max (L~ (Cage C,n))) `1 <> E-bound (L~ (Cage C,n)) by A2, A3, A16, A69, A73, JORDAN1G:7;
hence contradiction by EUCLID:56; :: thesis: verum
end;
x in L~ go by A70, XBOOLE_0:def 4;
then x in (L~ (Upper_Seq C,n)) /\ (L~ (Lower_Seq C,n)) by A48, A55, A71, 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 A61, A72, TARSKI:def 1; :: thesis: verum
end;
then A74: (L~ go) /\ (L~ do) = {(go /. 1)} by A65, XBOOLE_0:def 10;
set W2 = go /. 2;
A75: 2 in dom go by A35, FINSEQ_3:27;
A76: now
assume ((Gauge C,n) * j,i) `1 = W-bound (L~ (Cage C,n)) ; :: thesis: contradiction
then ((Gauge C,n) * 1,j) `1 = ((Gauge C,n) * j,i) `1 by A1, A11, A13, JORDAN1A:94;
hence contradiction by A1, A16, A24, JORDAN1G:7; :: thesis: verum
end;
go = mid (Upper_Seq C,n),1,(((Gauge C,n) * k,i) .. (Upper_Seq C,n)) by A36, JORDAN1G:57
.= (Upper_Seq C,n) | (((Gauge C,n) * k,i) .. (Upper_Seq C,n)) by A36, FINSEQ_4:31, FINSEQ_6:122 ;
then A77: go /. 2 = (Upper_Seq C,n) /. 2 by A75, FINSEQ_4:85;
A78: W-min (L~ (Cage C,n)) in rng go by A61, FINSEQ_6:46;
set pion = <*((Gauge C,n) * k,i),((Gauge C,n) * j,i)*>;
A79: now
let n be Element of NAT ; :: thesis: ( n in dom <*((Gauge C,n) * k,i),((Gauge C,n) * j,i)*> implies ex j, i being Element of NAT st
( [j,i] in Indices (Gauge C,n) & <*((Gauge C,n) * k,i),((Gauge C,n) * j,i)*> /. n = (Gauge C,n) * j,i ) )
assume n in dom <*((Gauge C,n) * k,i),((Gauge C,n) * j,i)*> ; :: thesis: ex j, i being Element of NAT st
( [j,i] in Indices (Gauge C,n) & <*((Gauge C,n) * k,i),((Gauge C,n) * j,i)*> /. n = (Gauge C,n) * j,i )
then n in {1,2} by FINSEQ_1:4, FINSEQ_3:29;
then ( n = 1 or n = 2 ) by TARSKI:def 2;
hence ex j, i being Element of NAT st
( [j,i] in Indices (Gauge C,n) & <*((Gauge C,n) * k,i),((Gauge C,n) * j,i)*> /. n = (Gauge C,n) * j,i ) by A16, A17, FINSEQ_4:26; :: thesis: verum
end;
A80: (Gauge C,n) * k,i <> (Gauge C,n) * j,i by A12, A16, A17, GOBOARD1:21;
((Gauge C,n) * k,i) `2 = ((Gauge C,n) * 1,i) `2 by A3, A4, A5, A14, GOBOARD5:2
.= ((Gauge C,n) * j,i) `2 by A1, A4, A5, A11, A13, GOBOARD5:2 ;
then LSeg ((Gauge C,n) * k,i),((Gauge C,n) * j,i) is horizontal by SPPOL_1:36;
then <*((Gauge C,n) * k,i),((Gauge C,n) * j,i)*> is being_S-Seq by A80, JORDAN1B:9;
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) * k,i),((Gauge C,n) * j,i)*> = L~ pion1 and
A84: <*((Gauge C,n) * k,i),((Gauge C,n) * j,i)*> /. 1 = pion1 /. 1 and
A85: <*((Gauge C,n) * k,i),((Gauge C,n) * j,i)*> /. (len <*((Gauge C,n) * k,i),((Gauge C,n) * j,i)*>) = pion1 /. (len pion1) and
A86: len <*((Gauge C,n) * k,i),((Gauge C,n) * j,i)*> <= len pion1 by A79, 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 A35, 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 A43, A84, FINSEQ_4:26;
then A95: go ^' pion1 is_sequence_on Gauge C,n by A37, A81, TOPREAL8:12;
A96: (go ^' pion1) /. (len (go ^' pion1)) = <*((Gauge C,n) * k,i),((Gauge C,n) * j,i)*> /. (len <*((Gauge C,n) * k,i),((Gauge C,n) * j,i)*>) by A85, GRAPH_2:58
.= <*((Gauge C,n) * k,i),((Gauge C,n) * j,i)*> /. 2 by FINSEQ_1:61
.= do /. 1 by A44, FINSEQ_4:26 ;
then A97: (go ^' pion1) ^' do is_sequence_on Gauge C,n by A40, A95, TOPREAL8:12;
LSeg pion1,1 c= L~ <*((Gauge C,n) * k,i),((Gauge C,n) * j,i)*> by A83, TOPREAL3:26;
then LSeg pion1,1 c= LSeg ((Gauge C,n) * k,i),((Gauge C,n) * j,i) by SPPOL_2:21;
then A98: (LSeg go,((len go) -' 1)) /\ (LSeg pion1,1) c= {((Gauge C,n) * k,i)} by A46, A53, XBOOLE_1:27;
A99: len pion1 >= 1 + 1 by A86, FINSEQ_1:61;
{((Gauge C,n) * k,i)} c= (LSeg go,m) /\ (LSeg pion1,1)
proof
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in {((Gauge C,n) * k,i)} or x in (LSeg go,m) /\ (LSeg pion1,1) )
assume x in {((Gauge C,n) * k,i)} ; :: thesis: x in (LSeg go,m) /\ (LSeg pion1,1)
then A100: x = (Gauge C,n) * k,i by TARSKI:def 1;
A101: (Gauge C,n) * k,i in LSeg go,m by A50, RLTOPSP1:69;
(Gauge C,n) * k,i in LSeg pion1,1 by A43, 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 A43, A46, 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) * k,i),((Gauge C,n) * j,i)*> by A83, TOPREAL3:26;
then LSeg pion1,((len pion1) -' 1) c= LSeg ((Gauge C,n) * k,i),((Gauge C,n) * j,i) by SPPOL_2:21;
then A109: (LSeg pion1,((len pion1) -' 1)) /\ (LSeg do,1) c= {((Gauge C,n) * j,i)} by A60, XBOOLE_1:27;
{((Gauge C,n) * j,i)} 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) * j,i)} or x in (LSeg pion1,((len pion1) -' 1)) /\ (LSeg do,1) )
assume x in {((Gauge C,n) * j,i)} ; :: thesis: x in (LSeg pion1,((len pion1) -' 1)) /\ (LSeg do,1)
then A110: x = (Gauge C,n) * j,i by TARSKI:def 1;
pion1 /. (((len pion1) -' 1) + 1) = <*((Gauge C,n) * k,i),((Gauge C,n) * j,i)*> /. 2 by A85, A106, FINSEQ_1:61
.= (Gauge C,n) * j,i by FINSEQ_4:26 ;
then A111: (Gauge C,n) * j,i in LSeg pion1,((len pion1) -' 1) by A105, A106, TOPREAL1:27;
(Gauge C,n) * j,i in LSeg do,1 by A57, RLTOPSP1:69;
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) * j,i)} 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 A44, 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 pion1 by FINSEQ_6:46;
x in rng go by A94, A116, REVROT_1:3;
hence x in (L~ go) /\ (L~ pion1) by A63, 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~ pion1 by XBOOLE_0:def 4;
x in L~ go by A118, XBOOLE_0:def 4;
then x in (L~ pion1) /\ (L~ (Upper_Seq C,n)) by A48, A119, XBOOLE_0:def 4;
hence x in {(pion1 /. 1)} by A6, A43, 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 A63, A114, A120, XBOOLE_1:27;
then A122: go ^' pion1 is one-to-one by JORDAN1J:55;
A123: <*((Gauge C,n) * k,i),((Gauge C,n) * j,i)*> /. (len <*((Gauge C,n) * k,i),((Gauge C,n) * j,i)*>) = <*((Gauge C,n) * k,i),((Gauge C,n) * j,i)*> /. 2 by FINSEQ_1:61
.= do /. 1 by A44, 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 pion1 by REVROT_1:3;
x in rng do by A85, A123, A125, FINSEQ_6:46;
hence x in (L~ do) /\ (L~ pion1) by A64, 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~ pion1 by XBOOLE_0:def 4;
x in L~ do by A127, XBOOLE_0:def 4;
then x in (L~ pion1) /\ (L~ (Lower_Seq C,n)) by A55, A128, XBOOLE_0:def 4;
hence x in {(pion1 /. (len pion1))} by A7, A44, 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 A74, 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 A62, 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: Upper_Arc C is_an_arc_of W-min C, E-max C by JORDAN6:def 8;
then A132: Upper_Arc C is connected by JORDAN6:11;
A133: W-min C in Upper_Arc C by A131, TOPREAL1:4;
A134: E-max C in Upper_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) * k,i) .. (Upper_Seq C,n) <= 1 ; :: thesis: contradiction
((Gauge C,n) * k,i) .. (Upper_Seq C,n) >= 1 by A36, FINSEQ_4:31;
then ((Gauge C,n) * k,i) .. (Upper_Seq C,n) = 1 by A139, XXREAL_0:1;
then (Gauge C,n) * k,i = (Upper_Seq C,n) /. 1 by A36, FINSEQ_5:41;
hence contradiction by A19, A23, 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 A48, A145, XBOOLE_1:1;
A148: L~ do c= L~ (Cage C,n) by A55, A146, XBOOLE_1:1;
A149: W-min C in C by SPRECT_1:15;
A150: L~ <*((Gauge C,n) * k,i),((Gauge C,n) * j,i)*> = LSeg ((Gauge C,n) * k,i),((Gauge C,n) * j,i) 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) * k,i)),1,(Gauge C,n) by A36, A93, A138, JORDAN1J:52
.= right_cell (go ^' pion1),1,(Gauge C,n) by A41, 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 A61, GRAPH_2:57 ;
A155: len (Upper_Seq C,n) >= 2 by A18, XXREAL_0:2;
A156: godo /. 2 = (go ^' pion1) /. 2 by A89, GRAPH_2:61
.= (Upper_Seq C,n) /. 2 by A35, A77, 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 A63, A78, 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;
A161: ((Gauge C,n) * j,i) `1 <= ((Gauge C,n) * k,i) `1 by A1, A2, A3, A4, A5, 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 A162: W-bound (L~ pion1) = ((Gauge C,n) * j,i) `1 by A83, SPPOL_2:21;
((Gauge C,n) * j,i) `1 >= W-bound (L~ (Cage C,n)) by A9, A146, PSCOMP_1:71;
then ((Gauge C,n) * j,i) `1 > W-bound (L~ (Cage C,n)) by A76, XXREAL_0:1;
then W-min (((L~ go) \/ (L~ do)) \/ (L~ pion1)) = W-min ((L~ go) \/ (L~ do)) by A157, A158, A159, A160, A162, JORDAN1J:33;
then A163: W-min (L~ godo) = W-min (L~ (Cage C,n)) by A143, A158, XBOOLE_1:4;
A164: rng godo c= L~ godo by A89, A91, SPPOL_2:18, XXREAL_0:2;
2 in dom godo by A92, FINSEQ_3:27;
then A165: 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 A163, PSCOMP_1:88
.= W-bound (L~ godo) by EUCLID:56 ;
then godo /. 2 in W-most (L~ godo) by A164, A165, SPRECT_2:16;
then (Rotate godo,(W-min (L~ godo))) /. 2 in W-most (L~ godo) by A154, A163, 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 A166: (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 ;
A167: 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
A168: p in east_halfline (E-max C) and
A169: p in L~ go by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A168;
p in L~ (Upper_Seq C,n) by A48, A169;
then p in (east_halfline (E-max C)) /\ (L~ (Cage C,n)) by A145, A168, XBOOLE_0:def 4;
then A170: p `1 = E-bound (L~ (Cage C,n)) by JORDAN1A:104, PSCOMP_1:111;
then A171: p = E-max (L~ (Cage C,n)) by A48, A169, JORDAN1J:46;
then E-max (L~ (Cage C,n)) = (Gauge C,n) * k,i by A10, A166, A169, JORDAN1J:43;
then ((Gauge C,n) * k,i) `1 = ((Gauge C,n) * (len (Gauge C,n)),k) `1 by A3, A14, A170, A171, JORDAN1A:92;
hence contradiction by A3, A17, A32, JORDAN1G:7; :: thesis: verum
end;
now
assume east_halfline (E-max C) meets L~ godo ; :: thesis: contradiction
then A172: ( 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 A172, XBOOLE_1:70;
suppose east_halfline (E-max C) meets L~ pion1 ; :: thesis: contradiction
then consider p being set such that
A173: p in east_halfline (E-max C) and
A174: p in L~ pion1 by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A173;
A175: p `2 = (E-max C) `2 by A173, TOPREAL1:def 13;
k + 1 <= len (Gauge C,n) by A3, NAT_1:13;
then (k + 1) - 1 <= (len (Gauge C,n)) - 1 by XREAL_1:11;
then A176: 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 A177: ((Gauge C,n) * k,i) `1 <= ((Gauge C,n) * ((len (Gauge C,n)) -' 1),1) `1 by A4, A5, A11, A14, A21, A176, JORDAN1A:39;
p `1 <= ((Gauge C,n) * k,i) `1 by A83, A150, A161, A174, TOPREAL1:9;
then p `1 <= ((Gauge C,n) * ((len (Gauge C,n)) -' 1),1) `1 by A177, XXREAL_0:2;
then p `1 <= E-bound C by A21, JORDAN8:15;
then A178: p `1 <= (E-max C) `1 by EUCLID:56;
p `1 >= (E-max C) `1 by A173, TOPREAL1:def 13;
then p `1 = (E-max C) `1 by A178, XXREAL_0:1;
then p = E-max C by A175, TOPREAL3:11;
hence contradiction by A8, A83, A134, A150, A174, XBOOLE_0:3; :: thesis: verum
end;
suppose east_halfline (E-max C) meets L~ do ; :: thesis: contradiction
then consider p being set such that
A179: p in east_halfline (E-max C) and
A180: p in L~ do by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A179;
A181: p in LSeg do,(Index p,do) by A180, JORDAN3:42;
consider t being Nat such that
A182: t in dom (Lower_Seq C,n) and
A183: (Lower_Seq C,n) . t = (Gauge C,n) * j,i by A39, FINSEQ_2:11;
1 <= t by A182, FINSEQ_3:27;
then A184: 1 < t by A34, A183, XXREAL_0:1;
t <= len (Lower_Seq C,n) by A182, FINSEQ_3:27;
then (Index ((Gauge C,n) * j,i),(Lower_Seq C,n)) + 1 = t by A183, A184, JORDAN3:45;
then A185: len (L_Cut (Lower_Seq C,n),((Gauge C,n) * j,i)) = (len (Lower_Seq C,n)) - (Index ((Gauge C,n) * j,i),(Lower_Seq C,n)) by A9, A183, JORDAN3:61;
Index p,do < len do by A180, JORDAN3:41;
then Index p,do < (len (Lower_Seq C,n)) -' (Index ((Gauge C,n) * j,i),(Lower_Seq C,n)) by A185, XREAL_0:def 2;
then (Index p,do) + 1 <= (len (Lower_Seq C,n)) -' (Index ((Gauge C,n) * j,i),(Lower_Seq C,n)) by NAT_1:13;
then A186: Index p,do <= ((len (Lower_Seq C,n)) -' (Index ((Gauge C,n) * j,i),(Lower_Seq C,n))) - 1 by XREAL_1:21;
A187: do = mid (Lower_Seq C,n),(((Gauge C,n) * j,i) .. (Lower_Seq C,n)),(len (Lower_Seq C,n)) by A39, JORDAN1J:37;
p in L~ (Lower_Seq C,n) by A55, A180;
then p in (east_halfline (E-max C)) /\ (L~ (Cage C,n)) by A146, A179, XBOOLE_0:def 4;
then A188: p `1 = E-bound (L~ (Cage C,n)) by JORDAN1A:104, PSCOMP_1:111;
A189: (Index ((Gauge C,n) * j,i),(Lower_Seq C,n)) + 1 = ((Gauge C,n) * j,i) .. (Lower_Seq C,n) by A34, A39, JORDAN1J:56;
0 + (Index ((Gauge C,n) * j,i),(Lower_Seq C,n)) < len (Lower_Seq C,n) by A9, JORDAN3:41;
then (len (Lower_Seq C,n)) - (Index ((Gauge C,n) * j,i),(Lower_Seq C,n)) > 0 by XREAL_1:22;
then Index p,do <= ((len (Lower_Seq C,n)) - (Index ((Gauge C,n) * j,i),(Lower_Seq C,n))) - 1 by A186, XREAL_0:def 2;
then Index p,do <= (len (Lower_Seq C,n)) - (((Gauge C,n) * j,i) .. (Lower_Seq C,n)) by A189;
then Index p,do <= (len (Lower_Seq C,n)) -' (((Gauge C,n) * j,i) .. (Lower_Seq C,n)) by XREAL_0:def 2;
then A190: Index p,do < ((len (Lower_Seq C,n)) -' (((Gauge C,n) * j,i) .. (Lower_Seq C,n))) + 1 by NAT_1:13;
A191: 1 <= Index p,do by A180, JORDAN3:41;
A192: ((Gauge C,n) * j,i) .. (Lower_Seq C,n) <= len (Lower_Seq C,n) by A39, FINSEQ_4:31;
((Gauge C,n) * j,i) .. (Lower_Seq C,n) <> len (Lower_Seq C,n) by A31, A39, FINSEQ_4:29;
then A193: ((Gauge C,n) * j,i) .. (Lower_Seq C,n) < len (Lower_Seq C,n) by A192, XXREAL_0:1;
A194: 1 + 1 <= len (Lower_Seq C,n) by A25, XXREAL_0:2;
then A195: 2 in dom (Lower_Seq C,n) by FINSEQ_3:27;
set tt = ((Index p,do) + (((Gauge C,n) * j,i) .. (Lower_Seq C,n))) -' 1;
set RC = Rotate (Cage C,n),(E-max (L~ (Cage C,n)));
A196: E-max C in right_cell (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))),1 by JORDAN1I:9;
A197: 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 ;
A198: 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
A199: 1 <= jj2 and
A200: jj2 <= width (Gauge C,n) and
A201: E-max (L~ (Cage C,n)) = (Gauge C,n) * (len (Gauge C,n)),jj2 by JORDAN1D:29;
A202: len (Gauge C,n) >= 4 by JORDAN8:13;
then len (Gauge C,n) >= 1 by XXREAL_0:2;
then A203: [(len (Gauge C,n)),jj2] in Indices (Gauge C,n) by A199, A200, MATRIX_1:37;
A204: 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 A205: LSeg (Lower_Seq C,n),1 = LSeg (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))),1 by A194, SPPOL_2:9;
A206: 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 A140, REVROT_1:34;
then consider ii, jj being Element of NAT such that
A207: [ii,(jj + 1)] in Indices (Gauge C,n) and
A208: [ii,jj] in Indices (Gauge C,n) and
A209: (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) /. 1 = (Gauge C,n) * ii,(jj + 1) and
A210: (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) /. (1 + 1) = (Gauge C,n) * ii,jj by A87, A198, A204, A206, FINSEQ_6:98, JORDAN1I:25;
A211: (jj + 1) + 1 <> jj ;
A212: 1 <= jj by A208, 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 A198, A206, FINSEQ_6:98;
then A213: ii = len (Gauge C,n) by A198, A207, A209, A201, A203, GOBOARD1:21;
then ii - 1 >= 4 - 1 by A202, XREAL_1:11;
then A214: ii - 1 >= 1 by XXREAL_0:2;
then A215: 1 <= ii -' 1 by XREAL_0:def 2;
A216: jj <= width (Gauge C,n) by A208, MATRIX_1:39;
then A217: ((Gauge C,n) * (len (Gauge C,n)),jj) `1 = E-bound (L~ (Cage C,n)) by A11, A212, JORDAN1A:92;
A218: jj + 1 <= width (Gauge C,n) by A207, MATRIX_1:39;
ii + 1 <> ii ;
then A219: right_cell (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))),1 = cell (Gauge C,n),(ii -' 1),jj by A87, A204, A197, A207, A208, A209, A210, A211, GOBOARD5:def 6;
A220: ii <= len (Gauge C,n) by A208, MATRIX_1:39;
A221: 1 <= ii by A208, MATRIX_1:39;
A222: ii <= len (Gauge C,n) by A207, MATRIX_1:39;
A223: 1 <= jj + 1 by A207, MATRIX_1:39;
then A224: E-bound (L~ (Cage C,n)) = ((Gauge C,n) * (len (Gauge C,n)),(jj + 1)) `1 by A11, A218, JORDAN1A:92;
A225: 1 <= ii by A207, MATRIX_1:39;
then A226: (ii -' 1) + 1 = ii by XREAL_1:237;
then A227: ii -' 1 < len (Gauge C,n) by A222, NAT_1:13;
then A228: ((Gauge C,n) * (ii -' 1),(jj + 1)) `2 = ((Gauge C,n) * 1,(jj + 1)) `2 by A223, A218, A215, GOBOARD5:2
.= ((Gauge C,n) * ii,(jj + 1)) `2 by A225, A222, A223, A218, GOBOARD5:2 ;
A229: (E-max C) `2 = p `2 by A179, TOPREAL1:def 13;
then A230: p `2 <= ((Gauge C,n) * (ii -' 1),(jj + 1)) `2 by A196, A222, A218, A212, A219, A226, A214, JORDAN9:19;
A231: ((Gauge C,n) * (ii -' 1),jj) `2 = ((Gauge C,n) * 1,jj) `2 by A212, A216, A215, A227, GOBOARD5:2
.= ((Gauge C,n) * ii,jj) `2 by A221, A220, A212, A216, GOBOARD5:2 ;
((Gauge C,n) * (ii -' 1),jj) `2 <= p `2 by A229, A196, A222, A218, A212, A219, A226, A214, 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 A188, A209, A210, A213, A230, A231, A228, A217, A224, GOBOARD7:8;
then A232: p in LSeg (Lower_Seq C,n),1 by A87, A205, A204, TOPREAL1:def 5;
1 <= ((Gauge C,n) * j,i) .. (Lower_Seq C,n) by A39, FINSEQ_4:31;
then A233: LSeg (mid (Lower_Seq C,n),(((Gauge C,n) * j,i) .. (Lower_Seq C,n)),(len (Lower_Seq C,n))),(Index p,do) = LSeg (Lower_Seq C,n),(((Index p,do) + (((Gauge C,n) * j,i) .. (Lower_Seq C,n))) -' 1) by A193, A191, A190, JORDAN4:31;
1 <= Index ((Gauge C,n) * j,i),(Lower_Seq C,n) by A9, JORDAN3:41;
then A234: 1 + 1 <= ((Gauge C,n) * j,i) .. (Lower_Seq C,n) by A189, XREAL_1:9;
then (Index p,do) + (((Gauge C,n) * j,i) .. (Lower_Seq C,n)) >= (1 + 1) + 1 by A191, XREAL_1:9;
then ((Index p,do) + (((Gauge C,n) * j,i) .. (Lower_Seq C,n))) - 1 >= ((1 + 1) + 1) - 1 by XREAL_1:11;
then A235: ((Index p,do) + (((Gauge C,n) * j,i) .. (Lower_Seq C,n))) -' 1 >= 1 + 1 by XREAL_0:def 2;
now
per cases ( ((Index p,do) + (((Gauge C,n) * j,i) .. (Lower_Seq C,n))) -' 1 > 1 + 1 or ((Index p,do) + (((Gauge C,n) * j,i) .. (Lower_Seq C,n))) -' 1 = 1 + 1 ) by A235, XXREAL_0:1;
suppose ((Index p,do) + (((Gauge C,n) * j,i) .. (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) * j,i) .. (Lower_Seq C,n))) -' 1) by TOPREAL1:def 9;
hence contradiction by A232, A181, A187, A233, XBOOLE_0:3; :: thesis: verum
end;
suppose A236: ((Index p,do) + (((Gauge C,n) * j,i) .. (Lower_Seq C,n))) -' 1 = 1 + 1 ; :: thesis: contradiction
then 1 + 1 = ((Index p,do) + (((Gauge C,n) * j,i) .. (Lower_Seq C,n))) - 1 by XREAL_0:def 2;
then (1 + 1) + 1 = (Index p,do) + (((Gauge C,n) * j,i) .. (Lower_Seq C,n)) ;
then A237: ((Gauge C,n) * j,i) .. (Lower_Seq C,n) = 2 by A191, A234, JORDAN1E:10;
(LSeg (Lower_Seq C,n),1) /\ (LSeg (Lower_Seq C,n),(((Index p,do) + (((Gauge C,n) * j,i) .. (Lower_Seq C,n))) -' 1)) = {((Lower_Seq C,n) /. 2)} by A25, A236, TOPREAL1:def 8;
then p in {((Lower_Seq C,n) /. 2)} by A232, A181, A187, A233, XBOOLE_0:def 4;
then A238: p = (Lower_Seq C,n) /. 2 by TARSKI:def 1;
then A239: p in rng (Lower_Seq C,n) by A195, PARTFUN2:4;
p .. (Lower_Seq C,n) = 2 by A195, A238, FINSEQ_5:44;
then p = (Gauge C,n) * j,i by A39, A237, A239, FINSEQ_5:10;
then ((Gauge C,n) * j,i) `1 = E-bound (L~ (Cage C,n)) by A238, JORDAN1G:40;
then ((Gauge C,n) * j,i) `1 = ((Gauge C,n) * (len (Gauge C,n)),j) `1 by A1, A11, A13, JORDAN1A:92;
hence contradiction by A2, A3, A16, A69, 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
A240: W is_a_component_of (L~ godo) ` and
A241: east_halfline (E-max C) c= W by GOBOARD9:5;
not W is Bounded by A241, JORDAN2C:16, JORDAN2C:129;
then W is_outside_component_of L~ godo by A240, JORDAN2C:def 4;
then W c= UBD (L~ godo) by JORDAN2C:27;
then A242: east_halfline (E-max C) c= UBD (L~ godo) by A241, XBOOLE_1:1;
E-max C in east_halfline (E-max C) by TOPREAL1:45;
then E-max C in UBD (L~ godo) by A242;
then E-max C in LeftComp godo by GOBRD14:46;
then Upper_Arc C meets L~ godo by A132, A133, A134, A142, A153, JORDAN1J:36;
then A243: ( Upper_Arc C meets (L~ go) \/ (L~ pion1) or Upper_Arc C meets L~ do ) by A143, XBOOLE_1:70;
A244: Upper_Arc C c= C by JORDAN6:76;
per cases ( Upper_Arc C meets L~ go or Upper_Arc C meets L~ pion1 or Upper_Arc C meets L~ do ) by A243, XBOOLE_1:70;
suppose Upper_Arc C meets L~ go ; :: thesis: contradiction
then Upper_Arc C meets L~ (Cage C,n) by A48, A145, XBOOLE_1:1, XBOOLE_1:63;
hence contradiction by A244, JORDAN10:5, XBOOLE_1:63; :: thesis: verum
end;
suppose Upper_Arc C meets L~ pion1 ; :: thesis: contradiction
hence contradiction by A8, A83, A150; :: thesis: verum
end;
suppose Upper_Arc C meets L~ do ; :: thesis: contradiction
then Upper_Arc C meets L~ (Cage C,n) by A55, A146, XBOOLE_1:1, XBOOLE_1:63;
hence contradiction by A244, JORDAN10:5, XBOOLE_1:63; :: thesis: verum
end;
end;