let n be Nat; :: thesis: for C being Simple_closed_curve
for i1, i2, j, k being Nat st 1 < i1 & i1 <= i2 & i2 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} holds
(LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Lower_Arc C
let C be Simple_closed_curve; :: thesis: for i1, i2, j, k being Nat st 1 < i1 & i1 <= i2 & i2 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} holds
(LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Lower_Arc C
let i1, i2, j, k be Nat; :: thesis: ( 1 < i1 & i1 <= i2 & i2 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} implies (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Lower_Arc C )
set G = Gauge (C,n);
set pio = LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)));
set poz = LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)));
set US = Upper_Seq (C,n);
set LS = Lower_Seq (C,n);
assume that
A1: 1 < i1 and
A2: i1 <= i2 and
A3: i2 < len (Gauge (C,n)) and
A4: 1 <= j and
A5: j <= k and
A6: k <= width (Gauge (C,n)) and
A7: ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} and
A8: ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} and
A9: (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) misses Lower_Arc C ; :: thesis: contradiction
set Gij = (Gauge (C,n)) * (i1,j);
A10: j <= width (Gauge (C,n)) by A5, A6, XXREAL_0:2;
A11: i1 < len (Gauge (C,n)) by A2, A3, XXREAL_0:2;
then A12: [i1,j] in Indices (Gauge (C,n)) by A1, A4, A10, MATRIX_0:30;
set Gi1k = (Gauge (C,n)) * (i1,k);
set Gik = (Gauge (C,n)) * (i2,k);
A13: L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> = (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) by TOPREAL3:16;
len (Gauge (C,n)) >= 4 by JORDAN8:10;
then A14: len (Gauge (C,n)) >= 1 by XXREAL_0:2;
then A15: [(len (Gauge (C,n))),j] in Indices (Gauge (C,n)) by A4, A10, MATRIX_0:30;
A16: 1 <= k by A4, A5, XXREAL_0:2;
then A17: [1,k] in Indices (Gauge (C,n)) by A6, A14, MATRIX_0:30;
A18: 1 < i2 by A1, A2, XXREAL_0:2;
then A19: [i2,k] in Indices (Gauge (C,n)) by A3, A6, A16, MATRIX_0:30;
A20: ((Gauge (C,n)) * (i1,k)) `2 = ((Gauge (C,n)) * (1,k)) `2 by A1, A6, A11, A16, GOBOARD5:1
.= ((Gauge (C,n)) * (i2,k)) `2 by A3, A6, A18, A16, GOBOARD5:1 ;
((Gauge (C,n)) * (i1,k)) `1 = ((Gauge (C,n)) * (i1,1)) `1 by A1, A6, A11, A16, GOBOARD5:2
.= ((Gauge (C,n)) * (i1,j)) `1 by A1, A4, A11, A10, GOBOARD5:2 ;
then A21: (Gauge (C,n)) * (i1,k) = |[(((Gauge (C,n)) * (i1,j)) `1),(((Gauge (C,n)) * (i2,k)) `2)]| by A20, EUCLID:53;
A22: [(len (Gauge (C,n))),k] in Indices (Gauge (C,n)) by A6, A16, A14, MATRIX_0:30;
A23: [i1,j] in Indices (Gauge (C,n)) by A1, A4, A11, A10, MATRIX_0:30;
set Wbo = W-bound (L~ (Cage (C,n)));
set Wmin = W-min (L~ (Cage (C,n)));
A24: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def 1;
set Ebo = E-bound (L~ (Cage (C,n)));
set Emax = E-max (L~ (Cage (C,n)));
A25: len (Lower_Seq (C,n)) >= 1 + 2 by JORDAN1E:15;
then A26: len (Lower_Seq (C,n)) >= 1 by XXREAL_0:2;
then A27: 1 in dom (Lower_Seq (C,n)) by FINSEQ_3:25;
then A28: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by PARTFUN1:def 6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;
len (Lower_Seq (C,n)) in dom (Lower_Seq (C,n)) by A26, FINSEQ_3:25;
then A29: (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by PARTFUN1:def 6
.= W-min (L~ (Cage (C,n))) by JORDAN1F:8 ;
set co = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i1,j)));
(Gauge (C,n)) * (i1,j) in {((Gauge (C,n)) * (i1,j))} by TARSKI:def 1;
then A30: (Gauge (C,n)) * (i1,j) in L~ (Lower_Seq (C,n)) by A8, XBOOLE_0:def 4;
(W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * (1,k)) `1 by A6, A16, A24, JORDAN1A:73 ;
then A31: (Gauge (C,n)) * (i1,j) <> (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) by A1, A17, A29, A12, JORDAN1G:7;
then reconsider co = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i1,j))) as being_S-Seq FinSequence of (TOP-REAL 2) by A30, JORDAN3:34;
A32: (Gauge (C,n)) * (i1,j) in rng (Lower_Seq (C,n)) by A1, A4, A11, A30, A10, JORDAN1G:5, JORDAN1J:40;
then A33: co is_sequence_on Gauge (C,n) by JORDAN1G:5, JORDAN1J:39;
(E-max (L~ (Cage (C,n)))) `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by A6, A16, A24, JORDAN1A:71 ;
then A34: (Gauge (C,n)) * (i1,j) <> (Lower_Seq (C,n)) . 1 by A2, A3, A12, A22, A28, JORDAN1G:7;
A35: len co >= 1 + 1 by TOPREAL1:def 8;
then reconsider co = co as V29() being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A33, JGRAPH_1:12, JORDAN8:5;
A36: L~ co c= L~ (Lower_Seq (C,n)) by A30, JORDAN3:42;
A37: [1,j] in Indices (Gauge (C,n)) by A4, A10, A14, MATRIX_0:30;
A38: now :: thesis: not ((Gauge (C,n)) * (i1,j)) `1 = W-bound (L~ (Cage (C,n)))
assume ((Gauge (C,n)) * (i1,j)) `1 = W-bound (L~ (Cage (C,n))) ; :: thesis: contradiction
then ((Gauge (C,n)) * (1,j)) `1 = ((Gauge (C,n)) * (i1,j)) `1 by A4, A10, A24, JORDAN1A:73;
hence contradiction by A1, A23, A37, JORDAN1G:7; :: thesis: verum
end;
set pion = <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>;
A39: (Gauge (C,n)) * (i1,k) in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) by RLTOPSP1:68;
set LA = Lower_Arc C;
A40: (Gauge (C,n)) * (i1,k) in LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) by RLTOPSP1:68;
set go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i2,k)));
A41: len (Upper_Seq (C,n)) >= 3 by JORDAN1E:15;
then len (Upper_Seq (C,n)) >= 1 by XXREAL_0:2;
then 1 in dom (Upper_Seq (C,n)) by FINSEQ_3:25;
then A42: (Upper_Seq (C,n)) . 1 = (Upper_Seq (C,n)) /. 1 by PARTFUN1:def 6
.= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ;
A43: [i1,k] in Indices (Gauge (C,n)) by A1, A6, A11, A16, MATRIX_0:30;
A44: now :: thesis: for n being Nat st n in dom <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> holds
ex i, j being Nat st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. n = (Gauge (C,n)) * (i,j) )
let n be Nat; :: thesis: ( n in dom <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> implies ex i, j being Nat st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. n = (Gauge (C,n)) * (i,j) ) )
assume n in dom <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> ; :: thesis: ex i, j being Nat st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. n = (Gauge (C,n)) * (i,j) )
then n in {1,2,3} by FINSEQ_1:89, FINSEQ_3:1;
then ( n = 1 or n = 2 or n = 3 ) by ENUMSET1:def 1;
hence ex i, j being Nat st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. n = (Gauge (C,n)) * (i,j) ) by A23, A19, A43, FINSEQ_4:18; :: thesis: verum
end;
(Gauge (C,n)) * (i2,k) in {((Gauge (C,n)) * (i2,k))} by TARSKI:def 1;
then A45: (Gauge (C,n)) * (i2,k) in L~ (Upper_Seq (C,n)) by A7, XBOOLE_0:def 4;
(W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * (1,k)) `1 by A6, A16, A24, JORDAN1A:73 ;
then A46: (Gauge (C,n)) * (i2,k) <> (Upper_Seq (C,n)) . 1 by A1, A2, A19, A42, A17, JORDAN1G:7;
then reconsider go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i2,k))) as being_S-Seq FinSequence of (TOP-REAL 2) by A45, JORDAN3:35;
A47: (Gauge (C,n)) * (i2,k) in rng (Upper_Seq (C,n)) by A3, A6, A18, A45, A16, JORDAN1G:4, JORDAN1J:40;
then A48: go is_sequence_on Gauge (C,n) by JORDAN1G:4, JORDAN1J:38;
len co >= 1 by A35, XXREAL_0:2;
then 1 in dom co by FINSEQ_3:25;
then A49: co /. 1 = co . 1 by PARTFUN1:def 6
.= (Gauge (C,n)) * (i1,j) by A30, JORDAN3:23 ;
then A50: LSeg (co,1) = LSeg (((Gauge (C,n)) * (i1,j)),(co /. (1 + 1))) by A35, TOPREAL1:def 3;
A51: {((Gauge (C,n)) * (i1,j))} c= (LSeg (co,1)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>)
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in {((Gauge (C,n)) * (i1,j))} or x in (LSeg (co,1)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) )
assume x in {((Gauge (C,n)) * (i1,j))} ; :: thesis: x in (LSeg (co,1)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>)
then A52: x = (Gauge (C,n)) * (i1,j) by TARSKI:def 1;
(Gauge (C,n)) * (i1,j) in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))) by RLTOPSP1:68;
then (Gauge (C,n)) * (i1,j) in (LSeg (((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j)))) by XBOOLE_0:def 3;
then A53: (Gauge (C,n)) * (i1,j) in L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> by SPRECT_1:8;
(Gauge (C,n)) * (i1,j) in LSeg (co,1) by A50, RLTOPSP1:68;
hence x in (LSeg (co,1)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) by A52, A53, XBOOLE_0:def 4; :: thesis: verum
end;
LSeg (co,1) c= L~ co by TOPREAL3:19;
then LSeg (co,1) c= L~ (Lower_Seq (C,n)) by A36;
then (LSeg (co,1)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) c= {((Gauge (C,n)) * (i1,j))} by A8, A13, XBOOLE_1:26;
then A54: (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) /\ (LSeg (co,1)) = {((Gauge (C,n)) * (i1,j))} by A51;
A55: rng co c= L~ co by A35, SPPOL_2:18;
A56: len go >= 1 + 1 by TOPREAL1:def 8;
then reconsider go = go as V29() being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A48, JGRAPH_1:12, JORDAN8:5;
A57: L~ go c= L~ (Upper_Seq (C,n)) by A45, JORDAN3:41;
A58: len go > 1 by A56, NAT_1:13;
then A59: len go in dom go by FINSEQ_3:25;
then A60: go /. (len go) = go . (len go) by PARTFUN1:def 6
.= (Gauge (C,n)) * (i2,k) by A45, JORDAN3:24 ;
reconsider m = (len go) - 1 as Nat by A59, FINSEQ_3:26;
A61: m + 1 = len go ;
then A62: (len go) -' 1 = m by NAT_D:34;
m >= 1 by A56, XREAL_1:19;
then A63: LSeg (go,m) = LSeg ((go /. m),((Gauge (C,n)) * (i2,k))) by A60, A61, TOPREAL1:def 3;
A64: {((Gauge (C,n)) * (i2,k))} c= (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>)
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in {((Gauge (C,n)) * (i2,k))} or x in (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) )
assume x in {((Gauge (C,n)) * (i2,k))} ; :: thesis: x in (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>)
then A65: x = (Gauge (C,n)) * (i2,k) by TARSKI:def 1;
(Gauge (C,n)) * (i2,k) in LSeg (((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k))) by RLTOPSP1:68;
then (Gauge (C,n)) * (i2,k) in (LSeg (((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j)))) by XBOOLE_0:def 3;
then A66: (Gauge (C,n)) * (i2,k) in L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> by SPRECT_1:8;
(Gauge (C,n)) * (i2,k) in LSeg (go,m) by A63, RLTOPSP1:68;
hence x in (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) by A65, A66, XBOOLE_0:def 4; :: thesis: verum
end;
LSeg (go,m) c= L~ go by TOPREAL3:19;
then LSeg (go,m) c= L~ (Upper_Seq (C,n)) by A57;
then (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) c= {((Gauge (C,n)) * (i2,k))} by A7, A13, XBOOLE_1:26;
then A67: (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) = {((Gauge (C,n)) * (i2,k))} by A64;
A68: go /. 1 = (Upper_Seq (C,n)) /. 1 by A45, SPRECT_3:22
.= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ;
then A69: W-min (L~ (Cage (C,n))) in rng go by FINSEQ_6:42;
A70: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by A27, PARTFUN1:def 6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;
A71: (L~ go) /\ (L~ co) c= {(go /. 1)}
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in (L~ go) /\ (L~ co) or x in {(go /. 1)} )
assume A72: x in (L~ go) /\ (L~ co) ; :: thesis: x in {(go /. 1)}
then A73: x in L~ co by XBOOLE_0:def 4;
A74: now :: thesis: not x = E-max (L~ (Cage (C,n)))
assume x = E-max (L~ (Cage (C,n))) ; :: thesis: contradiction
then A75: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i1,j) by A30, A70, A73, JORDAN1E:7;
((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 = E-bound (L~ (Cage (C,n))) by A4, A10, A24, JORDAN1A:71;
then (E-max (L~ (Cage (C,n)))) `1 <> E-bound (L~ (Cage (C,n))) by A2, A3, A23, A15, A75, JORDAN1G:7;
hence contradiction by EUCLID:52; :: thesis: verum
end;
x in L~ go by A72, XBOOLE_0:def 4;
then x in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A57, A36, A73, XBOOLE_0:def 4;
then x in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16;
then ( x = W-min (L~ (Cage (C,n))) or x = E-max (L~ (Cage (C,n))) ) by TARSKI:def 2;
hence x in {(go /. 1)} by A68, A74, TARSKI:def 1; :: thesis: verum
end;
set W2 = go /. 2;
A76: 2 in dom go by A56, FINSEQ_3:25;
go = mid ((Upper_Seq (C,n)),1,(((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n)))) by A47, JORDAN1G:49
.= (Upper_Seq (C,n)) | (((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n))) by A47, FINSEQ_4:21, FINSEQ_6:116 ;
then A77: go /. 2 = (Upper_Seq (C,n)) /. 2 by A76, FINSEQ_4:70;
A78: rng go c= L~ go by A56, SPPOL_2:18;
A79: go /. 1 = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by A68, JORDAN1F:8
.= co /. (len co) by A30, JORDAN1J:35 ;
{(go /. 1)} c= (L~ go) /\ (L~ co)
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in {(go /. 1)} or x in (L~ go) /\ (L~ co) )
assume x in {(go /. 1)} ; :: thesis: x in (L~ go) /\ (L~ co)
then A80: x = go /. 1 by TARSKI:def 1;
then A81: x in rng go by FINSEQ_6:42;
x in rng co by A79, A80, FINSEQ_6:168;
hence x in (L~ go) /\ (L~ co) by A78, A55, A81, XBOOLE_0:def 4; :: thesis: verum
end;
then A82: (L~ go) /\ (L~ co) = {(go /. 1)} by A71;
now :: thesis: contradiction
per cases ( ( ((Gauge (C,n)) * (i1,j)) `1 <> ((Gauge (C,n)) * (i2,k)) `1 & ((Gauge (C,n)) * (i1,j)) `2 <> ((Gauge (C,n)) * (i2,k)) `2 ) or ((Gauge (C,n)) * (i1,j)) `1 = ((Gauge (C,n)) * (i2,k)) `1 or ((Gauge (C,n)) * (i1,j)) `2 = ((Gauge (C,n)) * (i2,k)) `2 ) ;
suppose ( ((Gauge (C,n)) * (i1,j)) `1 <> ((Gauge (C,n)) * (i2,k)) `1 & ((Gauge (C,n)) * (i1,j)) `2 <> ((Gauge (C,n)) * (i2,k)) `2 ) ; :: thesis: contradiction
then <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> is being_S-Seq by A21, TOPREAL3:35;
then consider pion1 being FinSequence of (TOP-REAL 2) such that
A83: pion1 is_sequence_on Gauge (C,n) and
A84: pion1 is being_S-Seq and
A85: L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> = L~ pion1 and
A86: <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. 1 = pion1 /. 1 and
A87: <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. (len <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) = pion1 /. (len pion1) and
A88: len <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> <= len pion1 by A44, GOBOARD3:2;
reconsider pion1 = pion1 as being_S-Seq FinSequence of (TOP-REAL 2) by A84;
A89: (go ^' pion1) /. (len (go ^' pion1)) = <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. (len <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) by A87, FINSEQ_6:156
.= <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. 3 by FINSEQ_1:45
.= co /. 1 by A49, FINSEQ_4:18 ;
A90: go /. (len go) = pion1 /. 1 by A60, A86, FINSEQ_4:18;
A91: (L~ go) /\ (L~ pion1) c= {(pion1 /. 1)}
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in (L~ go) /\ (L~ pion1) or x in {(pion1 /. 1)} )
assume A92: x in (L~ go) /\ (L~ pion1) ; :: thesis: x in {(pion1 /. 1)}
then A93: x in L~ pion1 by XBOOLE_0:def 4;
x in L~ go by A92, XBOOLE_0:def 4;
hence x in {(pion1 /. 1)} by A7, A13, A60, A57, A85, A90, A93, XBOOLE_0:def 4; :: thesis: verum
end;
len pion1 >= 2 + 1 by A88, FINSEQ_1:45;
then A94: len pion1 > 1 + 1 by NAT_1:13;
then A95: rng pion1 c= L~ pion1 by SPPOL_2:18;
{(pion1 /. 1)} c= (L~ go) /\ (L~ pion1)
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in {(pion1 /. 1)} or x in (L~ go) /\ (L~ pion1) )
assume x in {(pion1 /. 1)} ; :: thesis: x in (L~ go) /\ (L~ pion1)
then A96: x = pion1 /. 1 by TARSKI:def 1;
then A97: x in rng pion1 by FINSEQ_6:42;
x in rng go by A90, A96, FINSEQ_6:168;
hence x in (L~ go) /\ (L~ pion1) by A78, A95, A97, XBOOLE_0:def 4; :: thesis: verum
end;
then A98: (L~ go) /\ (L~ pion1) = {(pion1 /. 1)} by A91;
then A99: go ^' pion1 is s.n.c. by A90, JORDAN1J:54;
A100: <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. (len <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) = <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. 3 by FINSEQ_1:45
.= co /. 1 by A49, FINSEQ_4:18 ;
A101: {(pion1 /. (len pion1))} c= (L~ co) /\ (L~ pion1)
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in {(pion1 /. (len pion1))} or x in (L~ co) /\ (L~ pion1) )
assume x in {(pion1 /. (len pion1))} ; :: thesis: x in (L~ co) /\ (L~ pion1)
then A102: x = pion1 /. (len pion1) by TARSKI:def 1;
then A103: x in rng pion1 by FINSEQ_6:168;
x in rng co by A87, A100, A102, FINSEQ_6:42;
hence x in (L~ co) /\ (L~ pion1) by A55, A95, A103, XBOOLE_0:def 4; :: thesis: verum
end;
(L~ co) /\ (L~ pion1) c= {(pion1 /. (len pion1))}
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in (L~ co) /\ (L~ pion1) or x in {(pion1 /. (len pion1))} )
assume A104: x in (L~ co) /\ (L~ pion1) ; :: thesis: x in {(pion1 /. (len pion1))}
then A105: x in L~ pion1 by XBOOLE_0:def 4;
x in L~ co by A104, XBOOLE_0:def 4;
hence x in {(pion1 /. (len pion1))} by A8, A13, A49, A36, A85, A87, A100, A105, XBOOLE_0:def 4; :: thesis: verum
end;
then A106: (L~ co) /\ (L~ pion1) = {(pion1 /. (len pion1))} by A101;
A107: (L~ (go ^' pion1)) /\ (L~ co) = ((L~ go) \/ (L~ pion1)) /\ (L~ co) by A90, TOPREAL8:35
.= {(go /. 1)} \/ {(co /. 1)} by A82, A87, A100, A106, XBOOLE_1:23
.= {((go ^' pion1) /. 1)} \/ {(co /. 1)} by FINSEQ_6:155
.= {((go ^' pion1) /. 1),(co /. 1)} by ENUMSET1:1 ;
A108: Lower_Arc C is_an_arc_of E-max C, W-min C by JORDAN6:def 9;
then A109: Lower_Arc C is connected by JORDAN6:10;
set godo = (go ^' pion1) ^' co;
A110: co /. (len co) = (go ^' pion1) /. 1 by A79, FINSEQ_6:155;
A111: go ^' pion1 is_sequence_on Gauge (C,n) by A48, A83, A90, TOPREAL8:12;
then A112: (go ^' pion1) ^' co is_sequence_on Gauge (C,n) by A33, A89, TOPREAL8:12;
A113: (len pion1) - 1 >= 1 by A94, XREAL_1:19;
then A114: (len pion1) -' 1 = (len pion1) - 1 by XREAL_0:def 2;
A115: {((Gauge (C,n)) * (i1,j))} c= (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (co,1))
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in {((Gauge (C,n)) * (i1,j))} or x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (co,1)) )
assume x in {((Gauge (C,n)) * (i1,j))} ; :: thesis: x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (co,1))
then A116: x = (Gauge (C,n)) * (i1,j) by TARSKI:def 1;
pion1 /. (((len pion1) -' 1) + 1) = <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. 3 by A87, A114, FINSEQ_1:45
.= (Gauge (C,n)) * (i1,j) by FINSEQ_4:18 ;
then A117: (Gauge (C,n)) * (i1,j) in LSeg (pion1,((len pion1) -' 1)) by A113, A114, TOPREAL1:21;
(Gauge (C,n)) * (i1,j) in LSeg (co,1) by A50, RLTOPSP1:68;
hence x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (co,1)) by A116, A117, XBOOLE_0:def 4; :: thesis: verum
end;
LSeg (pion1,((len pion1) -' 1)) c= L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> by A85, TOPREAL3:19;
then (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (co,1)) c= {((Gauge (C,n)) * (i1,j))} by A54, XBOOLE_1:27;
then A118: (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (co,1)) = {((Gauge (C,n)) * (i1,j))} by A115;
((len pion1) - 1) + 1 <= len pion1 ;
then A119: (len pion1) -' 1 < len pion1 by A114, NAT_1:13;
len pion1 >= 2 + 1 by A88, FINSEQ_1:45;
then A120: (len pion1) - 2 >= 0 by XREAL_1:19;
then ((len pion1) -' 2) + 1 = ((len pion1) - 2) + 1 by XREAL_0:def 2
.= (len pion1) -' 1 by A113, XREAL_0:def 2 ;
then A121: (LSeg ((go ^' pion1),((len go) + ((len pion1) -' 2)))) /\ (LSeg (co,1)) = {((go ^' pion1) /. (len (go ^' pion1)))} by A49, A90, A89, A119, A118, TOPREAL8:31;
(rng go) /\ (rng pion1) c= {(pion1 /. 1)} by A78, A95, A98, XBOOLE_1:27;
then A122: go ^' pion1 is one-to-one by JORDAN1J:55;
((len (go ^' pion1)) + 1) - 1 = ((len go) + (len pion1)) - 1 by FINSEQ_6:139;
then (len (go ^' pion1)) - 1 = (len go) + ((len pion1) - 2)
.= (len go) + ((len pion1) -' 2) by A120, XREAL_0:def 2 ;
then A123: (len (go ^' pion1)) -' 1 = (len go) + ((len pion1) -' 2) by XREAL_0:def 2;
A124: L~ (Cage (C,n)) = (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n))) by JORDAN1E:13;
then A125: L~ (Upper_Seq (C,n)) c= L~ (Cage (C,n)) by XBOOLE_1:7;
then A126: L~ go c= L~ (Cage (C,n)) by A57;
A127: {((Gauge (C,n)) * (i2,k))} c= (LSeg (go,m)) /\ (LSeg (pion1,1))
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in {((Gauge (C,n)) * (i2,k))} or x in (LSeg (go,m)) /\ (LSeg (pion1,1)) )
assume x in {((Gauge (C,n)) * (i2,k))} ; :: thesis: x in (LSeg (go,m)) /\ (LSeg (pion1,1))
then A128: x = (Gauge (C,n)) * (i2,k) by TARSKI:def 1;
A129: (Gauge (C,n)) * (i2,k) in LSeg (go,m) by A63, RLTOPSP1:68;
(Gauge (C,n)) * (i2,k) in LSeg (pion1,1) by A60, A90, A94, TOPREAL1:21;
hence x in (LSeg (go,m)) /\ (LSeg (pion1,1)) by A128, A129, XBOOLE_0:def 4; :: thesis: verum
end;
LSeg (pion1,1) c= L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> by A85, TOPREAL3:19;
then (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) c= {((Gauge (C,n)) * (i2,k))} by A62, A67, XBOOLE_1:27;
then (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) = {(go /. (len go))} by A60, A62, A127;
then A130: go ^' pion1 is unfolded by A90, TOPREAL8:34;
len (go ^' pion1) >= len go by TOPREAL8:7;
then A131: len (go ^' pion1) >= 1 + 1 by A56, XXREAL_0:2;
then A132: len (go ^' pion1) > 1 + 0 by NAT_1:13;
A133: now :: thesis: not ((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n)) <= 1
assume A134: ((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n)) <= 1 ; :: thesis: contradiction
((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n)) >= 1 by A47, FINSEQ_4:21;
then ((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n)) = 1 by A134, XXREAL_0:1;
then (Gauge (C,n)) * (i2,k) = (Upper_Seq (C,n)) /. 1 by A47, FINSEQ_5:38;
hence contradiction by A42, A46, JORDAN1F:5; :: thesis: verum
end;
A135: Upper_Seq (C,n) is_sequence_on Gauge (C,n) by JORDAN1G:4;
A136: (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52;
set ff = Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))));
A137: 1 + 1 <= len (Cage (C,n)) by GOBOARD7:34, XXREAL_0:2;
A138: len ((go ^' pion1) ^' co) >= len (go ^' pion1) by TOPREAL8:7;
then A139: 1 + 1 <= len ((go ^' pion1) ^' co) by A131, XXREAL_0:2;
not go ^' pion1 is trivial by A131, NAT_D:60;
then reconsider godo = (go ^' pion1) ^' co as V29() standard special_circular_sequence by A139, A89, A112, A130, A123, A121, A99, A122, A107, A110, JORDAN8:4, JORDAN8:5, TOPREAL8:11, TOPREAL8:33, TOPREAL8:34;
A140: L~ godo = (L~ (go ^' pion1)) \/ (L~ co) by A89, TOPREAL8:35
.= ((L~ go) \/ (L~ pion1)) \/ (L~ co) by A90, TOPREAL8:35 ;
A141: (right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo) c= RightComp godo by A139, A112, JORDAN9:27;
2 in dom godo by A139, FINSEQ_3:25;
then A142: godo /. 2 in rng godo by PARTFUN2:2;
A143: W-min C in Lower_Arc C by A108, TOPREAL1:1;
W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
then A144: (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 = W-min (L~ (Cage (C,n))) by FINSEQ_6:92;
A145: L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33;
then (W-max (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A144, SPRECT_5:22;
then (N-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A144, A145, SPRECT_5:23, XXREAL_0:2;
then (N-max (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A144, A145, SPRECT_5:24, XXREAL_0:2;
then A146: (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A144, A145, SPRECT_5:25, XXREAL_0:2;
A147: Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def 1;
then A148: Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))) is_sequence_on Gauge (C,n) by REVROT_1:34;
A149: ((Gauge (C,n)) * (i1,k)) `1 = ((Gauge (C,n)) * (i1,1)) `1 by A1, A6, A11, A16, GOBOARD5:2
.= ((Gauge (C,n)) * (i1,j)) `1 by A1, A4, A11, A10, GOBOARD5:2 ;
then A150: W-bound (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) = ((Gauge (C,n)) * (i1,j)) `1 by SPRECT_1:54;
A151: L~ (Lower_Seq (C,n)) c= L~ (Cage (C,n)) by A124, XBOOLE_1:7;
then A152: L~ co c= L~ (Cage (C,n)) by A36;
A153: W-min C in C by SPRECT_1:13;
A154: now :: thesis: not W-min C in L~ godo
assume W-min C in L~ godo ; :: thesis: contradiction
then A155: ( W-min C in (L~ go) \/ (L~ pion1) or W-min C in L~ co ) by A140, XBOOLE_0:def 3;
end;
A156: len (Upper_Seq (C,n)) >= 2 by A41, XXREAL_0:2;
A157: (L~ go) \/ (L~ co) is compact by COMPTS_1:10;
1 + 1 <= len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by GOBOARD7:34, XXREAL_0:2;
then right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1) = right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(GoB (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) by JORDAN1H:23
.= 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:44
.= right_cell (((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n))))),1,(Gauge (C,n))) by A146, A148, JORDAN1J:53
.= right_cell ((Upper_Seq (C,n)),1,(Gauge (C,n))) by JORDAN1E:def 1
.= right_cell ((R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i2,k)))),1,(Gauge (C,n))) by A47, A135, A133, JORDAN1J:52
.= right_cell ((go ^' pion1),1,(Gauge (C,n))) by A58, A111, JORDAN1J:51
.= right_cell (godo,1,(Gauge (C,n))) by A132, A112, JORDAN1J:51 ;
then W-min C in right_cell (godo,1,(Gauge (C,n))) by JORDAN1I:6;
then A158: W-min C in (right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo) by A154, XBOOLE_0:def 5;
A159: rng godo c= L~ godo by A131, A138, SPPOL_2:18, XXREAL_0:2;
A160: godo /. 1 = (go ^' pion1) /. 1 by FINSEQ_6:155
.= W-min (L~ (Cage (C,n))) by A68, FINSEQ_6:155 ;
A161: ((Gauge (C,n)) * (i1,k)) `1 <= ((Gauge (C,n)) * (i2,k)) `1 by A1, A2, A3, A6, A16, JORDAN1A:18;
then A162: W-bound (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) = ((Gauge (C,n)) * (i1,k)) `1 by SPRECT_1:54;
W-bound ((LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))))) = min ((W-bound (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))),(W-bound (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))))) by SPRECT_1:47
.= ((Gauge (C,n)) * (i1,j)) `1 by A149, A162, A150 ;
then A163: W-bound (L~ pion1) = ((Gauge (C,n)) * (i1,j)) `1 by A85, TOPREAL3:16;
A164: Lower_Arc C c= C by JORDAN6:61;
((Gauge (C,n)) * (i1,j)) `1 >= W-bound (L~ (Cage (C,n))) by A30, A151, PSCOMP_1:24;
then A165: ((Gauge (C,n)) * (i1,j)) `1 > W-bound (L~ (Cage (C,n))) by A38, XXREAL_0:1;
A166: E-max C in Lower_Arc C by A108, TOPREAL1:1;
W-min (L~ (Cage (C,n))) in (L~ go) \/ (L~ co) by A78, A69, XBOOLE_0:def 3;
then A167: W-min ((L~ go) \/ (L~ co)) = W-min (L~ (Cage (C,n))) by A126, A152, A157, JORDAN1J:21, XBOOLE_1:8;
(W-min ((L~ go) \/ (L~ co))) `1 = W-bound ((L~ go) \/ (L~ co)) by EUCLID:52;
then W-min (((L~ go) \/ (L~ co)) \/ (L~ pion1)) = W-min ((L~ go) \/ (L~ co)) by A163, A157, A167, A136, A165, JORDAN1J:33;
then A168: W-min (L~ godo) = W-min (L~ (Cage (C,n))) by A140, A167, XBOOLE_1:4;
godo /. 2 = (go ^' pion1) /. 2 by A131, FINSEQ_6:159
.= (Upper_Seq (C,n)) /. 2 by A56, A77, FINSEQ_6:159
.= ((Upper_Seq (C,n)) ^' (Lower_Seq (C,n))) /. 2 by A156, FINSEQ_6:159
.= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 2 by JORDAN1E:11 ;
then godo /. 2 in W-most (L~ (Cage (C,n))) by JORDAN1I:25;
then (godo /. 2) `1 = (W-min (L~ godo)) `1 by A168, PSCOMP_1:31
.= W-bound (L~ godo) by EUCLID:52 ;
then godo /. 2 in W-most (L~ godo) by A159, A142, SPRECT_2:12;
then (Rotate (godo,(W-min (L~ godo)))) /. 2 in W-most (L~ godo) by A160, A168, FINSEQ_6:89;
then reconsider godo = godo as V29() standard clockwise_oriented special_circular_sequence by JORDAN1I:25;
len (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by FINSEQ_5:6;
then A169: (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) = (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) by PARTFUN1:def 6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:7 ;
A170: east_halfline (E-max C) misses L~ go
proof
assume east_halfline (E-max C) meets L~ go ; :: thesis: contradiction
then consider p being object such that
A171: p in east_halfline (E-max C) and
A172: p in L~ go by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A171;
p in L~ (Upper_Seq (C,n)) by A57, A172;
then p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A125, A171, XBOOLE_0:def 4;
then A173: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;
then A174: p = E-max (L~ (Cage (C,n))) by A57, A172, JORDAN1J:46;
then E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i2,k) by A45, A169, A172, JORDAN1J:43;
then ((Gauge (C,n)) * (i2,k)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by A6, A16, A24, A173, A174, JORDAN1A:71;
hence contradiction by A3, A19, A22, JORDAN1G:7; :: thesis: verum
end;
now :: thesis: not east_halfline (E-max C) meets L~ godo
assume east_halfline (E-max C) meets L~ godo ; :: thesis: contradiction
then A175: ( east_halfline (E-max C) meets (L~ go) \/ (L~ pion1) or east_halfline (E-max C) meets L~ co ) by A140, XBOOLE_1:70;
per cases ( east_halfline (E-max C) meets L~ go or east_halfline (E-max C) meets L~ pion1 or east_halfline (E-max C) meets L~ co ) by A175, XBOOLE_1:70;
suppose east_halfline (E-max C) meets L~ pion1 ; :: thesis: contradiction
then consider p being object such that
A176: p in east_halfline (E-max C) and
A177: p in L~ pion1 by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A176;
A178: p `2 = (E-max C) `2 by A176, TOPREAL1:def 11;
A179: now :: thesis: p `1 <= ((Gauge (C,n)) * (i2,k)) `1
per cases ( p in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) or p in LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) ) by A13, A85, A177, XBOOLE_0:def 3;
suppose p in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) ; :: thesis: p `1 <= ((Gauge (C,n)) * (i2,k)) `1
hence p `1 <= ((Gauge (C,n)) * (i2,k)) `1 by A161, TOPREAL1:3; :: thesis: verum
end;
suppose p in LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) ; :: thesis: p `1 <= ((Gauge (C,n)) * (i2,k)) `1
hence p `1 <= ((Gauge (C,n)) * (i2,k)) `1 by A149, A161, GOBOARD7:5; :: thesis: verum
end;
end;
end;
i2 + 1 <= len (Gauge (C,n)) by A3, NAT_1:13;
then (i2 + 1) - 1 <= (len (Gauge (C,n))) - 1 by XREAL_1:9;
then A180: i2 <= (len (Gauge (C,n))) -' 1 by XREAL_0:def 2;
(len (Gauge (C,n))) -' 1 <= len (Gauge (C,n)) by NAT_D:35;
then ((Gauge (C,n)) * (i2,k)) `1 <= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),1)) `1 by A6, A18, A16, A24, A14, A180, JORDAN1A:18;
then p `1 <= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),1)) `1 by A179, XXREAL_0:2;
then p `1 <= E-bound C by A14, JORDAN8:12;
then A181: p `1 <= (E-max C) `1 by EUCLID:52;
p `1 >= (E-max C) `1 by A176, TOPREAL1:def 11;
then p `1 = (E-max C) `1 by A181, XXREAL_0:1;
then p = E-max C by A178, TOPREAL3:6;
hence contradiction by A9, A13, A85, A166, A177, XBOOLE_0:3; :: thesis: verum
end;
suppose east_halfline (E-max C) meets L~ co ; :: thesis: contradiction
then consider p being object such that
A182: p in east_halfline (E-max C) and
A183: p in L~ co by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A182;
A184: p in LSeg (co,(Index (p,co))) by A183, JORDAN3:9;
consider t being Nat such that
A185: t in dom (Lower_Seq (C,n)) and
A186: (Lower_Seq (C,n)) . t = (Gauge (C,n)) * (i1,j) by A32, FINSEQ_2:10;
1 <= t by A185, FINSEQ_3:25;
then A187: 1 < t by A34, A186, XXREAL_0:1;
t <= len (Lower_Seq (C,n)) by A185, FINSEQ_3:25;
then (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) + 1 = t by A186, A187, JORDAN3:12;
then A188: len (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i1,j)))) = (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) by A30, A186, JORDAN3:26;
Index (p,co) < len co by A183, JORDAN3:8;
then Index (p,co) < (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) by A188, XREAL_0:def 2;
then (Index (p,co)) + 1 <= (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) by NAT_1:13;
then A189: Index (p,co) <= ((len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n))))) - 1 by XREAL_1:19;
A190: co = mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n)))) by A32, JORDAN1J:37;
p in L~ (Lower_Seq (C,n)) by A36, A183;
then p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A151, A182, XBOOLE_0:def 4;
then A191: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;
A192: (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) + 1 = ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) by A34, A32, JORDAN1J:56;
0 + (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) < len (Lower_Seq (C,n)) by A30, JORDAN3:8;
then (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) > 0 by XREAL_1:20;
then Index (p,co) <= ((len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n))))) - 1 by A189, XREAL_0:def 2;
then Index (p,co) <= (len (Lower_Seq (C,n))) - (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))) by A192;
then Index (p,co) <= (len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))) by XREAL_0:def 2;
then A193: Index (p,co) < ((len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) + 1 by NAT_1:13;
A194: 1 <= Index (p,co) by A183, JORDAN3:8;
A195: ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) <= len (Lower_Seq (C,n)) by A32, FINSEQ_4:21;
((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) <> len (Lower_Seq (C,n)) by A31, A32, FINSEQ_4:19;
then A196: ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) < len (Lower_Seq (C,n)) by A195, XXREAL_0:1;
A197: 1 + 1 <= len (Lower_Seq (C,n)) by A25, XXREAL_0:2;
then A198: 2 in dom (Lower_Seq (C,n)) by FINSEQ_3:25;
set tt = ((Index (p,co)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1;
set RC = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))));
A199: E-max C in right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by JORDAN1I:7;
A200: GoB (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = GoB (Cage (C,n)) by REVROT_1:28
.= Gauge (C,n) by JORDAN1H:44 ;
A201: L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33;
consider jj2 being Nat such that
A202: 1 <= jj2 and
A203: jj2 <= width (Gauge (C,n)) and
A204: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * ((len (Gauge (C,n))),jj2) by JORDAN1D:25;
A205: len (Gauge (C,n)) >= 4 by JORDAN8:10;
then len (Gauge (C,n)) >= 1 by XXREAL_0:2;
then A206: [(len (Gauge (C,n))),jj2] in Indices (Gauge (C,n)) by A202, A203, MATRIX_0:30;
A207: len (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = len (Cage (C,n)) by FINSEQ_6:179;
Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) by JORDAN1G:18;
then A208: LSeg ((Lower_Seq (C,n)),1) = LSeg ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by A197, SPPOL_2:9;
A209: E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))) is_sequence_on Gauge (C,n) by A147, REVROT_1:34;
then consider ii, jj being Nat such that
A210: [ii,(jj + 1)] in Indices (Gauge (C,n)) and
A211: [ii,jj] in Indices (Gauge (C,n)) and
A212: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = (Gauge (C,n)) * (ii,(jj + 1)) and
A213: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1) = (Gauge (C,n)) * (ii,jj) by A137, A201, A207, A209, FINSEQ_6:92, JORDAN1I:23;
A214: (jj + 1) + 1 <> jj ;
A215: 1 <= jj by A211, MATRIX_0:32;
(Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = E-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))))) by A201, A209, FINSEQ_6:92;
then A216: ii = len (Gauge (C,n)) by A201, A210, A212, A204, A206, GOBOARD1:5;
then ii - 1 >= 4 - 1 by A205, XREAL_1:9;
then A217: ii - 1 >= 1 by XXREAL_0:2;
then A218: 1 <= ii -' 1 by XREAL_0:def 2;
A219: jj <= width (Gauge (C,n)) by A211, MATRIX_0:32;
then A220: ((Gauge (C,n)) * ((len (Gauge (C,n))),jj)) `1 = E-bound (L~ (Cage (C,n))) by A24, A215, JORDAN1A:71;
A221: jj + 1 <= width (Gauge (C,n)) by A210, MATRIX_0:32;
ii + 1 <> ii ;
then A222: right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) = cell ((Gauge (C,n)),(ii -' 1),jj) by A137, A207, A200, A210, A211, A212, A213, A214, GOBOARD5:def 6;
A223: ii <= len (Gauge (C,n)) by A211, MATRIX_0:32;
A224: 1 <= ii by A211, MATRIX_0:32;
A225: ii <= len (Gauge (C,n)) by A210, MATRIX_0:32;
A226: 1 <= jj + 1 by A210, MATRIX_0:32;
then A227: E-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * ((len (Gauge (C,n))),(jj + 1))) `1 by A24, A221, JORDAN1A:71;
A228: 1 <= ii by A210, MATRIX_0:32;
then A229: (ii -' 1) + 1 = ii by XREAL_1:235;
then A230: ii -' 1 < len (Gauge (C,n)) by A225, NAT_1:13;
then A231: ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 = ((Gauge (C,n)) * (1,(jj + 1))) `2 by A226, A221, A218, GOBOARD5:1
.= ((Gauge (C,n)) * (ii,(jj + 1))) `2 by A228, A225, A226, A221, GOBOARD5:1 ;
A232: (E-max C) `2 = p `2 by A182, TOPREAL1:def 11;
then A233: p `2 <= ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 by A199, A225, A221, A215, A222, A229, A217, JORDAN9:17;
A234: ((Gauge (C,n)) * ((ii -' 1),jj)) `2 = ((Gauge (C,n)) * (1,jj)) `2 by A215, A219, A218, A230, GOBOARD5:1
.= ((Gauge (C,n)) * (ii,jj)) `2 by A224, A223, A215, A219, GOBOARD5:1 ;
((Gauge (C,n)) * ((ii -' 1),jj)) `2 <= p `2 by A232, A199, A225, A221, A215, A222, A229, A217, JORDAN9:17;
then p in LSeg (((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1),((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1))) by A191, A212, A213, A216, A233, A234, A231, A220, A227, GOBOARD7:7;
then A235: p in LSeg ((Lower_Seq (C,n)),1) by A137, A208, A207, TOPREAL1:def 3;
1 <= ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) by A32, FINSEQ_4:21;
then A236: LSeg ((mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n))))),(Index (p,co))) = LSeg ((Lower_Seq (C,n)),(((Index (p,co)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1)) by A196, A194, A193, JORDAN4:19;
1 <= Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n))) by A30, JORDAN3:8;
then A237: 1 + 1 <= ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) by A192, XREAL_1:7;
then (Index (p,co)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))) >= (1 + 1) + 1 by A194, XREAL_1:7;
then ((Index (p,co)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) - 1 >= ((1 + 1) + 1) - 1 by XREAL_1:9;
then A238: ((Index (p,co)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1 >= 1 + 1 by XREAL_0:def 2;
now :: thesis: contradiction
per cases ( ((Index (p,co)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1 > 1 + 1 or ((Index (p,co)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ) by A238, XXREAL_0:1;
suppose ((Index (p,co)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1 > 1 + 1 ; :: thesis: contradiction
then LSeg ((Lower_Seq (C,n)),1) misses LSeg ((Lower_Seq (C,n)),(((Index (p,co)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1)) by TOPREAL1:def 7;
hence contradiction by A235, A184, A190, A236, XBOOLE_0:3; :: thesis: verum
end;
suppose A239: ((Index (p,co)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ; :: thesis: contradiction
then 1 + 1 = ((Index (p,co)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) - 1 by XREAL_0:def 2;
then (1 + 1) + 1 = (Index (p,co)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))) ;
then A240: ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) = 2 by A194, A237, JORDAN1E:6;
(LSeg ((Lower_Seq (C,n)),1)) /\ (LSeg ((Lower_Seq (C,n)),(((Index (p,co)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1))) = {((Lower_Seq (C,n)) /. 2)} by A25, A239, TOPREAL1:def 6;
then p in {((Lower_Seq (C,n)) /. 2)} by A235, A184, A190, A236, XBOOLE_0:def 4;
then A241: p = (Lower_Seq (C,n)) /. 2 by TARSKI:def 1;
then A242: p in rng (Lower_Seq (C,n)) by A198, PARTFUN2:2;
p .. (Lower_Seq (C,n)) = 2 by A198, A241, FINSEQ_5:41;
then p = (Gauge (C,n)) * (i1,j) by A32, A240, A242, FINSEQ_5:9;
then ((Gauge (C,n)) * (i1,j)) `1 = E-bound (L~ (Cage (C,n))) by A241, JORDAN1G:32;
then ((Gauge (C,n)) * (i1,j)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 by A4, A10, A24, JORDAN1A:71;
hence contradiction by A2, A3, A23, A15, JORDAN1G:7; :: thesis: verum
end;
end;
end;
hence contradiction ; :: thesis: verum
end;
end;
end;
then east_halfline (E-max C) c= (L~ godo) ` by SUBSET_1:23;
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 GOBOARD9:3;
not W is bounded by A244, JORDAN2C:121, RLTOPSP1:42;
then W is_outside_component_of L~ godo by A243, JORDAN2C:def 3;
then W c= UBD (L~ godo) by JORDAN2C:23;
then A245: east_halfline (E-max C) c= UBD (L~ godo) by A244;
E-max C in east_halfline (E-max C) by TOPREAL1:38;
then E-max C in UBD (L~ godo) by A245;
then E-max C in LeftComp godo by GOBRD14:36;
then Lower_Arc C meets L~ godo by A109, A143, A166, A141, A158, JORDAN1J:36;
then A246: ( Lower_Arc C meets (L~ go) \/ (L~ pion1) or Lower_Arc C meets L~ co ) by A140, XBOOLE_1:70;
hence contradiction ; :: thesis: verum
end;
suppose ((Gauge (C,n)) * (i1,j)) `1 = ((Gauge (C,n)) * (i2,k)) `1 ; :: thesis: contradiction
then A247: i1 = i2 by A23, A19, JORDAN1G:7;
then LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) = {((Gauge (C,n)) * (i1,k))} by RLTOPSP1:70;
then LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) c= LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) by A40, ZFMISC_1:31;
then (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) = LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) by XBOOLE_1:12;
hence contradiction by A1, A3, A4, A5, A6, A7, A8, A9, A247, JORDAN1J:58; :: thesis: verum
end;
suppose ((Gauge (C,n)) * (i1,j)) `2 = ((Gauge (C,n)) * (i2,k)) `2 ; :: thesis: contradiction
then A248: j = k by A23, A19, JORDAN1G:6;
then LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) = {((Gauge (C,n)) * (i1,k))} by RLTOPSP1:70;
then LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) c= LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) by A39, ZFMISC_1:31;
then (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) = LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) by XBOOLE_1:12;
hence contradiction by A1, A2, A3, A4, A6, A7, A8, A9, A248, Th28; :: thesis: verum
end;
end;
end;
hence contradiction ; :: thesis: verum