let n be Nat; :: thesis:
let C be Simple_closed_curve; :: thesis:
let i, j, k be Nat; :: thesis:
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 Gik = (Gauge (C,n)) * (i,k);
set Gij = (Gauge (C,n)) * (i,j);
assume that
A1: 1 < i and
A2: i < len (Gauge (C,n)) and
A3: 1 <= k and
A4: k <= j and
A5: j <= width (Gauge (C,n)) and
A6: (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k))} and
A7: (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j))} and
A8: LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) misses Upper_Arc C ; :: thesis:
(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, JORDAN1J:57;
A12: 1 <= j by A3, A4, XXREAL_0:2;
A13: k <= width (Gauge (C,n)) by A4, A5, XXREAL_0:2;
A14: [i,j] in Indices (Gauge (C,n)) by A1, A2, A5, A12, MATRIX_0:30;
A15: [i,k] in Indices (Gauge (C,n)) by A1, A2, A3, A13, MATRIX_0:30;
set co = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)));
set go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,k)));
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)))
.= ((Gauge (C,n)) * (1,k)) `1 by A3, 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 A3, 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)))
.= ((Gauge (C,n)) * (1,k)) `1 by A3, A13, A16, JORDAN1A:73 ;
A28: [i,j] in Indices (Gauge (C,n)) by A1, A2, A5, 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 A3, 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)))
.= ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by A3, 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, A3, A10, A13, JORDAN1G:4, JORDAN1J:40;
then A35: go is_sequence_on Gauge (C,n) by JORDAN1G:4, JORDAN1J:38;
A36: len co >= 1 + 1 by TOPREAL1:def 8;
A37: (Gauge (C,n)) * (i,j) in rng (Lower_Seq (C,n)) by A1, A2, A5, A9, A12, JORDAN1G:5, JORDAN1J:40;
then A38: 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 A33, A35, JGRAPH_1:12, JORDAN8:5;
reconsider co = co as constant 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 ; :: according to TARSKI:def 3 :: thesis:
assume x in {((Gauge (C,n)) * (i,k))} ; :: thesis:
then A49: x = (Gauge (C,n)) * (i,k) by TARSKI:def 1;
A50: (Gauge (C,n)) * (i,k) in LSeg (go,m) by A48, RLTOPSP1:68;
(Gauge (C,n)) * (i,k) in LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) by RLTOPSP1:68;
hence x in (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) by A49, A50, XBOOLE_0:def 4; :: thesis:

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 ; :: according to TARSKI:def 3 :: thesis:
assume x in {((Gauge (C,n)) * (i,j))} ; :: thesis:
then A56: x = (Gauge (C,n)) * (i,j) by TARSKI:def 1;
A57: (Gauge (C,n)) * (i,j) in LSeg (co,1) by A55, RLTOPSP1:68;
(Gauge (C,n)) * (i,j) in LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) by RLTOPSP1:68;
hence x in (LSeg (co,1)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) by A56, A57, XBOOLE_0:def 4; :: thesis:

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, JORDAN1J:35 ;
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)

proof

let x be object ; :: according to TARSKI:def 3 :: thesis:
assume x in {(go /. 1)} ; :: thesis:
then A64: x = go /. 1 by TARSKI:def 1;
then A65: x in rng go by FINSEQ_6:42;
x in rng co by A60, A64, FINSEQ_6:168;
hence x in (L~ go) /\ (L~ co) by A61, A62, A65, XBOOLE_0:def 4; :: thesis:

end;

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 A5, A12, A20, MATRIX_0:30;
(L~ go) /\ (L~ co) c= {(go /. 1)}

proof

let x be object ; :: according to TARSKI:def 3 :: thesis:
assume A68: x in (L~ go) /\ (L~ co) ; :: thesis:
then A69: x in L~ go by XBOOLE_0:def 4;
A70: x in L~ co 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 A71: ( x = W-min (L~ (Cage (C,n))) or x = E-max (L~ (Cage (C,n))) ) by TARSKI:def 2;

now :: thesis:

assume x = E-max (L~ (Cage (C,n))) ; :: thesis:
then A72: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i,j) by A9, A66, A70, JORDAN1E:7;
((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 = E-bound (L~ (Cage (C,n))) by A5, A12, A16, JORDAN1A:71;
then (E-max (L~ (Cage (C,n)))) `1 <> E-bound (L~ (Cage (C,n))) by A2, A14, A67, A72, JORDAN1G:7;
hence contradiction ; :: thesis:

end;

hence x in {(go /. 1)} by A59, A71, TARSKI:def 1; :: thesis:

end;

then A73: (L~ go) /\ (L~ co) = {(go /. 1)} by A63;
set W2 = go /. 2;
A74: 2 in dom go by A33, FINSEQ_3:25;

A75: now :: thesis:

assume ((Gauge (C,n)) * (i,k)) `1 = W-bound (L~ (Cage (C,n))) ; :: thesis:
then ((Gauge (C,n)) * (1,k)) `1 = ((Gauge (C,n)) * (i,k)) `1 by A3, A13, A16, JORDAN1A:73;
hence contradiction by A1, A15, A21, JORDAN1G:7; :: thesis:

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 A76: go /. 2 = (Upper_Seq (C,n)) /. 2 by A74, FINSEQ_4:70;
A77: W-min (L~ (Cage (C,n))) in rng go by A59, FINSEQ_6:42;
set pion = <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*>;

A78: now :: thesis:

let n be Nat; :: thesis:
assume n in dom <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> ; :: thesis:
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; :: thesis:

end;

A79: (Gauge (C,n)) * (i,k) <> (Gauge (C,n)) * (i,j) by A11, A14, A15, GOBOARD1:5;
A80: ((Gauge (C,n)) * (i,k)) `1 = ((Gauge (C,n)) * (i,1)) `1 by A1, A2, A3, A13, GOBOARD5:2
.= ((Gauge (C,n)) * (i,j)) `1 by A1, A2, A5, 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 A79, JORDAN1B:7;
then consider pion1 being FinSequence of (TOP-REAL 2) such that
A81: pion1 is_sequence_on Gauge (C,n) and
A82: pion1 is being_S-Seq and
A83: L~ <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> = L~ pion1 and
A84: <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. 1 = pion1 /. 1 and
A85: <*((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
A86: len <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> <= len pion1 by A78, GOBOARD3:2;
reconsider pion1 = pion1 as being_S-Seq FinSequence of (TOP-REAL 2) by A82;
set godo = (go ^' pion1) ^' co;
A87: 1 + 1 <= len (Cage (C,n)) by GOBOARD7:34, XXREAL_0:2;
A88: 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 A89: len (go ^' pion1) >= 1 + 1 by A33, XXREAL_0:2;
then A90: len (go ^' pion1) > 1 + 0 by NAT_1:13;
A91: len ((go ^' pion1) ^' co) >= len (go ^' pion1) by TOPREAL8:7;
then A92: 1 + 1 <= len ((go ^' pion1) ^' co) by A89, XXREAL_0:2;
A93: Upper_Seq (C,n) is_sequence_on Gauge (C,n) by JORDAN1G:4;
A94: go /. (len go) = pion1 /. 1 by A41, A84, FINSEQ_4:17;
then A95: go ^' pion1 is_sequence_on Gauge (C,n) by A35, A81, TOPREAL8:12;
A96: (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 A85, 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 A97: (go ^' pion1) ^' co is_sequence_on Gauge (C,n) by A38, A95, TOPREAL8:12;
LSeg (pion1,1) c= L~ <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> by A83, TOPREAL3:19;
then LSeg (pion1,1) c= LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) by SPPOL_2:21;
then A98: (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) c= {((Gauge (C,n)) * (i,k))} by A44, A51, XBOOLE_1:27;
A99: len pion1 >= 1 + 1 by A86, FINSEQ_1:44;
{((Gauge (C,n)) * (i,k))} c= (LSeg (go,m)) /\ (LSeg (pion1,1))

proof

let x be object ; :: according to TARSKI:def 3 :: thesis:
assume x in {((Gauge (C,n)) * (i,k))} ; :: thesis:
then A100: x = (Gauge (C,n)) * (i,k) by TARSKI:def 1;
A101: (Gauge (C,n)) * (i,k) in LSeg (go,m) by A48, RLTOPSP1:68;
(Gauge (C,n)) * (i,k) in LSeg (pion1,1) by A41, A94, A99, TOPREAL1:21;
hence x in (LSeg (go,m)) /\ (LSeg (pion1,1)) by A100, A101, XBOOLE_0:def 4; :: thesis:

end;

then (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) = {(go /. (len go))} by A41, A44, A98;
then A102: go ^' pion1 is unfolded by A94, TOPREAL8:34;
len pion1 >= 2 + 0 by A86, FINSEQ_1:44;
then A103: (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 A103, XREAL_0:def 2 ;
then A104: (len (go ^' pion1)) -' 1 = (len go) + ((len pion1) -' 2) by XREAL_0:def 2;
A105: (len pion1) - 1 >= 1 by A99, XREAL_1:19;
then A106: (len pion1) -' 1 = (len pion1) - 1 by XREAL_0:def 2;
A107: ((len pion1) -' 2) + 1 = ((len pion1) - 2) + 1 by A103, XREAL_0:def 2
.= (len pion1) -' 1 by A105, XREAL_0:def 2 ;
((len pion1) - 1) + 1 <= len pion1 ;
then A108: (len pion1) -' 1 < len pion1 by A106, NAT_1:13;
LSeg (pion1,((len pion1) -' 1)) c= L~ <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> by A83, 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 A109: (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 ; :: according to TARSKI:def 3 :: thesis:
assume x in {((Gauge (C,n)) * (i,j))} ; :: thesis:
then A110: x = (Gauge (C,n)) * (i,j) by TARSKI:def 1;
A111: (Gauge (C,n)) * (i,j) in LSeg (co,1) by A55, RLTOPSP1:68;
pion1 /. (((len pion1) -' 1) + 1) = <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. 2 by A85, A106, FINSEQ_1:44
.= (Gauge (C,n)) * (i,j) by FINSEQ_4:17 ;
then (Gauge (C,n)) * (i,j) in LSeg (pion1,((len pion1) -' 1)) by A105, A106, TOPREAL1:21;
hence x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (co,1)) by A110, A111, XBOOLE_0:def 4; :: thesis:

end;

then (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (co,1)) = {((Gauge (C,n)) * (i,j))} by A109;
then A112: (LSeg ((go ^' pion1),((len go) + ((len pion1) -' 2)))) /\ (LSeg (co,1)) = {((go ^' pion1) /. (len (go ^' pion1)))} by A42, A94, A96, A107, A108, TOPREAL8:31;
A113: not go ^' pion1 is trivial by A89, NAT_D:60;
A114: rng pion1 c= L~ pion1 by A99, SPPOL_2:18;
A115: {(pion1 /. 1)} c= (L~ go) /\ (L~ pion1)

proof

let x be object ; :: according to TARSKI:def 3 :: thesis:
assume x in {(pion1 /. 1)} ; :: thesis:
then A116: x = pion1 /. 1 by TARSKI:def 1;
then A117: x in rng go by A94, FINSEQ_6:168;
x in rng pion1 by A116, FINSEQ_6:42;
hence x in (L~ go) /\ (L~ pion1) by A61, A114, A117, XBOOLE_0:def 4; :: thesis:

end;

(L~ go) /\ (L~ pion1) c= {(pion1 /. 1)}

proof

let x be object ; :: according to TARSKI:def 3 :: thesis:
assume A118: x in (L~ go) /\ (L~ pion1) ; :: thesis:
then A119: x in L~ go by XBOOLE_0:def 4;
x in L~ pion1 by A118, XBOOLE_0:def 4;
then x in (L~ pion1) /\ (L~ (Upper_Seq (C,n))) by A46, A119, XBOOLE_0:def 4;
hence x in {(pion1 /. 1)} by A6, A41, A83, A94, SPPOL_2:21; :: thesis:

end;

then A120: (L~ go) /\ (L~ pion1) = {(pion1 /. 1)} by A115;
then A121: go ^' pion1 is s.n.c. by A94, JORDAN1J:54;
(rng go) /\ (rng pion1) c= {(pion1 /. 1)} by A61, A114, A120, XBOOLE_1:27;
then A122: go ^' pion1 is one-to-one by JORDAN1J:55;
A123: <*((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 ;
A124: {(pion1 /. (len pion1))} c= (L~ co) /\ (L~ pion1)

proof

let x be object ; :: according to TARSKI:def 3 :: thesis:
assume x in {(pion1 /. (len pion1))} ; :: thesis:
then A125: x = pion1 /. (len pion1) by TARSKI:def 1;
then A126: x in rng co by A85, A123, FINSEQ_6:42;
x in rng pion1 by A125, FINSEQ_6:168;
hence x in (L~ co) /\ (L~ pion1) by A62, A114, A126, XBOOLE_0:def 4; :: thesis:

end;

(L~ co) /\ (L~ pion1) c= {(pion1 /. (len pion1))}

proof

let x be object ; :: according to TARSKI:def 3 :: thesis:
assume A127: x in (L~ co) /\ (L~ pion1) ; :: thesis:
then A128: x in L~ co by XBOOLE_0:def 4;
x in L~ pion1 by A127, XBOOLE_0:def 4;
then x in (L~ pion1) /\ (L~ (Lower_Seq (C,n))) by A53, A128, XBOOLE_0:def 4;
hence x in {(pion1 /. (len pion1))} by A7, A42, A83, A85, A123, SPPOL_2:21; :: thesis:

end;

A138: now :: thesis:

assume A139: ((Gauge (C,n)) * (i,k)) .. (Upper_Seq (C,n)) <= 1 ; :: thesis:
((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 A139, 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; :: thesis:

end;

A140: Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def 1;
then A141: Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))) is_sequence_on Gauge (C,n) by REVROT_1:34;
A142: (right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo) c= RightComp godo by A92, A97, JORDAN9:27;
A143: L~ godo = (L~ (go ^' pion1)) \/ (L~ co) by A96, TOPREAL8:35
.= ((L~ go) \/ (L~ pion1)) \/ (L~ co) by A94, TOPREAL8:35 ;
A144: L~ (Cage (C,n)) = (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n))) by JORDAN1E:13;
then A145: L~ (Upper_Seq (C,n)) c= L~ (Cage (C,n)) by XBOOLE_1:7;
A146: L~ (Lower_Seq (C,n)) c= L~ (Cage (C,n)) by A144, XBOOLE_1:7;
A147: L~ go c= L~ (Cage (C,n)) by A46, A145;
A148: L~ co c= L~ (Cage (C,n)) by A53, A146;
A149: W-min C in C by SPRECT_1:13;
A150: 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;

A151: now :: thesis:

assume W-min C in L~ godo ; :: thesis:
then A152: ( W-min C in (L~ go) \/ (L~ pion1) or W-min C in L~ co ) by A143, XBOOLE_0:def 3;

per cases by A152, XBOOLE_0:def 3;

suppose W-min C in L~ go ; :: thesis:

then C meets L~ (Cage (C,n)) by A147, A149, XBOOLE_0:3;
hence contradiction by JORDAN10:5; :: thesis:

end;

suppose W-min C in L~ pion1 ; :: thesis:

hence contradiction by A8, A83, A133, A150, XBOOLE_0:3; :: thesis:

end;

suppose W-min C in L~ co ; :: thesis:

then C meets L~ (Cage (C,n)) by A148, A149, XBOOLE_0:3;
hence contradiction by JORDAN10:5; :: thesis:

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 A88, 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 A137, A141, 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)) * (i,k)))),1,(Gauge (C,n))) by A34, A93, A138, JORDAN1J:52
.= right_cell ((go ^' pion1),1,(Gauge (C,n))) by A39, A95, JORDAN1J:51
.= right_cell (godo,1,(Gauge (C,n))) by A90, A97, JORDAN1J:51 ;
then W-min C in right_cell (godo,1,(Gauge (C,n))) by JORDAN1I:6;
then A153: W-min C in (right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo) by A151, XBOOLE_0:def 5;
A154: godo /. 1 = (go ^' pion1) /. 1 by FINSEQ_6:155
.= W-min (L~ (Cage (C,n))) by A59, FINSEQ_6:155 ;
A155: len (Upper_Seq (C,n)) >= 2 by A17, XXREAL_0:2;
A156: godo /. 2 = (go ^' pion1) /. 2 by A89, FINSEQ_6:159
.= (Upper_Seq (C,n)) /. 2 by A33, A76, FINSEQ_6:159
.= ((Upper_Seq (C,n)) ^' (Lower_Seq (C,n))) /. 2 by A155, FINSEQ_6:159
.= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 2 by JORDAN1E:11 ;
A157: (L~ go) \/ (L~ co) is compact by COMPTS_1:10;
W-min (L~ (Cage (C,n))) in (L~ go) \/ (L~ co) by A61, A77, XBOOLE_0:def 3;
then A158: W-min ((L~ go) \/ (L~ co)) = W-min (L~ (Cage (C,n))) by A147, A148, A157, JORDAN1J:21, XBOOLE_1:8;
A160: (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) ;
W-bound (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) = ((Gauge (C,n)) * (i,k)) `1 by A80, SPRECT_1:54;
then A161: W-bound (L~ pion1) = ((Gauge (C,n)) * (i,k)) `1 by A83, SPPOL_2:21;
((Gauge (C,n)) * (i,k)) `1 >= W-bound (L~ (Cage (C,n))) by A10, A145, PSCOMP_1:24;
then ((Gauge (C,n)) * (i,k)) `1 > W-bound (L~ (Cage (C,n))) by A75, XXREAL_0:1;
then W-min (((L~ go) \/ (L~ co)) \/ (L~ pion1)) = W-min ((L~ go) \/ (L~ co)) by A157, A158, A160, A161, JORDAN1J:33;
then A162: W-min (L~ godo) = W-min (L~ (Cage (C,n))) by A143, A158, XBOOLE_1:4;
A163: rng godo c= L~ godo by A89, A91, SPPOL_2:18, XXREAL_0:2;
2 in dom godo by A92, FINSEQ_3:25;
then A164: godo /. 2 in rng godo by PARTFUN2:2;
godo /. 2 in W-most (L~ (Cage (C,n))) by A156, JORDAN1I:25;
then (godo /. 2) `1 = (W-min (L~ godo)) `1 by A162, PSCOMP_1:31
.= W-bound (L~ godo) ;
then godo /. 2 in W-most (L~ godo) by A163, A164, SPRECT_2:12;
then (Rotate (godo,(W-min (L~ godo)))) /. 2 in W-most (L~ godo) by A154, A162, 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 A165: (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 ;
A166: east_halfline (E-max C) misses L~ go

proof

assume east_halfline (E-max C) meets L~ go ; :: thesis:
then consider p being object such that
A167: p in east_halfline (E-max C) and
A168: p in L~ go by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A167;
p in L~ (Upper_Seq (C,n)) by A46, A168;
then p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A145, A167, 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 A46, A168, JORDAN1J:46;
then E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i,k) by A10, A165, A168, JORDAN1J:43;
then ((Gauge (C,n)) * (i,k)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by A3, A13, A16, JORDAN1A:71;
hence contradiction by A2, A15, A30, JORDAN1G:7; :: thesis:

end;

now :: thesis:

assume east_halfline (E-max C) meets L~ godo ; :: thesis:
then A171: ( east_halfline (E-max C) meets (L~ go) \/ (L~ pion1) or east_halfline (E-max C) meets L~ co ) by A143, XBOOLE_1:70;

per cases by A171, XBOOLE_1:70;

suppose east_halfline (E-max C) meets L~ go ; :: thesis:

hence contradiction by A166; :: thesis:

end;

suppose east_halfline (E-max C) meets L~ pion1 ; :: thesis:

then consider p being object such that
A172: p in east_halfline (E-max C) and
A173: p in L~ pion1 by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A172;
A174: p `1 = ((Gauge (C,n)) * (i,k)) `1 by A80, A83, A150, A173, GOBOARD7:5;
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 A175: i <= (len (Gauge (C,n))) -' 1 by XREAL_0:def 2;
(len (Gauge (C,n))) -' 1 <= len (Gauge (C,n)) by NAT_D:35;
then p `1 <= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),1)) `1 by A1, A3, A13, A16, A20, A174, A175, JORDAN1A:18;
then p `1 <= E-bound C by A20, JORDAN8:12;
then A176: p `1 <= (E-max C) `1 ;
p `1 >= (E-max C) `1 by A172, TOPREAL1:def 11;
then A177: p `1 = (E-max C) `1 by A176, XXREAL_0:1;
p `2 = (E-max C) `2 by A172, TOPREAL1:def 11;
then p = E-max C by A177, TOPREAL3:6;
hence contradiction by A8, A83, A134, A150, A173, XBOOLE_0:3; :: thesis:

end;

suppose east_halfline (E-max C) meets L~ co ; :: thesis:

now :: thesis:

per cases ) + (((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 A237, XXREAL_0:1;

suppose ((Index (p,co)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1 > 1 + 1 ; :: thesis:

then 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 A219, A220, A221, A235, XBOOLE_0:3; :: thesis:

end;

suppose A239: ((Index (p,co)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ; :: thesis:

then (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, TOPREAL1:def 6;
then p in {((Lower_Seq (C,n)) /. 2)} by A219, A220, A221, A235, XBOOLE_0:def 4;
then A240: p = (Lower_Seq (C,n)) /. 2 by TARSKI:def 1;
then A241: p .. (Lower_Seq (C,n)) = 2 by A238, FINSEQ_5:41;
1 + 1 = ((Index (p,co)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) - 1 by A239, XREAL_0:def 2;
then (1 + 1) + 1 = (Index (p,co)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))) ;
then A242: ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) = 2 by A225, A236, JORDAN1E:6;
p in rng (Lower_Seq (C,n)) by A238, A240, PARTFUN2:2;
then p = (Gauge (C,n)) * (i,j) by A37, A241, A242, FINSEQ_5:9;
then ((Gauge (C,n)) * (i,j)) `1 = E-bound (L~ (Cage (C,n))) by A240, JORDAN1G:32;
then ((Gauge (C,n)) * (i,j)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 by A5, A12, A16, JORDAN1A:71;
hence contradiction by A2, A14, A67, JORDAN1G:7; :: thesis:

end;

hence contradiction ; :: thesis:

end;

then east_halfline (E-max C) c= (L~ godo) ` by SUBSET_1:23;
then consider W being Subset of (TOP-REAL 2) such that
A243: W is_a_component_of (L~ godo) ` and
A244: east_halfline (E-max C) c= W by GOBOARD9:3;
not W is bounded by A244, JORDAN2C:121, RLTOPSP1:42;
then W is_outside_component_of L~ godo by A243, JORDAN2C:def 3;
then W c= UBD (L~ godo) by JORDAN2C:23;
then A245: east_halfline (E-max C) c= UBD (L~ godo) by A244;
E-max C in east_halfline (E-max C) by TOPREAL1:38;
then E-max C in UBD (L~ godo) by A245;
then E-max C in LeftComp godo by GOBRD14:36;
then Upper_Arc C meets L~ godo by A132, A133, A134, A142, A153, JORDAN1J:36;
then A246: ( Upper_Arc C meets (L~ go) \/ (L~ pion1) or Upper_Arc C meets L~ co ) by A143, XBOOLE_1:70;
A247: Upper_Arc C c= C by JORDAN6:61;

per cases by A246, XBOOLE_1:70;

suppose Upper_Arc C meets L~ go ; :: thesis:

then Upper_Arc C meets L~ (Cage (C,n)) by A46, A145, XBOOLE_1:1, XBOOLE_1:63;
hence contradiction by A247, JORDAN10:5, XBOOLE_1:63; :: thesis:

end;

suppose Upper_Arc C meets L~ pion1 ; :: thesis:

hence contradiction by A8, A83, A150; :: thesis:

end;

suppose Upper_Arc C meets L~ co ; :: thesis:

then Upper_Arc C meets L~ (Cage (C,n)) by A53, A146, XBOOLE_1:1, XBOOLE_1:63;
hence contradiction by A247, JORDAN10:5, XBOOLE_1:63; :: thesis:

end;