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) * j,i)} & (LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n)) = {((Gauge C,n) * k,i)} holds
LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i) meets Lower_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) * j,i)} & (LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n)) = {((Gauge C,n) * k,i)} holds
LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i) meets Lower_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) * j,i)} & (LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n)) = {((Gauge C,n) * k,i)} implies LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i) 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 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) * j,i)} and
A7: (LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i)) /\ (L~ (Lower_Seq C,n)) = {((Gauge C,n) * k,i)} and
A8: LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i) misses Lower_Arc C ; :: thesis: contradiction
(Gauge C,n) * k,i in {((Gauge C,n) * k,i)} by TARSKI:def 1;
then A9: (Gauge C,n) * k,i in L~ (Lower_Seq C,n) by A7, XBOOLE_0:def 4;
(Gauge C,n) * j,i in {((Gauge C,n) * j,i)} by TARSKI:def 1;
then A10: (Gauge C,n) * j,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) * j,i);
set do = L_Cut (Lower_Seq C,n),((Gauge C,n) * k,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,i) `1 by A4, A5, A11, 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;
A23: [1,i] in Indices (Gauge C,n) by A4, A5, A21, MATRIX_1:37;
then A24: (Gauge C,n) * j,i <> (Upper_Seq C,n) . 1 by A1, A16, A19, A20, JORDAN1G:7;
then reconsider go = R_Cut (Upper_Seq C,n),((Gauge C,n) * j,i) as being_S-Seq FinSequence of (TOP-REAL 2) by A10, JORDAN3:70;
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 ;
(W-min (L~ (Cage C,n))) `1 = W-bound (L~ (Cage C,n)) by EUCLID:56
.= ((Gauge C,n) * 1,i) `1 by A4, A5, A11, JORDAN1A:94 ;
then A29: (Gauge C,n) * k,i <> (Lower_Seq C,n) . (len (Lower_Seq C,n)) by A1, A2, A17, A23, A28, JORDAN1G:7;
then reconsider do = L_Cut (Lower_Seq C,n),((Gauge C,n) * k,i) as being_S-Seq FinSequence of (TOP-REAL 2) by A9, JORDAN3:69;
A30: [(len (Gauge C,n)),k] in Indices (Gauge C,n) by A14, A15, A21, MATRIX_1:37;
A31: (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 A32: (Gauge C,n) * k,i <> (Lower_Seq C,n) . 1 by A3, A17, A30, A31, JORDAN1G:7;
A33: len go >= 1 + 1 by TOPREAL1:def 10;
A34: (Gauge C,n) * j,i in rng (Upper_Seq C,n) by A1, A4, A5, A10, A11, 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) * k,i in rng (Lower_Seq C,n) by A4, A5, A9, A11, A14, A15, 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) * j,i 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) * k,i 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) * k,i),((Gauge C,n) * j,i)) c= {((Gauge C,n) * j,i)} by A6, XBOOLE_1:26;
m >= 1 by A33, XREAL_1:21;
then A48: LSeg go,m = LSeg (go /. m),((Gauge C,n) * j,i) by A41, A43, TOPREAL1:def 5;
{((Gauge C,n) * j,i)} c= (LSeg go,m) /\ (LSeg ((Gauge C,n) * k,i),((Gauge C,n) * j,i))
proof
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in {((Gauge C,n) * j,i)} or x in (LSeg go,m) /\ (LSeg ((Gauge C,n) * k,i),((Gauge C,n) * j,i)) )
A49: (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 go,m) /\ (LSeg ((Gauge C,n) * k,i),((Gauge C,n) * j,i))
then A50: x = (Gauge C,n) * j,i by TARSKI:def 1;
(Gauge C,n) * j,i in LSeg go,m by A48, RLTOPSP1:69;
hence x in (LSeg go,m) /\ (LSeg ((Gauge C,n) * k,i),((Gauge C,n) * j,i)) by A50, A49, XBOOLE_0:def 4; :: thesis: verum
end;
then A51: (LSeg go,m) /\ (LSeg ((Gauge C,n) * k,i),((Gauge C,n) * j,i)) = {((Gauge C,n) * j,i)} 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) * k,i),((Gauge C,n) * j,i)) c= {((Gauge C,n) * k,i)} by A7, XBOOLE_1:26;
A55: LSeg do,1 = LSeg ((Gauge C,n) * k,i),(do /. (1 + 1)) by A36, A42, TOPREAL1:def 5;
{((Gauge C,n) * k,i)} c= (LSeg do,1) /\ (LSeg ((Gauge C,n) * k,i),((Gauge C,n) * j,i))
proof
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in {((Gauge C,n) * k,i)} or x in (LSeg do,1) /\ (LSeg ((Gauge C,n) * k,i),((Gauge C,n) * j,i)) )
A56: (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 do,1) /\ (LSeg ((Gauge C,n) * k,i),((Gauge C,n) * j,i))
then A57: x = (Gauge C,n) * k,i by TARSKI:def 1;
(Gauge C,n) * k,i in LSeg do,1 by A55, RLTOPSP1:69;
hence x in (LSeg do,1) /\ (LSeg ((Gauge C,n) * k,i),((Gauge C,n) * j,i)) by A57, A56, XBOOLE_0:def 4; :: thesis: verum
end;
then A58: (LSeg ((Gauge C,n) * k,i),((Gauge C,n) * j,i)) /\ (LSeg do,1) = {((Gauge C,n) * k,i)} 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 A27, PARTFUN1:def 8
.= E-max (L~ (Cage C,n)) by JORDAN1F:6 ;
A67: [(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 A68: x in (L~ go) /\ (L~ do) ; :: thesis: x in {(go /. 1)}
then A69: x in L~ do by XBOOLE_0:def 4;
A70: now
assume x = E-max (L~ (Cage C,n)) ; :: thesis: contradiction
then A71: E-max (L~ (Cage C,n)) = (Gauge C,n) * k,i by A9, A66, A69, 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 A3, A17, A67, A71, JORDAN1G:7;
hence contradiction by EUCLID:56; :: thesis: verum
end;
x in L~ go 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 ( 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 A59, A70, TARSKI:def 1; :: thesis: verum
end;
then A72: (L~ go) /\ (L~ do) = {(go /. 1)} by A63, XBOOLE_0:def 10;
set W2 = go /. 2;
A73: 2 in dom go by A33, FINSEQ_3:27;
A74: now
assume ((Gauge C,n) * j,i) `1 = W-bound (L~ (Cage C,n)) ; :: thesis: contradiction
then ((Gauge C,n) * 1,k) `1 = ((Gauge C,n) * j,i) `1 by A3, A14, JORDAN1A:94;
hence contradiction by A1, A16, A22, JORDAN1G:7; :: thesis: verum
end;
go = mid (Upper_Seq C,n),1,(((Gauge C,n) * j,i) .. (Upper_Seq C,n)) by A34, JORDAN1G:57
.= (Upper_Seq C,n) | (((Gauge C,n) * j,i) .. (Upper_Seq C,n)) by A34, FINSEQ_4:31, JORDAN3:25 ;
then A75: go /. 2 = (Upper_Seq C,n) /. 2 by A73, FINSEQ_4:85;
A76: W-min (L~ (Cage C,n)) in rng go by A59, FINSEQ_6:46;
set pion = <*((Gauge C,n) * j,i),((Gauge C,n) * k,i)*>;
A77: now
let n be Element of NAT ; :: thesis: ( n in dom <*((Gauge C,n) * j,i),((Gauge C,n) * k,i)*> implies ex j, i being Element of NAT st
( [j,i] in Indices (Gauge C,n) & <*((Gauge C,n) * j,i),((Gauge C,n) * k,i)*> /. n = (Gauge C,n) * j,i ) )
assume n in dom <*((Gauge C,n) * j,i),((Gauge C,n) * k,i)*> ; :: thesis: ex j, i being Element of NAT st
( [j,i] in Indices (Gauge C,n) & <*((Gauge C,n) * j,i),((Gauge C,n) * k,i)*> /. n = (Gauge C,n) * j,i )
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) * j,i),((Gauge C,n) * k,i)*> /. n = (Gauge C,n) * j,i ) by A16, A17, FINSEQ_4:26; :: thesis: verum
end;
A78: (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) * j,i),((Gauge C,n) * k,i)*> is being_S-Seq by A78, JORDAN1B:9;
then consider pion1 being FinSequence of (TOP-REAL 2) such that
A79: pion1 is_sequence_on Gauge C,n and
A80: pion1 is being_S-Seq and
A81: L~ <*((Gauge C,n) * j,i),((Gauge C,n) * k,i)*> = L~ pion1 and
A82: <*((Gauge C,n) * j,i),((Gauge C,n) * k,i)*> /. 1 = pion1 /. 1 and
A83: <*((Gauge C,n) * j,i),((Gauge C,n) * k,i)*> /. (len <*((Gauge C,n) * j,i),((Gauge C,n) * k,i)*>) = pion1 /. (len pion1) and
A84: len <*((Gauge C,n) * j,i),((Gauge C,n) * k,i)*> <= len pion1 by A77, GOBOARD3:2;
reconsider pion1 = pion1 as being_S-Seq FinSequence of (TOP-REAL 2) by A80;
set godo = (go ^' pion1) ^' do;
A85: 1 + 1 <= len (Cage C,n) by GOBOARD7:36, XXREAL_0:2;
A86: 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 A87: len (go ^' pion1) >= 1 + 1 by A33, XXREAL_0:2;
then A88: len (go ^' pion1) > 1 + 0 by NAT_1:13;
A89: len ((go ^' pion1) ^' do) >= len (go ^' pion1) by TOPREAL8:7;
then A90: 1 + 1 <= len ((go ^' pion1) ^' do) by A87, XXREAL_0:2;
A91: Upper_Seq C,n is_sequence_on Gauge C,n by JORDAN1G:4;
A92: go /. (len go) = pion1 /. 1 by A41, A82, FINSEQ_4:26;
then A93: go ^' pion1 is_sequence_on Gauge C,n by A35, A79, TOPREAL8:12;
A94: (go ^' pion1) /. (len (go ^' pion1)) = <*((Gauge C,n) * j,i),((Gauge C,n) * k,i)*> /. (len <*((Gauge C,n) * j,i),((Gauge C,n) * k,i)*>) by A83, GRAPH_2:58
.= <*((Gauge C,n) * j,i),((Gauge C,n) * k,i)*> /. 2 by FINSEQ_1:61
.= do /. 1 by A42, FINSEQ_4:26 ;
then A95: (go ^' pion1) ^' do is_sequence_on Gauge C,n by A38, A93, TOPREAL8:12;
LSeg pion1,1 c= L~ <*((Gauge C,n) * j,i),((Gauge C,n) * k,i)*> by A81, TOPREAL3:26;
then LSeg pion1,1 c= LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i) by SPPOL_2:21;
then A96: (LSeg go,((len go) -' 1)) /\ (LSeg pion1,1) c= {((Gauge C,n) * j,i)} by A44, A51, XBOOLE_1:27;
A97: len pion1 >= 1 + 1 by A84, FINSEQ_1:61;
{((Gauge C,n) * j,i)} c= (LSeg go,m) /\ (LSeg pion1,1)
proof
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in {((Gauge C,n) * j,i)} or x in (LSeg go,m) /\ (LSeg pion1,1) )
assume x in {((Gauge C,n) * j,i)} ; :: thesis: x in (LSeg go,m) /\ (LSeg pion1,1)
then A98: x = (Gauge C,n) * j,i by TARSKI:def 1;
A99: (Gauge C,n) * j,i in LSeg go,m by A48, RLTOPSP1:69;
(Gauge C,n) * j,i in LSeg pion1,1 by A41, A92, A97, TOPREAL1:27;
hence x in (LSeg go,m) /\ (LSeg pion1,1) by A98, A99, XBOOLE_0:def 4; :: thesis: verum
end;
then (LSeg go,((len go) -' 1)) /\ (LSeg pion1,1) = {(go /. (len go))} by A41, A44, A96, XBOOLE_0:def 10;
then A100: go ^' pion1 is unfolded by A92, TOPREAL8:34;
len pion1 >= 2 + 0 by A84, FINSEQ_1:61;
then A101: (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 A101, XREAL_0:def 2 ;
then A102: (len (go ^' pion1)) -' 1 = (len go) + ((len pion1) -' 2) by XREAL_0:def 2;
A103: (len pion1) - 1 >= 1 by A97, XREAL_1:21;
then A104: (len pion1) -' 1 = (len pion1) - 1 by XREAL_0:def 2;
A105: ((len pion1) -' 2) + 1 = ((len pion1) - 2) + 1 by A101, XREAL_0:def 2
.= (len pion1) -' 1 by A103, XREAL_0:def 2 ;
((len pion1) - 1) + 1 <= len pion1 ;
then A106: (len pion1) -' 1 < len pion1 by A104, NAT_1:13;
LSeg pion1,((len pion1) -' 1) c= L~ <*((Gauge C,n) * j,i),((Gauge C,n) * k,i)*> by A81, TOPREAL3:26;
then LSeg pion1,((len pion1) -' 1) c= LSeg ((Gauge C,n) * j,i),((Gauge C,n) * k,i) by SPPOL_2:21;
then A107: (LSeg pion1,((len pion1) -' 1)) /\ (LSeg do,1) c= {((Gauge C,n) * k,i)} by A58, XBOOLE_1:27;
{((Gauge C,n) * k,i)} c= (LSeg pion1,((len pion1) -' 1)) /\ (LSeg do,1)
proof
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in {((Gauge C,n) * k,i)} or x in (LSeg pion1,((len pion1) -' 1)) /\ (LSeg do,1) )
assume x in {((Gauge C,n) * k,i)} ; :: thesis: x in (LSeg pion1,((len pion1) -' 1)) /\ (LSeg do,1)
then A108: x = (Gauge C,n) * k,i by TARSKI:def 1;
pion1 /. (((len pion1) -' 1) + 1) = <*((Gauge C,n) * j,i),((Gauge C,n) * k,i)*> /. 2 by A83, A104, FINSEQ_1:61
.= (Gauge C,n) * k,i by FINSEQ_4:26 ;
then A109: (Gauge C,n) * k,i in LSeg pion1,((len pion1) -' 1) by A103, A104, TOPREAL1:27;
(Gauge C,n) * k,i in LSeg do,1 by A55, RLTOPSP1:69;
hence x in (LSeg pion1,((len pion1) -' 1)) /\ (LSeg do,1) by A108, A109, XBOOLE_0:def 4; :: thesis: verum
end;
then (LSeg pion1,((len pion1) -' 1)) /\ (LSeg do,1) = {((Gauge C,n) * k,i)} by A107, XBOOLE_0:def 10;
then A110: (LSeg (go ^' pion1),((len go) + ((len pion1) -' 2))) /\ (LSeg do,1) = {((go ^' pion1) /. (len (go ^' pion1)))} by A42, A92, A94, A105, A106, TOPREAL8:31;
A111: not go ^' pion1 is trivial by A87, REALSET1:13;
A112: rng pion1 c= L~ pion1 by A97, SPPOL_2:18;
A113: {(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 A114: x = pion1 /. 1 by TARSKI:def 1;
then A115: x in rng pion1 by FINSEQ_6:46;
x in rng go by A92, A114, REVROT_1:3;
hence x in (L~ go) /\ (L~ pion1) by A61, A112, A115, 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 A116: x in (L~ go) /\ (L~ pion1) ; :: thesis: x in {(pion1 /. 1)}
then A117: x in L~ pion1 by XBOOLE_0:def 4;
x in L~ go by A116, XBOOLE_0:def 4;
then x in (L~ pion1) /\ (L~ (Upper_Seq C,n)) by A46, A117, XBOOLE_0:def 4;
hence x in {(pion1 /. 1)} by A6, A41, A81, A92, SPPOL_2:21; :: thesis: verum
end;
then A118: (L~ go) /\ (L~ pion1) = {(pion1 /. 1)} by A113, XBOOLE_0:def 10;
then A119: go ^' pion1 is s.n.c. by A92, JORDAN1J:54;
(rng go) /\ (rng pion1) c= {(pion1 /. 1)} by A61, A112, A118, XBOOLE_1:27;
then A120: go ^' pion1 is one-to-one by JORDAN1J:55;
A121: <*((Gauge C,n) * j,i),((Gauge C,n) * k,i)*> /. (len <*((Gauge C,n) * j,i),((Gauge C,n) * k,i)*>) = <*((Gauge C,n) * j,i),((Gauge C,n) * k,i)*> /. 2 by FINSEQ_1:61
.= do /. 1 by A42, FINSEQ_4:26 ;
A122: {(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 A123: x = pion1 /. (len pion1) by TARSKI:def 1;
then A124: x in rng pion1 by REVROT_1:3;
x in rng do by A83, A121, A123, FINSEQ_6:46;
hence x in (L~ do) /\ (L~ pion1) by A62, A112, A124, 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 A125: x in (L~ do) /\ (L~ pion1) ; :: thesis: x in {(pion1 /. (len pion1))}
then A126: x in L~ pion1 by XBOOLE_0:def 4;
x in L~ do by A125, XBOOLE_0:def 4;
then x in (L~ pion1) /\ (L~ (Lower_Seq C,n)) by A53, A126, XBOOLE_0:def 4;
hence x in {(pion1 /. (len pion1))} by A7, A42, A81, A83, A121, SPPOL_2:21; :: thesis: verum
end;
then A127: (L~ do) /\ (L~ pion1) = {(pion1 /. (len pion1))} by A122, XBOOLE_0:def 10;
A128: (L~ (go ^' pion1)) /\ (L~ do) = ((L~ go) \/ (L~ pion1)) /\ (L~ do) by A92, TOPREAL8:35
.= {(go /. 1)} \/ {(do /. 1)} by A72, A83, A121, A127, 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 A90, A94, A95, A100, A102, A110, A111, A119, A120, A128, JORDAN8:7, JORDAN8:8, TOPREAL8:11, TOPREAL8:33, TOPREAL8:34;
A129: Lower_Arc C is_an_arc_of E-max C, W-min C by JORDAN6:def 9;
then A130: Lower_Arc C is connected by JORDAN6:11;
A131: W-min C in Lower_Arc C by A129, TOPREAL1:4;
A132: E-max C in Lower_Arc C by A129, 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 A133: (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) /. 1 = W-min (L~ (Cage C,n)) by FINSEQ_6:98;
A134: 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 A133, 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 A133, A134, 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 A133, A134, SPRECT_5:25, XXREAL_0:2;
then A135: (E-max (L~ (Cage C,n))) .. (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) > 1 by A133, A134, SPRECT_5:26, XXREAL_0:2;
A136: now
assume A137: ((Gauge C,n) * j,i) .. (Upper_Seq C,n) <= 1 ; :: thesis: contradiction
((Gauge C,n) * j,i) .. (Upper_Seq C,n) >= 1 by A34, FINSEQ_4:31;
then ((Gauge C,n) * j,i) .. (Upper_Seq C,n) = 1 by A137, XXREAL_0:1;
then (Gauge C,n) * j,i = (Upper_Seq C,n) /. 1 by A34, FINSEQ_5:41;
hence contradiction by A19, A24, JORDAN1F:5; :: thesis: verum
end;
A138: Cage C,n is_sequence_on Gauge C,n by JORDAN9:def 1;
then A139: Rotate (Cage C,n),(W-min (L~ (Cage C,n))) is_sequence_on Gauge C,n by REVROT_1:34;
A140: (right_cell godo,1,(Gauge C,n)) \ (L~ godo) c= RightComp godo by A90, A95, JORDAN9:29;
A141: L~ godo = (L~ (go ^' pion1)) \/ (L~ do) by A94, TOPREAL8:35
.= ((L~ go) \/ (L~ pion1)) \/ (L~ do) by A92, TOPREAL8:35 ;
A142: L~ (Cage C,n) = (L~ (Upper_Seq C,n)) \/ (L~ (Lower_Seq C,n)) by JORDAN1E:17;
then A143: L~ (Upper_Seq C,n) c= L~ (Cage C,n) by XBOOLE_1:7;
A144: L~ (Lower_Seq C,n) c= L~ (Cage C,n) by A142, XBOOLE_1:7;
A145: L~ go c= L~ (Cage C,n) by A46, A143, XBOOLE_1:1;
A146: L~ do c= L~ (Cage C,n) by A53, A144, XBOOLE_1:1;
A147: W-min C in C by SPRECT_1:15;
A148: L~ <*((Gauge C,n) * j,i),((Gauge C,n) * k,i)*> = LSeg ((Gauge C,n) * k,i),((Gauge C,n) * j,i) by SPPOL_2:21;
A149: now
assume W-min C in L~ godo ; :: thesis: contradiction
then A150: ( W-min C in (L~ go) \/ (L~ pion1) or W-min C in L~ do ) by A141, 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 A150, 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 A86, 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 A135, A139, JORDAN1J:53
.= right_cell (Upper_Seq C,n),1,(Gauge C,n) by JORDAN1E:def 1
.= right_cell (R_Cut (Upper_Seq C,n),((Gauge C,n) * j,i)),1,(Gauge C,n) by A34, A91, A136, JORDAN1J:52
.= right_cell (go ^' pion1),1,(Gauge C,n) by A39, A93, JORDAN1J:51
.= right_cell godo,1,(Gauge C,n) by A88, A95, JORDAN1J:51 ;
then W-min C in right_cell godo,1,(Gauge C,n) by JORDAN1I:8;
then A151: W-min C in (right_cell godo,1,(Gauge C,n)) \ (L~ godo) by A149, XBOOLE_0:def 5;
A152: godo /. 1 = (go ^' pion1) /. 1 by GRAPH_2:57
.= W-min (L~ (Cage C,n)) by A59, GRAPH_2:57 ;
A153: len (Upper_Seq C,n) >= 2 by A18, XXREAL_0:2;
A154: godo /. 2 = (go ^' pion1) /. 2 by A87, GRAPH_2:61
.= (Upper_Seq C,n) /. 2 by A33, A75, GRAPH_2:61
.= ((Upper_Seq C,n) ^' (Lower_Seq C,n)) /. 2 by A153, GRAPH_2:61
.= (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) /. 2 by JORDAN1E:15 ;
A155: (L~ go) \/ (L~ do) is compact by COMPTS_1:19;
W-min (L~ (Cage C,n)) in (L~ go) \/ (L~ do) by A61, A76, XBOOLE_0:def 3;
then A156: W-min ((L~ go) \/ (L~ do)) = W-min (L~ (Cage C,n)) by A145, A146, A155, JORDAN1J:21, XBOOLE_1:8;
A157: (W-min ((L~ go) \/ (L~ do))) `1 = W-bound ((L~ go) \/ (L~ do)) by EUCLID:56;
A158: (W-min (L~ (Cage C,n))) `1 = W-bound (L~ (Cage C,n)) by EUCLID:56;
A159: ((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 A160: W-bound (L~ pion1) = ((Gauge C,n) * j,i) `1 by A81, SPPOL_2:21;
((Gauge C,n) * j,i) `1 >= W-bound (L~ (Cage C,n)) by A10, A143, PSCOMP_1:71;
then ((Gauge C,n) * j,i) `1 > W-bound (L~ (Cage C,n)) by A74, XXREAL_0:1;
then W-min (((L~ go) \/ (L~ do)) \/ (L~ pion1)) = W-min ((L~ go) \/ (L~ do)) by A155, A156, A157, A158, A160, JORDAN1J:33;
then A161: W-min (L~ godo) = W-min (L~ (Cage C,n)) by A141, A156, XBOOLE_1:4;
A162: rng godo c= L~ godo by A87, A89, SPPOL_2:18, XXREAL_0:2;
2 in dom godo by A90, FINSEQ_3:27;
then A163: godo /. 2 in rng godo by PARTFUN2:4;
godo /. 2 in W-most (L~ (Cage C,n)) by A154, JORDAN1I:27;
then (godo /. 2) `1 = (W-min (L~ godo)) `1 by A161, PSCOMP_1:88
.= W-bound (L~ godo) by EUCLID:56 ;
then godo /. 2 in W-most (L~ godo) by A162, A163, SPRECT_2:16;
then (Rotate godo,(W-min (L~ godo))) /. 2 in W-most (L~ godo) by A152, A161, 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 A164: (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 ;
A165: 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
A166: p in east_halfline (E-max C) and
A167: p in L~ go by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A166;
p in L~ (Upper_Seq C,n) by A46, A167;
then p in (east_halfline (E-max C)) /\ (L~ (Cage C,n)) by A143, A166, XBOOLE_0:def 4;
then A168: p `1 = E-bound (L~ (Cage C,n)) by JORDAN1A:104, PSCOMP_1:111;
then A169: p = E-max (L~ (Cage C,n)) by A46, A167, JORDAN1J:46;
then E-max (L~ (Cage C,n)) = (Gauge C,n) * j,i by A10, A164, A167, JORDAN1J:43;
then ((Gauge C,n) * j,i) `1 = ((Gauge C,n) * (len (Gauge C,n)),k) `1 by A3, A14, A168, A169, JORDAN1A:92;
hence contradiction by A2, A3, A16, A30, JORDAN1G:7; :: thesis: verum
end;
now
assume east_halfline (E-max C) meets L~ godo ; :: thesis: contradiction
then A170: ( east_halfline (E-max C) meets (L~ go) \/ (L~ pion1) or east_halfline (E-max C) meets L~ do ) by A141, 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 A170, XBOOLE_1:70;
suppose east_halfline (E-max C) meets L~ pion1 ; :: thesis: contradiction
then consider p being set such that
A171: p in east_halfline (E-max C) and
A172: p in L~ pion1 by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A171;
A173: p `2 = (E-max C) `2 by A171, 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 A174: 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 A175: ((Gauge C,n) * k,i) `1 <= ((Gauge C,n) * ((len (Gauge C,n)) -' 1),1) `1 by A4, A5, A11, A14, A21, A174, JORDAN1A:39;
p `1 <= ((Gauge C,n) * k,i) `1 by A81, A148, A159, A172, TOPREAL1:9;
then p `1 <= ((Gauge C,n) * ((len (Gauge C,n)) -' 1),1) `1 by A175, XXREAL_0:2;
then p `1 <= E-bound C by A21, JORDAN8:15;
then A176: p `1 <= (E-max C) `1 by EUCLID:56;
p `1 >= (E-max C) `1 by A171, TOPREAL1:def 13;
then p `1 = (E-max C) `1 by A176, XXREAL_0:1;
then p = E-max C by A173, TOPREAL3:11;
hence contradiction by A8, A81, A132, A148, A172, XBOOLE_0:3; :: thesis: verum
end;
suppose east_halfline (E-max C) meets L~ do ; :: thesis: contradiction
then consider p being set such that
A177: p in east_halfline (E-max C) and
A178: p in L~ do by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A177;
A179: p in LSeg do,(Index p,do) by A178, JORDAN3:42;
consider t being Nat such that
A180: t in dom (Lower_Seq C,n) and
A181: (Lower_Seq C,n) . t = (Gauge C,n) * k,i by A37, FINSEQ_2:11;
1 <= t by A180, FINSEQ_3:27;
then A182: 1 < t by A32, A181, XXREAL_0:1;
t <= len (Lower_Seq C,n) by A180, FINSEQ_3:27;
then (Index ((Gauge C,n) * k,i),(Lower_Seq C,n)) + 1 = t by A181, A182, JORDAN3:45;
then A183: len (L_Cut (Lower_Seq C,n),((Gauge C,n) * k,i)) = (len (Lower_Seq C,n)) - (Index ((Gauge C,n) * k,i),(Lower_Seq C,n)) by A9, A181, JORDAN3:61;
Index p,do < len do by A178, JORDAN3:41;
then Index p,do < (len (Lower_Seq C,n)) -' (Index ((Gauge C,n) * k,i),(Lower_Seq C,n)) by A183, XREAL_0:def 2;
then (Index p,do) + 1 <= (len (Lower_Seq C,n)) -' (Index ((Gauge C,n) * k,i),(Lower_Seq C,n)) by NAT_1:13;
then A184: Index p,do <= ((len (Lower_Seq C,n)) -' (Index ((Gauge C,n) * k,i),(Lower_Seq C,n))) - 1 by XREAL_1:21;
A185: do = mid (Lower_Seq C,n),(((Gauge C,n) * k,i) .. (Lower_Seq C,n)),(len (Lower_Seq C,n)) by A37, JORDAN1J:37;
p in L~ (Lower_Seq C,n) by A53, A178;
then p in (east_halfline (E-max C)) /\ (L~ (Cage C,n)) by A144, A177, XBOOLE_0:def 4;
then A186: p `1 = E-bound (L~ (Cage C,n)) by JORDAN1A:104, PSCOMP_1:111;
A187: (Index ((Gauge C,n) * k,i),(Lower_Seq C,n)) + 1 = ((Gauge C,n) * k,i) .. (Lower_Seq C,n) by A32, A37, JORDAN1J:56;
0 + (Index ((Gauge C,n) * k,i),(Lower_Seq C,n)) < len (Lower_Seq C,n) by A9, JORDAN3:41;
then (len (Lower_Seq C,n)) - (Index ((Gauge C,n) * k,i),(Lower_Seq C,n)) > 0 by XREAL_1:22;
then Index p,do <= ((len (Lower_Seq C,n)) - (Index ((Gauge C,n) * k,i),(Lower_Seq C,n))) - 1 by A184, XREAL_0:def 2;
then Index p,do <= (len (Lower_Seq C,n)) - (((Gauge C,n) * k,i) .. (Lower_Seq C,n)) by A187;
then Index p,do <= (len (Lower_Seq C,n)) -' (((Gauge C,n) * k,i) .. (Lower_Seq C,n)) by XREAL_0:def 2;
then A188: Index p,do < ((len (Lower_Seq C,n)) -' (((Gauge C,n) * k,i) .. (Lower_Seq C,n))) + 1 by NAT_1:13;
A189: 1 <= Index p,do by A178, JORDAN3:41;
A190: ((Gauge C,n) * k,i) .. (Lower_Seq C,n) <= len (Lower_Seq C,n) by A37, FINSEQ_4:31;
((Gauge C,n) * k,i) .. (Lower_Seq C,n) <> len (Lower_Seq C,n) by A29, A37, FINSEQ_4:29;
then A191: ((Gauge C,n) * k,i) .. (Lower_Seq C,n) < len (Lower_Seq C,n) by A190, XXREAL_0:1;
A192: 1 + 1 <= len (Lower_Seq C,n) by A25, XXREAL_0:2;
then A193: 2 in dom (Lower_Seq C,n) by FINSEQ_3:27;
set tt = ((Index p,do) + (((Gauge C,n) * k,i) .. (Lower_Seq C,n))) -' 1;
set RC = Rotate (Cage C,n),(E-max (L~ (Cage C,n)));
A194: E-max C in right_cell (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))),1 by JORDAN1I:9;
A195: 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 ;
A196: L~ (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) = L~ (Cage C,n) by REVROT_1:33;
consider g2 being Element of NAT such that
A197: 1 <= g2 and
A198: g2 <= width (Gauge C,n) and
A199: E-max (L~ (Cage C,n)) = (Gauge C,n) * (len (Gauge C,n)),g2 by JORDAN1D:29;
A200: len (Gauge C,n) >= 4 by JORDAN8:13;
then len (Gauge C,n) >= 1 by XXREAL_0:2;
then A201: [(len (Gauge C,n)),g2] in Indices (Gauge C,n) by A197, A198, MATRIX_1:37;
A202: 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 A203: LSeg (Lower_Seq C,n),1 = LSeg (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))),1 by A192, SPPOL_2:9;
A204: 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 A138, REVROT_1:34;
then consider ii, g being Element of NAT such that
A205: [ii,(g + 1)] in Indices (Gauge C,n) and
A206: [ii,g] in Indices (Gauge C,n) and
A207: (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) /. 1 = (Gauge C,n) * ii,(g + 1) and
A208: (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) /. (1 + 1) = (Gauge C,n) * ii,g by A85, A196, A202, A204, FINSEQ_6:98, JORDAN1I:25;
A209: (g + 1) + 1 <> g ;
A210: 1 <= g by A206, 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 A196, A204, FINSEQ_6:98;
then A211: ii = len (Gauge C,n) by A196, A205, A207, A199, A201, GOBOARD1:21;
then ii - 1 >= 4 - 1 by A200, XREAL_1:11;
then A212: ii - 1 >= 1 by XXREAL_0:2;
then A213: 1 <= ii -' 1 by XREAL_0:def 2;
A214: g <= width (Gauge C,n) by A206, MATRIX_1:39;
then A215: ((Gauge C,n) * (len (Gauge C,n)),g) `1 = E-bound (L~ (Cage C,n)) by A11, A210, JORDAN1A:92;
A216: g + 1 <= width (Gauge C,n) by A205, MATRIX_1:39;
ii + 1 <> ii ;
then A217: right_cell (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))),1 = cell (Gauge C,n),(ii -' 1),g by A85, A202, A195, A205, A206, A207, A208, A209, GOBOARD5:def 6;
A218: ii <= len (Gauge C,n) by A206, MATRIX_1:39;
A219: 1 <= ii by A206, MATRIX_1:39;
A220: ii <= len (Gauge C,n) by A205, MATRIX_1:39;
A221: 1 <= g + 1 by A205, MATRIX_1:39;
then A222: E-bound (L~ (Cage C,n)) = ((Gauge C,n) * (len (Gauge C,n)),(g + 1)) `1 by A11, A216, JORDAN1A:92;
A223: 1 <= ii by A205, MATRIX_1:39;
then A224: (ii -' 1) + 1 = ii by XREAL_1:237;
then A225: ii -' 1 < len (Gauge C,n) by A220, NAT_1:13;
then A226: ((Gauge C,n) * (ii -' 1),(g + 1)) `2 = ((Gauge C,n) * 1,(g + 1)) `2 by A221, A216, A213, GOBOARD5:2
.= ((Gauge C,n) * ii,(g + 1)) `2 by A223, A220, A221, A216, GOBOARD5:2 ;
A227: (E-max C) `2 = p `2 by A177, TOPREAL1:def 13;
then A228: p `2 <= ((Gauge C,n) * (ii -' 1),(g + 1)) `2 by A194, A220, A216, A210, A217, A224, A212, JORDAN9:19;
A229: ((Gauge C,n) * (ii -' 1),g) `2 = ((Gauge C,n) * 1,g) `2 by A210, A214, A213, A225, GOBOARD5:2
.= ((Gauge C,n) * ii,g) `2 by A219, A218, A210, A214, GOBOARD5:2 ;
((Gauge C,n) * (ii -' 1),g) `2 <= p `2 by A227, A194, A220, A216, A210, A217, A224, A212, 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 A186, A207, A208, A211, A228, A229, A226, A215, A222, GOBOARD7:8;
then A230: p in LSeg (Lower_Seq C,n),1 by A85, A203, A202, TOPREAL1:def 5;
1 <= ((Gauge C,n) * k,i) .. (Lower_Seq C,n) by A37, FINSEQ_4:31;
then A231: LSeg (mid (Lower_Seq C,n),(((Gauge C,n) * k,i) .. (Lower_Seq C,n)),(len (Lower_Seq C,n))),(Index p,do) = LSeg (Lower_Seq C,n),(((Index p,do) + (((Gauge C,n) * k,i) .. (Lower_Seq C,n))) -' 1) by A191, A189, A188, JORDAN4:31;
1 <= Index ((Gauge C,n) * k,i),(Lower_Seq C,n) by A9, JORDAN3:41;
then A232: 1 + 1 <= ((Gauge C,n) * k,i) .. (Lower_Seq C,n) by A187, XREAL_1:9;
then (Index p,do) + (((Gauge C,n) * k,i) .. (Lower_Seq C,n)) >= (1 + 1) + 1 by A189, XREAL_1:9;
then ((Index p,do) + (((Gauge C,n) * k,i) .. (Lower_Seq C,n))) - 1 >= ((1 + 1) + 1) - 1 by XREAL_1:11;
then A233: ((Index p,do) + (((Gauge C,n) * k,i) .. (Lower_Seq C,n))) -' 1 >= 1 + 1 by XREAL_0:def 2;
now
per cases ( ((Index p,do) + (((Gauge C,n) * k,i) .. (Lower_Seq C,n))) -' 1 > 1 + 1 or ((Index p,do) + (((Gauge C,n) * k,i) .. (Lower_Seq C,n))) -' 1 = 1 + 1 ) by A233, XXREAL_0:1;
suppose ((Index p,do) + (((Gauge C,n) * k,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) * k,i) .. (Lower_Seq C,n))) -' 1) by TOPREAL1:def 9;
hence contradiction by A230, A179, A185, A231, XBOOLE_0:3; :: thesis: verum
end;
suppose A234: ((Index p,do) + (((Gauge C,n) * k,i) .. (Lower_Seq C,n))) -' 1 = 1 + 1 ; :: thesis: contradiction
then 1 + 1 = ((Index p,do) + (((Gauge C,n) * k,i) .. (Lower_Seq C,n))) - 1 by XREAL_0:def 2;
then (1 + 1) + 1 = (Index p,do) + (((Gauge C,n) * k,i) .. (Lower_Seq C,n)) ;
then A235: ((Gauge C,n) * k,i) .. (Lower_Seq C,n) = 2 by A189, A232, JORDAN1E:10;
(LSeg (Lower_Seq C,n),1) /\ (LSeg (Lower_Seq C,n),(((Index p,do) + (((Gauge C,n) * k,i) .. (Lower_Seq C,n))) -' 1)) = {((Lower_Seq C,n) /. 2)} by A25, A234, TOPREAL1:def 8;
then p in {((Lower_Seq C,n) /. 2)} by A230, A179, A185, A231, XBOOLE_0:def 4;
then A236: p = (Lower_Seq C,n) /. 2 by TARSKI:def 1;
then A237: p in rng (Lower_Seq C,n) by A193, PARTFUN2:4;
p .. (Lower_Seq C,n) = 2 by A193, A236, FINSEQ_5:44;
then p = (Gauge C,n) * k,i by A37, A235, A237, FINSEQ_5:10;
then ((Gauge C,n) * k,i) `1 = E-bound (L~ (Cage C,n)) by A236, JORDAN1G:40;
then ((Gauge C,n) * k,i) `1 = ((Gauge C,n) * (len (Gauge C,n)),j) `1 by A1, A11, A13, JORDAN1A:92;
hence contradiction by A3, A17, 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
A238: W is_a_component_of (L~ godo) ` and
A239: east_halfline (E-max C) c= W by GOBOARD9:5;
not W is Bounded by A239, JORDAN2C:16, JORDAN2C:129;
then W is_outside_component_of L~ godo by A238, JORDAN2C:def 4;
then W c= UBD (L~ godo) by JORDAN2C:27;
then A240: east_halfline (E-max C) c= UBD (L~ godo) by A239, XBOOLE_1:1;
E-max C in east_halfline (E-max C) by TOPREAL1:45;
then E-max C in UBD (L~ godo) by A240;
then E-max C in LeftComp godo by GOBRD14:46;
then Lower_Arc C meets L~ godo by A130, A131, A132, A140, A151, JORDAN1J:36;
then A241: ( Lower_Arc C meets (L~ go) \/ (L~ pion1) or Lower_Arc C meets L~ do ) by A141, XBOOLE_1:70;
A242: 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 A241, XBOOLE_1:70;
suppose Lower_Arc C meets L~ go ; :: thesis: contradiction
then Lower_Arc C meets L~ (Cage C,n) by A46, A143, XBOOLE_1:1, XBOOLE_1:63;
hence contradiction by A242, JORDAN10:5, XBOOLE_1:63; :: thesis: verum
end;
suppose Lower_Arc C meets L~ pion1 ; :: thesis: contradiction
hence contradiction by A8, A81, A148; :: thesis: verum
end;
suppose Lower_Arc C meets L~ do ; :: thesis: contradiction
then Lower_Arc C meets L~ (Cage C,n) by A53, A144, XBOOLE_1:1, XBOOLE_1:63;
hence contradiction by A242, JORDAN10:5, XBOOLE_1:63; :: thesis: verum
end;
end;