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)) * (i1,j))} & ((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)) * (i2,k))} 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 < 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)) * (i1,j))} & ((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)) * (i2,k))} 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 < 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)) * (i1,j))} & ((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)) * (i2,k))} 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 < 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)) * (i1,j))} 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)) * (i2,k))} 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 UA = Upper_Arc C;
set Wmin = W-min (L~ (Cage (C,n)));
set Emax = E-max (L~ (Cage (C,n)));
set Wbo = W-bound (L~ (Cage (C,n)));
set Ebo = E-bound (L~ (Cage (C,n)));
set Gik = (Gauge (C,n)) * (i2,k);
set Gij = (Gauge (C,n)) * (i1,j);
set Gi1k = (Gauge (C,n)) * (i1,k);
A10: i1 < len (Gauge (C,n)) by A2, A3, XXREAL_0:2;
A11: 1 < i2 by A1, A2, XXREAL_0:2;
A12: L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> = (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;
(Gauge (C,n)) * (i2,k) in {((Gauge (C,n)) * (i2,k))} by TARSKI:def 1;
then A13: (Gauge (C,n)) * (i2,k) in L~ (Lower_Seq (C,n)) by A8, XBOOLE_0:def 4;
(Gauge (C,n)) * (i1,j) in {((Gauge (C,n)) * (i1,j))} by TARSKI:def 1;
then A14: (Gauge (C,n)) * (i1,j) in L~ (Upper_Seq (C,n)) by A7, XBOOLE_0:def 4;
A15: j <= width (Gauge (C,n)) by A5, A6, XXREAL_0:2;
A16: 1 <= k by A4, A5, XXREAL_0:2;
A17: [i1,j] in Indices (Gauge (C,n)) by A1, A4, A10, A15, MATRIX_0:30;
A18: [i2,k] in Indices (Gauge (C,n)) by A3, A6, A11, A16, MATRIX_0:30;
A19: [i1,k] in Indices (Gauge (C,n)) by A1, A6, A10, A16, MATRIX_0:30;
set go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i1,j)));
set co = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i2,k)));
A20: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def 1;
A21: 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 A22: (Upper_Seq (C,n)) . 1 = (Upper_Seq (C,n)) /. 1 by PARTFUN1:def 6
.= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ;
A23: (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n)))
.= ((Gauge (C,n)) * (1,k)) `1 by A6, A16, A20, JORDAN1A:73 ;
len (Gauge (C,n)) >= 4 by JORDAN8:10;
then A24: len (Gauge (C,n)) >= 1 by XXREAL_0:2;
then A25: [1,k] in Indices (Gauge (C,n)) by A6, A16, MATRIX_0:30;
then A26: (Gauge (C,n)) * (i1,j) <> (Upper_Seq (C,n)) . 1 by A1, A17, A22, A23, JORDAN1G:7;
then reconsider go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i1,j))) as being_S-Seq FinSequence of (TOP-REAL 2) by A14, JORDAN3:35;
A27: [1,j] in Indices (Gauge (C,n)) by A4, A15, A24, MATRIX_0:30;
A28: len (Lower_Seq (C,n)) >= 1 + 2 by JORDAN1E:15;
then A29: len (Lower_Seq (C,n)) >= 1 by XXREAL_0:2;
then A30: 1 in dom (Lower_Seq (C,n)) by FINSEQ_3:25;
len (Lower_Seq (C,n)) in dom (Lower_Seq (C,n)) by A29, FINSEQ_3:25;
then A31: (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 ;
(W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n)))
.= ((Gauge (C,n)) * (1,k)) `1 by A6, A16, A20, JORDAN1A:73 ;
then A32: (Gauge (C,n)) * (i2,k) <> (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) by A1, A2, A18, A25, A31, JORDAN1G:7;
then reconsider co = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i2,k))) as being_S-Seq FinSequence of (TOP-REAL 2) by A13, JORDAN3:34;
A33: [(len (Gauge (C,n))),k] in Indices (Gauge (C,n)) by A6, A16, A24, MATRIX_0:30;
A34: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by A30, PARTFUN1:def 6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;
(E-max (L~ (Cage (C,n)))) `1 = E-bound (L~ (Cage (C,n)))
.= ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by A6, A16, A20, JORDAN1A:71 ;
then A35: (Gauge (C,n)) * (i2,k) <> (Lower_Seq (C,n)) . 1 by A3, A18, A33, A34, JORDAN1G:7;
A36: len go >= 1 + 1 by TOPREAL1:def 8;
A37: (Gauge (C,n)) * (i1,j) in rng (Upper_Seq (C,n)) by A1, A4, A10, A14, A15, JORDAN1G:4, JORDAN1J:40;
then A38: go is_sequence_on Gauge (C,n) by JORDAN1G:4, JORDAN1J:38;
A39: len co >= 1 + 1 by TOPREAL1:def 8;
A40: (Gauge (C,n)) * (i2,k) in rng (Lower_Seq (C,n)) by A3, A6, A11, A13, A16, JORDAN1G:5, JORDAN1J:40;
then A41: co is_sequence_on Gauge (C,n) by JORDAN1G:5, JORDAN1J:39;
reconsider go = go as constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A36, A38, JGRAPH_1:12, JORDAN8:5;
reconsider co = co as constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A39, A41, JGRAPH_1:12, JORDAN8:5;
A42: len go > 1 by A36, NAT_1:13;
then A43: len go in dom go by FINSEQ_3:25;
then A44: go /. (len go) = go . (len go) by PARTFUN1:def 6
.= (Gauge (C,n)) * (i1,j) by A14, JORDAN3:24 ;
len co >= 1 by A39, XXREAL_0:2;
then 1 in dom co by FINSEQ_3:25;
then A45: co /. 1 = co . 1 by PARTFUN1:def 6
.= (Gauge (C,n)) * (i2,k) by A13, JORDAN3:23 ;
reconsider m = (len go) - 1 as Nat by A43, FINSEQ_3:26;
A46: m + 1 = len go ;
then A47: (len go) -' 1 = m by NAT_D:34;
A48: LSeg (go,m) c= L~ go by TOPREAL3:19;
A49: L~ go c= L~ (Upper_Seq (C,n)) by A14, JORDAN3:41;
then LSeg (go,m) c= L~ (Upper_Seq (C,n)) by A48;
then A50: (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) c= {((Gauge (C,n)) * (i1,j))} by A7, A12, XBOOLE_1:26;
m >= 1 by A36, XREAL_1:19;
then A51: LSeg (go,m) = LSeg ((go /. m),((Gauge (C,n)) * (i1,j))) by A44, A46, TOPREAL1:def 3;
{((Gauge (C,n)) * (i1,j))} c= (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>)
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in {((Gauge (C,n)) * (i1,j))} or x in (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) )
assume x in {((Gauge (C,n)) * (i1,j))} ; :: thesis: x in (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>)
then A52: x = (Gauge (C,n)) * (i1,j) by TARSKI:def 1;
A53: (Gauge (C,n)) * (i1,j) in LSeg (go,m) by A51, RLTOPSP1:68;
(Gauge (C,n)) * (i1,j) in LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) by RLTOPSP1:68;
then (Gauge (C,n)) * (i1,j) in (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) by XBOOLE_0:def 3;
then (Gauge (C,n)) * (i1,j) in L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> by SPRECT_1:8;
hence x in (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) by A52, A53, XBOOLE_0:def 4; :: thesis: verum
end;
then A54: (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) = {((Gauge (C,n)) * (i1,j))} by A50;
A55: LSeg (co,1) c= L~ co by TOPREAL3:19;
A56: L~ co c= L~ (Lower_Seq (C,n)) by A13, JORDAN3:42;
then LSeg (co,1) c= L~ (Lower_Seq (C,n)) by A55;
then A57: (LSeg (co,1)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) c= {((Gauge (C,n)) * (i2,k))} by A8, A12, XBOOLE_1:26;
A58: LSeg (co,1) = LSeg (((Gauge (C,n)) * (i2,k)),(co /. (1 + 1))) by A39, A45, TOPREAL1:def 3;
{((Gauge (C,n)) * (i2,k))} c= (LSeg (co,1)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>)
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in {((Gauge (C,n)) * (i2,k))} or x in (LSeg (co,1)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) )
assume x in {((Gauge (C,n)) * (i2,k))} ; :: thesis: x in (LSeg (co,1)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>)
then A59: x = (Gauge (C,n)) * (i2,k) by TARSKI:def 1;
A60: (Gauge (C,n)) * (i2,k) in LSeg (co,1) by A58, RLTOPSP1:68;
(Gauge (C,n)) * (i2,k) in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) by RLTOPSP1:68;
then (Gauge (C,n)) * (i2,k) in (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) by XBOOLE_0:def 3;
then (Gauge (C,n)) * (i2,k) in L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> by SPRECT_1:8;
hence x in (LSeg (co,1)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) by A59, A60, XBOOLE_0:def 4; :: thesis: verum
end;
then A61: (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) /\ (LSeg (co,1)) = {((Gauge (C,n)) * (i2,k))} by A57;
A62: go /. 1 = (Upper_Seq (C,n)) /. 1 by A14, SPRECT_3:22
.= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ;
then A63: go /. 1 = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by JORDAN1F:8
.= co /. (len co) by A13, JORDAN1J:35 ;
A64: rng go c= L~ go by A36, SPPOL_2:18;
A65: rng co c= L~ co by A39, SPPOL_2:18;
A66: {(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 A67: x = go /. 1 by TARSKI:def 1;
then A68: x in rng go by FINSEQ_6:42;
x in rng co by A63, A67, FINSEQ_6:168;
hence x in (L~ go) /\ (L~ co) by A64, A65, A68, XBOOLE_0:def 4; :: thesis: verum
end;
A69: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by A30, PARTFUN1:def 6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;
A70: [(len (Gauge (C,n))),j] in Indices (Gauge (C,n)) by A4, A15, A24, MATRIX_0:30;
(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~ go by XBOOLE_0:def 4;
A73: x in L~ co by A71, XBOOLE_0:def 4;
then x in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A49, A56, A72, XBOOLE_0:def 4;
then x in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16;
then A74: ( x = W-min (L~ (Cage (C,n))) or x = E-max (L~ (Cage (C,n))) ) by TARSKI:def 2;
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)) * (i2,k) by A13, A69, A73, JORDAN1E:7;
((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 = E-bound (L~ (Cage (C,n))) by A4, A15, A20, JORDAN1A:71;
then (E-max (L~ (Cage (C,n)))) `1 <> E-bound (L~ (Cage (C,n))) by A3, A18, A70, A75, JORDAN1G:7;
hence contradiction ; :: thesis: verum
end;
hence x in {(go /. 1)} by A62, A74, TARSKI:def 1; :: thesis: verum
end;
then A76: (L~ go) /\ (L~ co) = {(go /. 1)} by A66;
set W2 = go /. 2;
A77: 2 in dom go by A36, FINSEQ_3:25;
A78: 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, A15, A20, JORDAN1A:73;
hence contradiction by A1, A17, A27, JORDAN1G:7; :: thesis: verum
end;
go = mid ((Upper_Seq (C,n)),1,(((Gauge (C,n)) * (i1,j)) .. (Upper_Seq (C,n)))) by A37, JORDAN1G:49
.= (Upper_Seq (C,n)) | (((Gauge (C,n)) * (i1,j)) .. (Upper_Seq (C,n))) by A37, FINSEQ_4:21, FINSEQ_6:116 ;
then A79: go /. 2 = (Upper_Seq (C,n)) /. 2 by A77, FINSEQ_4:70;
A80: W-min (L~ (Cage (C,n))) in rng go by A62, FINSEQ_6:42;
set pion = <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>;
A81: now :: thesis: for n being Nat st n in dom <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> holds
ex i, j being Nat st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. n = (Gauge (C,n)) * (i,j) )
let n be Nat; :: thesis: ( n in dom <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> implies ex i, j being Nat st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. n = (Gauge (C,n)) * (i,j) ) )
assume n in dom <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> ; :: thesis: ex i, j being Nat st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. 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)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. n = (Gauge (C,n)) * (i,j) ) by A17, A18, A19, FINSEQ_4:18; :: thesis: verum
end;
A82: ((Gauge (C,n)) * (i1,k)) `1 = ((Gauge (C,n)) * (i1,1)) `1 by A1, A6, A10, A16, GOBOARD5:2
.= ((Gauge (C,n)) * (i1,j)) `1 by A1, A4, A10, A15, GOBOARD5:2 ;
((Gauge (C,n)) * (i1,k)) `2 = ((Gauge (C,n)) * (1,k)) `2 by A1, A6, A10, A16, GOBOARD5:1
.= ((Gauge (C,n)) * (i2,k)) `2 by A3, A6, A11, A16, GOBOARD5:1 ;
then A83: (Gauge (C,n)) * (i1,k) = |[(((Gauge (C,n)) * (i1,j)) `1),(((Gauge (C,n)) * (i2,k)) `2)]| by A82, EUCLID:53;
A84: (Gauge (C,n)) * (i1,k) in LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) by RLTOPSP1:68;
A85: (Gauge (C,n)) * (i1,k) in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) by RLTOPSP1:68;
now :: thesis: contradiction
per cases ( ( ((Gauge (C,n)) * (i2,k)) `1 <> ((Gauge (C,n)) * (i1,j)) `1 & ((Gauge (C,n)) * (i2,k)) `2 <> ((Gauge (C,n)) * (i1,j)) `2 ) or ((Gauge (C,n)) * (i2,k)) `1 = ((Gauge (C,n)) * (i1,j)) `1 or ((Gauge (C,n)) * (i2,k)) `2 = ((Gauge (C,n)) * (i1,j)) `2 ) ;
suppose ( ((Gauge (C,n)) * (i2,k)) `1 <> ((Gauge (C,n)) * (i1,j)) `1 & ((Gauge (C,n)) * (i2,k)) `2 <> ((Gauge (C,n)) * (i1,j)) `2 ) ; :: thesis: contradiction
then <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> is being_S-Seq by A83, TOPREAL3:34;
then consider pion1 being FinSequence of (TOP-REAL 2) such that
A86: pion1 is_sequence_on Gauge (C,n) and
A87: pion1 is being_S-Seq and
A88: L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> = L~ pion1 and
A89: <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. 1 = pion1 /. 1 and
A90: <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. (len <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) = pion1 /. (len pion1) and
A91: len <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> <= len pion1 by A81, GOBOARD3:2;
reconsider pion1 = pion1 as being_S-Seq FinSequence of (TOP-REAL 2) by A87;
set godo = (go ^' pion1) ^' co;
A92: ((Gauge (C,n)) * (i1,k)) `1 = ((Gauge (C,n)) * (i1,1)) `1 by A1, A6, A10, A16, GOBOARD5:2
.= ((Gauge (C,n)) * (i1,j)) `1 by A1, A4, A10, A15, GOBOARD5:2 ;
A93: ((Gauge (C,n)) * (i1,k)) `1 <= ((Gauge (C,n)) * (i2,k)) `1 by A1, A2, A3, A6, A16, JORDAN1A:18;
then A94: W-bound (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) = ((Gauge (C,n)) * (i1,k)) `1 by SPRECT_1:54;
A95: W-bound (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) = ((Gauge (C,n)) * (i1,j)) `1 by A92, 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 A92, A94, A95 ;
then A96: W-bound (L~ pion1) = ((Gauge (C,n)) * (i1,j)) `1 by A88, TOPREAL3:16;
A97: 1 + 1 <= len (Cage (C,n)) by GOBOARD7:34, XXREAL_0:2;
A98: 1 + 1 <= len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by GOBOARD7:34, XXREAL_0:2;
len (go ^' pion1) >= len go by TOPREAL8:7;
then A99: len (go ^' pion1) >= 1 + 1 by A36, XXREAL_0:2;
then A100: len (go ^' pion1) > 1 + 0 by NAT_1:13;
A101: len ((go ^' pion1) ^' co) >= len (go ^' pion1) by TOPREAL8:7;
then A102: 1 + 1 <= len ((go ^' pion1) ^' co) by A99, XXREAL_0:2;
A103: Upper_Seq (C,n) is_sequence_on Gauge (C,n) by JORDAN1G:4;
A104: go /. (len go) = pion1 /. 1 by A44, A89, FINSEQ_4:18;
then A105: go ^' pion1 is_sequence_on Gauge (C,n) by A38, A86, TOPREAL8:12;
A106: (go ^' pion1) /. (len (go ^' pion1)) = <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. (len <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) by A90, FINSEQ_6:156
.= <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. 3 by FINSEQ_1:45
.= co /. 1 by A45, FINSEQ_4:18 ;
then A107: (go ^' pion1) ^' co is_sequence_on Gauge (C,n) by A41, A105, TOPREAL8:12;
LSeg (pion1,1) c= L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> by A88, TOPREAL3:19;
then A108: (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) c= {((Gauge (C,n)) * (i1,j))} by A47, A54, XBOOLE_1:27;
len pion1 >= 2 + 1 by A91, FINSEQ_1:45;
then A109: len pion1 > 1 + 1 by NAT_1:13;
{((Gauge (C,n)) * (i1,j))} c= (LSeg (go,m)) /\ (LSeg (pion1,1))
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in {((Gauge (C,n)) * (i1,j))} or x in (LSeg (go,m)) /\ (LSeg (pion1,1)) )
assume x in {((Gauge (C,n)) * (i1,j))} ; :: thesis: x in (LSeg (go,m)) /\ (LSeg (pion1,1))
then A110: x = (Gauge (C,n)) * (i1,j) by TARSKI:def 1;
A111: (Gauge (C,n)) * (i1,j) in LSeg (go,m) by A51, RLTOPSP1:68;
(Gauge (C,n)) * (i1,j) in LSeg (pion1,1) by A44, A104, A109, TOPREAL1:21;
hence x in (LSeg (go,m)) /\ (LSeg (pion1,1)) by A110, A111, XBOOLE_0:def 4; :: thesis: verum
end;
then (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) = {(go /. (len go))} by A44, A47, A108;
then A112: go ^' pion1 is unfolded by A104, TOPREAL8:34;
len pion1 >= 2 + 1 by A91, FINSEQ_1:45;
then A113: (len pion1) - 2 >= 0 by XREAL_1:19;
((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 A113, XREAL_0:def 2 ;
then A114: (len (go ^' pion1)) -' 1 = (len go) + ((len pion1) -' 2) by XREAL_0:def 2;
A115: (len pion1) - 1 >= 1 by A109, XREAL_1:19;
then A116: (len pion1) -' 1 = (len pion1) - 1 by XREAL_0:def 2;
A117: ((len pion1) -' 2) + 1 = ((len pion1) - 2) + 1 by A113, XREAL_0:def 2
.= (len pion1) -' 1 by A115, XREAL_0:def 2 ;
((len pion1) - 1) + 1 <= len pion1 ;
then A118: (len pion1) -' 1 < len pion1 by A116, NAT_1:13;
LSeg (pion1,((len pion1) -' 1)) c= L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> by A88, TOPREAL3:19;
then A119: (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (co,1)) c= {((Gauge (C,n)) * (i2,k))} by A61, XBOOLE_1:27;
{((Gauge (C,n)) * (i2,k))} 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)) * (i2,k))} or x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (co,1)) )
assume x in {((Gauge (C,n)) * (i2,k))} ; :: thesis: x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (co,1))
then A120: x = (Gauge (C,n)) * (i2,k) by TARSKI:def 1;
A121: (Gauge (C,n)) * (i2,k) in LSeg (co,1) by A58, RLTOPSP1:68;
pion1 /. (((len pion1) -' 1) + 1) = <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. 3 by A90, A116, FINSEQ_1:45
.= (Gauge (C,n)) * (i2,k) by FINSEQ_4:18 ;
then (Gauge (C,n)) * (i2,k) in LSeg (pion1,((len pion1) -' 1)) by A115, A116, TOPREAL1:21;
hence x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (co,1)) by A120, A121, XBOOLE_0:def 4; :: thesis: verum
end;
then (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (co,1)) = {((Gauge (C,n)) * (i2,k))} by A119;
then A122: (LSeg ((go ^' pion1),((len go) + ((len pion1) -' 2)))) /\ (LSeg (co,1)) = {((go ^' pion1) /. (len (go ^' pion1)))} by A45, A104, A106, A117, A118, TOPREAL8:31;
A123: not go ^' pion1 is trivial by A99, NAT_D:60;
A124: rng pion1 c= L~ pion1 by A109, SPPOL_2:18;
A125: {(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 A126: x = pion1 /. 1 by TARSKI:def 1;
then A127: x in rng go by A104, FINSEQ_6:168;
x in rng pion1 by A126, FINSEQ_6:42;
hence x in (L~ go) /\ (L~ pion1) by A64, A124, A127, XBOOLE_0:def 4; :: thesis: verum
end;
(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 A128: x in (L~ go) /\ (L~ pion1) ; :: thesis: x in {(pion1 /. 1)}
then A129: x in L~ go by XBOOLE_0:def 4;
x in L~ pion1 by A128, XBOOLE_0:def 4;
hence x in {(pion1 /. 1)} by A7, A12, A44, A49, A88, A104, A129, XBOOLE_0:def 4; :: thesis: verum
end;
then A130: (L~ go) /\ (L~ pion1) = {(pion1 /. 1)} by A125;
then A131: go ^' pion1 is s.n.c. by A104, JORDAN1J:54;
(rng go) /\ (rng pion1) c= {(pion1 /. 1)} by A64, A124, A130, XBOOLE_1:27;
then A132: go ^' pion1 is one-to-one by JORDAN1J:55;
A133: <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. (len <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) = <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. 3 by FINSEQ_1:45
.= co /. 1 by A45, FINSEQ_4:18 ;
A134: {(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 A135: x = pion1 /. (len pion1) by TARSKI:def 1;
then A136: x in rng co by A90, A133, FINSEQ_6:42;
x in rng pion1 by A135, FINSEQ_6:168;
hence x in (L~ co) /\ (L~ pion1) by A65, A124, A136, 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 A137: x in (L~ co) /\ (L~ pion1) ; :: thesis: x in {(pion1 /. (len pion1))}
then A138: x in L~ co by XBOOLE_0:def 4;
x in L~ pion1 by A137, XBOOLE_0:def 4;
hence x in {(pion1 /. (len pion1))} by A8, A12, A45, A56, A88, A90, A133, A138, XBOOLE_0:def 4; :: thesis: verum
end;
then A139: (L~ co) /\ (L~ pion1) = {(pion1 /. (len pion1))} by A134;
A140: (L~ (go ^' pion1)) /\ (L~ co) = ((L~ go) \/ (L~ pion1)) /\ (L~ co) by A104, TOPREAL8:35
.= {(go /. 1)} \/ {(co /. 1)} by A76, A90, A133, A139, XBOOLE_1:23
.= {((go ^' pion1) /. 1)} \/ {(co /. 1)} by FINSEQ_6:155
.= {((go ^' pion1) /. 1),(co /. 1)} by ENUMSET1:1 ;
co /. (len co) = (go ^' pion1) /. 1 by A63, FINSEQ_6:155;
then reconsider godo = (go ^' pion1) ^' co as constant standard special_circular_sequence by A102, A106, A107, A112, A114, A122, A123, A131, A132, A140, JORDAN8:4, JORDAN8:5, TOPREAL8:11, TOPREAL8:33, TOPREAL8:34;
A141: Upper_Arc C is_an_arc_of W-min C, E-max C by JORDAN6:def 8;
then A142: Upper_Arc C is connected by JORDAN6:10;
A143: W-min C in Upper_Arc C by A141, TOPREAL1:1;
A144: E-max C in Upper_Arc C by A141, TOPREAL1:1;
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_2:43;
then A145: (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 = W-min (L~ (Cage (C,n))) by FINSEQ_6:92;
A146: 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 A145, 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 A145, A146, 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 A145, A146, SPRECT_5:24, XXREAL_0:2;
then A147: (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A145, A146, SPRECT_5:25, XXREAL_0:2;
A148: now :: thesis: not ((Gauge (C,n)) * (i1,j)) .. (Upper_Seq (C,n)) <= 1
assume A149: ((Gauge (C,n)) * (i1,j)) .. (Upper_Seq (C,n)) <= 1 ; :: thesis: contradiction
((Gauge (C,n)) * (i1,j)) .. (Upper_Seq (C,n)) >= 1 by A37, FINSEQ_4:21;
then ((Gauge (C,n)) * (i1,j)) .. (Upper_Seq (C,n)) = 1 by A149, XXREAL_0:1;
then (Gauge (C,n)) * (i1,j) = (Upper_Seq (C,n)) /. 1 by A37, FINSEQ_5:38;
hence contradiction by A22, A26, JORDAN1F:5; :: thesis: verum
end;
A150: Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def 1;
then A151: Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))) is_sequence_on Gauge (C,n) by REVROT_1:34;
A152: (right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo) c= RightComp godo by A102, A107, JORDAN9:27;
A153: L~ godo = (L~ (go ^' pion1)) \/ (L~ co) by A106, TOPREAL8:35
.= ((L~ go) \/ (L~ pion1)) \/ (L~ co) by A104, TOPREAL8:35 ;
A154: L~ (Cage (C,n)) = (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n))) by JORDAN1E:13;
then A155: L~ (Upper_Seq (C,n)) c= L~ (Cage (C,n)) by XBOOLE_1:7;
A156: L~ (Lower_Seq (C,n)) c= L~ (Cage (C,n)) by A154, XBOOLE_1:7;
A157: L~ go c= L~ (Cage (C,n)) by A49, A155;
A158: L~ co c= L~ (Cage (C,n)) by A56, A156;
A159: W-min C in C by SPRECT_1:13;
A160: now :: thesis: not W-min C in L~ godo
assume W-min C in L~ godo ; :: thesis: contradiction
then A161: ( W-min C in (L~ go) \/ (L~ pion1) or W-min C in L~ co ) by A153, XBOOLE_0:def 3;
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 A98, 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 A147, A151, 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)) * (i1,j)))),1,(Gauge (C,n))) by A37, A103, A148, JORDAN1J:52
.= right_cell ((go ^' pion1),1,(Gauge (C,n))) by A42, A105, JORDAN1J:51
.= right_cell (godo,1,(Gauge (C,n))) by A100, A107, JORDAN1J:51 ;
then W-min C in right_cell (godo,1,(Gauge (C,n))) by JORDAN1I:6;
then A162: W-min C in (right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo) by A160, XBOOLE_0:def 5;
A163: godo /. 1 = (go ^' pion1) /. 1 by FINSEQ_6:155
.= W-min (L~ (Cage (C,n))) by A62, FINSEQ_6:155 ;
A164: len (Upper_Seq (C,n)) >= 2 by A21, XXREAL_0:2;
A165: godo /. 2 = (go ^' pion1) /. 2 by A99, FINSEQ_6:159
.= (Upper_Seq (C,n)) /. 2 by A36, A79, FINSEQ_6:159
.= ((Upper_Seq (C,n)) ^' (Lower_Seq (C,n))) /. 2 by A164, FINSEQ_6:159
.= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 2 by JORDAN1E:11 ;
A166: (L~ go) \/ (L~ co) is compact by COMPTS_1:10;
W-min (L~ (Cage (C,n))) in (L~ go) \/ (L~ co) by A64, A80, XBOOLE_0:def 3;
then A167: W-min ((L~ go) \/ (L~ co)) = W-min (L~ (Cage (C,n))) by A157, A158, A166, JORDAN1J:21, XBOOLE_1:8;
A169: (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) ;
((Gauge (C,n)) * (i1,j)) `1 >= W-bound (L~ (Cage (C,n))) by A14, A155, PSCOMP_1:24;
then ((Gauge (C,n)) * (i1,j)) `1 > W-bound (L~ (Cage (C,n))) by A78, XXREAL_0:1;
then W-min (((L~ go) \/ (L~ co)) \/ (L~ pion1)) = W-min ((L~ go) \/ (L~ co)) by A96, A166, A167, A169, JORDAN1J:33;
then A170: W-min (L~ godo) = W-min (L~ (Cage (C,n))) by A153, A167, XBOOLE_1:4;
A171: rng godo c= L~ godo by A99, A101, SPPOL_2:18, XXREAL_0:2;
2 in dom godo by A102, FINSEQ_3:25;
then A172: godo /. 2 in rng godo by PARTFUN2:2;
godo /. 2 in W-most (L~ (Cage (C,n))) by A165, JORDAN1I:25;
then (godo /. 2) `1 = (W-min (L~ godo)) `1 by A170, PSCOMP_1:31
.= W-bound (L~ godo) ;
then godo /. 2 in W-most (L~ godo) by A171, A172, SPRECT_2:12;
then (Rotate (godo,(W-min (L~ godo)))) /. 2 in W-most (L~ godo) by A163, A170, 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 A173: (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 ;
A174: 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
A175: p in east_halfline (E-max C) and
A176: p in L~ go by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A175;
p in L~ (Upper_Seq (C,n)) by A49, A176;
then p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A155, A175, XBOOLE_0:def 4;
then p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;
then p = E-max (L~ (Cage (C,n))) by A49, A176, JORDAN1J:46;
then E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i1,j) by A14, A173, A176, JORDAN1J:43;
then ((Gauge (C,n)) * (i1,j)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by A6, A16, A20, JORDAN1A:71;
hence contradiction by A2, A3, A17, A33, 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 A179: ( east_halfline (E-max C) meets (L~ go) \/ (L~ pion1) or east_halfline (E-max C) meets L~ co ) by A153, 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 A179, XBOOLE_1:70;
suppose east_halfline (E-max C) meets L~ pion1 ; :: thesis: contradiction
then consider p being object such that
A180: p in east_halfline (E-max C) and
A181: p in L~ pion1 by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A180;
A182: 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 A12, A88, A181, 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 A93, 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 A92, A93, GOBOARD7:5; :: thesis: verum
end;
end;
end;
i2 + 1 <= len (Gauge (C,n)) by A3, NAT_1:13;
then i2 <= (len (Gauge (C,n))) - 1 by XREAL_1:19;
then A183: 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, A11, A16, A20, A24, A183, JORDAN1A:18;
then p `1 <= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),1)) `1 by A182, XXREAL_0:2;
then p `1 <= E-bound C by A24, JORDAN8:12;
then A184: p `1 <= (E-max C) `1 ;
p `1 >= (E-max C) `1 by A180, TOPREAL1:def 11;
then A185: p `1 = (E-max C) `1 by A184, XXREAL_0:1;
p `2 = (E-max C) `2 by A180, TOPREAL1:def 11;
then p = E-max C by A185, TOPREAL3:6;
hence contradiction by A9, A12, A88, A144, A181, XBOOLE_0:3; :: thesis: verum
end;
suppose east_halfline (E-max C) meets L~ co ; :: thesis: contradiction
then consider p being object such that
A186: p in east_halfline (E-max C) and
A187: p in L~ co by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A186;
p in L~ (Lower_Seq (C,n)) by A56, A187;
then p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A156, A186, XBOOLE_0:def 4;
then A188: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;
A189: (E-max C) `2 = p `2 by A186, TOPREAL1:def 11;
set RC = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))));
A190: E-max C in right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by JORDAN1I:7;
A191: 1 + 1 <= len (Lower_Seq (C,n)) by A28, XXREAL_0:2;
Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) by JORDAN1G:18;
then A192: LSeg ((Lower_Seq (C,n)),1) = LSeg ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by A191, SPPOL_2:9;
A193: L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33;
A194: len (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = len (Cage (C,n)) by FINSEQ_6:179;
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:44 ;
A196: E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
A197: Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))) is_sequence_on Gauge (C,n) by A150, REVROT_1:34;
A198: (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 A193, A196, FINSEQ_6:92;
consider ii, jj being Nat such that
A199: [ii,(jj + 1)] in Indices (Gauge (C,n)) and
A200: [ii,jj] in Indices (Gauge (C,n)) and
A201: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = (Gauge (C,n)) * (ii,(jj + 1)) and
A202: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1) = (Gauge (C,n)) * (ii,jj) by A97, A193, A194, A196, A197, FINSEQ_6:92, JORDAN1I:23;
consider jj2 being Nat such that
A203: 1 <= jj2 and
A204: jj2 <= width (Gauge (C,n)) and
A205: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * ((len (Gauge (C,n))),jj2) by JORDAN1D:25;
A206: len (Gauge (C,n)) >= 4 by JORDAN8:10;
then len (Gauge (C,n)) >= 1 by XXREAL_0:2;
then [(len (Gauge (C,n))),jj2] in Indices (Gauge (C,n)) by A203, A204, MATRIX_0:30;
then A207: ii = len (Gauge (C,n)) by A193, A198, A199, A201, A205, GOBOARD1:5;
A208: 1 <= ii by A199, MATRIX_0:32;
A209: ii <= len (Gauge (C,n)) by A199, MATRIX_0:32;
A210: 1 <= jj + 1 by A199, MATRIX_0:32;
A211: jj + 1 <= width (Gauge (C,n)) by A199, MATRIX_0:32;
A212: 1 <= ii by A200, MATRIX_0:32;
A213: ii <= len (Gauge (C,n)) by A200, MATRIX_0:32;
A214: 1 <= jj by A200, MATRIX_0:32;
A215: jj <= width (Gauge (C,n)) by A200, MATRIX_0:32;
A216: ii + 1 <> ii ;
(jj + 1) + 1 <> jj ;
then A217: right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) = cell ((Gauge (C,n)),(ii -' 1),jj) by A97, A194, A195, A199, A200, A201, A202, A216, GOBOARD5:def 6;
A218: (ii -' 1) + 1 = ii by A208, XREAL_1:235;
ii - 1 >= 4 - 1 by A206, A207, XREAL_1:9;
then A219: ii - 1 >= 1 by XXREAL_0:2;
then A220: 1 <= ii -' 1 by XREAL_0:def 2;
A221: ((Gauge (C,n)) * ((ii -' 1),jj)) `2 <= p `2 by A189, A190, A209, A211, A214, A217, A218, A219, JORDAN9:17;
A222: p `2 <= ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 by A189, A190, A209, A211, A214, A217, A218, A219, JORDAN9:17;
A223: ii -' 1 < len (Gauge (C,n)) by A209, A218, NAT_1:13;
then A224: ((Gauge (C,n)) * ((ii -' 1),jj)) `2 = ((Gauge (C,n)) * (1,jj)) `2 by A214, A215, A220, GOBOARD5:1
.= ((Gauge (C,n)) * (ii,jj)) `2 by A212, A213, A214, A215, GOBOARD5:1 ;
A225: ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 = ((Gauge (C,n)) * (1,(jj + 1))) `2 by A210, A211, A220, A223, GOBOARD5:1
.= ((Gauge (C,n)) * (ii,(jj + 1))) `2 by A208, A209, A210, A211, GOBOARD5:1 ;
A226: ((Gauge (C,n)) * ((len (Gauge (C,n))),jj)) `1 = E-bound (L~ (Cage (C,n))) by A20, A214, A215, JORDAN1A:71;
E-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * ((len (Gauge (C,n))),(jj + 1))) `1 by A20, A210, A211, JORDAN1A:71;
then p in LSeg (((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1),((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1))) by A188, A201, A202, A207, A221, A222, A224, A225, A226, GOBOARD7:7;
then A227: p in LSeg ((Lower_Seq (C,n)),1) by A97, A192, A194, TOPREAL1:def 3;
A228: p in LSeg (co,(Index (p,co))) by A187, JORDAN3:9;
A229: co = mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n)))) by A40, JORDAN1J:37;
A230: 1 <= ((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)) by A40, FINSEQ_4:21;
A231: ((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)) <= len (Lower_Seq (C,n)) by A40, FINSEQ_4:21;
((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)) <> len (Lower_Seq (C,n)) by A32, A40, FINSEQ_4:19;
then A232: ((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)) < len (Lower_Seq (C,n)) by A231, XXREAL_0:1;
A233: 1 <= Index (p,co) by A187, JORDAN3:8;
A234: Index (p,co) < len co by A187, JORDAN3:8;
A235: (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n)))) + 1 = ((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)) by A35, A40, JORDAN1J:56;
consider t being Nat such that
A236: t in dom (Lower_Seq (C,n)) and
A237: (Lower_Seq (C,n)) . t = (Gauge (C,n)) * (i2,k) by A40, FINSEQ_2:10;
A238: 1 <= t by A236, FINSEQ_3:25;
A239: t <= len (Lower_Seq (C,n)) by A236, FINSEQ_3:25;
1 < t by A35, A237, A238, XXREAL_0:1;
then (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n)))) + 1 = t by A237, A239, JORDAN3:12;
then A240: len (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i2,k)))) = (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n)))) by A13, A237, JORDAN3:26;
set tt = ((Index (p,co)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1;
A241: 1 <= Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n))) by A13, JORDAN3:8;
0 + (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n)))) < len (Lower_Seq (C,n)) by A13, JORDAN3:8;
then A242: (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n)))) > 0 by XREAL_1:20;
Index (p,co) < (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n)))) by A234, A240, XREAL_0:def 2;
then (Index (p,co)) + 1 <= (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n)))) by NAT_1:13;
then Index (p,co) <= ((len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n))))) - 1 by XREAL_1:19;
then Index (p,co) <= ((len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n))))) - 1 by A242, XREAL_0:def 2;
then Index (p,co) <= (len (Lower_Seq (C,n))) - (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n))) by A235;
then Index (p,co) <= (len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n))) by XREAL_0:def 2;
then Index (p,co) < ((len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) + 1 by NAT_1:13;
then A243: LSeg ((mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n))))),(Index (p,co))) = LSeg ((Lower_Seq (C,n)),(((Index (p,co)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1)) by A230, A232, A233, JORDAN4:19;
A244: 1 + 1 <= ((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)) by A235, A241, XREAL_1:7;
then (Index (p,co)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n))) >= (1 + 1) + 1 by A233, XREAL_1:7;
then ((Index (p,co)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) - 1 >= ((1 + 1) + 1) - 1 by XREAL_1:9;
then A245: ((Index (p,co)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1 >= 1 + 1 by XREAL_0:def 2;
A246: 2 in dom (Lower_Seq (C,n)) by A191, FINSEQ_3:25;
now :: thesis: contradiction
per cases ( ((Index (p,co)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1 > 1 + 1 or ((Index (p,co)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ) by A245, XXREAL_0:1;
suppose ((Index (p,co)) + (((Gauge (C,n)) * (i2,k)) .. (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)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1)) by TOPREAL1:def 7;
hence contradiction by A227, A228, A229, A243, XBOOLE_0:3; :: thesis: verum
end;
suppose A247: ((Index (p,co)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ; :: thesis: contradiction
then (LSeg ((Lower_Seq (C,n)),1)) /\ (LSeg ((Lower_Seq (C,n)),(((Index (p,co)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1))) = {((Lower_Seq (C,n)) /. 2)} by A28, TOPREAL1:def 6;
then p in {((Lower_Seq (C,n)) /. 2)} by A227, A228, A229, A243, XBOOLE_0:def 4;
then A248: p = (Lower_Seq (C,n)) /. 2 by TARSKI:def 1;
then A249: p .. (Lower_Seq (C,n)) = 2 by A246, FINSEQ_5:41;
1 + 1 = ((Index (p,co)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) - 1 by A247, XREAL_0:def 2;
then (1 + 1) + 1 = (Index (p,co)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n))) ;
then A250: ((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)) = 2 by A233, A244, JORDAN1E:6;
p in rng (Lower_Seq (C,n)) by A246, A248, PARTFUN2:2;
then p = (Gauge (C,n)) * (i2,k) by A40, A249, A250, FINSEQ_5:9;
then ((Gauge (C,n)) * (i2,k)) `1 = E-bound (L~ (Cage (C,n))) by A248, JORDAN1G:32;
then ((Gauge (C,n)) * (i2,k)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 by A4, A15, A20, JORDAN1A:71;
hence contradiction by A3, A18, A70, 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
A251: W is_a_component_of (L~ godo) ` and
A252: east_halfline (E-max C) c= W by GOBOARD9:3;
not W is bounded by A252, JORDAN2C:121, RLTOPSP1:42;
then W is_outside_component_of L~ godo by A251, JORDAN2C:def 3;
then W c= UBD (L~ godo) by JORDAN2C:23;
then A253: east_halfline (E-max C) c= UBD (L~ godo) by A252;
E-max C in east_halfline (E-max C) by TOPREAL1:38;
then E-max C in UBD (L~ godo) by A253;
then E-max C in LeftComp godo by GOBRD14:36;
then Upper_Arc C meets L~ godo by A142, A143, A144, A152, A162, JORDAN1J:36;
then A254: ( Upper_Arc C meets (L~ go) \/ (L~ pion1) or Upper_Arc C meets L~ co ) by A153, XBOOLE_1:70;
A255: Upper_Arc C c= C by JORDAN6:61;
hence contradiction ; :: thesis: verum
end;
suppose ((Gauge (C,n)) * (i2,k)) `1 = ((Gauge (C,n)) * (i1,j)) `1 ; :: thesis: contradiction
then A256: i1 = i2 by A17, A18, 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 A84, 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, A256, Th12; :: thesis: verum
end;
suppose ((Gauge (C,n)) * (i2,k)) `2 = ((Gauge (C,n)) * (i1,j)) `2 ; :: thesis: contradiction
then A257: j = k by A17, A18, 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 A85, 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, A257, JORDAN15:37; :: thesis: verum
end;
end;
end;
hence contradiction ; :: thesis: verum