let n be Nat; :: thesis: for C being Simple_closed_curve
for i1, i2, j, k being Nat st 1 < i2 & i2 <= i1 & i1 < 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 Upper_Arc C
let C be Simple_closed_curve; :: thesis: for i1, i2, j, k being Nat st 1 < i2 & i2 <= i1 & i1 < 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 Upper_Arc C
let i1, i2, j, k be Nat; :: thesis: ( 1 < i2 & i2 <= i1 & i1 < 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 Upper_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 < i2 and
A2: i2 <= i1 and
A3: i1 < 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 Upper_Arc C ; :: thesis: contradiction
set Gi1k = (Gauge (C,n)) * (i1,k);
set Gik = (Gauge (C,n)) * (i2,k);
A10: 1 <= k by A4, A5, XXREAL_0:2;
A11: i2 < len (Gauge (C,n)) by A2, A3, XXREAL_0:2;
then A12: [i2,k] in Indices (Gauge (C,n)) by A1, A6, A10, MATRIX_0:30;
set Wmin = W-min (L~ (Cage (C,n)));
set Wbo = W-bound (L~ (Cage (C,n)));
A13: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def 1;
set go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i2,k)));
A14: 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 A15: (Upper_Seq (C,n)) . 1 = (Upper_Seq (C,n)) /. 1 by PARTFUN1:def 6
.= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ;
set Gij = (Gauge (C,n)) * (i1,j);
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 A16: (Gauge (C,n)) * (i1,j) in L~ (Lower_Seq (C,n)) by A8, XBOOLE_0:def 4;
A17: 1 < i1 by A1, A2, XXREAL_0:2;
then A18: ((Gauge (C,n)) * (i1,k)) `2 = ((Gauge (C,n)) * (1,k)) `2 by A3, A6, A10, GOBOARD5:1
.= ((Gauge (C,n)) * (i2,k)) `2 by A1, A6, A11, A10, GOBOARD5:1 ;
A19: j <= width (Gauge (C,n)) by A5, A6, XXREAL_0:2;
then A20: [i1,j] in Indices (Gauge (C,n)) by A3, A4, A17, MATRIX_0:30;
len (Gauge (C,n)) >= 4 by JORDAN8:10;
then A21: len (Gauge (C,n)) >= 1 by XXREAL_0:2;
then A22: [(len (Gauge (C,n))),j] in Indices (Gauge (C,n)) by A4, A19, MATRIX_0:30;
A23: [1,k] in Indices (Gauge (C,n)) by A6, A10, A21, MATRIX_0:30;
A24: now :: thesis: not ((Gauge (C,n)) * (i2,k)) `1 = W-bound (L~ (Cage (C,n)))
assume ((Gauge (C,n)) * (i2,k)) `1 = W-bound (L~ (Cage (C,n))) ; :: thesis: contradiction
then ((Gauge (C,n)) * (1,k)) `1 = ((Gauge (C,n)) * (i2,k)) `1 by A6, A10, A13, JORDAN1A:73;
hence contradiction by A1, A12, A23, JORDAN1G:7; :: thesis: verum
end;
A25: [i1,j] in Indices (Gauge (C,n)) by A3, A4, A17, A19, MATRIX_0:30;
set pion = <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>;
A26: (Gauge (C,n)) * (i1,k) in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) by RLTOPSP1:68;
set UA = Upper_Arc C;
A27: (Gauge (C,n)) * (i1,k) in LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) by RLTOPSP1:68;
A28: [i1,k] in Indices (Gauge (C,n)) by A3, A6, A17, A10, MATRIX_0:30;
A29: 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 A25, A12, A28, FINSEQ_4:18; :: thesis: verum
end;
(Gauge (C,n)) * (i2,k) in {((Gauge (C,n)) * (i2,k))} by TARSKI:def 1;
then A30: (Gauge (C,n)) * (i2,k) in L~ (Upper_Seq (C,n)) by A7, XBOOLE_0:def 4;
set Emax = E-max (L~ (Cage (C,n)));
A31: len (Lower_Seq (C,n)) >= 1 + 2 by JORDAN1E:15;
then A32: len (Lower_Seq (C,n)) >= 1 by XXREAL_0:2;
then A33: 1 in dom (Lower_Seq (C,n)) by FINSEQ_3:25;
then A34: (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 A32, FINSEQ_3:25;
then A35: (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 Ebo = E-bound (L~ (Cage (C,n)));
A36: 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;
(W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * (1,k)) `1 by A6, A10, A13, JORDAN1A:73 ;
then A37: (Gauge (C,n)) * (i2,k) <> (Upper_Seq (C,n)) . 1 by A1, A12, A15, A23, 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 A30, JORDAN3:35;
A38: (Gauge (C,n)) * (i2,k) in rng (Upper_Seq (C,n)) by A1, A6, A11, A30, A10, JORDAN1G:4, JORDAN1J:40;
then A39: go is_sequence_on Gauge (C,n) by JORDAN1G:4, JORDAN1J:38;
((Gauge (C,n)) * (i1,k)) `1 = ((Gauge (C,n)) * (i1,1)) `1 by A3, A6, A17, A10, GOBOARD5:2
.= ((Gauge (C,n)) * (i1,j)) `1 by A3, A4, A17, A19, GOBOARD5:2 ;
then A40: (Gauge (C,n)) * (i1,k) = |[(((Gauge (C,n)) * (i1,j)) `1),(((Gauge (C,n)) * (i2,k)) `2)]| by A18, EUCLID:53;
A41: [(len (Gauge (C,n))),k] in Indices (Gauge (C,n)) by A6, A10, A21, MATRIX_0:30;
A42: len go >= 1 + 1 by TOPREAL1:def 8;
(W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * (1,k)) `1 by A6, A10, A13, JORDAN1A:73 ;
then A43: (Gauge (C,n)) * (i1,j) <> (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) by A1, A2, A23, A35, A20, 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 A16, JORDAN3:34;
A44: (Gauge (C,n)) * (i1,j) in rng (Lower_Seq (C,n)) by A3, A4, A17, A16, A19, JORDAN1G:5, JORDAN1J:40;
then A45: 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, A10, A13, JORDAN1A:71 ;
then A46: (Gauge (C,n)) * (i1,j) <> (Lower_Seq (C,n)) . 1 by A3, A20, A41, A34, JORDAN1G:7;
A47: len co >= 1 + 1 by TOPREAL1:def 8;
then reconsider co = co as constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A45, JGRAPH_1:12, JORDAN8:5;
A48: L~ co c= L~ (Lower_Seq (C,n)) by A16, JORDAN3:42;
len co >= 1 by A47, 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 A16, JORDAN3:23 ;
then A50: LSeg (co,1) = LSeg (((Gauge (C,n)) * (i1,j)),(co /. (1 + 1))) by A47, 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 A48;
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, A36, 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 A47, SPPOL_2:18;
reconsider go = go as constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A42, A39, JGRAPH_1:12, JORDAN8:5;
A56: L~ go c= L~ (Upper_Seq (C,n)) by A30, JORDAN3:41;
A57: len go > 1 by A42, NAT_1:13;
then A58: len go in dom go by FINSEQ_3:25;
then A59: go /. (len go) = go . (len go) by PARTFUN1:def 6
.= (Gauge (C,n)) * (i2,k) by A30, JORDAN3:24 ;
reconsider m = (len go) - 1 as Nat by A58, FINSEQ_3:26;
A60: m + 1 = len go ;
then A61: (len go) -' 1 = m by NAT_D:34;
m >= 1 by A42, XREAL_1:19;
then A62: LSeg (go,m) = LSeg ((go /. m),((Gauge (C,n)) * (i2,k))) by A59, A60, TOPREAL1:def 3;
A63: {((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 A64: 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 A65: (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 A62, 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 A64, A65, 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 A56;
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, A36, XBOOLE_1:26;
then A66: (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 A63;
A67: go /. 1 = (Upper_Seq (C,n)) /. 1 by A30, SPRECT_3:22
.= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ;
then A68: W-min (L~ (Cage (C,n))) in rng go by FINSEQ_6:42;
A69: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by A33, PARTFUN1:def 6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;
A70: (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 A71: x in (L~ go) /\ (L~ co) ; :: thesis: x in {(go /. 1)}
then A72: x in L~ co by XBOOLE_0:def 4;
A73: now :: thesis: not x = E-max (L~ (Cage (C,n)))
assume x = E-max (L~ (Cage (C,n))) ; :: thesis: contradiction
then A74: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i1,j) by A16, A69, A72, JORDAN1E:7;
((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 = E-bound (L~ (Cage (C,n))) by A4, A19, A13, JORDAN1A:71;
then (E-max (L~ (Cage (C,n)))) `1 <> E-bound (L~ (Cage (C,n))) by A3, A25, A22, A74, JORDAN1G:7;
hence contradiction by EUCLID:52; :: thesis: verum
end;
x in L~ go by A71, XBOOLE_0:def 4;
then x in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A56, A48, A72, 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 A67, A73, TARSKI:def 1; :: thesis: verum
end;
set W2 = go /. 2;
A75: 2 in dom go by A42, FINSEQ_3:25;
go = mid ((Upper_Seq (C,n)),1,(((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n)))) by A38, JORDAN1G:49
.= (Upper_Seq (C,n)) | (((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n))) by A38, FINSEQ_4:21, FINSEQ_6:116 ;
then A76: go /. 2 = (Upper_Seq (C,n)) /. 2 by A75, FINSEQ_4:70;
A77: rng go c= L~ go by A42, SPPOL_2:18;
A78: go /. 1 = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by A67, JORDAN1F:8
.= co /. (len co) by A16, 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 A79: x = go /. 1 by TARSKI:def 1;
then A80: x in rng go by FINSEQ_6:42;
x in rng co by A78, A79, FINSEQ_6:168;
hence x in (L~ go) /\ (L~ co) by A77, A55, A80, XBOOLE_0:def 4; :: thesis: verum
end;
then A81: (L~ go) /\ (L~ co) = {(go /. 1)} by A70;
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 A40, TOPREAL3:35;
then consider pion1 being FinSequence of (TOP-REAL 2) such that
A82: pion1 is_sequence_on Gauge (C,n) and
A83: pion1 is being_S-Seq and
A84: L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> = L~ pion1 and
A85: <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. 1 = pion1 /. 1 and
A86: <*((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
A87: len <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> <= len pion1 by A29, GOBOARD3:2;
reconsider pion1 = pion1 as being_S-Seq FinSequence of (TOP-REAL 2) by A83;
A88: (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 A86, 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 ;
A89: go /. (len go) = pion1 /. 1 by A59, A85, FINSEQ_4:18;
A90: (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 A91: x in (L~ go) /\ (L~ pion1) ; :: thesis: x in {(pion1 /. 1)}
then A92: x in L~ pion1 by XBOOLE_0:def 4;
x in L~ go by A91, XBOOLE_0:def 4;
hence x in {(pion1 /. 1)} by A7, A36, A59, A56, A84, A89, A92, XBOOLE_0:def 4; :: thesis: verum
end;
len pion1 >= 2 + 1 by A87, FINSEQ_1:45;
then A93: len pion1 > 1 + 1 by NAT_1:13;
then A94: 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 A95: x = pion1 /. 1 by TARSKI:def 1;
then A96: x in rng pion1 by FINSEQ_6:42;
x in rng go by A89, A95, FINSEQ_6:168;
hence x in (L~ go) /\ (L~ pion1) by A77, A94, A96, XBOOLE_0:def 4; :: thesis: verum
end;
then A97: (L~ go) /\ (L~ pion1) = {(pion1 /. 1)} by A90;
then A98: go ^' pion1 is s.n.c. by A89, JORDAN1J:54;
A99: {((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 A100: x = (Gauge (C,n)) * (i2,k) by TARSKI:def 1;
A101: (Gauge (C,n)) * (i2,k) in LSeg (go,m) by A62, RLTOPSP1:68;
(Gauge (C,n)) * (i2,k) in LSeg (pion1,1) by A59, A89, A93, TOPREAL1:21;
hence x in (LSeg (go,m)) /\ (LSeg (pion1,1)) by A100, A101, 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 A84, TOPREAL3:19;
then (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) c= {((Gauge (C,n)) * (i2,k))} by A61, A66, XBOOLE_1:27;
then (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) = {(go /. (len go))} by A59, A61, A99;
then A102: go ^' pion1 is unfolded by A89, TOPREAL8:34;
len (go ^' pion1) >= len go by TOPREAL8:7;
then A103: len (go ^' pion1) >= 1 + 1 by A42, XXREAL_0:2;
then A104: len (go ^' pion1) > 1 + 0 by NAT_1:13;
A105: <*((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 ;
A106: {(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 A107: x = pion1 /. (len pion1) by TARSKI:def 1;
then A108: x in rng pion1 by FINSEQ_6:168;
x in rng co by A86, A105, A107, FINSEQ_6:42;
hence x in (L~ co) /\ (L~ pion1) by A55, A94, A108, 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 A109: x in (L~ co) /\ (L~ pion1) ; :: thesis: x in {(pion1 /. (len pion1))}
then A110: x in L~ pion1 by XBOOLE_0:def 4;
x in L~ co by A109, XBOOLE_0:def 4;
hence x in {(pion1 /. (len pion1))} by A8, A36, A49, A48, A84, A86, A105, A110, XBOOLE_0:def 4; :: thesis: verum
end;
then A111: (L~ co) /\ (L~ pion1) = {(pion1 /. (len pion1))} by A106;
A112: (L~ (go ^' pion1)) /\ (L~ co) = ((L~ go) \/ (L~ pion1)) /\ (L~ co) by A89, TOPREAL8:35
.= {(go /. 1)} \/ {(co /. 1)} by A81, A86, A105, A111, XBOOLE_1:23
.= {((go ^' pion1) /. 1)} \/ {(co /. 1)} by FINSEQ_6:155
.= {((go ^' pion1) /. 1),(co /. 1)} by ENUMSET1:1 ;
A113: Upper_Arc C is_an_arc_of W-min C, E-max C by JORDAN6:def 8;
then A114: Upper_Arc C is connected by JORDAN6:10;
set godo = (go ^' pion1) ^' co;
A115: co /. (len co) = (go ^' pion1) /. 1 by A78, FINSEQ_6:155;
A116: go ^' pion1 is_sequence_on Gauge (C,n) by A39, A82, A89, TOPREAL8:12;
then A117: (go ^' pion1) ^' co is_sequence_on Gauge (C,n) by A45, A88, TOPREAL8:12;
A118: (len pion1) - 1 >= 1 by A93, XREAL_1:19;
then A119: (len pion1) -' 1 = (len pion1) - 1 by XREAL_0:def 2;
A120: {((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 A121: 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 A86, A119, FINSEQ_1:45
.= (Gauge (C,n)) * (i1,j) by FINSEQ_4:18 ;
then A122: (Gauge (C,n)) * (i1,j) in LSeg (pion1,((len pion1) -' 1)) by A118, A119, 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 A121, A122, 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 A84, TOPREAL3:19;
then (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (co,1)) c= {((Gauge (C,n)) * (i1,j))} by A54, XBOOLE_1:27;
then A123: (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (co,1)) = {((Gauge (C,n)) * (i1,j))} by A120;
((len pion1) - 1) + 1 <= len pion1 ;
then A124: (len pion1) -' 1 < len pion1 by A119, NAT_1:13;
len pion1 >= 2 + 1 by A87, FINSEQ_1:45;
then A125: (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 A118, XREAL_0:def 2 ;
then A126: (LSeg ((go ^' pion1),((len go) + ((len pion1) -' 2)))) /\ (LSeg (co,1)) = {((go ^' pion1) /. (len (go ^' pion1)))} by A49, A89, A88, A124, A123, TOPREAL8:31;
(rng go) /\ (rng pion1) c= {(pion1 /. 1)} by A77, A94, A97, XBOOLE_1:27;
then A127: 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 A125, XREAL_0:def 2 ;
then A128: (len (go ^' pion1)) -' 1 = (len go) + ((len pion1) -' 2) by XREAL_0:def 2;
A129: L~ (Cage (C,n)) = (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n))) by JORDAN1E:13;
then A130: L~ (Upper_Seq (C,n)) c= L~ (Cage (C,n)) by XBOOLE_1:7;
then A131: L~ go c= L~ (Cage (C,n)) by A56;
A132: len ((go ^' pion1) ^' co) >= len (go ^' pion1) by TOPREAL8:7;
then A133: 1 + 1 <= len ((go ^' pion1) ^' co) by A103, XXREAL_0:2;
not go ^' pion1 is trivial by A103, NAT_D:60;
then reconsider godo = (go ^' pion1) ^' co as constant standard special_circular_sequence by A133, A88, A117, A102, A128, A126, A98, A127, A112, A115, JORDAN8:4, JORDAN8:5, TOPREAL8:11, TOPREAL8:33, TOPREAL8:34;
A134: L~ godo = (L~ (go ^' pion1)) \/ (L~ co) by A88, TOPREAL8:35
.= ((L~ go) \/ (L~ pion1)) \/ (L~ co) by A89, TOPREAL8:35 ;
A135: now :: thesis: not ((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n)) <= 1
assume A136: ((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n)) <= 1 ; :: thesis: contradiction
((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n)) >= 1 by A38, FINSEQ_4:21;
then ((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n)) = 1 by A136, XXREAL_0:1;
then (Gauge (C,n)) * (i2,k) = (Upper_Seq (C,n)) /. 1 by A38, FINSEQ_5:38;
hence contradiction by A15, A37, JORDAN1F:5; :: thesis: verum
end;
A137: Upper_Seq (C,n) is_sequence_on Gauge (C,n) by JORDAN1G:4;
A138: ((Gauge (C,n)) * (i2,k)) `1 <= ((Gauge (C,n)) * (i1,k)) `1 by A1, A2, A3, A6, A10, JORDAN1A:18;
then A139: W-bound (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) = ((Gauge (C,n)) * (i2,k)) `1 by SPRECT_1:54;
A140: ((Gauge (C,n)) * (i1,k)) `1 = ((Gauge (C,n)) * (i1,1)) `1 by A3, A6, A17, A10, GOBOARD5:2
.= ((Gauge (C,n)) * (i1,j)) `1 by A3, A4, A17, A19, GOBOARD5:2 ;
then A141: W-bound (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) = ((Gauge (C,n)) * (i1,j)) `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)) * (i2,k)) `1 by A140, A138, A139, A141, XXREAL_0:def 9 ;
then A142: W-bound (L~ pion1) = ((Gauge (C,n)) * (i2,k)) `1 by A84, TOPREAL3:16;
A143: Upper_Arc C c= C by JORDAN6:61;
((Gauge (C,n)) * (i2,k)) `1 >= W-bound (L~ (Cage (C,n))) by A30, A130, PSCOMP_1:24;
then A144: ((Gauge (C,n)) * (i2,k)) `1 > W-bound (L~ (Cage (C,n))) by A24, XXREAL_0:1;
A145: len (Upper_Seq (C,n)) >= 2 by A14, XXREAL_0:2;
A146: (L~ go) \/ (L~ co) is compact by COMPTS_1:10;
A147: L~ (Lower_Seq (C,n)) c= L~ (Cage (C,n)) by A129, XBOOLE_1:7;
then A148: L~ co c= L~ (Cage (C,n)) by A48;
A149: (right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo) c= RightComp godo by A133, A117, JORDAN9:27;
2 in dom godo by A133, FINSEQ_3:25;
then A150: godo /. 2 in rng godo by PARTFUN2:2;
A151: rng godo c= L~ godo by A103, A132, SPPOL_2:18, XXREAL_0:2;
A152: godo /. 1 = (go ^' pion1) /. 1 by FINSEQ_6:155
.= W-min (L~ (Cage (C,n))) by A67, FINSEQ_6:155 ;
A153: W-min C in Upper_Arc C by A113, TOPREAL1:1;
A154: W-min C in C by SPRECT_1:13;
A155: now :: thesis: not W-min C in L~ godo
assume W-min C in L~ godo ; :: thesis: contradiction
then A156: ( W-min C in (L~ go) \/ (L~ pion1) or W-min C in L~ co ) by A134, XBOOLE_0:def 3;
end;
A157: (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)))));
A158: 1 + 1 <= len (Cage (C,n)) by GOBOARD7:34, XXREAL_0:2;
W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
then A159: (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 = W-min (L~ (Cage (C,n))) by FINSEQ_6:92;
A160: 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 A159, 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 A159, A160, 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 A159, A160, SPRECT_5:24, XXREAL_0:2;
then A161: (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A159, A160, SPRECT_5:25, XXREAL_0:2;
A162: Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def 1;
then A163: Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))) is_sequence_on Gauge (C,n) by REVROT_1:34;
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 A161, A163, 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 A38, A137, A135, JORDAN1J:52
.= right_cell ((go ^' pion1),1,(Gauge (C,n))) by A57, A116, JORDAN1J:51
.= right_cell (godo,1,(Gauge (C,n))) by A104, A117, JORDAN1J:51 ;
then W-min C in right_cell (godo,1,(Gauge (C,n))) by JORDAN1I:6;
then A164: W-min C in (right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo) by A155, XBOOLE_0:def 5;
A165: E-max C in Upper_Arc C by A113, TOPREAL1:1;
W-min (L~ (Cage (C,n))) in (L~ go) \/ (L~ co) by A77, A68, XBOOLE_0:def 3;
then A166: W-min ((L~ go) \/ (L~ co)) = W-min (L~ (Cage (C,n))) by A131, A148, A146, 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 A142, A146, A166, A157, A144, JORDAN1J:33;
then A167: W-min (L~ godo) = W-min (L~ (Cage (C,n))) by A134, A166, XBOOLE_1:4;
godo /. 2 = (go ^' pion1) /. 2 by A103, FINSEQ_6:159
.= (Upper_Seq (C,n)) /. 2 by A42, A76, FINSEQ_6:159
.= ((Upper_Seq (C,n)) ^' (Lower_Seq (C,n))) /. 2 by A145, 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 A167, PSCOMP_1:31
.= W-bound (L~ godo) by EUCLID:52 ;
then godo /. 2 in W-most (L~ godo) by A151, A150, SPRECT_2:12;
then (Rotate (godo,(W-min (L~ godo)))) /. 2 in W-most (L~ godo) by A152, A167, FINSEQ_6:89;
then reconsider godo = godo as constant 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 A168: (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 ;
A169: 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
A170: p in east_halfline (E-max C) and
A171: p in L~ go by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A170;
p in L~ (Upper_Seq (C,n)) by A56, A171;
then p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A130, A170, XBOOLE_0:def 4;
then A172: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;
then A173: p = E-max (L~ (Cage (C,n))) by A56, A171, JORDAN1J:46;
then E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i2,k) by A30, A168, A171, JORDAN1J:43;
then ((Gauge (C,n)) * (i2,k)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by A6, A10, A13, A172, A173, JORDAN1A:71;
hence contradiction by A2, A3, A12, A41, 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 A174: ( east_halfline (E-max C) meets (L~ go) \/ (L~ pion1) or east_halfline (E-max C) meets L~ co ) by A134, 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 A174, XBOOLE_1:70;
suppose east_halfline (E-max C) meets L~ pion1 ; :: thesis: contradiction
then consider p being object such that
A175: p in east_halfline (E-max C) and
A176: p in L~ pion1 by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A175;
A177: p `2 = (E-max C) `2 by A175, TOPREAL1:def 11;
A178: now :: thesis: p `1 <= ((Gauge (C,n)) * (i1,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 A36, A84, A176, XBOOLE_0:def 3;
suppose p in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) ; :: thesis: p `1 <= ((Gauge (C,n)) * (i1,k)) `1
hence p `1 <= ((Gauge (C,n)) * (i1,k)) `1 by A138, 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)) * (i1,k)) `1
hence p `1 <= ((Gauge (C,n)) * (i1,k)) `1 by A140, GOBOARD7:5; :: thesis: verum
end;
end;
end;
i1 + 1 <= len (Gauge (C,n)) by A3, NAT_1:13;
then (i1 + 1) - 1 <= (len (Gauge (C,n))) - 1 by XREAL_1:9;
then A179: i1 <= (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)) * (i1,k)) `1 <= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),1)) `1 by A6, A17, A10, A13, A21, A179, JORDAN1A:18;
then p `1 <= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),1)) `1 by A178, XXREAL_0:2;
then p `1 <= E-bound C by A21, JORDAN8:12;
then A180: p `1 <= (E-max C) `1 by EUCLID:52;
p `1 >= (E-max C) `1 by A175, TOPREAL1:def 11;
then p `1 = (E-max C) `1 by A180, XXREAL_0:1;
then p = E-max C by A177, TOPREAL3:6;
hence contradiction by A9, A36, A84, A165, A176, XBOOLE_0:3; :: thesis: verum
end;
suppose east_halfline (E-max C) meets L~ co ; :: thesis: contradiction
then consider p being object such that
A181: p in east_halfline (E-max C) and
A182: p in L~ co by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A181;
A183: p in LSeg (co,(Index (p,co))) by A182, JORDAN3:9;
consider t being Nat such that
A184: t in dom (Lower_Seq (C,n)) and
A185: (Lower_Seq (C,n)) . t = (Gauge (C,n)) * (i1,j) by A44, FINSEQ_2:10;
1 <= t by A184, FINSEQ_3:25;
then A186: 1 < t by A46, A185, XXREAL_0:1;
t <= len (Lower_Seq (C,n)) by A184, FINSEQ_3:25;
then (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) + 1 = t by A185, A186, JORDAN3:12;
then A187: 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 A16, A185, JORDAN3:26;
Index (p,co) < len co by A182, JORDAN3:8;
then Index (p,co) < (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) by A187, 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 A188: Index (p,co) <= ((len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n))))) - 1 by XREAL_1:19;
A189: co = mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n)))) by A44, JORDAN1J:37;
p in L~ (Lower_Seq (C,n)) by A48, A182;
then p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A147, A181, XBOOLE_0:def 4;
then A190: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;
A191: (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) + 1 = ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) by A46, A44, JORDAN1J:56;
0 + (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) < len (Lower_Seq (C,n)) by A16, 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 A188, XREAL_0:def 2;
then Index (p,co) <= (len (Lower_Seq (C,n))) - (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))) by A191;
then Index (p,co) <= (len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))) by XREAL_0:def 2;
then A192: Index (p,co) < ((len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) + 1 by NAT_1:13;
A193: 1 <= Index (p,co) by A182, JORDAN3:8;
A194: ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) <= len (Lower_Seq (C,n)) by A44, FINSEQ_4:21;
((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) <> len (Lower_Seq (C,n)) by A43, A44, FINSEQ_4:19;
then A195: ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) < len (Lower_Seq (C,n)) by A194, XXREAL_0:1;
A196: 1 + 1 <= len (Lower_Seq (C,n)) by A31, XXREAL_0:2;
then A197: 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)))));
A198: E-max C in right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by JORDAN1I:7;
A199: 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 ;
A200: 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
A201: 1 <= jj2 and
A202: jj2 <= width (Gauge (C,n)) and
A203: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * ((len (Gauge (C,n))),jj2) by JORDAN1D:25;
A204: len (Gauge (C,n)) >= 4 by JORDAN8:10;
then len (Gauge (C,n)) >= 1 by XXREAL_0:2;
then A205: [(len (Gauge (C,n))),jj2] in Indices (Gauge (C,n)) by A201, A202, MATRIX_0:30;
A206: 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 A207: LSeg ((Lower_Seq (C,n)),1) = LSeg ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by A196, SPPOL_2:9;
A208: 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 A162, REVROT_1:34;
then consider ii, jj being Nat such that
A209: [ii,(jj + 1)] in Indices (Gauge (C,n)) and
A210: [ii,jj] in Indices (Gauge (C,n)) and
A211: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = (Gauge (C,n)) * (ii,(jj + 1)) and
A212: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1) = (Gauge (C,n)) * (ii,jj) by A158, A200, A206, A208, FINSEQ_6:92, JORDAN1I:23;
A213: (jj + 1) + 1 <> jj ;
A214: 1 <= jj by A210, 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 A200, A208, FINSEQ_6:92;
then A215: ii = len (Gauge (C,n)) by A200, A209, A211, A203, A205, GOBOARD1:5;
then ii - 1 >= 4 - 1 by A204, XREAL_1:9;
then A216: ii - 1 >= 1 by XXREAL_0:2;
then A217: 1 <= ii -' 1 by XREAL_0:def 2;
A218: jj <= width (Gauge (C,n)) by A210, MATRIX_0:32;
then A219: ((Gauge (C,n)) * ((len (Gauge (C,n))),jj)) `1 = E-bound (L~ (Cage (C,n))) by A13, A214, JORDAN1A:71;
A220: jj + 1 <= width (Gauge (C,n)) by A209, MATRIX_0:32;
ii + 1 <> ii ;
then A221: right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) = cell ((Gauge (C,n)),(ii -' 1),jj) by A158, A206, A199, A209, A210, A211, A212, A213, GOBOARD5:def 6;
A222: ii <= len (Gauge (C,n)) by A210, MATRIX_0:32;
A223: 1 <= ii by A210, MATRIX_0:32;
A224: ii <= len (Gauge (C,n)) by A209, MATRIX_0:32;
A225: 1 <= jj + 1 by A209, MATRIX_0:32;
then A226: E-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * ((len (Gauge (C,n))),(jj + 1))) `1 by A13, A220, JORDAN1A:71;
A227: 1 <= ii by A209, MATRIX_0:32;
then A228: (ii -' 1) + 1 = ii by XREAL_1:235;
then A229: ii -' 1 < len (Gauge (C,n)) by A224, NAT_1:13;
then A230: ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 = ((Gauge (C,n)) * (1,(jj + 1))) `2 by A225, A220, A217, GOBOARD5:1
.= ((Gauge (C,n)) * (ii,(jj + 1))) `2 by A227, A224, A225, A220, GOBOARD5:1 ;
A231: (E-max C) `2 = p `2 by A181, TOPREAL1:def 11;
then A232: p `2 <= ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 by A198, A224, A220, A214, A221, A228, A216, JORDAN9:17;
A233: ((Gauge (C,n)) * ((ii -' 1),jj)) `2 = ((Gauge (C,n)) * (1,jj)) `2 by A214, A218, A217, A229, GOBOARD5:1
.= ((Gauge (C,n)) * (ii,jj)) `2 by A223, A222, A214, A218, GOBOARD5:1 ;
((Gauge (C,n)) * ((ii -' 1),jj)) `2 <= p `2 by A231, A198, A224, A220, A214, A221, A228, A216, 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 A190, A211, A212, A215, A232, A233, A230, A219, A226, GOBOARD7:7;
then A234: p in LSeg ((Lower_Seq (C,n)),1) by A158, A207, A206, TOPREAL1:def 3;
1 <= ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) by A44, FINSEQ_4:21;
then A235: 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 A195, A193, A192, JORDAN4:19;
1 <= Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n))) by A16, JORDAN3:8;
then A236: 1 + 1 <= ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) by A191, XREAL_1:7;
then (Index (p,co)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))) >= (1 + 1) + 1 by A193, 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 A237: ((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 A237, 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 A234, A183, A189, A235, XBOOLE_0:3; :: thesis: verum
end;
suppose A238: ((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 A239: ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) = 2 by A193, A236, 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 A31, A238, TOPREAL1:def 6;
then p in {((Lower_Seq (C,n)) /. 2)} by A234, A183, A189, A235, XBOOLE_0:def 4;
then A240: p = (Lower_Seq (C,n)) /. 2 by TARSKI:def 1;
then A241: p in rng (Lower_Seq (C,n)) by A197, PARTFUN2:2;
p .. (Lower_Seq (C,n)) = 2 by A197, A240, FINSEQ_5:41;
then p = (Gauge (C,n)) * (i1,j) by A44, A239, A241, FINSEQ_5:9;
then ((Gauge (C,n)) * (i1,j)) `1 = E-bound (L~ (Cage (C,n))) by A240, JORDAN1G:32;
then ((Gauge (C,n)) * (i1,j)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 by A4, A19, A13, JORDAN1A:71;
hence contradiction by A3, A25, A22, 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
A242: W is_a_component_of (L~ godo) ` and
A243: east_halfline (E-max C) c= W by GOBOARD9:3;
not W is bounded by A243, JORDAN2C:121, RLTOPSP1:42;
then W is_outside_component_of L~ godo by A242, JORDAN2C:def 3;
then W c= UBD (L~ godo) by JORDAN2C:23;
then A244: east_halfline (E-max C) c= UBD (L~ godo) by A243;
E-max C in east_halfline (E-max C) by TOPREAL1:38;
then E-max C in UBD (L~ godo) by A244;
then E-max C in LeftComp godo by GOBRD14:36;
then Upper_Arc C meets L~ godo by A114, A153, A165, A149, A164, JORDAN1J:36;
then A245: ( Upper_Arc C meets (L~ go) \/ (L~ pion1) or Upper_Arc C meets L~ co ) by A134, XBOOLE_1:70;
hence contradiction ; :: thesis: verum
end;
suppose ((Gauge (C,n)) * (i1,j)) `1 = ((Gauge (C,n)) * (i2,k)) `1 ; :: thesis: contradiction
then A246: i1 = i2 by A25, A12, 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 A27, 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, A246, JORDAN1J:59; :: thesis: verum
end;
suppose ((Gauge (C,n)) * (i1,j)) `2 = ((Gauge (C,n)) * (i2,k)) `2 ; :: thesis: contradiction
then A247: j = k by A25, A12, 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 A26, 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, A247, Th37; :: thesis: verum
end;
end;
end;
hence contradiction ; :: thesis: verum