let n be Nat; for C being Simple_closed_curve
for i, j, k being Nat st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j))} holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C
let C be Simple_closed_curve; for i, j, k being Nat st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j))} holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C
let i, j, k be Nat; ( 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j))} implies LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C )
set Ga = Gauge (C,n);
set US = Upper_Seq (C,n);
set LS = Lower_Seq (C,n);
set UA = Upper_Arc C;
set Wmin = W-min (L~ (Cage (C,n)));
set Emax = E-max (L~ (Cage (C,n)));
set Wbo = W-bound (L~ (Cage (C,n)));
set Ebo = E-bound (L~ (Cage (C,n)));
set Gij = (Gauge (C,n)) * (i,j);
set Gik = (Gauge (C,n)) * (i,k);
assume that
A1:
1 < i
and
A2:
i < len (Gauge (C,n))
and
A3:
1 <= j
and
A4:
j <= k
and
A5:
k <= width (Gauge (C,n))
and
A6:
(LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k))}
and
A7:
(LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j))}
and
A8:
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) misses Upper_Arc C
; contradiction
(Gauge (C,n)) * (i,j) in {((Gauge (C,n)) * (i,j))}
by TARSKI:def 1;
then A9:
(Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n))
by A7, XBOOLE_0:def 4;
(Gauge (C,n)) * (i,k) in {((Gauge (C,n)) * (i,k))}
by TARSKI:def 1;
then A10:
(Gauge (C,n)) * (i,k) in L~ (Upper_Seq (C,n))
by A6, XBOOLE_0:def 4;
then A11:
j <> k
by A1, A2, A3, A5, A9, Th57;
A12:
j <= width (Gauge (C,n))
by A4, A5, XXREAL_0:2;
A13:
1 <= k
by A3, A4, XXREAL_0:2;
A14:
[i,j] in Indices (Gauge (C,n))
by A1, A2, A3, A12, MATRIX_0:30;
A15:
[i,k] in Indices (Gauge (C,n))
by A1, A2, A5, A13, MATRIX_0:30;
set go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,k)));
set co = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)));
A16:
len (Gauge (C,n)) = width (Gauge (C,n))
by JORDAN8:def 1;
A17:
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 A18: (Upper_Seq (C,n)) . 1 =
(Upper_Seq (C,n)) /. 1
by PARTFUN1:def 6
.=
W-min (L~ (Cage (C,n)))
by JORDAN1F:5
;
A19: (W-min (L~ (Cage (C,n)))) `1 =
W-bound (L~ (Cage (C,n)))
by EUCLID:52
.=
((Gauge (C,n)) * (1,k)) `1
by A5, A13, A16, JORDAN1A:73
;
len (Gauge (C,n)) >= 4
by JORDAN8:10;
then A20:
len (Gauge (C,n)) >= 1
by XXREAL_0:2;
then A21:
[1,k] in Indices (Gauge (C,n))
by A5, A13, MATRIX_0:30;
then A22:
(Gauge (C,n)) * (i,k) <> (Upper_Seq (C,n)) . 1
by A1, A15, A18, A19, JORDAN1G:7;
then reconsider go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,k))) as being_S-Seq FinSequence of (TOP-REAL 2) by A10, JORDAN3:35;
A23:
len (Lower_Seq (C,n)) >= 1 + 2
by JORDAN1E:15;
then A24:
len (Lower_Seq (C,n)) >= 1
by XXREAL_0:2;
then A25:
1 in dom (Lower_Seq (C,n))
by FINSEQ_3:25;
len (Lower_Seq (C,n)) in dom (Lower_Seq (C,n))
by A24, FINSEQ_3:25;
then A26: (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
;
A27: (W-min (L~ (Cage (C,n)))) `1 =
W-bound (L~ (Cage (C,n)))
by EUCLID:52
.=
((Gauge (C,n)) * (1,k)) `1
by A5, A13, A16, JORDAN1A:73
;
A28:
[i,j] in Indices (Gauge (C,n))
by A1, A2, A3, A12, MATRIX_0:30;
then A29:
(Gauge (C,n)) * (i,j) <> (Lower_Seq (C,n)) . (len (Lower_Seq (C,n)))
by A1, A21, A26, A27, JORDAN1G:7;
then reconsider co = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))) as being_S-Seq FinSequence of (TOP-REAL 2) by A9, JORDAN3:34;
A30:
[(len (Gauge (C,n))),k] in Indices (Gauge (C,n))
by A5, A13, A20, MATRIX_0:30;
A31: (Lower_Seq (C,n)) . 1 =
(Lower_Seq (C,n)) /. 1
by A25, 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)))
by EUCLID:52
.=
((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1
by A5, A13, A16, JORDAN1A:71
;
then A32:
(Gauge (C,n)) * (i,j) <> (Lower_Seq (C,n)) . 1
by A2, A28, A30, A31, JORDAN1G:7;
A33:
len go >= 1 + 1
by TOPREAL1:def 8;
A34:
(Gauge (C,n)) * (i,k) in rng (Upper_Seq (C,n))
by A1, A2, A5, A10, A13, Th40, JORDAN1G:4;
then A35:
go is_sequence_on Gauge (C,n)
by Th38, JORDAN1G:4;
A36:
len co >= 1 + 1
by TOPREAL1:def 8;
A37:
(Gauge (C,n)) * (i,j) in rng (Lower_Seq (C,n))
by A1, A2, A3, A9, A12, Th40, JORDAN1G:5;
then A38:
co is_sequence_on Gauge (C,n)
by Th39, JORDAN1G:5;
reconsider go = go as V22() being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A33, A35, JGRAPH_1:12, JORDAN8:5;
reconsider co = co as V22() being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A36, A38, JGRAPH_1:12, JORDAN8:5;
A39:
len go > 1
by A33, NAT_1:13;
then A40:
len go in dom go
by FINSEQ_3:25;
then A41: go /. (len go) =
go . (len go)
by PARTFUN1:def 6
.=
(Gauge (C,n)) * (i,k)
by A10, JORDAN3:24
;
len co >= 1
by A36, XXREAL_0:2;
then
1 in dom co
by FINSEQ_3:25;
then A42: co /. 1 =
co . 1
by PARTFUN1:def 6
.=
(Gauge (C,n)) * (i,j)
by A9, JORDAN3:23
;
reconsider m = (len go) - 1 as Nat by A40, FINSEQ_3:26;
A43:
m + 1 = len go
;
then A44:
(len go) -' 1 = m
by NAT_D:34;
A45:
LSeg (go,m) c= L~ go
by TOPREAL3:19;
A46:
L~ go c= L~ (Upper_Seq (C,n))
by A10, JORDAN3:41;
then
LSeg (go,m) c= L~ (Upper_Seq (C,n))
by A45;
then A47:
(LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) c= {((Gauge (C,n)) * (i,k))}
by A6, XBOOLE_1:26;
m >= 1
by A33, XREAL_1:19;
then A48:
LSeg (go,m) = LSeg ((go /. m),((Gauge (C,n)) * (i,k)))
by A41, A43, TOPREAL1:def 3;
{((Gauge (C,n)) * (i,k))} c= (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))))
proof
let x be
object ;
TARSKI:def 3 ( not x in {((Gauge (C,n)) * (i,k))} or x in (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) )
A49:
(Gauge (C,n)) * (
i,
k)
in LSeg (
((Gauge (C,n)) * (i,k)),
((Gauge (C,n)) * (i,j)))
by RLTOPSP1:68;
assume
x in {((Gauge (C,n)) * (i,k))}
;
x in (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))))
then A50:
x = (Gauge (C,n)) * (
i,
k)
by TARSKI:def 1;
(Gauge (C,n)) * (
i,
k)
in LSeg (
go,
m)
by A48, RLTOPSP1:68;
hence
x in (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))))
by A50, A49, XBOOLE_0:def 4;
verum
end;
then A51:
(LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) = {((Gauge (C,n)) * (i,k))}
by A47;
A52:
LSeg (co,1) c= L~ co
by TOPREAL3:19;
A53:
L~ co c= L~ (Lower_Seq (C,n))
by A9, JORDAN3:42;
then
LSeg (co,1) c= L~ (Lower_Seq (C,n))
by A52;
then A54:
(LSeg (co,1)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) c= {((Gauge (C,n)) * (i,j))}
by A7, XBOOLE_1:26;
A55:
LSeg (co,1) = LSeg (((Gauge (C,n)) * (i,j)),(co /. (1 + 1)))
by A36, A42, TOPREAL1:def 3;
{((Gauge (C,n)) * (i,j))} c= (LSeg (co,1)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))))
proof
let x be
object ;
TARSKI:def 3 ( not x in {((Gauge (C,n)) * (i,j))} or x in (LSeg (co,1)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) )
A56:
(Gauge (C,n)) * (
i,
j)
in LSeg (
((Gauge (C,n)) * (i,k)),
((Gauge (C,n)) * (i,j)))
by RLTOPSP1:68;
assume
x in {((Gauge (C,n)) * (i,j))}
;
x in (LSeg (co,1)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))))
then A57:
x = (Gauge (C,n)) * (
i,
j)
by TARSKI:def 1;
(Gauge (C,n)) * (
i,
j)
in LSeg (
co,1)
by A55, RLTOPSP1:68;
hence
x in (LSeg (co,1)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))))
by A57, A56, XBOOLE_0:def 4;
verum
end;
then A58:
(LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) /\ (LSeg (co,1)) = {((Gauge (C,n)) * (i,j))}
by A54;
A59: go /. 1 =
(Upper_Seq (C,n)) /. 1
by A10, SPRECT_3:22
.=
W-min (L~ (Cage (C,n)))
by JORDAN1F:5
;
then A60: go /. 1 =
(Lower_Seq (C,n)) /. (len (Lower_Seq (C,n)))
by JORDAN1F:8
.=
co /. (len co)
by A9, Th35
;
A61:
rng go c= L~ go
by A33, SPPOL_2:18;
A62:
rng co c= L~ co
by A36, SPPOL_2:18;
A63:
{(go /. 1)} c= (L~ go) /\ (L~ co)
A66: (Lower_Seq (C,n)) . 1 =
(Lower_Seq (C,n)) /. 1
by A25, PARTFUN1:def 6
.=
E-max (L~ (Cage (C,n)))
by JORDAN1F:6
;
A67:
[(len (Gauge (C,n))),j] in Indices (Gauge (C,n))
by A3, A12, A20, MATRIX_0:30;
(L~ go) /\ (L~ co) c= {(go /. 1)}
proof
let x be
object ;
TARSKI:def 3 ( not x in (L~ go) /\ (L~ co) or x in {(go /. 1)} )
assume A68:
x in (L~ go) /\ (L~ co)
;
x in {(go /. 1)}
then A69:
x in L~ co
by XBOOLE_0:def 4;
A70:
now not x = E-max (L~ (Cage (C,n)))assume
x = E-max (L~ (Cage (C,n)))
;
contradictionthen A71:
E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (
i,
j)
by A9, A66, A69, JORDAN1E:7;
((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 = E-bound (L~ (Cage (C,n)))
by A3, A12, A16, JORDAN1A:71;
then
(E-max (L~ (Cage (C,n)))) `1 <> E-bound (L~ (Cage (C,n)))
by A2, A14, A67, A71, JORDAN1G:7;
hence
contradiction
by EUCLID:52;
verum end;
x in L~ go
by A68, XBOOLE_0:def 4;
then
x in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n)))
by A46, A53, A69, XBOOLE_0:def 4;
then
x in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))}
by JORDAN1E: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 A59, A70, TARSKI:def 1;
verum
end;
then A72:
(L~ go) /\ (L~ co) = {(go /. 1)}
by A63;
set W2 = go /. 2;
A73:
2 in dom go
by A33, FINSEQ_3:25;
A74:
now not ((Gauge (C,n)) * (i,k)) `1 = W-bound (L~ (Cage (C,n)))assume
((Gauge (C,n)) * (i,k)) `1 = W-bound (L~ (Cage (C,n)))
;
contradictionthen
((Gauge (C,n)) * (1,k)) `1 = ((Gauge (C,n)) * (i,k)) `1
by A5, A13, A16, JORDAN1A:73;
hence
contradiction
by A1, A15, A21, JORDAN1G:7;
verum end;
go =
mid ((Upper_Seq (C,n)),1,(((Gauge (C,n)) * (i,k)) .. (Upper_Seq (C,n))))
by A34, JORDAN1G:49
.=
(Upper_Seq (C,n)) | (((Gauge (C,n)) * (i,k)) .. (Upper_Seq (C,n)))
by A34, FINSEQ_4:21, FINSEQ_6:116
;
then A75:
go /. 2 = (Upper_Seq (C,n)) /. 2
by A73, FINSEQ_4:70;
set pion = <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*>;
A76:
now for n being Nat st n in dom <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> holds
ex i, j being Nat st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. n = (Gauge (C,n)) * (i,j) )let n be
Nat;
( n in dom <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> implies ex i, j being Nat st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. n = (Gauge (C,n)) * (i,j) ) )assume
n in dom <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*>
;
ex i, j being Nat st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. n = (Gauge (C,n)) * (i,j) )then
n in Seg 2
by FINSEQ_1:89;
then
(
n = 1 or
n = 2 )
by FINSEQ_1:2, TARSKI:def 2;
hence
ex
i,
j being
Nat st
(
[i,j] in Indices (Gauge (C,n)) &
<*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. n = (Gauge (C,n)) * (
i,
j) )
by A14, A15, FINSEQ_4:17;
verum end;
A77:
(Gauge (C,n)) * (i,k) <> (Gauge (C,n)) * (i,j)
by A11, A14, A15, GOBOARD1:5;
A78: ((Gauge (C,n)) * (i,k)) `1 =
((Gauge (C,n)) * (i,1)) `1
by A1, A2, A5, A13, GOBOARD5:2
.=
((Gauge (C,n)) * (i,j)) `1
by A1, A2, A3, A12, GOBOARD5:2
;
then
LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) is vertical
by SPPOL_1:16;
then
<*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> is being_S-Seq
by A77, JORDAN1B:7;
then consider pion1 being FinSequence of (TOP-REAL 2) such that
A79:
pion1 is_sequence_on Gauge (C,n)
and
A80:
pion1 is being_S-Seq
and
A81:
L~ <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> = L~ pion1
and
A82:
<*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. 1 = pion1 /. 1
and
A83:
<*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. (len <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*>) = pion1 /. (len pion1)
and
A84:
len <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> <= len pion1
by A76, GOBOARD3:2;
reconsider pion1 = pion1 as being_S-Seq FinSequence of (TOP-REAL 2) by A80;
set godo = (go ^' pion1) ^' co;
A85:
1 + 1 <= len (Cage (C,n))
by GOBOARD7:34, XXREAL_0:2;
A86:
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 A87:
len (go ^' pion1) >= 1 + 1
by A33, XXREAL_0:2;
then A88:
len (go ^' pion1) > 1 + 0
by NAT_1:13;
A89:
len ((go ^' pion1) ^' co) >= len (go ^' pion1)
by TOPREAL8:7;
then A90:
1 + 1 <= len ((go ^' pion1) ^' co)
by A87, XXREAL_0:2;
A91:
Upper_Seq (C,n) is_sequence_on Gauge (C,n)
by JORDAN1G:4;
A92:
go /. (len go) = pion1 /. 1
by A41, A82, FINSEQ_4:17;
then A93:
go ^' pion1 is_sequence_on Gauge (C,n)
by A35, A79, TOPREAL8:12;
A94: (go ^' pion1) /. (len (go ^' pion1)) =
<*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. (len <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*>)
by A83, FINSEQ_6:156
.=
<*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. 2
by FINSEQ_1:44
.=
co /. 1
by A42, FINSEQ_4:17
;
then A95:
(go ^' pion1) ^' co is_sequence_on Gauge (C,n)
by A38, A93, TOPREAL8:12;
LSeg (pion1,1) c= L~ <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*>
by A81, TOPREAL3:19;
then
LSeg (pion1,1) c= LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))
by SPPOL_2:21;
then A96:
(LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) c= {((Gauge (C,n)) * (i,k))}
by A44, A51, XBOOLE_1:27;
A97:
len pion1 >= 1 + 1
by A84, FINSEQ_1:44;
{((Gauge (C,n)) * (i,k))} c= (LSeg (go,m)) /\ (LSeg (pion1,1))
proof
let x be
object ;
TARSKI:def 3 ( not x in {((Gauge (C,n)) * (i,k))} or x in (LSeg (go,m)) /\ (LSeg (pion1,1)) )
assume
x in {((Gauge (C,n)) * (i,k))}
;
x in (LSeg (go,m)) /\ (LSeg (pion1,1))
then A98:
x = (Gauge (C,n)) * (
i,
k)
by TARSKI:def 1;
A99:
(Gauge (C,n)) * (
i,
k)
in LSeg (
go,
m)
by A48, RLTOPSP1:68;
(Gauge (C,n)) * (
i,
k)
in LSeg (
pion1,1)
by A41, A92, A97, TOPREAL1:21;
hence
x in (LSeg (go,m)) /\ (LSeg (pion1,1))
by A98, A99, XBOOLE_0:def 4;
verum
end;
then
(LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) = {(go /. (len go))}
by A41, A44, A96;
then A100:
go ^' pion1 is unfolded
by A92, TOPREAL8:34;
len pion1 >= 2 + 0
by A84, FINSEQ_1:44;
then A101:
(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 A101, XREAL_0:def 2
;
then A102:
(len (go ^' pion1)) -' 1 = (len go) + ((len pion1) -' 2)
by XREAL_0:def 2;
A103:
(len pion1) - 1 >= 1
by A97, XREAL_1:19;
then A104:
(len pion1) -' 1 = (len pion1) - 1
by XREAL_0:def 2;
A105: ((len pion1) -' 2) + 1 =
((len pion1) - 2) + 1
by A101, XREAL_0:def 2
.=
(len pion1) -' 1
by A103, XREAL_0:def 2
;
((len pion1) - 1) + 1 <= len pion1
;
then A106:
(len pion1) -' 1 < len pion1
by A104, NAT_1:13;
LSeg (pion1,((len pion1) -' 1)) c= L~ <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*>
by A81, TOPREAL3:19;
then
LSeg (pion1,((len pion1) -' 1)) c= LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))
by SPPOL_2:21;
then A107:
(LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (co,1)) c= {((Gauge (C,n)) * (i,j))}
by A58, XBOOLE_1:27;
{((Gauge (C,n)) * (i,j))} c= (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (co,1))
proof
let x be
object ;
TARSKI:def 3 ( not x in {((Gauge (C,n)) * (i,j))} or x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (co,1)) )
assume
x in {((Gauge (C,n)) * (i,j))}
;
x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (co,1))
then A108:
x = (Gauge (C,n)) * (
i,
j)
by TARSKI:def 1;
pion1 /. (((len pion1) -' 1) + 1) =
<*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. 2
by A83, A104, FINSEQ_1:44
.=
(Gauge (C,n)) * (
i,
j)
by FINSEQ_4:17
;
then A109:
(Gauge (C,n)) * (
i,
j)
in LSeg (
pion1,
((len pion1) -' 1))
by A103, A104, TOPREAL1:21;
(Gauge (C,n)) * (
i,
j)
in LSeg (
co,1)
by A55, RLTOPSP1:68;
hence
x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (co,1))
by A108, A109, XBOOLE_0:def 4;
verum
end;
then
(LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (co,1)) = {((Gauge (C,n)) * (i,j))}
by A107;
then A110:
(LSeg ((go ^' pion1),((len go) + ((len pion1) -' 2)))) /\ (LSeg (co,1)) = {((go ^' pion1) /. (len (go ^' pion1)))}
by A42, A92, A94, A105, A106, TOPREAL8:31;
A111:
not go ^' pion1 is trivial
by A87, NAT_D:60;
A112:
rng pion1 c= L~ pion1
by A97, SPPOL_2:18;
A113:
{(pion1 /. 1)} c= (L~ go) /\ (L~ pion1)
(L~ go) /\ (L~ pion1) c= {(pion1 /. 1)}
then A118:
(L~ go) /\ (L~ pion1) = {(pion1 /. 1)}
by A113;
then A119:
go ^' pion1 is s.n.c.
by A92, Th54;
(rng go) /\ (rng pion1) c= {(pion1 /. 1)}
by A61, A112, A118, XBOOLE_1:27;
then A120:
go ^' pion1 is one-to-one
by Th55;
A121: <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. (len <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*>) =
<*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. 2
by FINSEQ_1:44
.=
co /. 1
by A42, FINSEQ_4:17
;
A122:
{(pion1 /. (len pion1))} c= (L~ co) /\ (L~ pion1)
(L~ co) /\ (L~ pion1) c= {(pion1 /. (len pion1))}
then A127:
(L~ co) /\ (L~ pion1) = {(pion1 /. (len pion1))}
by A122;
A128: (L~ (go ^' pion1)) /\ (L~ co) =
((L~ go) \/ (L~ pion1)) /\ (L~ co)
by A92, TOPREAL8:35
.=
{(go /. 1)} \/ {(co /. 1)}
by A72, A83, A121, A127, 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 A60, FINSEQ_6:155;
then reconsider godo = (go ^' pion1) ^' co as V22() standard special_circular_sequence by A90, A94, A95, A100, A102, A110, A111, A119, A120, A128, JORDAN8:4, JORDAN8:5, TOPREAL8:11, TOPREAL8:33, TOPREAL8:34;
A129:
Upper_Arc C is_an_arc_of W-min C, E-max C
by JORDAN6:def 8;
then A130:
Upper_Arc C is connected
by JORDAN6:10;
A131:
W-min C in Upper_Arc C
by A129, TOPREAL1:1;
A132:
E-max C in Upper_Arc C
by A129, 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 A133:
(Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 = W-min (L~ (Cage (C,n)))
by FINSEQ_6:92;
A134:
L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) = L~ (Cage (C,n))
by REVROT_1:33;
then
(W-max (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1
by A133, SPRECT_5:22;
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 A133, A134, 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 A133, A134, SPRECT_5:24, XXREAL_0:2;
then A135:
(E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1
by A133, A134, SPRECT_5:25, XXREAL_0:2;
A136:
now not ((Gauge (C,n)) * (i,k)) .. (Upper_Seq (C,n)) <= 1assume A137:
((Gauge (C,n)) * (i,k)) .. (Upper_Seq (C,n)) <= 1
;
contradiction
((Gauge (C,n)) * (i,k)) .. (Upper_Seq (C,n)) >= 1
by A34, FINSEQ_4:21;
then
((Gauge (C,n)) * (i,k)) .. (Upper_Seq (C,n)) = 1
by A137, XXREAL_0:1;
then
(Gauge (C,n)) * (
i,
k)
= (Upper_Seq (C,n)) /. 1
by A34, FINSEQ_5:38;
hence
contradiction
by A18, A22, JORDAN1F:5;
verum end;
A138:
Cage (C,n) is_sequence_on Gauge (C,n)
by JORDAN9:def 1;
then A139:
Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))) is_sequence_on Gauge (C,n)
by REVROT_1:34;
A140:
(right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo) c= RightComp godo
by A90, A95, JORDAN9:27;
A141: L~ godo =
(L~ (go ^' pion1)) \/ (L~ co)
by A94, TOPREAL8:35
.=
((L~ go) \/ (L~ pion1)) \/ (L~ co)
by A92, TOPREAL8:35
;
A142:
L~ (Cage (C,n)) = (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n)))
by JORDAN1E:13;
then A143:
L~ (Upper_Seq (C,n)) c= L~ (Cage (C,n))
by XBOOLE_1:7;
A144:
L~ (Lower_Seq (C,n)) c= L~ (Cage (C,n))
by A142, XBOOLE_1:7;
A145:
L~ go c= L~ (Cage (C,n))
by A46, A143;
A146:
L~ co c= L~ (Cage (C,n))
by A53, A144;
A147:
W-min C in C
by SPRECT_1:13;
A148:
L~ <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> = LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))
by SPPOL_2:21;
right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1) =
right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(GoB (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))))
by A86, JORDAN1H: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 A135, A139, Th53
.=
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)) * (i,k)))),1,(Gauge (C,n)))
by A34, A91, A136, Th52
.=
right_cell ((go ^' pion1),1,(Gauge (C,n)))
by A39, A93, Th51
.=
right_cell (godo,1,(Gauge (C,n)))
by A88, A95, Th51
;
then
W-min C in right_cell (godo,1,(Gauge (C,n)))
by JORDAN1I:6;
then A151:
W-min C in (right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo)
by A149, XBOOLE_0:def 5;
A152: godo /. 1 =
(go ^' pion1) /. 1
by FINSEQ_6:155
.=
W-min (L~ (Cage (C,n)))
by A59, FINSEQ_6:155
;
A153:
len (Upper_Seq (C,n)) >= 2
by A17, XXREAL_0:2;
A154: godo /. 2 =
(go ^' pion1) /. 2
by A87, FINSEQ_6:159
.=
(Upper_Seq (C,n)) /. 2
by A33, A75, FINSEQ_6:159
.=
((Upper_Seq (C,n)) ^' (Lower_Seq (C,n))) /. 2
by A153, FINSEQ_6:159
.=
(Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 2
by JORDAN1E:11
;
A155:
(L~ go) \/ (L~ co) is compact
by COMPTS_1:10;
W-min (L~ (Cage (C,n))) in rng go
by A59, FINSEQ_6:42;
then
W-min (L~ (Cage (C,n))) in (L~ go) \/ (L~ co)
by A61, XBOOLE_0:def 3;
then A156:
W-min ((L~ go) \/ (L~ co)) = W-min (L~ (Cage (C,n)))
by A145, A146, A155, Th21, XBOOLE_1:8;
A157:
(W-min ((L~ go) \/ (L~ co))) `1 = W-bound ((L~ go) \/ (L~ co))
by EUCLID:52;
A158:
(W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n)))
by EUCLID:52;
W-bound (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) = ((Gauge (C,n)) * (i,k)) `1
by A78, SPRECT_1:54;
then A159:
W-bound (L~ pion1) = ((Gauge (C,n)) * (i,k)) `1
by A81, SPPOL_2:21;
((Gauge (C,n)) * (i,k)) `1 >= W-bound (L~ (Cage (C,n)))
by A10, A143, PSCOMP_1:24;
then
((Gauge (C,n)) * (i,k)) `1 > W-bound (L~ (Cage (C,n)))
by A74, XXREAL_0:1;
then
W-min (((L~ go) \/ (L~ co)) \/ (L~ pion1)) = W-min ((L~ go) \/ (L~ co))
by A155, A156, A157, A158, A159, Th33;
then A160:
W-min (L~ godo) = W-min (L~ (Cage (C,n)))
by A141, A156, XBOOLE_1:4;
A161:
rng godo c= L~ godo
by A87, A89, SPPOL_2:18, XXREAL_0:2;
2 in dom godo
by A90, FINSEQ_3:25;
then A162:
godo /. 2 in rng godo
by PARTFUN2:2;
godo /. 2 in W-most (L~ (Cage (C,n)))
by A154, JORDAN1I:25;
then (godo /. 2) `1 =
(W-min (L~ godo)) `1
by A160, PSCOMP_1:31
.=
W-bound (L~ godo)
by EUCLID:52
;
then
godo /. 2 in W-most (L~ godo)
by A161, A162, SPRECT_2:12;
then
(Rotate (godo,(W-min (L~ godo)))) /. 2 in W-most (L~ godo)
by A152, A160, FINSEQ_6:89;
then reconsider godo = godo as V22() 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 A163: (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
;
A164:
east_halfline (E-max C) misses L~ go
proof
assume
east_halfline (E-max C) meets L~ go
;
contradiction
then consider p being
object such that A165:
p in east_halfline (E-max C)
and A166:
p in L~ go
by XBOOLE_0:3;
reconsider p =
p as
Point of
(TOP-REAL 2) by A165;
p in L~ (Upper_Seq (C,n))
by A46, A166;
then
p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n)))
by A143, A165, XBOOLE_0:def 4;
then A167:
p `1 = E-bound (L~ (Cage (C,n)))
by JORDAN1A:83, PSCOMP_1:50;
then A168:
p = E-max (L~ (Cage (C,n)))
by A46, A166, Th46;
then
E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (
i,
k)
by A10, A163, A166, Th43;
then
((Gauge (C,n)) * (i,k)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1
by A5, A13, A16, A167, A168, JORDAN1A:71;
hence
contradiction
by A2, A15, A30, JORDAN1G:7;
verum
end;
now not east_halfline (E-max C) meets L~ godoassume
east_halfline (E-max C) meets L~ godo
;
contradictionthen A169:
(
east_halfline (E-max C) meets (L~ go) \/ (L~ pion1) or
east_halfline (E-max C) meets L~ co )
by A141, XBOOLE_1:70;
per cases
( east_halfline (E-max C) meets L~ go or east_halfline (E-max C) meets L~ pion1 or east_halfline (E-max C) meets L~ co )
by A169, XBOOLE_1:70;
suppose
east_halfline (E-max C) meets L~ pion1
;
contradictionthen consider p being
object such that A170:
p in east_halfline (E-max C)
and A171:
p in L~ pion1
by XBOOLE_0:3;
reconsider p =
p as
Point of
(TOP-REAL 2) by A170;
A172:
p `2 = (E-max C) `2
by A170, TOPREAL1:def 11;
i + 1
<= len (Gauge (C,n))
by A2, NAT_1:13;
then
(i + 1) - 1
<= (len (Gauge (C,n))) - 1
by XREAL_1:9;
then A173:
i <= (len (Gauge (C,n))) -' 1
by XREAL_0:def 2;
A174:
(len (Gauge (C,n))) -' 1
<= len (Gauge (C,n))
by NAT_D:35;
p `1 = ((Gauge (C,n)) * (i,k)) `1
by A78, A81, A148, A171, GOBOARD7:5;
then
p `1 <= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),1)) `1
by A1, A5, A13, A16, A20, A173, A174, JORDAN1A:18;
then
p `1 <= E-bound C
by A20, JORDAN8:12;
then A175:
p `1 <= (E-max C) `1
by EUCLID:52;
p `1 >= (E-max C) `1
by A170, TOPREAL1:def 11;
then
p `1 = (E-max C) `1
by A175, XXREAL_0:1;
then
p = E-max C
by A172, TOPREAL3:6;
hence
contradiction
by A8, A81, A132, A148, A171, XBOOLE_0:3;
verum end; suppose
east_halfline (E-max C) meets L~ co
;
contradictionthen consider p being
object such that A176:
p in east_halfline (E-max C)
and A177:
p in L~ co
by XBOOLE_0:3;
reconsider p =
p as
Point of
(TOP-REAL 2) by A176;
A178:
(E-max C) `2 = p `2
by A176, TOPREAL1:def 11;
set tt =
((Index (p,co)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1;
set RC =
Rotate (
(Cage (C,n)),
(E-max (L~ (Cage (C,n)))));
A179:
L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = L~ (Cage (C,n))
by REVROT_1:33;
consider t being
Nat such that A180:
t in dom (Lower_Seq (C,n))
and A181:
(Lower_Seq (C,n)) . t = (Gauge (C,n)) * (
i,
j)
by A37, FINSEQ_2:10;
1
<= t
by A180, FINSEQ_3:25;
then A182:
1
< t
by A32, A181, XXREAL_0:1;
t <= len (Lower_Seq (C,n))
by A180, FINSEQ_3:25;
then
(Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1
= t
by A181, A182, JORDAN3:12;
then A183:
len (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) = (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n))))
by A9, A181, JORDAN3:26;
Index (
p,
co)
< len co
by A177, JORDAN3:8;
then
Index (
p,
co)
< (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n))))
by A183, XREAL_0:def 2;
then
(Index (p,co)) + 1
<= (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n))))
by NAT_1:13;
then A184:
Index (
p,
co)
<= ((len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n))))) - 1
by XREAL_1:19;
A185:
co = mid (
(Lower_Seq (C,n)),
(((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))),
(len (Lower_Seq (C,n))))
by A37, Th37;
A186:
len (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = len (Cage (C,n))
by FINSEQ_6:179;
p in L~ (Lower_Seq (C,n))
by A53, A177;
then
p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n)))
by A144, A176, XBOOLE_0:def 4;
then A187:
p `1 = E-bound (L~ (Cage (C,n)))
by JORDAN1A:83, PSCOMP_1:50;
A188:
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
;
A189:
1
+ 1
<= len (Lower_Seq (C,n))
by A23, XXREAL_0:2;
then A190:
2
in dom (Lower_Seq (C,n))
by FINSEQ_3:25;
consider jj2 being
Nat such that A191:
1
<= jj2
and A192:
jj2 <= width (Gauge (C,n))
and A193:
E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (
(len (Gauge (C,n))),
jj2)
by JORDAN1D:25;
A194:
len (Gauge (C,n)) >= 4
by JORDAN8:10;
then
len (Gauge (C,n)) >= 1
by XXREAL_0:2;
then A195:
[(len (Gauge (C,n))),jj2] in Indices (Gauge (C,n))
by A191, A192, MATRIX_0:30;
A196:
1
<= Index (
p,
co)
by A177, JORDAN3:8;
Lower_Seq (
C,
n)
= (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n))))
by JORDAN1G:18;
then A197:
LSeg (
(Lower_Seq (C,n)),1)
= LSeg (
(Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1)
by A189, SPPOL_2:9;
A198:
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 A138, REVROT_1:34;
then 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 A85, A179, A186, A198, FINSEQ_6:92, JORDAN1I:23;
A203:
(jj + 1) + 1
<> jj
;
A204:
1
<= jj
by A200, 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 A179, A198, FINSEQ_6:92;
then A205:
ii = len (Gauge (C,n))
by A179, A199, A201, A193, A195, GOBOARD1:5;
then
ii - 1
>= 4
- 1
by A194, XREAL_1:9;
then A206:
ii - 1
>= 1
by XXREAL_0:2;
then A207:
1
<= ii -' 1
by XREAL_0:def 2;
A208:
jj <= width (Gauge (C,n))
by A200, MATRIX_0:32;
then A209:
((Gauge (C,n)) * ((len (Gauge (C,n))),jj)) `1 = E-bound (L~ (Cage (C,n)))
by A16, A204, JORDAN1A:71;
A210:
jj + 1
<= width (Gauge (C,n))
by A199, MATRIX_0:32;
ii + 1
<> ii
;
then A211:
right_cell (
(Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1)
= cell (
(Gauge (C,n)),
(ii -' 1),
jj)
by A85, A186, A188, A199, A200, A201, A202, A203, GOBOARD5:def 6;
A212:
ii <= len (Gauge (C,n))
by A200, MATRIX_0:32;
A213:
1
<= ii
by A200, MATRIX_0:32;
A214:
ii <= len (Gauge (C,n))
by A199, MATRIX_0:32;
A215:
1
<= jj + 1
by A199, MATRIX_0:32;
then A216:
E-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * ((len (Gauge (C,n))),(jj + 1))) `1
by A16, A210, JORDAN1A:71;
A217:
1
<= ii
by A199, MATRIX_0:32;
then A218:
(ii -' 1) + 1
= ii
by XREAL_1:235;
then A219:
ii -' 1
< len (Gauge (C,n))
by A214, NAT_1:13;
then A220:
((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 =
((Gauge (C,n)) * (1,(jj + 1))) `2
by A215, A210, A207, GOBOARD5:1
.=
((Gauge (C,n)) * (ii,(jj + 1))) `2
by A217, A214, A215, A210, GOBOARD5:1
;
A221:
E-max C in right_cell (
(Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1)
by JORDAN1I:7;
then A222:
((Gauge (C,n)) * ((ii -' 1),jj)) `2 <= (E-max C) `2
by A214, A210, A204, A211, A218, A206, JORDAN9:17;
A223:
(E-max C) `2 <= ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2
by A221, A214, A210, A204, A211, A218, A206, JORDAN9:17;
((Gauge (C,n)) * ((ii -' 1),jj)) `2 =
((Gauge (C,n)) * (1,jj)) `2
by A204, A208, A207, A219, GOBOARD5:1
.=
((Gauge (C,n)) * (ii,jj)) `2
by A213, A212, A204, A208, GOBOARD5:1
;
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 A187, A178, A201, A202, A205, A222, A223, A220, A209, A216, GOBOARD7:7;
then A224:
p in LSeg (
(Lower_Seq (C,n)),1)
by A85, A197, A186, TOPREAL1:def 3;
A225:
((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) <= len (Lower_Seq (C,n))
by A37, FINSEQ_4:21;
((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) <> len (Lower_Seq (C,n))
by A29, A37, FINSEQ_4:19;
then A226:
((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) < len (Lower_Seq (C,n))
by A225, XXREAL_0:1;
A227:
(Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1
= ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))
by A32, A37, Th56;
0 + (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) < len (Lower_Seq (C,n))
by A9, JORDAN3:8;
then
(len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) > 0
by XREAL_1:20;
then
Index (
p,
co)
<= ((len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n))))) - 1
by A184, XREAL_0:def 2;
then
Index (
p,
co)
<= (len (Lower_Seq (C,n))) - (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))
by A227;
then
Index (
p,
co)
<= (len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))
by XREAL_0:def 2;
then A228:
Index (
p,
co)
< ((len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) + 1
by NAT_1:13;
A229:
p in LSeg (
co,
(Index (p,co)))
by A177, JORDAN3:9;
1
<= ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))
by A37, FINSEQ_4:21;
then A230:
LSeg (
(mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n))))),
(Index (p,co)))
= LSeg (
(Lower_Seq (C,n)),
(((Index (p,co)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1))
by A226, A196, A228, JORDAN4:19;
1
<= Index (
((Gauge (C,n)) * (i,j)),
(Lower_Seq (C,n)))
by A9, JORDAN3:8;
then A231:
1
+ 1
<= ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))
by A227, XREAL_1:7;
then
(Index (p,co)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))) >= (1 + 1) + 1
by A196, XREAL_1:7;
then
((Index (p,co)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) - 1
>= ((1 + 1) + 1) - 1
by XREAL_1:9;
then A232:
((Index (p,co)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1
>= 1
+ 1
by XREAL_0:def 2;
now contradictionper cases
( ((Index (p,co)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1 > 1 + 1 or ((Index (p,co)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 )
by A232, XXREAL_0:1;
suppose
((Index (p,co)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1
> 1
+ 1
;
contradictionthen
LSeg (
(Lower_Seq (C,n)),1)
misses LSeg (
(Lower_Seq (C,n)),
(((Index (p,co)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1))
by TOPREAL1:def 7;
hence
contradiction
by A224, A229, A185, A230, XBOOLE_0:3;
verum end; suppose A233:
((Index (p,co)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1
= 1
+ 1
;
contradictionthen
1
+ 1
= ((Index (p,co)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) - 1
by XREAL_0:def 2;
then
(1 + 1) + 1
= (Index (p,co)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))
;
then A234:
((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) = 2
by A196, A231, JORDAN1E:6;
(LSeg ((Lower_Seq (C,n)),1)) /\ (LSeg ((Lower_Seq (C,n)),(((Index (p,co)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1))) = {((Lower_Seq (C,n)) /. 2)}
by A23, A233, TOPREAL1:def 6;
then
p in {((Lower_Seq (C,n)) /. 2)}
by A224, A229, A185, A230, XBOOLE_0:def 4;
then A235:
p = (Lower_Seq (C,n)) /. 2
by TARSKI:def 1;
then A236:
p in rng (Lower_Seq (C,n))
by A190, PARTFUN2:2;
p .. (Lower_Seq (C,n)) = 2
by A190, A235, FINSEQ_5:41;
then
p = (Gauge (C,n)) * (
i,
j)
by A37, A234, A236, FINSEQ_5:9;
then
((Gauge (C,n)) * (i,j)) `1 = E-bound (L~ (Cage (C,n)))
by A235, JORDAN1G:32;
then
((Gauge (C,n)) * (i,j)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1
by A3, A12, A16, JORDAN1A:71;
hence
contradiction
by A2, A14, A67, JORDAN1G:7;
verum end; end; end; hence
contradiction
;
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
A237:
W is_a_component_of (L~ godo) `
and
A238:
east_halfline (E-max C) c= W
by GOBOARD9:3;
not W is bounded
by A238, JORDAN2C:121, RLTOPSP1:42;
then
W is_outside_component_of L~ godo
by A237, JORDAN2C:def 3;
then
W c= UBD (L~ godo)
by JORDAN2C:23;
then A239:
east_halfline (E-max C) c= UBD (L~ godo)
by A238;
E-max C in east_halfline (E-max C)
by TOPREAL1:38;
then
E-max C in UBD (L~ godo)
by A239;
then
E-max C in LeftComp godo
by GOBRD14:36;
then
Upper_Arc C meets L~ godo
by A130, A131, A132, A140, A151, Th36;
then A240:
( Upper_Arc C meets (L~ go) \/ (L~ pion1) or Upper_Arc C meets L~ co )
by A141, XBOOLE_1:70;
A241:
Upper_Arc C c= C
by JORDAN6:61;