let n be Nat; :: thesis: for C being Simple_closed_curve
for i, j, k being Nat st 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (j,i))} & (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (k,i))} holds
LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Lower_Arc C

let C be Simple_closed_curve; :: thesis: for i, j, k being Nat st 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (j,i))} & (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (k,i))} holds
LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Lower_Arc C

let i, j, k be Nat; :: thesis: ( 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (j,i))} & (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (k,i))} implies LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Lower_Arc C )
set Ga = Gauge (C,n);
set US = Upper_Seq (C,n);
set LS = Lower_Seq (C,n);
set LA = Lower_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)) * (j,i);
set Gik = (Gauge (C,n)) * (k,i);
assume that
A1: 1 < j and
A2: j <= k and
A3: k < len (Gauge (C,n)) and
A4: 1 <= i and
A5: i <= width (Gauge (C,n)) and
A6: (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (j,i))} and
A7: (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (k,i))} and
A8: LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) misses Lower_Arc C ; :: thesis: contradiction
(Gauge (C,n)) * (k,i) in {((Gauge (C,n)) * (k,i))} by TARSKI:def 1;
then A9: (Gauge (C,n)) * (k,i) in L~ (Lower_Seq (C,n)) by ;
(Gauge (C,n)) * (j,i) in {((Gauge (C,n)) * (j,i))} by TARSKI:def 1;
then A10: (Gauge (C,n)) * (j,i) in L~ (Upper_Seq (C,n)) by ;
A11: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def 1;
A12: j <> k by A1, A3, A4, A5, A9, A10, Th27;
A13: j < width (Gauge (C,n)) by ;
A14: 1 < k by ;
A15: k < width (Gauge (C,n)) by ;
A16: [j,i] in Indices (Gauge (C,n)) by ;
A17: [k,i] in Indices (Gauge (C,n)) by ;
set go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (j,i)));
set co = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (k,i)));
A18: 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 A19: (Upper_Seq (C,n)) . 1 = (Upper_Seq (C,n)) /. 1 by PARTFUN1:def 6
.= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ;
A20: (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * (1,i)) `1 by ;
len (Gauge (C,n)) >= 4 by JORDAN8:10;
then A21: len (Gauge (C,n)) >= 1 by XXREAL_0:2;
then A22: [1,k] in Indices (Gauge (C,n)) by ;
A23: [1,i] in Indices (Gauge (C,n)) by ;
then A24: (Gauge (C,n)) * (j,i) <> (Upper_Seq (C,n)) . 1 by ;
then reconsider go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (j,i))) as being_S-Seq FinSequence of () by ;
A25: len (Lower_Seq (C,n)) >= 1 + 2 by JORDAN1E:15;
then A26: len (Lower_Seq (C,n)) >= 1 by XXREAL_0:2;
then A27: 1 in dom (Lower_Seq (C,n)) by FINSEQ_3:25;
len (Lower_Seq (C,n)) in dom (Lower_Seq (C,n)) by ;
then A28: (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))) by EUCLID:52
.= ((Gauge (C,n)) * (1,i)) `1 by ;
then A29: (Gauge (C,n)) * (k,i) <> (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) by ;
then reconsider co = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (k,i))) as being_S-Seq FinSequence of () by ;
A30: [(len (Gauge (C,n))),k] in Indices (Gauge (C,n)) by ;
A31: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by
.= 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 ;
then A32: (Gauge (C,n)) * (k,i) <> (Lower_Seq (C,n)) . 1 by ;
A33: len go >= 1 + 1 by TOPREAL1:def 8;
A34: (Gauge (C,n)) * (j,i) in rng (Upper_Seq (C,n)) by ;
then A35: go is_sequence_on Gauge (C,n) by ;
A36: len co >= 1 + 1 by TOPREAL1:def 8;
A37: (Gauge (C,n)) * (k,i) in rng (Lower_Seq (C,n)) by ;
then A38: co is_sequence_on Gauge (C,n) by ;
reconsider go = go as V29() being_S-Seq s.c.c. FinSequence of () by ;
reconsider co = co as V29() being_S-Seq s.c.c. FinSequence of () by ;
A39: len go > 1 by ;
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)) * (j,i) by ;
len co >= 1 by ;
then 1 in dom co by FINSEQ_3:25;
then A42: co /. 1 = co . 1 by PARTFUN1:def 6
.= (Gauge (C,n)) * (k,i) by ;
reconsider m = (len go) - 1 as Nat by ;
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 ;
then LSeg (go,m) c= L~ (Upper_Seq (C,n)) by A45;
then A47: (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i)))) c= {((Gauge (C,n)) * (j,i))} by ;
m >= 1 by ;
then A48: LSeg (go,m) = LSeg ((go /. m),((Gauge (C,n)) * (j,i))) by ;
{((Gauge (C,n)) * (j,i))} c= (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))))
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in {((Gauge (C,n)) * (j,i))} or x in (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i)))) )
A49: (Gauge (C,n)) * (j,i) in LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))) by RLTOPSP1:68;
assume x in {((Gauge (C,n)) * (j,i))} ; :: thesis: x in (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))))
then A50: x = (Gauge (C,n)) * (j,i) by TARSKI:def 1;
(Gauge (C,n)) * (j,i) in LSeg (go,m) by ;
hence x in (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i)))) by ; :: thesis: verum
end;
then A51: (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i)))) = {((Gauge (C,n)) * (j,i))} by A47;
A52: LSeg (co,1) c= L~ co by TOPREAL3:19;
A53: L~ co c= L~ (Lower_Seq (C,n)) by ;
then LSeg (co,1) c= L~ (Lower_Seq (C,n)) by A52;
then A54: (LSeg (co,1)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i)))) c= {((Gauge (C,n)) * (k,i))} by ;
A55: LSeg (co,1) = LSeg (((Gauge (C,n)) * (k,i)),(co /. (1 + 1))) by ;
{((Gauge (C,n)) * (k,i))} c= (LSeg (co,1)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))))
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in {((Gauge (C,n)) * (k,i))} or x in (LSeg (co,1)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i)))) )
A56: (Gauge (C,n)) * (k,i) in LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))) by RLTOPSP1:68;
assume x in {((Gauge (C,n)) * (k,i))} ; :: thesis: x in (LSeg (co,1)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))))
then A57: x = (Gauge (C,n)) * (k,i) by TARSKI:def 1;
(Gauge (C,n)) * (k,i) in LSeg (co,1) by ;
hence x in (LSeg (co,1)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i)))) by ; :: thesis: verum
end;
then A58: (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i)))) /\ (LSeg (co,1)) = {((Gauge (C,n)) * (k,i))} by A54;
A59: go /. 1 = (Upper_Seq (C,n)) /. 1 by
.= 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 ;
A61: rng go c= L~ go by ;
A62: rng co c= L~ co by ;
A63: {(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 A64: x = go /. 1 by TARSKI:def 1;
then A65: x in rng go by FINSEQ_6:42;
x in rng co by ;
hence x in (L~ go) /\ (L~ co) by ; :: thesis: verum
end;
A66: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by
.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;
A67: [(len (Gauge (C,n))),j] in Indices (Gauge (C,n)) by ;
(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 A68: x in (L~ go) /\ (L~ co) ; :: thesis: x in {(go /. 1)}
then A69: x in L~ co by XBOOLE_0:def 4;
A70: now :: thesis: not x = E-max (L~ (Cage (C,n)))
assume x = E-max (L~ (Cage (C,n))) ; :: thesis: contradiction
then A71: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (k,i) by ;
((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 = E-bound (L~ (Cage (C,n))) by ;
then (E-max (L~ (Cage (C,n)))) `1 <> E-bound (L~ (Cage (C,n))) by ;
hence contradiction by EUCLID:52; :: thesis: verum
end;
x in L~ go by ;
then x in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by ;
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 ; :: thesis: verum
end;
then A72: (L~ go) /\ (L~ co) = {(go /. 1)} by A63;
set W2 = go /. 2;
A73: 2 in dom go by ;
A74: now :: thesis: not ((Gauge (C,n)) * (j,i)) `1 = W-bound (L~ (Cage (C,n)))
assume ((Gauge (C,n)) * (j,i)) `1 = W-bound (L~ (Cage (C,n))) ; :: thesis: contradiction
then ((Gauge (C,n)) * (1,k)) `1 = ((Gauge (C,n)) * (j,i)) `1 by ;
hence contradiction by A1, A16, A22, JORDAN1G:7; :: thesis: verum
end;
go = mid ((Upper_Seq (C,n)),1,(((Gauge (C,n)) * (j,i)) .. (Upper_Seq (C,n)))) by
.= (Upper_Seq (C,n)) | (((Gauge (C,n)) * (j,i)) .. (Upper_Seq (C,n))) by ;
then A75: go /. 2 = (Upper_Seq (C,n)) /. 2 by ;
A76: W-min (L~ (Cage (C,n))) in rng go by ;
set pion = <*((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))*>;
A77: now :: thesis: for n being Nat st n in dom <*((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))*> holds
ex j, i being Nat st
( [j,i] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))*> /. n = (Gauge (C,n)) * (j,i) )
let n be Nat; :: thesis: ( n in dom <*((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))*> implies ex j, i being Nat st
( [j,i] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))*> /. n = (Gauge (C,n)) * (j,i) ) )

assume n in dom <*((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))*> ; :: thesis: ex j, i being Nat st
( [j,i] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))*> /. n = (Gauge (C,n)) * (j,i) )

then n in {1,2} by ;
then ( n = 1 or n = 2 ) by TARSKI:def 2;
hence ex j, i being Nat st
( [j,i] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))*> /. n = (Gauge (C,n)) * (j,i) ) by ; :: thesis: verum
end;
A78: (Gauge (C,n)) * (k,i) <> (Gauge (C,n)) * (j,i) by ;
((Gauge (C,n)) * (k,i)) `2 = ((Gauge (C,n)) * (1,i)) `2 by
.= ((Gauge (C,n)) * (j,i)) `2 by ;
then LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))) is horizontal by SPPOL_1:15;
then <*((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))*> is being_S-Seq by ;
then consider pion1 being FinSequence of () such that
A79: pion1 is_sequence_on Gauge (C,n) and
A80: pion1 is being_S-Seq and
A81: L~ <*((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))*> = L~ pion1 and
A82: <*((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))*> /. 1 = pion1 /. 1 and
A83: <*((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))*> /. (len <*((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))*>) = pion1 /. (len pion1) and
A84: len <*((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))*> <= len pion1 by ;
reconsider pion1 = pion1 as being_S-Seq FinSequence of () by A80;
set godo = (go ^' pion1) ^' co;
A85: 1 + 1 <= len (Cage (C,n)) by ;
A86: 1 + 1 <= len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by ;
len (go ^' pion1) >= len go by TOPREAL8:7;
then A87: len (go ^' pion1) >= 1 + 1 by ;
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 ;
A91: Upper_Seq (C,n) is_sequence_on Gauge (C,n) by JORDAN1G:4;
A92: go /. (len go) = pion1 /. 1 by ;
then A93: go ^' pion1 is_sequence_on Gauge (C,n) by ;
A94: (go ^' pion1) /. (len (go ^' pion1)) = <*((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))*> /. (len <*((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))*>) by
.= <*((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))*> /. 2 by FINSEQ_1:44
.= co /. 1 by ;
then A95: (go ^' pion1) ^' co is_sequence_on Gauge (C,n) by ;
LSeg (pion1,1) c= L~ <*((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))*> by ;
then LSeg (pion1,1) c= LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) by SPPOL_2:21;
then A96: (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) c= {((Gauge (C,n)) * (j,i))} by ;
A97: len pion1 >= 1 + 1 by ;
{((Gauge (C,n)) * (j,i))} c= (LSeg (go,m)) /\ (LSeg (pion1,1))
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in {((Gauge (C,n)) * (j,i))} or x in (LSeg (go,m)) /\ (LSeg (pion1,1)) )
assume x in {((Gauge (C,n)) * (j,i))} ; :: thesis: x in (LSeg (go,m)) /\ (LSeg (pion1,1))
then A98: x = (Gauge (C,n)) * (j,i) by TARSKI:def 1;
A99: (Gauge (C,n)) * (j,i) in LSeg (go,m) by ;
(Gauge (C,n)) * (j,i) in LSeg (pion1,1) by ;
hence x in (LSeg (go,m)) /\ (LSeg (pion1,1)) by ; :: thesis: verum
end;
then (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) = {(go /. (len go))} by ;
then A100: go ^' pion1 is unfolded by ;
len pion1 >= 2 + 0 by ;
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 ;
then A102: (len (go ^' pion1)) -' 1 = (len go) + ((len pion1) -' 2) by XREAL_0:def 2;
A103: (len pion1) - 1 >= 1 by ;
then A104: (len pion1) -' 1 = (len pion1) - 1 by XREAL_0:def 2;
A105: ((len pion1) -' 2) + 1 = ((len pion1) - 2) + 1 by
.= (len pion1) -' 1 by ;
((len pion1) - 1) + 1 <= len pion1 ;
then A106: (len pion1) -' 1 < len pion1 by ;
LSeg (pion1,((len pion1) -' 1)) c= L~ <*((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))*> by ;
then LSeg (pion1,((len pion1) -' 1)) c= LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) by SPPOL_2:21;
then A107: (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (co,1)) c= {((Gauge (C,n)) * (k,i))} by ;
{((Gauge (C,n)) * (k,i))} 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)) * (k,i))} or x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (co,1)) )
assume x in {((Gauge (C,n)) * (k,i))} ; :: thesis: x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (co,1))
then A108: x = (Gauge (C,n)) * (k,i) by TARSKI:def 1;
pion1 /. (((len pion1) -' 1) + 1) = <*((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))*> /. 2 by
.= (Gauge (C,n)) * (k,i) by FINSEQ_4:17 ;
then A109: (Gauge (C,n)) * (k,i) in LSeg (pion1,((len pion1) -' 1)) by ;
(Gauge (C,n)) * (k,i) in LSeg (co,1) by ;
hence x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (co,1)) by ; :: thesis: verum
end;
then (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (co,1)) = {((Gauge (C,n)) * (k,i))} by A107;
then A110: (LSeg ((go ^' pion1),((len go) + ((len pion1) -' 2)))) /\ (LSeg (co,1)) = {((go ^' pion1) /. (len (go ^' pion1)))} by ;
A111: not go ^' pion1 is trivial by ;
A112: rng pion1 c= L~ pion1 by ;
A113: {(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 A114: x = pion1 /. 1 by TARSKI:def 1;
then A115: x in rng pion1 by FINSEQ_6:42;
x in rng go by ;
hence x in (L~ go) /\ (L~ pion1) by ; :: 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 A116: x in (L~ go) /\ (L~ pion1) ; :: thesis: x in {(pion1 /. 1)}
then A117: x in L~ pion1 by XBOOLE_0:def 4;
x in L~ go by ;
then x in (L~ pion1) /\ (L~ (Upper_Seq (C,n))) by ;
hence x in {(pion1 /. 1)} by ; :: thesis: verum
end;
then A118: (L~ go) /\ (L~ pion1) = {(pion1 /. 1)} by A113;
then A119: go ^' pion1 is s.n.c. by ;
(rng go) /\ (rng pion1) c= {(pion1 /. 1)} by ;
then A120: go ^' pion1 is one-to-one by JORDAN1J:55;
A121: <*((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))*> /. (len <*((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))*>) = <*((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))*> /. 2 by FINSEQ_1:44
.= co /. 1 by ;
A122: {(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 A123: x = pion1 /. (len pion1) by TARSKI:def 1;
then A124: x in rng pion1 by FINSEQ_6:168;
x in rng co by ;
hence x in (L~ co) /\ (L~ pion1) by ; :: 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 A125: x in (L~ co) /\ (L~ pion1) ; :: thesis: x in {(pion1 /. (len pion1))}
then A126: x in L~ pion1 by XBOOLE_0:def 4;
x in L~ co by ;
then x in (L~ pion1) /\ (L~ (Lower_Seq (C,n))) by ;
hence x in {(pion1 /. (len pion1))} by ; :: thesis: verum
end;
then A127: (L~ co) /\ (L~ pion1) = {(pion1 /. (len pion1))} by A122;
A128: (L~ (go ^' pion1)) /\ (L~ co) = ((L~ go) \/ (L~ pion1)) /\ (L~ co) by
.= {(go /. 1)} \/ {(co /. 1)} by
.= {((go ^' pion1) /. 1)} \/ {(co /. 1)} by FINSEQ_6:155
.= {((go ^' pion1) /. 1),(co /. 1)} by ENUMSET1:1 ;
co /. (len co) = (go ^' pion1) /. 1 by ;
then reconsider godo = (go ^' pion1) ^' co as V29() standard special_circular_sequence by ;
A129: Lower_Arc C is_an_arc_of E-max C, W-min C by JORDAN6:def 9;
then A130: Lower_Arc C is connected by JORDAN6:10;
A131: W-min C in Lower_Arc C by ;
A132: E-max C in Lower_Arc C by ;
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 ;
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 ;
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 ;
then A135: (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by ;
A136: now :: thesis: not ((Gauge (C,n)) * (j,i)) .. (Upper_Seq (C,n)) <= 1
assume A137: ((Gauge (C,n)) * (j,i)) .. (Upper_Seq (C,n)) <= 1 ; :: thesis: contradiction
((Gauge (C,n)) * (j,i)) .. (Upper_Seq (C,n)) >= 1 by ;
then ((Gauge (C,n)) * (j,i)) .. (Upper_Seq (C,n)) = 1 by ;
then (Gauge (C,n)) * (j,i) = (Upper_Seq (C,n)) /. 1 by ;
hence contradiction by A19, A24, JORDAN1F:5; :: thesis: 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 ;
A141: L~ godo = (L~ (go ^' pion1)) \/ (L~ co) by
.= ((L~ go) \/ (L~ pion1)) \/ (L~ co) by ;
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 ;
A145: L~ go c= L~ (Cage (C,n)) by ;
A146: L~ co c= L~ (Cage (C,n)) by ;
A147: W-min C in C by SPRECT_1:13;
A148: L~ <*((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))*> = LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))) by SPPOL_2:21;
A149: now :: thesis: not W-min C in L~ godo
assume W-min C in L~ godo ; :: thesis: contradiction
then A150: ( W-min C in (L~ go) \/ (L~ pion1) or W-min C in L~ co ) by ;
per cases ( W-min C in L~ go or W-min C in L~ pion1 or W-min C in L~ co ) by ;
end;
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
.= 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
.= 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)) * (j,i)))),1,(Gauge (C,n))) by
.= right_cell ((go ^' pion1),1,(Gauge (C,n))) by
.= right_cell (godo,1,(Gauge (C,n))) by ;
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 ;
A152: godo /. 1 = (go ^' pion1) /. 1 by FINSEQ_6:155
.= W-min (L~ (Cage (C,n))) by ;
A153: len (Upper_Seq (C,n)) >= 2 by ;
A154: godo /. 2 = (go ^' pion1) /. 2 by
.= (Upper_Seq (C,n)) /. 2 by
.= ((Upper_Seq (C,n)) ^' (Lower_Seq (C,n))) /. 2 by
.= (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 (L~ go) \/ (L~ co) by ;
then A156: W-min ((L~ go) \/ (L~ co)) = W-min (L~ (Cage (C,n))) by ;
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;
A159: ((Gauge (C,n)) * (j,i)) `1 <= ((Gauge (C,n)) * (k,i)) `1 by ;
then W-bound (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i)))) = ((Gauge (C,n)) * (j,i)) `1 by SPRECT_1:54;
then A160: W-bound (L~ pion1) = ((Gauge (C,n)) * (j,i)) `1 by ;
((Gauge (C,n)) * (j,i)) `1 >= W-bound (L~ (Cage (C,n))) by ;
then ((Gauge (C,n)) * (j,i)) `1 > W-bound (L~ (Cage (C,n))) by ;
then W-min (((L~ go) \/ (L~ co)) \/ (L~ pion1)) = W-min ((L~ go) \/ (L~ co)) by ;
then A161: W-min (L~ godo) = W-min (L~ (Cage (C,n))) by ;
A162: rng godo c= L~ godo by ;
2 in dom godo by ;
then A163: godo /. 2 in rng godo by PARTFUN2:2;
godo /. 2 in W-most (L~ (Cage (C,n))) by ;
then (godo /. 2) `1 = (W-min (L~ godo)) `1 by
.= W-bound (L~ godo) by EUCLID:52 ;
then godo /. 2 in W-most (L~ godo) by ;
then (Rotate (godo,(W-min (L~ godo)))) /. 2 in W-most (L~ godo) by ;
then reconsider godo = godo as V29() standard clockwise_oriented special_circular_sequence by JORDAN1I:25;
len (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by FINSEQ_5:6;
then A164: (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 ;
A165: east_halfline () misses L~ go
proof
assume east_halfline () meets L~ go ; :: thesis: contradiction
then consider p being object such that
A166: p in east_halfline () and
A167: p in L~ go by XBOOLE_0:3;
reconsider p = p as Point of () by A166;
p in L~ (Upper_Seq (C,n)) by ;
then p in () /\ (L~ (Cage (C,n))) by ;
then A168: p `1 = E-bound (L~ (Cage (C,n))) by ;
then A169: p = E-max (L~ (Cage (C,n))) by ;
then E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (j,i) by ;
then ((Gauge (C,n)) * (j,i)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by ;
hence contradiction by A2, A3, A16, A30, JORDAN1G:7; :: thesis: verum
end;
now :: thesis: not east_halfline () meets L~ godo
assume east_halfline () meets L~ godo ; :: thesis: contradiction
then A170: ( east_halfline () meets (L~ go) \/ (L~ pion1) or east_halfline () meets L~ co ) by ;
per cases ( east_halfline () meets L~ go or east_halfline () meets L~ pion1 or east_halfline () meets L~ co ) by ;
suppose east_halfline () meets L~ pion1 ; :: thesis: contradiction
then consider p being object such that
A171: p in east_halfline () and
A172: p in L~ pion1 by XBOOLE_0:3;
reconsider p = p as Point of () by A171;
A173: p `2 = () `2 by ;
k + 1 <= len (Gauge (C,n)) by ;
then (k + 1) - 1 <= (len (Gauge (C,n))) - 1 by XREAL_1:9;
then A174: k <= (len (Gauge (C,n))) -' 1 by XREAL_0:def 2;
(len (Gauge (C,n))) -' 1 <= len (Gauge (C,n)) by NAT_D:35;
then A175: ((Gauge (C,n)) * (k,i)) `1 <= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),1)) `1 by ;
p `1 <= ((Gauge (C,n)) * (k,i)) `1 by ;
then p `1 <= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),1)) `1 by ;
then p `1 <= E-bound C by ;
then A176: p `1 <= () `1 by EUCLID:52;
p `1 >= () `1 by ;
then p `1 = () `1 by ;
then p = E-max C by ;
hence contradiction by A8, A81, A132, A148, A172, XBOOLE_0:3; :: thesis: verum
end;
suppose east_halfline () meets L~ co ; :: thesis: contradiction
then consider p being object such that
A177: p in east_halfline () and
A178: p in L~ co by XBOOLE_0:3;
reconsider p = p as Point of () by A177;
A179: p in LSeg (co,(Index (p,co))) by ;
consider t being Nat such that
A180: t in dom (Lower_Seq (C,n)) and
A181: (Lower_Seq (C,n)) . t = (Gauge (C,n)) * (k,i) by ;
1 <= t by ;
then A182: 1 < t by ;
t <= len (Lower_Seq (C,n)) by ;
then (Index (((Gauge (C,n)) * (k,i)),(Lower_Seq (C,n)))) + 1 = t by ;
then A183: len (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (k,i)))) = (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (k,i)),(Lower_Seq (C,n)))) by ;
Index (p,co) < len co by ;
then Index (p,co) < (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (k,i)),(Lower_Seq (C,n)))) by ;
then (Index (p,co)) + 1 <= (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (k,i)),(Lower_Seq (C,n)))) by NAT_1:13;
then A184: Index (p,co) <= ((len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (k,i)),(Lower_Seq (C,n))))) - 1 by XREAL_1:19;
A185: co = mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n)))) by ;
p in L~ (Lower_Seq (C,n)) by ;
then p in () /\ (L~ (Cage (C,n))) by ;
then A186: p `1 = E-bound (L~ (Cage (C,n))) by ;
A187: (Index (((Gauge (C,n)) * (k,i)),(Lower_Seq (C,n)))) + 1 = ((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)) by ;
0 + (Index (((Gauge (C,n)) * (k,i)),(Lower_Seq (C,n)))) < len (Lower_Seq (C,n)) by ;
then (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (k,i)),(Lower_Seq (C,n)))) > 0 by XREAL_1:20;
then Index (p,co) <= ((len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (k,i)),(Lower_Seq (C,n))))) - 1 by ;
then Index (p,co) <= (len (Lower_Seq (C,n))) - (((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n))) by A187;
then Index (p,co) <= (len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n))) by XREAL_0:def 2;
then A188: Index (p,co) < ((len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)))) + 1 by NAT_1:13;
A189: 1 <= Index (p,co) by ;
A190: ((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)) <= len (Lower_Seq (C,n)) by ;
((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)) <> len (Lower_Seq (C,n)) by ;
then A191: ((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)) < len (Lower_Seq (C,n)) by ;
A192: 1 + 1 <= len (Lower_Seq (C,n)) by ;
then A193: 2 in dom (Lower_Seq (C,n)) by FINSEQ_3:25;
set tt = ((Index (p,co)) + (((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)))) -' 1;
set RC = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))));
A194: E-max C in right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by JORDAN1I:7;
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: L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33;
consider g2 being Nat such that
A197: 1 <= g2 and
A198: g2 <= width (Gauge (C,n)) and
A199: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * ((len (Gauge (C,n))),g2) by JORDAN1D:25;
A200: len (Gauge (C,n)) >= 4 by JORDAN8:10;
then len (Gauge (C,n)) >= 1 by XXREAL_0:2;
then A201: [(len (Gauge (C,n))),g2] in Indices (Gauge (C,n)) by ;
A202: len (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = len (Cage (C,n)) by FINSEQ_6:179;
Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) by JORDAN1G:18;
then A203: LSeg ((Lower_Seq (C,n)),1) = LSeg ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by ;
A204: 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 ;
then consider ii, g being Nat such that
A205: [ii,(g + 1)] in Indices (Gauge (C,n)) and
A206: [ii,g] in Indices (Gauge (C,n)) and
A207: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = (Gauge (C,n)) * (ii,(g + 1)) and
A208: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1) = (Gauge (C,n)) * (ii,g) by ;
A209: (g + 1) + 1 <> g ;
A210: 1 <= g by ;
(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 ;
then A211: ii = len (Gauge (C,n)) by ;
then ii - 1 >= 4 - 1 by ;
then A212: ii - 1 >= 1 by XXREAL_0:2;
then A213: 1 <= ii -' 1 by XREAL_0:def 2;
A214: g <= width (Gauge (C,n)) by ;
then A215: ((Gauge (C,n)) * ((len (Gauge (C,n))),g)) `1 = E-bound (L~ (Cage (C,n))) by ;
A216: g + 1 <= width (Gauge (C,n)) by ;
ii + 1 <> ii ;
then A217: right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) = cell ((Gauge (C,n)),(ii -' 1),g) by ;
A218: ii <= len (Gauge (C,n)) by ;
A219: 1 <= ii by ;
A220: ii <= len (Gauge (C,n)) by ;
A221: 1 <= g + 1 by ;
then A222: E-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * ((len (Gauge (C,n))),(g + 1))) `1 by ;
A223: 1 <= ii by ;
then A224: (ii -' 1) + 1 = ii by XREAL_1:235;
then A225: ii -' 1 < len (Gauge (C,n)) by ;
then A226: ((Gauge (C,n)) * ((ii -' 1),(g + 1))) `2 = ((Gauge (C,n)) * (1,(g + 1))) `2 by
.= ((Gauge (C,n)) * (ii,(g + 1))) `2 by ;
A227: (E-max C) `2 = p `2 by ;
then A228: p `2 <= ((Gauge (C,n)) * ((ii -' 1),(g + 1))) `2 by ;
A229: ((Gauge (C,n)) * ((ii -' 1),g)) `2 = ((Gauge (C,n)) * (1,g)) `2 by
.= ((Gauge (C,n)) * (ii,g)) `2 by ;
((Gauge (C,n)) * ((ii -' 1),g)) `2 <= p `2 by ;
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 ;
then A230: p in LSeg ((Lower_Seq (C,n)),1) by ;
1 <= ((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)) by ;
then A231: LSeg ((mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n))))),(Index (p,co))) = LSeg ((Lower_Seq (C,n)),(((Index (p,co)) + (((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)))) -' 1)) by ;
1 <= Index (((Gauge (C,n)) * (k,i)),(Lower_Seq (C,n))) by ;
then A232: 1 + 1 <= ((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)) by ;
then (Index (p,co)) + (((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n))) >= (1 + 1) + 1 by ;
then ((Index (p,co)) + (((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)))) - 1 >= ((1 + 1) + 1) - 1 by XREAL_1:9;
then A233: ((Index (p,co)) + (((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)))) -' 1 >= 1 + 1 by XREAL_0:def 2;
per cases ( ((Index (p,co)) + (((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)))) -' 1 > 1 + 1 or ((Index (p,co)) + (((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ) by ;
suppose ((Index (p,co)) + (((Gauge (C,n)) * (k,i)) .. (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)) * (k,i)) .. (Lower_Seq (C,n)))) -' 1)) by TOPREAL1:def 7;
hence contradiction by A230, A179, A185, A231, XBOOLE_0:3; :: thesis: verum
end;
suppose A234: ((Index (p,co)) + (((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ; :: thesis: contradiction
then 1 + 1 = ((Index (p,co)) + (((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)))) - 1 by XREAL_0:def 2;
then (1 + 1) + 1 = (Index (p,co)) + (((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n))) ;
then A235: ((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)) = 2 by ;
(LSeg ((Lower_Seq (C,n)),1)) /\ (LSeg ((Lower_Seq (C,n)),(((Index (p,co)) + (((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)))) -' 1))) = {((Lower_Seq (C,n)) /. 2)} by ;
then p in {((Lower_Seq (C,n)) /. 2)} by ;
then A236: p = (Lower_Seq (C,n)) /. 2 by TARSKI:def 1;
then A237: p in rng (Lower_Seq (C,n)) by ;
p .. (Lower_Seq (C,n)) = 2 by ;
then p = (Gauge (C,n)) * (k,i) by ;
then ((Gauge (C,n)) * (k,i)) `1 = E-bound (L~ (Cage (C,n))) by ;
then ((Gauge (C,n)) * (k,i)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 by ;
hence contradiction by A3, A17, A67, JORDAN1G:7; :: thesis: verum
end;
end;
end;
hence contradiction ; :: thesis: verum
end;
end;
end;
then east_halfline () c= (L~ godo) ` by SUBSET_1:23;
then consider W being Subset of () such that
A238: W is_a_component_of (L~ godo) ` and
A239: east_halfline () c= W by GOBOARD9:3;
not W is bounded by ;
then W is_outside_component_of L~ godo by ;
then W c= UBD (L~ godo) by JORDAN2C:23;
then A240: east_halfline () c= UBD (L~ godo) by A239;
E-max C in east_halfline () by TOPREAL1:38;
then E-max C in UBD (L~ godo) by A240;
then E-max C in LeftComp godo by GOBRD14:36;
then Lower_Arc C meets L~ godo by ;
then A241: ( Lower_Arc C meets (L~ go) \/ (L~ pion1) or Lower_Arc C meets L~ co ) by ;
A242: Lower_Arc C c= C by JORDAN6:61;
per cases ( Lower_Arc C meets L~ go or Lower_Arc C meets L~ pion1 or Lower_Arc C meets L~ co ) by ;
end;