:: JORDAN15 semantic presentation
begin
theorem :: JORDAN15:1
for A, B being Subset of (TOP-REAL 2) st A meets B holds
proj1 .: A meets proj1 .: B
proof
let A, B be Subset of (TOP-REAL 2); ::_thesis: ( A meets B implies proj1 .: A meets proj1 .: B )
assume A meets B ; ::_thesis: proj1 .: A meets proj1 .: B
then consider e being set such that
A1: e in A and
A2: e in B by XBOOLE_0:3;
reconsider e = e as Point of (TOP-REAL 2) by A1;
A3: e `1 = proj1 . e by PSCOMP_1:def_5;
then A4: e `1 in proj1 .: B by A2, FUNCT_2:35;
e `1 in proj1 .: A by A1, A3, FUNCT_2:35;
hence proj1 .: A meets proj1 .: B by A4, XBOOLE_0:3; ::_thesis: verum
end;
theorem :: JORDAN15:2
for A, B being Subset of (TOP-REAL 2)
for s being real number st A misses B & A c= Horizontal_Line s & B c= Horizontal_Line s holds
proj1 .: A misses proj1 .: B
proof
let A, B be Subset of (TOP-REAL 2); ::_thesis: for s being real number st A misses B & A c= Horizontal_Line s & B c= Horizontal_Line s holds
proj1 .: A misses proj1 .: B
let s be real number ; ::_thesis: ( A misses B & A c= Horizontal_Line s & B c= Horizontal_Line s implies proj1 .: A misses proj1 .: B )
assume that
A1: A misses B and
A2: A c= Horizontal_Line s and
A3: B c= Horizontal_Line s ; ::_thesis: proj1 .: A misses proj1 .: B
assume proj1 .: A meets proj1 .: B ; ::_thesis: contradiction
then consider e being set such that
A4: e in proj1 .: A and
A5: e in proj1 .: B by XBOOLE_0:3;
reconsider e = e as Real by A4;
consider d being Point of (TOP-REAL 2) such that
A6: d in B and
A7: e = proj1 . d by A5, FUNCT_2:65;
A8: d `2 = s by A3, A6, JORDAN6:32;
consider c being Point of (TOP-REAL 2) such that
A9: c in A and
A10: e = proj1 . c by A4, FUNCT_2:65;
c `2 = s by A2, A9, JORDAN6:32;
then c = |[(c `1),(d `2)]| by A8, EUCLID:53
.= |[e,(d `2)]| by A10, PSCOMP_1:def_5
.= |[(d `1),(d `2)]| by A7, PSCOMP_1:def_5
.= d by EUCLID:53 ;
hence contradiction by A1, A9, A6, XBOOLE_0:3; ::_thesis: verum
end;
theorem Th3: :: JORDAN15:3
for S being closed Subset of (TOP-REAL 2) st S is bounded holds
proj1 .: S is closed
proof
let S be closed Subset of (TOP-REAL 2); ::_thesis: ( S is bounded implies proj1 .: S is closed )
assume S is bounded ; ::_thesis: proj1 .: S is closed
then Cl (proj1 .: S) = proj1 .: (Cl S) by TOPREAL6:83
.= proj1 .: S by PRE_TOPC:22 ;
hence proj1 .: S is closed ; ::_thesis: verum
end;
theorem Th4: :: JORDAN15:4
for S being compact Subset of (TOP-REAL 2) holds proj1 .: S is compact
proof
let S be compact Subset of (TOP-REAL 2); ::_thesis: proj1 .: S is compact
proj1 .: S is closed by Th3;
hence proj1 .: S is compact by JORDAN1C:3, RCOMP_1:11; ::_thesis: verum
end;
theorem Th5: :: JORDAN15:5
for G being Go-board
for i, j, k, j1, k1 being Element of NAT st 1 <= i & i <= len G & 1 <= j & j <= j1 & j1 <= k1 & k1 <= k & k <= width G holds
LSeg ((G * (i,j1)),(G * (i,k1))) c= LSeg ((G * (i,j)),(G * (i,k)))
proof
let G be Go-board; ::_thesis: for i, j, k, j1, k1 being Element of NAT st 1 <= i & i <= len G & 1 <= j & j <= j1 & j1 <= k1 & k1 <= k & k <= width G holds
LSeg ((G * (i,j1)),(G * (i,k1))) c= LSeg ((G * (i,j)),(G * (i,k)))
let i, j, k, j1, k1 be Element of NAT ; ::_thesis: ( 1 <= i & i <= len G & 1 <= j & j <= j1 & j1 <= k1 & k1 <= k & k <= width G implies LSeg ((G * (i,j1)),(G * (i,k1))) c= LSeg ((G * (i,j)),(G * (i,k))) )
assume that
A1: 1 <= i and
A2: i <= len G and
A3: 1 <= j and
A4: j <= j1 and
A5: j1 <= k1 and
A6: k1 <= k and
A7: k <= width G ; ::_thesis: LSeg ((G * (i,j1)),(G * (i,k1))) c= LSeg ((G * (i,j)),(G * (i,k)))
A8: j1 <= k by A5, A6, XXREAL_0:2;
j <= k1 by A4, A5, XXREAL_0:2;
then A9: 1 <= k1 by A3, XXREAL_0:2;
then A10: (G * (i,k1)) `2 <= (G * (i,k)) `2 by A1, A2, A6, A7, SPRECT_3:12;
A11: 1 <= j1 by A3, A4, XXREAL_0:2;
1 <= j1 by A3, A4, XXREAL_0:2;
then A12: 1 <= k by A8, XXREAL_0:2;
A13: k1 <= width G by A6, A7, XXREAL_0:2;
j <= k1 by A4, A5, XXREAL_0:2;
then A14: j <= width G by A13, XXREAL_0:2;
then (G * (i,j)) `1 = (G * (i,1)) `1 by A1, A2, A3, GOBOARD5:2
.= (G * (i,k)) `1 by A1, A2, A7, A12, GOBOARD5:2 ;
then A15: LSeg ((G * (i,j)),(G * (i,k))) is vertical by SPPOL_1:16;
j1 <= k by A5, A6, XXREAL_0:2;
then A16: j1 <= width G by A7, XXREAL_0:2;
then A17: (G * (i,j)) `2 <= (G * (i,j1)) `2 by A1, A2, A3, A4, SPRECT_3:12;
A18: k1 <= width G by A6, A7, XXREAL_0:2;
then A19: (G * (i,j1)) `2 <= (G * (i,k1)) `2 by A1, A2, A5, A11, SPRECT_3:12;
(G * (i,j1)) `1 = (G * (i,1)) `1 by A1, A2, A11, A16, GOBOARD5:2
.= (G * (i,k1)) `1 by A1, A2, A9, A18, GOBOARD5:2 ;
then A20: LSeg ((G * (i,j1)),(G * (i,k1))) is vertical by SPPOL_1:16;
(G * (i,j)) `1 = (G * (i,1)) `1 by A1, A2, A3, A14, GOBOARD5:2
.= (G * (i,j1)) `1 by A1, A2, A11, A16, GOBOARD5:2 ;
hence LSeg ((G * (i,j1)),(G * (i,k1))) c= LSeg ((G * (i,j)),(G * (i,k))) by A15, A20, A17, A19, A10, GOBOARD7:63; ::_thesis: verum
end;
theorem Th6: :: JORDAN15:6
for G being Go-board
for i, j, k, j1, k1 being Element of NAT st 1 <= i & i <= width G & 1 <= j & j <= j1 & j1 <= k1 & k1 <= k & k <= len G holds
LSeg ((G * (j1,i)),(G * (k1,i))) c= LSeg ((G * (j,i)),(G * (k,i)))
proof
let G be Go-board; ::_thesis: for i, j, k, j1, k1 being Element of NAT st 1 <= i & i <= width G & 1 <= j & j <= j1 & j1 <= k1 & k1 <= k & k <= len G holds
LSeg ((G * (j1,i)),(G * (k1,i))) c= LSeg ((G * (j,i)),(G * (k,i)))
let i, j, k, j1, k1 be Element of NAT ; ::_thesis: ( 1 <= i & i <= width G & 1 <= j & j <= j1 & j1 <= k1 & k1 <= k & k <= len G implies LSeg ((G * (j1,i)),(G * (k1,i))) c= LSeg ((G * (j,i)),(G * (k,i))) )
assume that
A1: 1 <= i and
A2: i <= width G and
A3: 1 <= j and
A4: j <= j1 and
A5: j1 <= k1 and
A6: k1 <= k and
A7: k <= len G ; ::_thesis: LSeg ((G * (j1,i)),(G * (k1,i))) c= LSeg ((G * (j,i)),(G * (k,i)))
A8: j1 <= k by A5, A6, XXREAL_0:2;
j <= k1 by A4, A5, XXREAL_0:2;
then A9: 1 <= k1 by A3, XXREAL_0:2;
then A10: (G * (k1,i)) `1 <= (G * (k,i)) `1 by A1, A2, A6, A7, SPRECT_3:13;
A11: 1 <= j1 by A3, A4, XXREAL_0:2;
1 <= j1 by A3, A4, XXREAL_0:2;
then A12: 1 <= k by A8, XXREAL_0:2;
A13: k1 <= len G by A6, A7, XXREAL_0:2;
j <= k1 by A4, A5, XXREAL_0:2;
then A14: j <= len G by A13, XXREAL_0:2;
then (G * (j,i)) `2 = (G * (1,i)) `2 by A1, A2, A3, GOBOARD5:1
.= (G * (k,i)) `2 by A1, A2, A7, A12, GOBOARD5:1 ;
then A15: LSeg ((G * (j,i)),(G * (k,i))) is horizontal by SPPOL_1:15;
j1 <= k by A5, A6, XXREAL_0:2;
then A16: j1 <= len G by A7, XXREAL_0:2;
then A17: (G * (j,i)) `1 <= (G * (j1,i)) `1 by A1, A2, A3, A4, SPRECT_3:13;
A18: k1 <= len G by A6, A7, XXREAL_0:2;
then A19: (G * (j1,i)) `1 <= (G * (k1,i)) `1 by A1, A2, A5, A11, SPRECT_3:13;
(G * (j1,i)) `2 = (G * (1,i)) `2 by A1, A2, A11, A16, GOBOARD5:1
.= (G * (k1,i)) `2 by A1, A2, A9, A18, GOBOARD5:1 ;
then A20: LSeg ((G * (j1,i)),(G * (k1,i))) is horizontal by SPPOL_1:15;
(G * (j,i)) `2 = (G * (1,i)) `2 by A1, A2, A3, A14, GOBOARD5:1
.= (G * (j1,i)) `2 by A1, A2, A11, A16, GOBOARD5:1 ;
hence LSeg ((G * (j1,i)),(G * (k1,i))) c= LSeg ((G * (j,i)),(G * (k,i))) by A15, A20, A17, A19, A10, GOBOARD7:64; ::_thesis: verum
end;
theorem :: JORDAN15:7
for G being Go-board
for j, k, j1, k1 being Element of NAT st 1 <= j & j <= j1 & j1 <= k1 & k1 <= k & k <= width G holds
LSeg ((G * ((Center G),j1)),(G * ((Center G),k1))) c= LSeg ((G * ((Center G),j)),(G * ((Center G),k)))
proof
let G be Go-board; ::_thesis: for j, k, j1, k1 being Element of NAT st 1 <= j & j <= j1 & j1 <= k1 & k1 <= k & k <= width G holds
LSeg ((G * ((Center G),j1)),(G * ((Center G),k1))) c= LSeg ((G * ((Center G),j)),(G * ((Center G),k)))
let j, k, j1, k1 be Element of NAT ; ::_thesis: ( 1 <= j & j <= j1 & j1 <= k1 & k1 <= k & k <= width G implies LSeg ((G * ((Center G),j1)),(G * ((Center G),k1))) c= LSeg ((G * ((Center G),j)),(G * ((Center G),k))) )
assume that
A1: 1 <= j and
A2: j <= j1 and
A3: j1 <= k1 and
A4: k1 <= k and
A5: k <= width G ; ::_thesis: LSeg ((G * ((Center G),j1)),(G * ((Center G),k1))) c= LSeg ((G * ((Center G),j)),(G * ((Center G),k)))
A6: Center G <= len G by JORDAN1B:13;
1 <= Center G by JORDAN1B:11;
hence LSeg ((G * ((Center G),j1)),(G * ((Center G),k1))) c= LSeg ((G * ((Center G),j)),(G * ((Center G),k))) by A1, A2, A3, A4, A5, A6, Th5; ::_thesis: verum
end;
theorem :: JORDAN15:8
for G being Go-board st len G = width G holds
for j, k, j1, k1 being Element of NAT st 1 <= j & j <= j1 & j1 <= k1 & k1 <= k & k <= len G holds
LSeg ((G * (j1,(Center G))),(G * (k1,(Center G)))) c= LSeg ((G * (j,(Center G))),(G * (k,(Center G))))
proof
let G be Go-board; ::_thesis: ( len G = width G implies for j, k, j1, k1 being Element of NAT st 1 <= j & j <= j1 & j1 <= k1 & k1 <= k & k <= len G holds
LSeg ((G * (j1,(Center G))),(G * (k1,(Center G)))) c= LSeg ((G * (j,(Center G))),(G * (k,(Center G)))) )
assume len G = width G ; ::_thesis: for j, k, j1, k1 being Element of NAT st 1 <= j & j <= j1 & j1 <= k1 & k1 <= k & k <= len G holds
LSeg ((G * (j1,(Center G))),(G * (k1,(Center G)))) c= LSeg ((G * (j,(Center G))),(G * (k,(Center G))))
then A1: Center G <= width G by JORDAN1B:13;
let j, k, j1, k1 be Element of NAT ; ::_thesis: ( 1 <= j & j <= j1 & j1 <= k1 & k1 <= k & k <= len G implies LSeg ((G * (j1,(Center G))),(G * (k1,(Center G)))) c= LSeg ((G * (j,(Center G))),(G * (k,(Center G)))) )
assume that
A2: 1 <= j and
A3: j <= j1 and
A4: j1 <= k1 and
A5: k1 <= k and
A6: k <= len G ; ::_thesis: LSeg ((G * (j1,(Center G))),(G * (k1,(Center G)))) c= LSeg ((G * (j,(Center G))),(G * (k,(Center G))))
1 <= Center G by JORDAN1B:11;
hence LSeg ((G * (j1,(Center G))),(G * (k1,(Center G)))) c= LSeg ((G * (j,(Center G))),(G * (k,(Center G)))) by A2, A3, A4, A5, A6, A1, Th6; ::_thesis: verum
end;
theorem Th9: :: JORDAN15:9
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i, j, k being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) holds
ex j1 being Element of NAT st
( j <= j1 & j1 <= k & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j1))} )
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i, j, k being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) holds
ex j1 being Element of NAT st
( j <= j1 & j1 <= k & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j1))} )
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for i, j, k being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) holds
ex j1 being Element of NAT st
( j <= j1 & j1 <= k & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j1))} )
let i, j, k be Element of NAT ; ::_thesis: ( 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) implies ex j1 being Element of NAT st
( j <= j1 & j1 <= k & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j1))} ) )
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: (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) ; ::_thesis: ex j1 being Element of NAT st
( j <= j1 & j1 <= k & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j1))} )
set G = Gauge (C,n);
A7: k >= 1 by A3, A4, XXREAL_0:2;
then A8: [i,k] in Indices (Gauge (C,n)) by A1, A2, A5, MATRIX_1:36;
set X = (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n)));
A9: (Gauge (C,n)) * (i,j) in LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) by RLTOPSP1:68;
then reconsider X1 = (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))) as non empty compact Subset of (TOP-REAL 2) by A6, XBOOLE_0:def_4;
A10: LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets L~ (Lower_Seq (C,n)) by A6, A9, XBOOLE_0:3;
set s = ((Gauge (C,n)) * (i,1)) `1 ;
set e = (Gauge (C,n)) * (i,k);
set f = (Gauge (C,n)) * (i,j);
set w2 = upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n)))));
A11: j <= width (Gauge (C,n)) by A4, A5, XXREAL_0:2;
then [i,j] in Indices (Gauge (C,n)) by A1, A2, A3, MATRIX_1:36;
then consider j1 being Element of NAT such that
A12: j <= j1 and
A13: j1 <= k and
A14: ((Gauge (C,n)) * (i,j1)) `2 = upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))) by A4, A10, A8, JORDAN1F:2, JORDAN1G:5;
set q = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|;
A15: j1 <= width (Gauge (C,n)) by A5, A13, XXREAL_0:2;
A16: ((Gauge (C,n)) * (i,k)) `1 = ((Gauge (C,n)) * (i,1)) `1 by A1, A2, A5, A7, GOBOARD5:2;
then ((Gauge (C,n)) * (i,j)) `1 = ((Gauge (C,n)) * (i,k)) `1 by A1, A2, A3, A11, GOBOARD5:2;
then A17: LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) is vertical by SPPOL_1:16;
take j1 ; ::_thesis: ( j <= j1 & j1 <= k & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j1))} )
thus ( j <= j1 & j1 <= k ) by A12, A13; ::_thesis: (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j1))}
consider pp being set such that
A18: pp in N-most X1 by XBOOLE_0:def_1;
reconsider pp = pp as Point of (TOP-REAL 2) by A18;
A19: pp in (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))) by A18, XBOOLE_0:def_4;
then A20: pp in L~ (Lower_Seq (C,n)) by XBOOLE_0:def_4;
A21: 1 <= j1 by A3, A12, XXREAL_0:2;
then A22: ((Gauge (C,n)) * (i,j1)) `1 = ((Gauge (C,n)) * (i,1)) `1 by A1, A2, A15, GOBOARD5:2;
then A23: |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| = (Gauge (C,n)) * (i,j1) by A14, EUCLID:53;
then A24: |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| `2 <= ((Gauge (C,n)) * (i,k)) `2 by A1, A2, A5, A13, A21, SPRECT_3:12;
A25: |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| `2 = N-bound ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n)))) by A14, A23, SPRECT_1:45
.= (N-min ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))) `2 by EUCLID:52
.= pp `2 by A18, PSCOMP_1:39 ;
pp in LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) by A19, XBOOLE_0:def_4;
then pp `1 = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| `1 by A16, A22, A23, A17, SPPOL_1:41;
then A26: |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| in L~ (Lower_Seq (C,n)) by A20, A25, TOPREAL3:6;
for x being set holds
( x in (LSeg (((Gauge (C,n)) * (i,k)),|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))) iff x = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| )
proof
let x be set ; ::_thesis: ( x in (LSeg (((Gauge (C,n)) * (i,k)),|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))) iff x = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| )
thus ( x in (LSeg (((Gauge (C,n)) * (i,k)),|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))) implies x = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| ) ::_thesis: ( x = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| implies x in (LSeg (((Gauge (C,n)) * (i,k)),|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))) )
proof
reconsider EE = (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))) as compact Subset of (TOP-REAL 2) ;
reconsider E0 = proj2 .: EE as compact Subset of REAL by JCT_MISC:15;
A27: (Gauge (C,n)) * (i,k) in LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) by RLTOPSP1:68;
A28: ((Gauge (C,n)) * (i,j)) `2 <= |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| `2 by A1, A2, A3, A12, A15, A23, SPRECT_3:12;
((Gauge (C,n)) * (i,j)) `1 = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| `1 by A1, A2, A3, A11, A22, A23, GOBOARD5:2;
then |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| in LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) by A16, A22, A23, A24, A28, GOBOARD7:7;
then A29: LSeg (((Gauge (C,n)) * (i,k)),|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|) c= LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) by A27, TOPREAL1:6;
assume A30: x in (LSeg (((Gauge (C,n)) * (i,k)),|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))) ; ::_thesis: x = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|
then reconsider pp = x as Point of (TOP-REAL 2) ;
A31: pp in LSeg (((Gauge (C,n)) * (i,k)),|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|) by A30, XBOOLE_0:def_4;
then A32: pp `2 >= |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| `2 by A24, TOPREAL1:4;
pp in L~ (Lower_Seq (C,n)) by A30, XBOOLE_0:def_4;
then pp in EE by A31, A29, XBOOLE_0:def_4;
then proj2 . pp in E0 by FUNCT_2:35;
then A33: pp `2 in E0 by PSCOMP_1:def_6;
E0 is real-bounded by RCOMP_1:10;
then E0 is bounded_above by XXREAL_2:def_11;
then |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| `2 >= pp `2 by A14, A23, A33, SEQ_4:def_1;
then A34: pp `2 = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| `2 by A32, XXREAL_0:1;
pp `1 = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| `1 by A16, A22, A23, A31, GOBOARD7:5;
hence x = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| by A34, TOPREAL3:6; ::_thesis: verum
end;
assume A35: x = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| ; ::_thesis: x in (LSeg (((Gauge (C,n)) * (i,k)),|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n)))
then x in LSeg (((Gauge (C,n)) * (i,k)),|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|) by RLTOPSP1:68;
hence x in (LSeg (((Gauge (C,n)) * (i,k)),|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))) by A26, A35, XBOOLE_0:def_4; ::_thesis: verum
end;
hence (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j1))} by A23, TARSKI:def_1; ::_thesis: verum
end;
theorem :: JORDAN15:10
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i, j, k being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Upper_Seq (C,n)) holds
ex k1 being Element of NAT st
( j <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))} )
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i, j, k being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Upper_Seq (C,n)) holds
ex k1 being Element of NAT st
( j <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))} )
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for i, j, k being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Upper_Seq (C,n)) holds
ex k1 being Element of NAT st
( j <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))} )
let i, j, k be Element of NAT ; ::_thesis: ( 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Upper_Seq (C,n)) implies ex k1 being Element of NAT st
( j <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))} ) )
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: (Gauge (C,n)) * (i,k) in L~ (Upper_Seq (C,n)) ; ::_thesis: ex k1 being Element of NAT st
( j <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))} )
set G = Gauge (C,n);
A7: k >= 1 by A3, A4, XXREAL_0:2;
then A8: [i,k] in Indices (Gauge (C,n)) by A1, A2, A5, MATRIX_1:36;
set X = (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n)));
A9: (Gauge (C,n)) * (i,k) in LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) by RLTOPSP1:68;
then reconsider X1 = (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))) as non empty compact Subset of (TOP-REAL 2) by A6, XBOOLE_0:def_4;
A10: LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets L~ (Upper_Seq (C,n)) by A6, A9, XBOOLE_0:3;
set s = ((Gauge (C,n)) * (i,1)) `1 ;
set e = (Gauge (C,n)) * (i,k);
set f = (Gauge (C,n)) * (i,j);
set w1 = lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n)))));
A11: j <= width (Gauge (C,n)) by A4, A5, XXREAL_0:2;
then [i,j] in Indices (Gauge (C,n)) by A1, A2, A3, MATRIX_1:36;
then consider k1 being Element of NAT such that
A12: j <= k1 and
A13: k1 <= k and
A14: ((Gauge (C,n)) * (i,k1)) `2 = lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))) by A4, A10, A8, JORDAN1F:1, JORDAN1G:4;
set p = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|;
A15: k1 <= width (Gauge (C,n)) by A5, A13, XXREAL_0:2;
((Gauge (C,n)) * (i,j)) `1 = ((Gauge (C,n)) * (i,1)) `1 by A1, A2, A3, A11, GOBOARD5:2
.= ((Gauge (C,n)) * (i,k)) `1 by A1, A2, A5, A7, GOBOARD5:2 ;
then A16: LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) is vertical by SPPOL_1:16;
take k1 ; ::_thesis: ( j <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))} )
thus ( j <= k1 & k1 <= k ) by A12, A13; ::_thesis: (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))}
consider pp being set such that
A17: pp in S-most X1 by XBOOLE_0:def_1;
A18: 1 <= k1 by A3, A12, XXREAL_0:2;
then A19: ((Gauge (C,n)) * (i,k1)) `1 = ((Gauge (C,n)) * (i,1)) `1 by A1, A2, A15, GOBOARD5:2;
then A20: |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| = (Gauge (C,n)) * (i,k1) by A14, EUCLID:53;
then A21: ((Gauge (C,n)) * (i,j)) `2 <= |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| `2 by A1, A2, A3, A12, A15, SPRECT_3:12;
A22: ((Gauge (C,n)) * (i,j)) `1 = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| `1 by A1, A2, A3, A11, A19, A20, GOBOARD5:2;
reconsider pp = pp as Point of (TOP-REAL 2) by A17;
A23: pp in (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))) by A17, XBOOLE_0:def_4;
then A24: pp in L~ (Upper_Seq (C,n)) by XBOOLE_0:def_4;
A25: |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| `2 = S-bound ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n)))) by A14, A20, SPRECT_1:44
.= (S-min ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))) `2 by EUCLID:52
.= pp `2 by A17, PSCOMP_1:55 ;
pp in LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) by A23, XBOOLE_0:def_4;
then pp `1 = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| `1 by A22, A16, SPPOL_1:41;
then A26: |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| in L~ (Upper_Seq (C,n)) by A24, A25, TOPREAL3:6;
for x being set holds
( x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|,((Gauge (C,n)) * (i,j)))) /\ (L~ (Upper_Seq (C,n))) iff x = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| )
proof
let x be set ; ::_thesis: ( x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|,((Gauge (C,n)) * (i,j)))) /\ (L~ (Upper_Seq (C,n))) iff x = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| )
thus ( x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|,((Gauge (C,n)) * (i,j)))) /\ (L~ (Upper_Seq (C,n))) implies x = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| ) ::_thesis: ( x = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| implies x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|,((Gauge (C,n)) * (i,j)))) /\ (L~ (Upper_Seq (C,n))) )
proof
reconsider EE = (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))) as compact Subset of (TOP-REAL 2) ;
reconsider E0 = proj2 .: EE as compact Subset of REAL by JCT_MISC:15;
A27: (Gauge (C,n)) * (i,j) in LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) by RLTOPSP1:68;
A28: ((Gauge (C,n)) * (i,k)) `1 = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| `1 by A1, A2, A5, A7, A19, A20, GOBOARD5:2;
A29: |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| `2 <= ((Gauge (C,n)) * (i,k)) `2 by A1, A2, A5, A13, A18, A20, SPRECT_3:12;
A30: ((Gauge (C,n)) * (i,j)) `2 <= |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| `2 by A1, A2, A3, A12, A15, A20, SPRECT_3:12;
((Gauge (C,n)) * (i,j)) `1 = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| `1 by A1, A2, A3, A11, A19, A20, GOBOARD5:2;
then |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| in LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) by A28, A30, A29, GOBOARD7:7;
then A31: LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|,((Gauge (C,n)) * (i,j))) c= LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) by A27, TOPREAL1:6;
assume A32: x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|,((Gauge (C,n)) * (i,j)))) /\ (L~ (Upper_Seq (C,n))) ; ::_thesis: x = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|
then reconsider pp = x as Point of (TOP-REAL 2) ;
A33: pp in LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|,((Gauge (C,n)) * (i,j))) by A32, XBOOLE_0:def_4;
then A34: pp `2 <= |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| `2 by A21, TOPREAL1:4;
pp in L~ (Upper_Seq (C,n)) by A32, XBOOLE_0:def_4;
then pp in EE by A33, A31, XBOOLE_0:def_4;
then proj2 . pp in E0 by FUNCT_2:35;
then A35: pp `2 in E0 by PSCOMP_1:def_6;
E0 is real-bounded by RCOMP_1:10;
then E0 is bounded_below by XXREAL_2:def_11;
then |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| `2 <= pp `2 by A14, A20, A35, SEQ_4:def_2;
then A36: pp `2 = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| `2 by A34, XXREAL_0:1;
pp `1 = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| `1 by A22, A33, GOBOARD7:5;
hence x = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| by A36, TOPREAL3:6; ::_thesis: verum
end;
assume A37: x = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| ; ::_thesis: x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|,((Gauge (C,n)) * (i,j)))) /\ (L~ (Upper_Seq (C,n)))
then x in LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|,((Gauge (C,n)) * (i,j))) by RLTOPSP1:68;
hence x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|,((Gauge (C,n)) * (i,j)))) /\ (L~ (Upper_Seq (C,n))) by A26, A37, XBOOLE_0:def_4; ::_thesis: verum
end;
hence (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))} by A20, TARSKI:def_1; ::_thesis: verum
end;
theorem Th11: :: JORDAN15:11
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i, j, k being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Upper_Seq (C,n)) holds
ex j1, k1 being Element of NAT st
( j <= j1 & j1 <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j1))} & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))} )
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i, j, k being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Upper_Seq (C,n)) holds
ex j1, k1 being Element of NAT st
( j <= j1 & j1 <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j1))} & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))} )
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for i, j, k being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Upper_Seq (C,n)) holds
ex j1, k1 being Element of NAT st
( j <= j1 & j1 <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j1))} & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))} )
let i, j, k be Element of NAT ; ::_thesis: ( 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Upper_Seq (C,n)) implies ex j1, k1 being Element of NAT st
( j <= j1 & j1 <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j1))} & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))} ) )
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: (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) and
A7: (Gauge (C,n)) * (i,k) in L~ (Upper_Seq (C,n)) ; ::_thesis: ex j1, k1 being Element of NAT st
( j <= j1 & j1 <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j1))} & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))} )
set G = Gauge (C,n);
A8: j <= width (Gauge (C,n)) by A4, A5, XXREAL_0:2;
then A9: [i,j] in Indices (Gauge (C,n)) by A1, A2, A3, MATRIX_1:36;
set s = ((Gauge (C,n)) * (i,1)) `1 ;
set e = (Gauge (C,n)) * (i,k);
set f = (Gauge (C,n)) * (i,j);
set w1 = lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n)))));
A10: (Gauge (C,n)) * (i,k) in LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) by RLTOPSP1:68;
then A11: LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets L~ (Upper_Seq (C,n)) by A7, XBOOLE_0:3;
A12: k >= 1 by A3, A4, XXREAL_0:2;
then [i,k] in Indices (Gauge (C,n)) by A1, A2, A5, MATRIX_1:36;
then consider k1 being Element of NAT such that
A13: j <= k1 and
A14: k1 <= k and
A15: ((Gauge (C,n)) * (i,k1)) `2 = lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))) by A4, A11, A9, JORDAN1F:1, JORDAN1G:4;
A16: k1 <= width (Gauge (C,n)) by A5, A14, XXREAL_0:2;
A17: (Gauge (C,n)) * (i,j) in LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k1))) by RLTOPSP1:68;
then A18: LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k1))) meets L~ (Lower_Seq (C,n)) by A6, XBOOLE_0:3;
set X = (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Lower_Seq (C,n)));
reconsider X1 = (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Lower_Seq (C,n))) as non empty compact Subset of (TOP-REAL 2) by A6, A17, XBOOLE_0:def_4;
consider pp being set such that
A19: pp in N-most X1 by XBOOLE_0:def_1;
reconsider pp = pp as Point of (TOP-REAL 2) by A19;
A20: pp in (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Lower_Seq (C,n))) by A19, XBOOLE_0:def_4;
then A21: pp in L~ (Lower_Seq (C,n)) by XBOOLE_0:def_4;
set p = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|;
set w2 = upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n)))));
set q = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]|;
A22: pp in LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k1))) by A20, XBOOLE_0:def_4;
A23: 1 <= k1 by A3, A13, XXREAL_0:2;
then A24: ((Gauge (C,n)) * (i,k1)) `1 = ((Gauge (C,n)) * (i,1)) `1 by A1, A2, A16, GOBOARD5:2;
then A25: |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| = (Gauge (C,n)) * (i,k1) by A15, EUCLID:53;
[i,k1] in Indices (Gauge (C,n)) by A1, A2, A23, A16, MATRIX_1:36;
then consider j1 being Element of NAT such that
A26: j <= j1 and
A27: j1 <= k1 and
A28: ((Gauge (C,n)) * (i,j1)) `2 = upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))) by A9, A13, A25, A18, JORDAN1F:2, JORDAN1G:5;
take j1 ; ::_thesis: ex k1 being Element of NAT st
( j <= j1 & j1 <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j1))} & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))} )
take k1 ; ::_thesis: ( j <= j1 & j1 <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j1))} & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))} )
thus ( j <= j1 & j1 <= k1 & k1 <= k ) by A14, A26, A27; ::_thesis: ( (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j1))} & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))} )
A29: j1 <= width (Gauge (C,n)) by A16, A27, XXREAL_0:2;
A30: 1 <= j1 by A3, A26, XXREAL_0:2;
then A31: ((Gauge (C,n)) * (i,j1)) `1 = ((Gauge (C,n)) * (i,1)) `1 by A1, A2, A29, GOBOARD5:2;
then A32: |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]| = (Gauge (C,n)) * (i,j1) by A28, EUCLID:53;
then A33: |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]| `2 <= |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| `2 by A1, A2, A16, A25, A27, A30, SPRECT_3:12;
A34: |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]| `2 = N-bound ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Lower_Seq (C,n)))) by A25, A28, A32, SPRECT_1:45
.= (N-min ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Lower_Seq (C,n))))) `2 by EUCLID:52
.= pp `2 by A19, PSCOMP_1:39 ;
A35: ((Gauge (C,n)) * (i,j)) `1 = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| `1 by A1, A2, A3, A8, A24, A25, GOBOARD5:2;
then LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|) is vertical by SPPOL_1:16;
then pp `1 = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]| `1 by A24, A25, A31, A32, A22, SPPOL_1:41;
then A36: |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]| in L~ (Lower_Seq (C,n)) by A21, A34, TOPREAL3:6;
for x being set holds
( x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|,|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))) iff x = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]| )
proof
let x be set ; ::_thesis: ( x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|,|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))) iff x = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]| )
thus ( x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|,|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))) implies x = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]| ) ::_thesis: ( x = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]| implies x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|,|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))) )
proof
reconsider EE = (LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))) as compact Subset of (TOP-REAL 2) ;
reconsider E0 = proj2 .: EE as compact Subset of REAL by JCT_MISC:15;
A37: |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| in LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|) by RLTOPSP1:68;
A38: ((Gauge (C,n)) * (i,j)) `2 <= |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]| `2 by A1, A2, A3, A26, A29, A32, SPRECT_3:12;
((Gauge (C,n)) * (i,j)) `1 = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]| `1 by A1, A2, A3, A8, A31, A32, GOBOARD5:2;
then |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]| in LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|,((Gauge (C,n)) * (i,j))) by A24, A25, A31, A32, A33, A38, GOBOARD7:7;
then A39: LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|,|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]|) c= LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|) by A37, TOPREAL1:6;
assume A40: x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|,|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))) ; ::_thesis: x = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]|
then reconsider pp = x as Point of (TOP-REAL 2) ;
A41: pp in LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|,|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]|) by A40, XBOOLE_0:def_4;
then A42: pp `2 >= |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]| `2 by A33, TOPREAL1:4;
pp in L~ (Lower_Seq (C,n)) by A40, XBOOLE_0:def_4;
then pp in EE by A41, A39, XBOOLE_0:def_4;
then proj2 . pp in E0 by FUNCT_2:35;
then A43: pp `2 in E0 by PSCOMP_1:def_6;
E0 is real-bounded by RCOMP_1:10;
then E0 is bounded_above by XXREAL_2:def_11;
then |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]| `2 >= pp `2 by A28, A32, A43, SEQ_4:def_1;
then A44: pp `2 = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]| `2 by A42, XXREAL_0:1;
pp `1 = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]| `1 by A24, A25, A31, A32, A41, GOBOARD7:5;
hence x = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]| by A44, TOPREAL3:6; ::_thesis: verum
end;
assume A45: x = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]| ; ::_thesis: x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|,|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n)))
then x in LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|,|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]|) by RLTOPSP1:68;
hence x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|,|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))) by A36, A45, XBOOLE_0:def_4; ::_thesis: verum
end;
hence (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j1))} by A25, A32, TARSKI:def_1; ::_thesis: (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))}
set X = (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n)));
reconsider X1 = (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))) as non empty compact Subset of (TOP-REAL 2) by A7, A10, XBOOLE_0:def_4;
consider pp being set such that
A46: pp in S-most X1 by XBOOLE_0:def_1;
reconsider pp = pp as Point of (TOP-REAL 2) by A46;
A47: pp in (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))) by A46, XBOOLE_0:def_4;
then A48: pp in L~ (Upper_Seq (C,n)) by XBOOLE_0:def_4;
((Gauge (C,n)) * (i,j)) `1 = ((Gauge (C,n)) * (i,1)) `1 by A1, A2, A3, A8, GOBOARD5:2
.= ((Gauge (C,n)) * (i,k)) `1 by A1, A2, A5, A12, GOBOARD5:2 ;
then A49: LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) is vertical by SPPOL_1:16;
pp in LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) by A47, XBOOLE_0:def_4;
then A50: pp `1 = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| `1 by A35, A49, SPPOL_1:41;
|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| `2 = S-bound ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n)))) by A15, A25, SPRECT_1:44
.= (S-min ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))) `2 by EUCLID:52
.= pp `2 by A46, PSCOMP_1:55 ;
then A51: |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| in L~ (Upper_Seq (C,n)) by A48, A50, TOPREAL3:6;
for x being set holds
( x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|,|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))) iff x = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| )
proof
let x be set ; ::_thesis: ( x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|,|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))) iff x = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| )
thus ( x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|,|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))) implies x = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| ) ::_thesis: ( x = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| implies x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|,|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))) )
proof
A52: |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| `2 <= ((Gauge (C,n)) * (i,k)) `2 by A1, A2, A5, A14, A23, A25, SPRECT_3:12;
A53: ((Gauge (C,n)) * (i,j)) `2 <= |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| `2 by A1, A2, A3, A13, A16, A25, SPRECT_3:12;
A54: ((Gauge (C,n)) * (i,k)) `1 = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| `1 by A1, A2, A5, A12, A24, A25, GOBOARD5:2;
((Gauge (C,n)) * (i,j)) `1 = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| `1 by A1, A2, A3, A8, A24, A25, GOBOARD5:2;
then A55: |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| in LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) by A54, A53, A52, GOBOARD7:7;
A56: ((Gauge (C,n)) * (i,k)) `1 = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]| `1 by A1, A2, A5, A12, A31, A32, GOBOARD5:2;
j1 <= k by A14, A27, XXREAL_0:2;
then A57: |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]| `2 <= ((Gauge (C,n)) * (i,k)) `2 by A1, A2, A5, A30, A32, SPRECT_3:12;
A58: ((Gauge (C,n)) * (i,j)) `2 <= |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]| `2 by A1, A2, A3, A26, A29, A32, SPRECT_3:12;
((Gauge (C,n)) * (i,j)) `1 = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]| `1 by A1, A2, A3, A8, A31, A32, GOBOARD5:2;
then |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]| in LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) by A56, A58, A57, GOBOARD7:7;
then A59: LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|,|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]|) c= LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) by A55, TOPREAL1:6;
reconsider EE = (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))) as compact Subset of (TOP-REAL 2) ;
reconsider E0 = proj2 .: EE as compact Subset of REAL by JCT_MISC:15;
assume A60: x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|,|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))) ; ::_thesis: x = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|
then reconsider pp = x as Point of (TOP-REAL 2) ;
A61: pp in LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|,|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]|) by A60, XBOOLE_0:def_4;
then A62: pp `2 <= |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| `2 by A33, TOPREAL1:4;
pp in L~ (Upper_Seq (C,n)) by A60, XBOOLE_0:def_4;
then pp in EE by A61, A59, XBOOLE_0:def_4;
then proj2 . pp in E0 by FUNCT_2:35;
then A63: pp `2 in E0 by PSCOMP_1:def_6;
E0 is real-bounded by RCOMP_1:10;
then E0 is bounded_below by XXREAL_2:def_11;
then |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| `2 <= pp `2 by A15, A25, A63, SEQ_4:def_2;
then A64: pp `2 = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| `2 by A62, XXREAL_0:1;
pp `1 = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| `1 by A24, A25, A31, A32, A61, GOBOARD7:5;
hence x = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| by A64, TOPREAL3:6; ::_thesis: verum
end;
assume A65: x = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| ; ::_thesis: x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|,|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n)))
then x in LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|,|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]|) by RLTOPSP1:68;
hence x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|,|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))) by A51, A65, XBOOLE_0:def_4; ::_thesis: verum
end;
hence (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))} by A25, A32, TARSKI:def_1; ::_thesis: verum
end;
theorem :: JORDAN15:12
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i, j, k being Element of NAT st 1 <= j & j <= k & k <= len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Lower_Seq (C,n)) holds
ex j1 being Element of NAT st
( j <= j1 & j1 <= k & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))} )
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i, j, k being Element of NAT st 1 <= j & j <= k & k <= len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Lower_Seq (C,n)) holds
ex j1 being Element of NAT st
( j <= j1 & j1 <= k & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))} )
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for i, j, k being Element of NAT st 1 <= j & j <= k & k <= len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Lower_Seq (C,n)) holds
ex j1 being Element of NAT st
( j <= j1 & j1 <= k & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))} )
let i, j, k be Element of NAT ; ::_thesis: ( 1 <= j & j <= k & k <= len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Lower_Seq (C,n)) implies ex j1 being Element of NAT st
( j <= j1 & j1 <= k & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (j1,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: (Gauge (C,n)) * (j,i) in L~ (Lower_Seq (C,n)) ; ::_thesis: ex j1 being Element of NAT st
( j <= j1 & j1 <= k & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))} )
set G = Gauge (C,n);
A7: k >= 1 by A1, A2, XXREAL_0:2;
then A8: [k,i] in Indices (Gauge (C,n)) by A3, A4, A5, MATRIX_1:36;
set X = (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)));
A9: (Gauge (C,n)) * (j,i) in LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) by RLTOPSP1:68;
then reconsider X1 = (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))) as non empty compact Subset of (TOP-REAL 2) by A6, XBOOLE_0:def_4;
A10: LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets L~ (Lower_Seq (C,n)) by A6, A9, XBOOLE_0:3;
set s = ((Gauge (C,n)) * (1,i)) `2 ;
set e = (Gauge (C,n)) * (k,i);
set f = (Gauge (C,n)) * (j,i);
set w2 = upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))));
A11: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
then A12: j <= width (Gauge (C,n)) by A2, A3, XXREAL_0:2;
then [j,i] in Indices (Gauge (C,n)) by A1, A4, A5, A11, MATRIX_1:36;
then consider j1 being Element of NAT such that
A13: j <= j1 and
A14: j1 <= k and
A15: ((Gauge (C,n)) * (j1,i)) `1 = upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))))) by A2, A10, A8, JORDAN1F:4, JORDAN1G:5;
set q = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|;
A16: 1 <= j1 by A1, A13, XXREAL_0:2;
take j1 ; ::_thesis: ( j <= j1 & j1 <= k & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))} )
thus ( j <= j1 & j1 <= k ) by A13, A14; ::_thesis: (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))}
consider pp being set such that
A17: pp in E-most X1 by XBOOLE_0:def_1;
reconsider pp = pp as Point of (TOP-REAL 2) by A17;
A18: pp in (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))) by A17, XBOOLE_0:def_4;
then A19: pp in L~ (Lower_Seq (C,n)) by XBOOLE_0:def_4;
A20: j1 <= width (Gauge (C,n)) by A3, A11, A14, XXREAL_0:2;
then A21: ((Gauge (C,n)) * (j1,i)) `2 = ((Gauge (C,n)) * (1,i)) `2 by A4, A5, A11, A16, GOBOARD5:1;
then A22: |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| = (Gauge (C,n)) * (j1,i) by A15, EUCLID:53;
then A23: |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 <= ((Gauge (C,n)) * (k,i)) `1 by A3, A4, A5, A14, A16, SPRECT_3:13;
A24: ((Gauge (C,n)) * (k,i)) `2 = ((Gauge (C,n)) * (1,i)) `2 by A3, A4, A5, A7, GOBOARD5:1;
then ((Gauge (C,n)) * (j,i)) `2 = ((Gauge (C,n)) * (k,i)) `2 by A1, A4, A5, A11, A12, GOBOARD5:1;
then A25: LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) is horizontal by SPPOL_1:15;
A26: |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 = E-bound ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))) by A15, A22, SPRECT_1:46
.= (E-min ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))))) `1 by EUCLID:52
.= pp `1 by A17, PSCOMP_1:47 ;
pp in LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) by A18, XBOOLE_0:def_4;
then pp `2 = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `2 by A24, A21, A22, A25, SPPOL_1:40;
then A27: |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| in L~ (Lower_Seq (C,n)) by A19, A26, TOPREAL3:6;
for x being set holds
( x in (LSeg (((Gauge (C,n)) * (k,i)),|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n))) iff x = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| )
proof
let x be set ; ::_thesis: ( x in (LSeg (((Gauge (C,n)) * (k,i)),|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n))) iff x = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| )
thus ( x in (LSeg (((Gauge (C,n)) * (k,i)),|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n))) implies x = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| ) ::_thesis: ( x = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| implies x in (LSeg (((Gauge (C,n)) * (k,i)),|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n))) )
proof
A28: ((Gauge (C,n)) * (j,i)) `1 <= |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 by A1, A4, A5, A11, A13, A20, A22, SPRECT_3:13;
((Gauge (C,n)) * (j,i)) `2 = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `2 by A1, A4, A5, A11, A12, A21, A22, GOBOARD5:1;
then A29: |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| in LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))) by A24, A21, A22, A23, A28, GOBOARD7:8;
(Gauge (C,n)) * (k,i) in LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) by RLTOPSP1:68;
then A30: LSeg (((Gauge (C,n)) * (k,i)),|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|) c= LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) by A29, TOPREAL1:6;
reconsider EE = (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))) as compact Subset of (TOP-REAL 2) ;
reconsider E0 = proj1 .: EE as compact Subset of REAL by Th4;
assume A31: x in (LSeg (((Gauge (C,n)) * (k,i)),|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n))) ; ::_thesis: x = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|
then reconsider pp = x as Point of (TOP-REAL 2) ;
A32: pp in LSeg (((Gauge (C,n)) * (k,i)),|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|) by A31, XBOOLE_0:def_4;
then A33: pp `1 >= |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 by A23, TOPREAL1:3;
pp in L~ (Lower_Seq (C,n)) by A31, XBOOLE_0:def_4;
then pp in EE by A32, A30, XBOOLE_0:def_4;
then proj1 . pp in E0 by FUNCT_2:35;
then A34: pp `1 in E0 by PSCOMP_1:def_5;
E0 is real-bounded by RCOMP_1:10;
then E0 is bounded_above by XXREAL_2:def_11;
then |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 >= pp `1 by A15, A22, A34, SEQ_4:def_1;
then A35: pp `1 = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 by A33, XXREAL_0:1;
pp `2 = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `2 by A24, A21, A22, A32, GOBOARD7:6;
hence x = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| by A35, TOPREAL3:6; ::_thesis: verum
end;
assume A36: x = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| ; ::_thesis: x in (LSeg (((Gauge (C,n)) * (k,i)),|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))
then x in LSeg (((Gauge (C,n)) * (k,i)),|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|) by RLTOPSP1:68;
hence x in (LSeg (((Gauge (C,n)) * (k,i)),|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n))) by A27, A36, XBOOLE_0:def_4; ::_thesis: verum
end;
hence (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))} by A22, TARSKI:def_1; ::_thesis: verum
end;
theorem Th13: :: JORDAN15:13
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i, j, k being Element of NAT st 1 <= j & j <= k & k <= len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (k,i) in L~ (Upper_Seq (C,n)) holds
ex k1 being Element of NAT st
( j <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (k1,i))} )
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i, j, k being Element of NAT st 1 <= j & j <= k & k <= len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (k,i) in L~ (Upper_Seq (C,n)) holds
ex k1 being Element of NAT st
( j <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (k1,i))} )
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for i, j, k being Element of NAT st 1 <= j & j <= k & k <= len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (k,i) in L~ (Upper_Seq (C,n)) holds
ex k1 being Element of NAT st
( j <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (k1,i))} )
let i, j, k be Element of NAT ; ::_thesis: ( 1 <= j & j <= k & k <= len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (k,i) in L~ (Upper_Seq (C,n)) implies ex k1 being Element of NAT st
( j <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (k1,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: (Gauge (C,n)) * (k,i) in L~ (Upper_Seq (C,n)) ; ::_thesis: ex k1 being Element of NAT st
( j <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (k1,i))} )
set G = Gauge (C,n);
A7: k >= 1 by A1, A2, XXREAL_0:2;
then A8: [k,i] in Indices (Gauge (C,n)) by A3, A4, A5, MATRIX_1:36;
set X = (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)));
A9: (Gauge (C,n)) * (k,i) in LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) by RLTOPSP1:68;
then reconsider X1 = (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n))) as non empty compact Subset of (TOP-REAL 2) by A6, XBOOLE_0:def_4;
A10: LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets L~ (Upper_Seq (C,n)) by A6, A9, XBOOLE_0:3;
set s = ((Gauge (C,n)) * (1,i)) `2 ;
set e = (Gauge (C,n)) * (k,i);
set f = (Gauge (C,n)) * (j,i);
set w1 = lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))));
A11: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
then A12: j <= width (Gauge (C,n)) by A2, A3, XXREAL_0:2;
then [j,i] in Indices (Gauge (C,n)) by A1, A4, A5, A11, MATRIX_1:36;
then consider k1 being Element of NAT such that
A13: j <= k1 and
A14: k1 <= k and
A15: ((Gauge (C,n)) * (k1,i)) `1 = lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n))))) by A2, A10, A8, JORDAN1F:3, JORDAN1G:4;
set p = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|;
A16: k1 <= width (Gauge (C,n)) by A3, A11, A14, XXREAL_0:2;
((Gauge (C,n)) * (j,i)) `2 = ((Gauge (C,n)) * (1,i)) `2 by A1, A4, A5, A11, A12, GOBOARD5:1
.= ((Gauge (C,n)) * (k,i)) `2 by A3, A4, A5, A7, GOBOARD5:1 ;
then A17: LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) is horizontal by SPPOL_1:15;
take k1 ; ::_thesis: ( j <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (k1,i))} )
thus ( j <= k1 & k1 <= k ) by A13, A14; ::_thesis: (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (k1,i))}
consider pp being set such that
A18: pp in W-most X1 by XBOOLE_0:def_1;
A19: 1 <= k1 by A1, A13, XXREAL_0:2;
then A20: ((Gauge (C,n)) * (k1,i)) `2 = ((Gauge (C,n)) * (1,i)) `2 by A4, A5, A11, A16, GOBOARD5:1;
then A21: |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| = (Gauge (C,n)) * (k1,i) by A15, EUCLID:53;
then A22: ((Gauge (C,n)) * (j,i)) `1 <= |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 by A1, A4, A5, A11, A13, A16, SPRECT_3:13;
A23: ((Gauge (C,n)) * (j,i)) `2 = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `2 by A1, A4, A5, A11, A12, A20, A21, GOBOARD5:1;
reconsider pp = pp as Point of (TOP-REAL 2) by A18;
A24: pp in (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n))) by A18, XBOOLE_0:def_4;
then A25: pp in L~ (Upper_Seq (C,n)) by XBOOLE_0:def_4;
A26: |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 = W-bound ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))) by A15, A21, SPRECT_1:43
.= (W-min ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n))))) `1 by EUCLID:52
.= pp `1 by A18, PSCOMP_1:31 ;
pp in LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) by A24, XBOOLE_0:def_4;
then pp `2 = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `2 by A23, A17, SPPOL_1:40;
then A27: |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| in L~ (Upper_Seq (C,n)) by A25, A26, TOPREAL3:6;
for x being set holds
( x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,((Gauge (C,n)) * (j,i)))) /\ (L~ (Upper_Seq (C,n))) iff x = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| )
proof
let x be set ; ::_thesis: ( x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,((Gauge (C,n)) * (j,i)))) /\ (L~ (Upper_Seq (C,n))) iff x = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| )
thus ( x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,((Gauge (C,n)) * (j,i)))) /\ (L~ (Upper_Seq (C,n))) implies x = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| ) ::_thesis: ( x = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| implies x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,((Gauge (C,n)) * (j,i)))) /\ (L~ (Upper_Seq (C,n))) )
proof
reconsider EE = (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n))) as compact Subset of (TOP-REAL 2) ;
assume A28: x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,((Gauge (C,n)) * (j,i)))) /\ (L~ (Upper_Seq (C,n))) ; ::_thesis: x = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|
then reconsider pp = x as Point of (TOP-REAL 2) ;
A29: pp in LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,((Gauge (C,n)) * (j,i))) by A28, XBOOLE_0:def_4;
then A30: pp `1 <= |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 by A22, TOPREAL1:3;
A31: |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 <= ((Gauge (C,n)) * (k,i)) `1 by A3, A4, A5, A14, A19, A21, SPRECT_3:13;
A32: ((Gauge (C,n)) * (j,i)) `1 <= |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 by A1, A4, A5, A11, A13, A16, A21, SPRECT_3:13;
A33: ((Gauge (C,n)) * (k,i)) `2 = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `2 by A3, A4, A5, A7, A20, A21, GOBOARD5:1;
reconsider E0 = proj1 .: EE as compact Subset of REAL by Th4;
A34: (Gauge (C,n)) * (j,i) in LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) by RLTOPSP1:68;
((Gauge (C,n)) * (j,i)) `2 = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `2 by A1, A4, A5, A11, A12, A20, A21, GOBOARD5:1;
then |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| in LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) by A33, A32, A31, GOBOARD7:8;
then A35: LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,((Gauge (C,n)) * (j,i))) c= LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) by A34, TOPREAL1:6;
pp in L~ (Upper_Seq (C,n)) by A28, XBOOLE_0:def_4;
then pp in EE by A29, A35, XBOOLE_0:def_4;
then proj1 . pp in E0 by FUNCT_2:35;
then A36: pp `1 in E0 by PSCOMP_1:def_5;
E0 is real-bounded by RCOMP_1:10;
then E0 is bounded_below by XXREAL_2:def_11;
then |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 <= pp `1 by A15, A21, A36, SEQ_4:def_2;
then A37: pp `1 = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 by A30, XXREAL_0:1;
pp `2 = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `2 by A23, A29, GOBOARD7:6;
hence x = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| by A37, TOPREAL3:6; ::_thesis: verum
end;
assume A38: x = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| ; ::_thesis: x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,((Gauge (C,n)) * (j,i)))) /\ (L~ (Upper_Seq (C,n)))
then x in LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,((Gauge (C,n)) * (j,i))) by RLTOPSP1:68;
hence x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,((Gauge (C,n)) * (j,i)))) /\ (L~ (Upper_Seq (C,n))) by A27, A38, XBOOLE_0:def_4; ::_thesis: verum
end;
hence (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (k1,i))} by A21, TARSKI:def_1; ::_thesis: verum
end;
theorem Th14: :: JORDAN15:14
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i, j, k being Element of NAT st 1 <= j & j <= k & k <= len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Lower_Seq (C,n)) & (Gauge (C,n)) * (k,i) in L~ (Upper_Seq (C,n)) holds
ex j1, k1 being Element of NAT st
( j <= j1 & j1 <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))} & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (k1,i))} )
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i, j, k being Element of NAT st 1 <= j & j <= k & k <= len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Lower_Seq (C,n)) & (Gauge (C,n)) * (k,i) in L~ (Upper_Seq (C,n)) holds
ex j1, k1 being Element of NAT st
( j <= j1 & j1 <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))} & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (k1,i))} )
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for i, j, k being Element of NAT st 1 <= j & j <= k & k <= len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Lower_Seq (C,n)) & (Gauge (C,n)) * (k,i) in L~ (Upper_Seq (C,n)) holds
ex j1, k1 being Element of NAT st
( j <= j1 & j1 <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))} & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (k1,i))} )
let i, j, k be Element of NAT ; ::_thesis: ( 1 <= j & j <= k & k <= len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Lower_Seq (C,n)) & (Gauge (C,n)) * (k,i) in L~ (Upper_Seq (C,n)) implies ex j1, k1 being Element of NAT st
( j <= j1 & j1 <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))} & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (k1,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: (Gauge (C,n)) * (j,i) in L~ (Lower_Seq (C,n)) and
A7: (Gauge (C,n)) * (k,i) in L~ (Upper_Seq (C,n)) ; ::_thesis: ex j1, k1 being Element of NAT st
( j <= j1 & j1 <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))} & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (k1,i))} )
set G = Gauge (C,n);
A8: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
then A9: j <= width (Gauge (C,n)) by A2, A3, XXREAL_0:2;
then A10: [j,i] in Indices (Gauge (C,n)) by A1, A4, A5, A8, MATRIX_1:36;
set s = ((Gauge (C,n)) * (1,i)) `2 ;
set e = (Gauge (C,n)) * (k,i);
set f = (Gauge (C,n)) * (j,i);
set w1 = lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))));
A11: (Gauge (C,n)) * (k,i) in LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) by RLTOPSP1:68;
then A12: LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets L~ (Upper_Seq (C,n)) by A7, XBOOLE_0:3;
A13: k >= 1 by A1, A2, XXREAL_0:2;
then [k,i] in Indices (Gauge (C,n)) by A3, A4, A5, MATRIX_1:36;
then consider k1 being Element of NAT such that
A14: j <= k1 and
A15: k1 <= k and
A16: ((Gauge (C,n)) * (k1,i)) `1 = lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n))))) by A2, A12, A10, JORDAN1F:3, JORDAN1G:4;
A17: k1 <= width (Gauge (C,n)) by A3, A8, A15, XXREAL_0:2;
set p = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|;
set w2 = upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))));
set q = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|;
A18: (Gauge (C,n)) * (j,i) in LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k1,i))) by RLTOPSP1:68;
then A19: LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k1,i))) meets L~ (Lower_Seq (C,n)) by A6, XBOOLE_0:3;
A20: 1 <= k1 by A1, A14, XXREAL_0:2;
then A21: ((Gauge (C,n)) * (k1,i)) `2 = ((Gauge (C,n)) * (1,i)) `2 by A4, A5, A8, A17, GOBOARD5:1;
then A22: |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| = (Gauge (C,n)) * (k1,i) by A16, EUCLID:53;
((Gauge (C,n)) * (j,i)) `2 = ((Gauge (C,n)) * (1,i)) `2 by A1, A4, A5, A8, A9, GOBOARD5:1
.= ((Gauge (C,n)) * (k,i)) `2 by A3, A4, A5, A13, GOBOARD5:1 ;
then A23: LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) is horizontal by SPPOL_1:15;
set X = (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Lower_Seq (C,n)));
reconsider X1 = (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Lower_Seq (C,n))) as non empty compact Subset of (TOP-REAL 2) by A6, A18, XBOOLE_0:def_4;
consider pp being set such that
A24: pp in E-most X1 by XBOOLE_0:def_1;
[k1,i] in Indices (Gauge (C,n)) by A4, A5, A8, A20, A17, MATRIX_1:36;
then consider j1 being Element of NAT such that
A25: j <= j1 and
A26: j1 <= k1 and
A27: ((Gauge (C,n)) * (j1,i)) `1 = upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n))))) by A10, A14, A22, A19, JORDAN1F:4, JORDAN1G:5;
A28: j1 <= width (Gauge (C,n)) by A17, A26, XXREAL_0:2;
reconsider pp = pp as Point of (TOP-REAL 2) by A24;
A29: pp in (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Lower_Seq (C,n))) by A24, XBOOLE_0:def_4;
then A30: pp in L~ (Lower_Seq (C,n)) by XBOOLE_0:def_4;
take j1 ; ::_thesis: ex k1 being Element of NAT st
( j <= j1 & j1 <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))} & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (k1,i))} )
take k1 ; ::_thesis: ( j <= j1 & j1 <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))} & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (k1,i))} )
thus ( j <= j1 & j1 <= k1 & k1 <= k ) by A15, A25, A26; ::_thesis: ( (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))} & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (k1,i))} )
A31: pp in LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k1,i))) by A29, XBOOLE_0:def_4;
A32: 1 <= j1 by A1, A25, XXREAL_0:2;
then A33: ((Gauge (C,n)) * (j1,i)) `2 = ((Gauge (C,n)) * (1,i)) `2 by A4, A5, A8, A28, GOBOARD5:1;
then A34: |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| = (Gauge (C,n)) * (j1,i) by A27, EUCLID:53;
then A35: |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 <= |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 by A4, A5, A8, A17, A22, A26, A32, SPRECT_3:13;
A36: |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 = E-bound ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Lower_Seq (C,n)))) by A22, A27, A34, SPRECT_1:46
.= (E-min ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Lower_Seq (C,n))))) `1 by EUCLID:52
.= pp `1 by A24, PSCOMP_1:47 ;
A37: ((Gauge (C,n)) * (j,i)) `2 = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `2 by A1, A4, A5, A8, A9, A21, A22, GOBOARD5:1;
then LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|) is horizontal by SPPOL_1:15;
then pp `2 = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `2 by A21, A22, A33, A34, A31, SPPOL_1:40;
then A38: |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| in L~ (Lower_Seq (C,n)) by A30, A36, TOPREAL3:6;
for x being set holds
( x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n))) iff x = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| )
proof
let x be set ; ::_thesis: ( x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n))) iff x = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| )
thus ( x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n))) implies x = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| ) ::_thesis: ( x = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| implies x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n))) )
proof
reconsider EE = (LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n))) as compact Subset of (TOP-REAL 2) ;
assume A39: x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n))) ; ::_thesis: x = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|
then reconsider pp = x as Point of (TOP-REAL 2) ;
A40: pp in LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|) by A39, XBOOLE_0:def_4;
then A41: pp `1 >= |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 by A35, TOPREAL1:3;
A42: ((Gauge (C,n)) * (j,i)) `1 <= |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 by A1, A4, A5, A8, A25, A28, A34, SPRECT_3:13;
reconsider E0 = proj1 .: EE as compact Subset of REAL by Th4;
A43: |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| in LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|) by RLTOPSP1:68;
((Gauge (C,n)) * (j,i)) `2 = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `2 by A1, A4, A5, A8, A9, A33, A34, GOBOARD5:1;
then |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| in LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,((Gauge (C,n)) * (j,i))) by A21, A22, A33, A34, A35, A42, GOBOARD7:8;
then A44: LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|) c= LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|) by A43, TOPREAL1:6;
pp in L~ (Lower_Seq (C,n)) by A39, XBOOLE_0:def_4;
then pp in EE by A40, A44, XBOOLE_0:def_4;
then proj1 . pp in E0 by FUNCT_2:35;
then A45: pp `1 in E0 by PSCOMP_1:def_5;
E0 is real-bounded by RCOMP_1:10;
then E0 is bounded_above by XXREAL_2:def_11;
then |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 >= pp `1 by A27, A34, A45, SEQ_4:def_1;
then A46: pp `1 = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 by A41, XXREAL_0:1;
pp `2 = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `2 by A21, A22, A33, A34, A40, GOBOARD7:6;
hence x = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| by A46, TOPREAL3:6; ::_thesis: verum
end;
assume A47: x = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| ; ::_thesis: x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))
then x in LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|) by RLTOPSP1:68;
hence x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n))) by A38, A47, XBOOLE_0:def_4; ::_thesis: verum
end;
hence (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))} by A22, A34, TARSKI:def_1; ::_thesis: (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (k1,i))}
set X = (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)));
reconsider X1 = (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n))) as non empty compact Subset of (TOP-REAL 2) by A7, A11, XBOOLE_0:def_4;
consider pp being set such that
A48: pp in W-most X1 by XBOOLE_0:def_1;
reconsider pp = pp as Point of (TOP-REAL 2) by A48;
A49: pp in (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n))) by A48, XBOOLE_0:def_4;
then A50: pp in L~ (Upper_Seq (C,n)) by XBOOLE_0:def_4;
pp in LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) by A49, XBOOLE_0:def_4;
then A51: pp `2 = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `2 by A37, A23, SPPOL_1:40;
|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 = W-bound ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))) by A16, A22, SPRECT_1:43
.= (W-min ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n))))) `1 by EUCLID:52
.= pp `1 by A48, PSCOMP_1:31 ;
then A52: |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| in L~ (Upper_Seq (C,n)) by A50, A51, TOPREAL3:6;
for x being set holds
( x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n))) iff x = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| )
proof
let x be set ; ::_thesis: ( x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n))) iff x = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| )
thus ( x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n))) implies x = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| ) ::_thesis: ( x = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| implies x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n))) )
proof
j1 <= k by A15, A26, XXREAL_0:2;
then A53: |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 <= ((Gauge (C,n)) * (k,i)) `1 by A3, A4, A5, A32, A34, SPRECT_3:13;
A54: ((Gauge (C,n)) * (k,i)) `2 = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `2 by A3, A4, A5, A13, A21, A22, GOBOARD5:1;
A55: ((Gauge (C,n)) * (j,i)) `1 <= |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 by A1, A4, A5, A8, A14, A17, A22, SPRECT_3:13;
A56: ((Gauge (C,n)) * (j,i)) `1 <= |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 by A1, A4, A5, A8, A25, A28, A34, SPRECT_3:13;
A57: |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 <= ((Gauge (C,n)) * (k,i)) `1 by A3, A4, A5, A15, A20, A22, SPRECT_3:13;
((Gauge (C,n)) * (j,i)) `2 = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `2 by A1, A4, A5, A8, A9, A21, A22, GOBOARD5:1;
then A58: |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| in LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) by A54, A55, A57, GOBOARD7:8;
A59: ((Gauge (C,n)) * (k,i)) `2 = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `2 by A3, A4, A5, A13, A33, A34, GOBOARD5:1;
((Gauge (C,n)) * (j,i)) `2 = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `2 by A1, A4, A5, A8, A9, A33, A34, GOBOARD5:1;
then |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| in LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) by A59, A56, A53, GOBOARD7:8;
then A60: LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|) c= LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) by A58, TOPREAL1:6;
reconsider EE = (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n))) as compact Subset of (TOP-REAL 2) ;
reconsider E0 = proj1 .: EE as compact Subset of REAL by Th4;
assume A61: x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n))) ; ::_thesis: x = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|
then reconsider pp = x as Point of (TOP-REAL 2) ;
A62: pp in LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|) by A61, XBOOLE_0:def_4;
then A63: pp `1 <= |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 by A35, TOPREAL1:3;
pp in L~ (Upper_Seq (C,n)) by A61, XBOOLE_0:def_4;
then pp in EE by A62, A60, XBOOLE_0:def_4;
then proj1 . pp in E0 by FUNCT_2:35;
then A64: pp `1 in E0 by PSCOMP_1:def_5;
E0 is real-bounded by RCOMP_1:10;
then E0 is bounded_below by XXREAL_2:def_11;
then |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 <= pp `1 by A16, A22, A64, SEQ_4:def_2;
then A65: pp `1 = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 by A63, XXREAL_0:1;
pp `2 = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `2 by A21, A22, A33, A34, A62, GOBOARD7:6;
hence x = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| by A65, TOPREAL3:6; ::_thesis: verum
end;
assume A66: x = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| ; ::_thesis: x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))
then x in LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|) by RLTOPSP1:68;
hence x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n))) by A52, A66, XBOOLE_0:def_4; ::_thesis: verum
end;
hence (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (k1,i))} by A22, A34, TARSKI:def_1; ::_thesis: verum
end;
theorem :: JORDAN15:15
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i, j, k being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Upper_Seq (C,n)) holds
ex j1 being Element of NAT st
( j <= j1 & j1 <= k & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,j1))} )
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i, j, k being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Upper_Seq (C,n)) holds
ex j1 being Element of NAT st
( j <= j1 & j1 <= k & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,j1))} )
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for i, j, k being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Upper_Seq (C,n)) holds
ex j1 being Element of NAT st
( j <= j1 & j1 <= k & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,j1))} )
let i, j, k be Element of NAT ; ::_thesis: ( 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Upper_Seq (C,n)) implies ex j1 being Element of NAT st
( j <= j1 & j1 <= k & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,j1))} ) )
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: (Gauge (C,n)) * (i,j) in L~ (Upper_Seq (C,n)) ; ::_thesis: ex j1 being Element of NAT st
( j <= j1 & j1 <= k & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,j1))} )
set G = Gauge (C,n);
A7: k >= 1 by A3, A4, XXREAL_0:2;
then A8: [i,k] in Indices (Gauge (C,n)) by A1, A2, A5, MATRIX_1:36;
set X = (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n)));
A9: (Gauge (C,n)) * (i,j) in LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) by RLTOPSP1:68;
then reconsider X1 = (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))) as non empty compact Subset of (TOP-REAL 2) by A6, XBOOLE_0:def_4;
A10: LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets L~ (Upper_Seq (C,n)) by A6, A9, XBOOLE_0:3;
set s = ((Gauge (C,n)) * (i,1)) `1 ;
set e = (Gauge (C,n)) * (i,k);
set f = (Gauge (C,n)) * (i,j);
set w2 = upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n)))));
A11: j <= width (Gauge (C,n)) by A4, A5, XXREAL_0:2;
then [i,j] in Indices (Gauge (C,n)) by A1, A2, A3, MATRIX_1:36;
then consider j1 being Element of NAT such that
A12: j <= j1 and
A13: j1 <= k and
A14: ((Gauge (C,n)) * (i,j1)) `2 = upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))) by A4, A10, A8, JORDAN1F:2, JORDAN1G:4;
set q = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|;
A15: j1 <= width (Gauge (C,n)) by A5, A13, XXREAL_0:2;
A16: ((Gauge (C,n)) * (i,k)) `1 = ((Gauge (C,n)) * (i,1)) `1 by A1, A2, A5, A7, GOBOARD5:2;
then ((Gauge (C,n)) * (i,j)) `1 = ((Gauge (C,n)) * (i,k)) `1 by A1, A2, A3, A11, GOBOARD5:2;
then A17: LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) is vertical by SPPOL_1:16;
take j1 ; ::_thesis: ( j <= j1 & j1 <= k & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,j1))} )
thus ( j <= j1 & j1 <= k ) by A12, A13; ::_thesis: (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,j1))}
consider pp being set such that
A18: pp in N-most X1 by XBOOLE_0:def_1;
reconsider pp = pp as Point of (TOP-REAL 2) by A18;
A19: pp in (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))) by A18, XBOOLE_0:def_4;
then A20: pp in L~ (Upper_Seq (C,n)) by XBOOLE_0:def_4;
A21: 1 <= j1 by A3, A12, XXREAL_0:2;
then A22: ((Gauge (C,n)) * (i,j1)) `1 = ((Gauge (C,n)) * (i,1)) `1 by A1, A2, A15, GOBOARD5:2;
then A23: |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| = (Gauge (C,n)) * (i,j1) by A14, EUCLID:53;
then A24: |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| `2 <= ((Gauge (C,n)) * (i,k)) `2 by A1, A2, A5, A13, A21, SPRECT_3:12;
A25: |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| `2 = N-bound ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n)))) by A14, A23, SPRECT_1:45
.= (N-min ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))) `2 by EUCLID:52
.= pp `2 by A18, PSCOMP_1:39 ;
pp in LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) by A19, XBOOLE_0:def_4;
then pp `1 = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| `1 by A16, A22, A23, A17, SPPOL_1:41;
then A26: |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| in L~ (Upper_Seq (C,n)) by A20, A25, TOPREAL3:6;
for x being set holds
( x in (LSeg (((Gauge (C,n)) * (i,k)),|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))) iff x = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| )
proof
let x be set ; ::_thesis: ( x in (LSeg (((Gauge (C,n)) * (i,k)),|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))) iff x = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| )
thus ( x in (LSeg (((Gauge (C,n)) * (i,k)),|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))) implies x = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| ) ::_thesis: ( x = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| implies x in (LSeg (((Gauge (C,n)) * (i,k)),|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))) )
proof
reconsider EE = (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))) as compact Subset of (TOP-REAL 2) ;
reconsider E0 = proj2 .: EE as compact Subset of REAL by JCT_MISC:15;
A27: (Gauge (C,n)) * (i,k) in LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) by RLTOPSP1:68;
A28: ((Gauge (C,n)) * (i,j)) `2 <= |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| `2 by A1, A2, A3, A12, A15, A23, SPRECT_3:12;
((Gauge (C,n)) * (i,j)) `1 = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| `1 by A1, A2, A3, A11, A22, A23, GOBOARD5:2;
then |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| in LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) by A16, A22, A23, A24, A28, GOBOARD7:7;
then A29: LSeg (((Gauge (C,n)) * (i,k)),|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|) c= LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) by A27, TOPREAL1:6;
assume A30: x in (LSeg (((Gauge (C,n)) * (i,k)),|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))) ; ::_thesis: x = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|
then reconsider pp = x as Point of (TOP-REAL 2) ;
A31: pp in LSeg (((Gauge (C,n)) * (i,k)),|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|) by A30, XBOOLE_0:def_4;
then A32: pp `2 >= |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| `2 by A24, TOPREAL1:4;
pp in L~ (Upper_Seq (C,n)) by A30, XBOOLE_0:def_4;
then pp in EE by A31, A29, XBOOLE_0:def_4;
then proj2 . pp in E0 by FUNCT_2:35;
then A33: pp `2 in E0 by PSCOMP_1:def_6;
E0 is real-bounded by RCOMP_1:10;
then E0 is bounded_above by XXREAL_2:def_11;
then |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| `2 >= pp `2 by A14, A23, A33, SEQ_4:def_1;
then A34: pp `2 = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| `2 by A32, XXREAL_0:1;
pp `1 = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| `1 by A16, A22, A23, A31, GOBOARD7:5;
hence x = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| by A34, TOPREAL3:6; ::_thesis: verum
end;
assume A35: x = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]| ; ::_thesis: x in (LSeg (((Gauge (C,n)) * (i,k)),|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n)))
then x in LSeg (((Gauge (C,n)) * (i,k)),|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|) by RLTOPSP1:68;
hence x in (LSeg (((Gauge (C,n)) * (i,k)),|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))) by A26, A35, XBOOLE_0:def_4; ::_thesis: verum
end;
hence (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,j1))} by A23, TARSKI:def_1; ::_thesis: verum
end;
theorem :: JORDAN15:16
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i, j, k being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Lower_Seq (C,n)) holds
ex k1 being Element of NAT st
( j <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))} )
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i, j, k being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Lower_Seq (C,n)) holds
ex k1 being Element of NAT st
( j <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))} )
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for i, j, k being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Lower_Seq (C,n)) holds
ex k1 being Element of NAT st
( j <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))} )
let i, j, k be Element of NAT ; ::_thesis: ( 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Lower_Seq (C,n)) implies ex k1 being Element of NAT st
( j <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))} ) )
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: (Gauge (C,n)) * (i,k) in L~ (Lower_Seq (C,n)) ; ::_thesis: ex k1 being Element of NAT st
( j <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))} )
set G = Gauge (C,n);
A7: k >= 1 by A3, A4, XXREAL_0:2;
then A8: [i,k] in Indices (Gauge (C,n)) by A1, A2, A5, MATRIX_1:36;
set X = (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n)));
A9: (Gauge (C,n)) * (i,k) in LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) by RLTOPSP1:68;
then reconsider X1 = (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))) as non empty compact Subset of (TOP-REAL 2) by A6, XBOOLE_0:def_4;
A10: LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets L~ (Lower_Seq (C,n)) by A6, A9, XBOOLE_0:3;
set s = ((Gauge (C,n)) * (i,1)) `1 ;
set e = (Gauge (C,n)) * (i,k);
set f = (Gauge (C,n)) * (i,j);
set w1 = lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n)))));
A11: j <= width (Gauge (C,n)) by A4, A5, XXREAL_0:2;
then [i,j] in Indices (Gauge (C,n)) by A1, A2, A3, MATRIX_1:36;
then consider k1 being Element of NAT such that
A12: j <= k1 and
A13: k1 <= k and
A14: ((Gauge (C,n)) * (i,k1)) `2 = lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))) by A4, A10, A8, JORDAN1F:1, JORDAN1G:5;
set p = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|;
A15: k1 <= width (Gauge (C,n)) by A5, A13, XXREAL_0:2;
((Gauge (C,n)) * (i,j)) `1 = ((Gauge (C,n)) * (i,1)) `1 by A1, A2, A3, A11, GOBOARD5:2
.= ((Gauge (C,n)) * (i,k)) `1 by A1, A2, A5, A7, GOBOARD5:2 ;
then A16: LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) is vertical by SPPOL_1:16;
take k1 ; ::_thesis: ( j <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))} )
thus ( j <= k1 & k1 <= k ) by A12, A13; ::_thesis: (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))}
consider pp being set such that
A17: pp in S-most X1 by XBOOLE_0:def_1;
A18: 1 <= k1 by A3, A12, XXREAL_0:2;
then A19: ((Gauge (C,n)) * (i,k1)) `1 = ((Gauge (C,n)) * (i,1)) `1 by A1, A2, A15, GOBOARD5:2;
then A20: |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| = (Gauge (C,n)) * (i,k1) by A14, EUCLID:53;
then A21: ((Gauge (C,n)) * (i,j)) `2 <= |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| `2 by A1, A2, A3, A12, A15, SPRECT_3:12;
A22: ((Gauge (C,n)) * (i,j)) `1 = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| `1 by A1, A2, A3, A11, A19, A20, GOBOARD5:2;
reconsider pp = pp as Point of (TOP-REAL 2) by A17;
A23: pp in (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))) by A17, XBOOLE_0:def_4;
then A24: pp in L~ (Lower_Seq (C,n)) by XBOOLE_0:def_4;
A25: |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| `2 = S-bound ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n)))) by A14, A20, SPRECT_1:44
.= (S-min ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))) `2 by EUCLID:52
.= pp `2 by A17, PSCOMP_1:55 ;
pp in LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) by A23, XBOOLE_0:def_4;
then pp `1 = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| `1 by A22, A16, SPPOL_1:41;
then A26: |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| in L~ (Lower_Seq (C,n)) by A24, A25, TOPREAL3:6;
for x being set holds
( x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|,((Gauge (C,n)) * (i,j)))) /\ (L~ (Lower_Seq (C,n))) iff x = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| )
proof
let x be set ; ::_thesis: ( x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|,((Gauge (C,n)) * (i,j)))) /\ (L~ (Lower_Seq (C,n))) iff x = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| )
thus ( x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|,((Gauge (C,n)) * (i,j)))) /\ (L~ (Lower_Seq (C,n))) implies x = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| ) ::_thesis: ( x = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| implies x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|,((Gauge (C,n)) * (i,j)))) /\ (L~ (Lower_Seq (C,n))) )
proof
reconsider EE = (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))) as compact Subset of (TOP-REAL 2) ;
reconsider E0 = proj2 .: EE as compact Subset of REAL by JCT_MISC:15;
A27: (Gauge (C,n)) * (i,j) in LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) by RLTOPSP1:68;
A28: ((Gauge (C,n)) * (i,k)) `1 = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| `1 by A1, A2, A5, A7, A19, A20, GOBOARD5:2;
A29: |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| `2 <= ((Gauge (C,n)) * (i,k)) `2 by A1, A2, A5, A13, A18, A20, SPRECT_3:12;
A30: ((Gauge (C,n)) * (i,j)) `2 <= |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| `2 by A1, A2, A3, A12, A15, A20, SPRECT_3:12;
((Gauge (C,n)) * (i,j)) `1 = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| `1 by A1, A2, A3, A11, A19, A20, GOBOARD5:2;
then |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| in LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) by A28, A30, A29, GOBOARD7:7;
then A31: LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|,((Gauge (C,n)) * (i,j))) c= LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) by A27, TOPREAL1:6;
assume A32: x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|,((Gauge (C,n)) * (i,j)))) /\ (L~ (Lower_Seq (C,n))) ; ::_thesis: x = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|
then reconsider pp = x as Point of (TOP-REAL 2) ;
A33: pp in LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|,((Gauge (C,n)) * (i,j))) by A32, XBOOLE_0:def_4;
then A34: pp `2 <= |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| `2 by A21, TOPREAL1:4;
pp in L~ (Lower_Seq (C,n)) by A32, XBOOLE_0:def_4;
then pp in EE by A33, A31, XBOOLE_0:def_4;
then proj2 . pp in E0 by FUNCT_2:35;
then A35: pp `2 in E0 by PSCOMP_1:def_6;
E0 is real-bounded by RCOMP_1:10;
then E0 is bounded_below by XXREAL_2:def_11;
then |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| `2 <= pp `2 by A14, A20, A35, SEQ_4:def_2;
then A36: pp `2 = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| `2 by A34, XXREAL_0:1;
pp `1 = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| `1 by A22, A33, GOBOARD7:5;
hence x = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| by A36, TOPREAL3:6; ::_thesis: verum
end;
assume A37: x = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| ; ::_thesis: x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|,((Gauge (C,n)) * (i,j)))) /\ (L~ (Lower_Seq (C,n)))
then x in LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|,((Gauge (C,n)) * (i,j))) by RLTOPSP1:68;
hence x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|,((Gauge (C,n)) * (i,j)))) /\ (L~ (Lower_Seq (C,n))) by A26, A37, XBOOLE_0:def_4; ::_thesis: verum
end;
hence (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))} by A20, TARSKI:def_1; ::_thesis: verum
end;
theorem :: JORDAN15:17
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i, j, k being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Lower_Seq (C,n)) holds
ex j1, k1 being Element of NAT st
( j <= j1 & j1 <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,j1))} & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))} )
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i, j, k being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Lower_Seq (C,n)) holds
ex j1, k1 being Element of NAT st
( j <= j1 & j1 <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,j1))} & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))} )
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for i, j, k being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Lower_Seq (C,n)) holds
ex j1, k1 being Element of NAT st
( j <= j1 & j1 <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,j1))} & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))} )
let i, j, k be Element of NAT ; ::_thesis: ( 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Lower_Seq (C,n)) implies ex j1, k1 being Element of NAT st
( j <= j1 & j1 <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,j1))} & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))} ) )
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: (Gauge (C,n)) * (i,j) in L~ (Upper_Seq (C,n)) and
A7: (Gauge (C,n)) * (i,k) in L~ (Lower_Seq (C,n)) ; ::_thesis: ex j1, k1 being Element of NAT st
( j <= j1 & j1 <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,j1))} & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))} )
set G = Gauge (C,n);
A8: j <= width (Gauge (C,n)) by A4, A5, XXREAL_0:2;
then A9: [i,j] in Indices (Gauge (C,n)) by A1, A2, A3, MATRIX_1:36;
set s = ((Gauge (C,n)) * (i,1)) `1 ;
set e = (Gauge (C,n)) * (i,k);
set f = (Gauge (C,n)) * (i,j);
set w1 = lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n)))));
A10: (Gauge (C,n)) * (i,k) in LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) by RLTOPSP1:68;
then A11: LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets L~ (Lower_Seq (C,n)) by A7, XBOOLE_0:3;
A12: k >= 1 by A3, A4, XXREAL_0:2;
then [i,k] in Indices (Gauge (C,n)) by A1, A2, A5, MATRIX_1:36;
then consider k1 being Element of NAT such that
A13: j <= k1 and
A14: k1 <= k and
A15: ((Gauge (C,n)) * (i,k1)) `2 = lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))) by A4, A11, A9, JORDAN1F:1, JORDAN1G:5;
A16: k1 <= width (Gauge (C,n)) by A5, A14, XXREAL_0:2;
A17: (Gauge (C,n)) * (i,j) in LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k1))) by RLTOPSP1:68;
then A18: LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k1))) meets L~ (Upper_Seq (C,n)) by A6, XBOOLE_0:3;
set X = (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Upper_Seq (C,n)));
reconsider X1 = (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Upper_Seq (C,n))) as non empty compact Subset of (TOP-REAL 2) by A6, A17, XBOOLE_0:def_4;
consider pp being set such that
A19: pp in N-most X1 by XBOOLE_0:def_1;
reconsider pp = pp as Point of (TOP-REAL 2) by A19;
A20: pp in (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Upper_Seq (C,n))) by A19, XBOOLE_0:def_4;
then A21: pp in L~ (Upper_Seq (C,n)) by XBOOLE_0:def_4;
set p = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|;
set w2 = upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n)))));
set q = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]|;
A22: pp in LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k1))) by A20, XBOOLE_0:def_4;
A23: 1 <= k1 by A3, A13, XXREAL_0:2;
then A24: ((Gauge (C,n)) * (i,k1)) `1 = ((Gauge (C,n)) * (i,1)) `1 by A1, A2, A16, GOBOARD5:2;
then A25: |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| = (Gauge (C,n)) * (i,k1) by A15, EUCLID:53;
[i,k1] in Indices (Gauge (C,n)) by A1, A2, A23, A16, MATRIX_1:36;
then consider j1 being Element of NAT such that
A26: j <= j1 and
A27: j1 <= k1 and
A28: ((Gauge (C,n)) * (i,j1)) `2 = upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))) by A9, A13, A25, A18, JORDAN1F:2, JORDAN1G:4;
take j1 ; ::_thesis: ex k1 being Element of NAT st
( j <= j1 & j1 <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,j1))} & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))} )
take k1 ; ::_thesis: ( j <= j1 & j1 <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,j1))} & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))} )
thus ( j <= j1 & j1 <= k1 & k1 <= k ) by A14, A26, A27; ::_thesis: ( (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,j1))} & (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))} )
A29: j1 <= width (Gauge (C,n)) by A16, A27, XXREAL_0:2;
A30: 1 <= j1 by A3, A26, XXREAL_0:2;
then A31: ((Gauge (C,n)) * (i,j1)) `1 = ((Gauge (C,n)) * (i,1)) `1 by A1, A2, A29, GOBOARD5:2;
then A32: |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]| = (Gauge (C,n)) * (i,j1) by A28, EUCLID:53;
then A33: |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]| `2 <= |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| `2 by A1, A2, A16, A25, A27, A30, SPRECT_3:12;
A34: |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]| `2 = N-bound ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Upper_Seq (C,n)))) by A25, A28, A32, SPRECT_1:45
.= (N-min ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Upper_Seq (C,n))))) `2 by EUCLID:52
.= pp `2 by A19, PSCOMP_1:39 ;
A35: ((Gauge (C,n)) * (i,j)) `1 = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| `1 by A1, A2, A3, A8, A24, A25, GOBOARD5:2;
then LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|) is vertical by SPPOL_1:16;
then pp `1 = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]| `1 by A24, A25, A31, A32, A22, SPPOL_1:41;
then A36: |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]| in L~ (Upper_Seq (C,n)) by A21, A34, TOPREAL3:6;
for x being set holds
( x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|,|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))) iff x = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]| )
proof
let x be set ; ::_thesis: ( x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|,|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))) iff x = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]| )
thus ( x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|,|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))) implies x = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]| ) ::_thesis: ( x = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]| implies x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|,|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))) )
proof
reconsider EE = (LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))) as compact Subset of (TOP-REAL 2) ;
reconsider E0 = proj2 .: EE as compact Subset of REAL by JCT_MISC:15;
A37: |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| in LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|) by RLTOPSP1:68;
A38: ((Gauge (C,n)) * (i,j)) `2 <= |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]| `2 by A1, A2, A3, A26, A29, A32, SPRECT_3:12;
((Gauge (C,n)) * (i,j)) `1 = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]| `1 by A1, A2, A3, A8, A31, A32, GOBOARD5:2;
then |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]| in LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|,((Gauge (C,n)) * (i,j))) by A24, A25, A31, A32, A33, A38, GOBOARD7:7;
then A39: LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|,|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]|) c= LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|) by A37, TOPREAL1:6;
assume A40: x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|,|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))) ; ::_thesis: x = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]|
then reconsider pp = x as Point of (TOP-REAL 2) ;
A41: pp in LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|,|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]|) by A40, XBOOLE_0:def_4;
then A42: pp `2 >= |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]| `2 by A33, TOPREAL1:4;
pp in L~ (Upper_Seq (C,n)) by A40, XBOOLE_0:def_4;
then pp in EE by A41, A39, XBOOLE_0:def_4;
then proj2 . pp in E0 by FUNCT_2:35;
then A43: pp `2 in E0 by PSCOMP_1:def_6;
E0 is real-bounded by RCOMP_1:10;
then E0 is bounded_above by XXREAL_2:def_11;
then |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]| `2 >= pp `2 by A28, A32, A43, SEQ_4:def_1;
then A44: pp `2 = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]| `2 by A42, XXREAL_0:1;
pp `1 = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]| `1 by A24, A25, A31, A32, A41, GOBOARD7:5;
hence x = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]| by A44, TOPREAL3:6; ::_thesis: verum
end;
assume A45: x = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]| ; ::_thesis: x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|,|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n)))
then x in LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|,|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]|) by RLTOPSP1:68;
hence x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|,|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))) by A36, A45, XBOOLE_0:def_4; ::_thesis: verum
end;
hence (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,j1))} by A25, A32, TARSKI:def_1; ::_thesis: (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))}
set X = (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n)));
reconsider X1 = (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))) as non empty compact Subset of (TOP-REAL 2) by A7, A10, XBOOLE_0:def_4;
consider pp being set such that
A46: pp in S-most X1 by XBOOLE_0:def_1;
reconsider pp = pp as Point of (TOP-REAL 2) by A46;
A47: pp in (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))) by A46, XBOOLE_0:def_4;
then A48: pp in L~ (Lower_Seq (C,n)) by XBOOLE_0:def_4;
((Gauge (C,n)) * (i,j)) `1 = ((Gauge (C,n)) * (i,1)) `1 by A1, A2, A3, A8, GOBOARD5:2
.= ((Gauge (C,n)) * (i,k)) `1 by A1, A2, A5, A12, GOBOARD5:2 ;
then A49: LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) is vertical by SPPOL_1:16;
pp in LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) by A47, XBOOLE_0:def_4;
then A50: pp `1 = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| `1 by A35, A49, SPPOL_1:41;
|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| `2 = S-bound ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n)))) by A15, A25, SPRECT_1:44
.= (S-min ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))) `2 by EUCLID:52
.= pp `2 by A46, PSCOMP_1:55 ;
then A51: |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| in L~ (Lower_Seq (C,n)) by A48, A50, TOPREAL3:6;
for x being set holds
( x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|,|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))) iff x = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| )
proof
let x be set ; ::_thesis: ( x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|,|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))) iff x = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| )
thus ( x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|,|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))) implies x = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| ) ::_thesis: ( x = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| implies x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|,|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))) )
proof
A52: |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| `2 <= ((Gauge (C,n)) * (i,k)) `2 by A1, A2, A5, A14, A23, A25, SPRECT_3:12;
A53: ((Gauge (C,n)) * (i,j)) `2 <= |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| `2 by A1, A2, A3, A13, A16, A25, SPRECT_3:12;
A54: ((Gauge (C,n)) * (i,k)) `1 = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| `1 by A1, A2, A5, A12, A24, A25, GOBOARD5:2;
((Gauge (C,n)) * (i,j)) `1 = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| `1 by A1, A2, A3, A8, A24, A25, GOBOARD5:2;
then A55: |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| in LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) by A54, A53, A52, GOBOARD7:7;
A56: ((Gauge (C,n)) * (i,k)) `1 = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]| `1 by A1, A2, A5, A12, A31, A32, GOBOARD5:2;
j1 <= k by A14, A27, XXREAL_0:2;
then A57: |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]| `2 <= ((Gauge (C,n)) * (i,k)) `2 by A1, A2, A5, A30, A32, SPRECT_3:12;
A58: ((Gauge (C,n)) * (i,j)) `2 <= |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]| `2 by A1, A2, A3, A26, A29, A32, SPRECT_3:12;
((Gauge (C,n)) * (i,j)) `1 = |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]| `1 by A1, A2, A3, A8, A31, A32, GOBOARD5:2;
then |[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]| in LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) by A56, A58, A57, GOBOARD7:7;
then A59: LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|,|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]|) c= LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) by A55, TOPREAL1:6;
reconsider EE = (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))) as compact Subset of (TOP-REAL 2) ;
reconsider E0 = proj2 .: EE as compact Subset of REAL by JCT_MISC:15;
assume A60: x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|,|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))) ; ::_thesis: x = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|
then reconsider pp = x as Point of (TOP-REAL 2) ;
A61: pp in LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|,|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]|) by A60, XBOOLE_0:def_4;
then A62: pp `2 <= |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| `2 by A33, TOPREAL1:4;
pp in L~ (Lower_Seq (C,n)) by A60, XBOOLE_0:def_4;
then pp in EE by A61, A59, XBOOLE_0:def_4;
then proj2 . pp in E0 by FUNCT_2:35;
then A63: pp `2 in E0 by PSCOMP_1:def_6;
E0 is real-bounded by RCOMP_1:10;
then E0 is bounded_below by XXREAL_2:def_11;
then |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| `2 <= pp `2 by A15, A25, A63, SEQ_4:def_2;
then A64: pp `2 = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| `2 by A62, XXREAL_0:1;
pp `1 = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| `1 by A24, A25, A31, A32, A61, GOBOARD7:5;
hence x = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| by A64, TOPREAL3:6; ::_thesis: verum
end;
assume A65: x = |[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]| ; ::_thesis: x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|,|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n)))
then x in LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|,|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]|) by RLTOPSP1:68;
hence x in (LSeg (|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|,|[(((Gauge (C,n)) * (i,1)) `1),(upper_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),|[(((Gauge (C,n)) * (i,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))))))]|)) /\ (L~ (Upper_Seq (C,n))))))]|)) /\ (L~ (Lower_Seq (C,n))) by A51, A65, XBOOLE_0:def_4; ::_thesis: verum
end;
hence (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))} by A25, A32, TARSKI:def_1; ::_thesis: verum
end;
theorem Th18: :: JORDAN15:18
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i, j, k being Element of NAT st 1 <= j & j <= k & k <= len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Upper_Seq (C,n)) holds
ex j1 being Element of NAT st
( j <= j1 & j1 <= k & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))} )
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i, j, k being Element of NAT st 1 <= j & j <= k & k <= len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Upper_Seq (C,n)) holds
ex j1 being Element of NAT st
( j <= j1 & j1 <= k & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))} )
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for i, j, k being Element of NAT st 1 <= j & j <= k & k <= len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Upper_Seq (C,n)) holds
ex j1 being Element of NAT st
( j <= j1 & j1 <= k & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))} )
let i, j, k be Element of NAT ; ::_thesis: ( 1 <= j & j <= k & k <= len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Upper_Seq (C,n)) implies ex j1 being Element of NAT st
( j <= j1 & j1 <= k & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (j1,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: (Gauge (C,n)) * (j,i) in L~ (Upper_Seq (C,n)) ; ::_thesis: ex j1 being Element of NAT st
( j <= j1 & j1 <= k & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))} )
set G = Gauge (C,n);
A7: k >= 1 by A1, A2, XXREAL_0:2;
then A8: [k,i] in Indices (Gauge (C,n)) by A3, A4, A5, MATRIX_1:36;
set X = (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)));
A9: (Gauge (C,n)) * (j,i) in LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) by RLTOPSP1:68;
then reconsider X1 = (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n))) as non empty compact Subset of (TOP-REAL 2) by A6, XBOOLE_0:def_4;
A10: LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets L~ (Upper_Seq (C,n)) by A6, A9, XBOOLE_0:3;
set s = ((Gauge (C,n)) * (1,i)) `2 ;
set e = (Gauge (C,n)) * (k,i);
set f = (Gauge (C,n)) * (j,i);
set w2 = upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))));
A11: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
then A12: j <= width (Gauge (C,n)) by A2, A3, XXREAL_0:2;
then [j,i] in Indices (Gauge (C,n)) by A1, A4, A5, A11, MATRIX_1:36;
then consider j1 being Element of NAT such that
A13: j <= j1 and
A14: j1 <= k and
A15: ((Gauge (C,n)) * (j1,i)) `1 = upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n))))) by A2, A10, A8, JORDAN1F:4, JORDAN1G:4;
set q = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|;
A16: 1 <= j1 by A1, A13, XXREAL_0:2;
take j1 ; ::_thesis: ( j <= j1 & j1 <= k & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))} )
thus ( j <= j1 & j1 <= k ) by A13, A14; ::_thesis: (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))}
consider pp being set such that
A17: pp in E-most X1 by XBOOLE_0:def_1;
reconsider pp = pp as Point of (TOP-REAL 2) by A17;
A18: pp in (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n))) by A17, XBOOLE_0:def_4;
then A19: pp in L~ (Upper_Seq (C,n)) by XBOOLE_0:def_4;
A20: j1 <= width (Gauge (C,n)) by A3, A11, A14, XXREAL_0:2;
then A21: ((Gauge (C,n)) * (j1,i)) `2 = ((Gauge (C,n)) * (1,i)) `2 by A4, A5, A11, A16, GOBOARD5:1;
then A22: |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| = (Gauge (C,n)) * (j1,i) by A15, EUCLID:53;
then A23: |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 <= ((Gauge (C,n)) * (k,i)) `1 by A3, A4, A5, A14, A16, SPRECT_3:13;
A24: ((Gauge (C,n)) * (k,i)) `2 = ((Gauge (C,n)) * (1,i)) `2 by A3, A4, A5, A7, GOBOARD5:1;
then ((Gauge (C,n)) * (j,i)) `2 = ((Gauge (C,n)) * (k,i)) `2 by A1, A4, A5, A11, A12, GOBOARD5:1;
then A25: LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) is horizontal by SPPOL_1:15;
A26: |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 = E-bound ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))) by A15, A22, SPRECT_1:46
.= (E-min ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n))))) `1 by EUCLID:52
.= pp `1 by A17, PSCOMP_1:47 ;
pp in LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) by A18, XBOOLE_0:def_4;
then pp `2 = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `2 by A24, A21, A22, A25, SPPOL_1:40;
then A27: |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| in L~ (Upper_Seq (C,n)) by A19, A26, TOPREAL3:6;
for x being set holds
( x in (LSeg (((Gauge (C,n)) * (k,i)),|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n))) iff x = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| )
proof
let x be set ; ::_thesis: ( x in (LSeg (((Gauge (C,n)) * (k,i)),|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n))) iff x = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| )
thus ( x in (LSeg (((Gauge (C,n)) * (k,i)),|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n))) implies x = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| ) ::_thesis: ( x = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| implies x in (LSeg (((Gauge (C,n)) * (k,i)),|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n))) )
proof
A28: ((Gauge (C,n)) * (j,i)) `1 <= |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 by A1, A4, A5, A11, A13, A20, A22, SPRECT_3:13;
((Gauge (C,n)) * (j,i)) `2 = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `2 by A1, A4, A5, A11, A12, A21, A22, GOBOARD5:1;
then A29: |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| in LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))) by A24, A21, A22, A23, A28, GOBOARD7:8;
(Gauge (C,n)) * (k,i) in LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) by RLTOPSP1:68;
then A30: LSeg (((Gauge (C,n)) * (k,i)),|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|) c= LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) by A29, TOPREAL1:6;
reconsider EE = (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n))) as compact Subset of (TOP-REAL 2) ;
reconsider E0 = proj1 .: EE as compact Subset of REAL by Th4;
assume A31: x in (LSeg (((Gauge (C,n)) * (k,i)),|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n))) ; ::_thesis: x = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|
then reconsider pp = x as Point of (TOP-REAL 2) ;
A32: pp in LSeg (((Gauge (C,n)) * (k,i)),|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|) by A31, XBOOLE_0:def_4;
then A33: pp `1 >= |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 by A23, TOPREAL1:3;
pp in L~ (Upper_Seq (C,n)) by A31, XBOOLE_0:def_4;
then pp in EE by A32, A30, XBOOLE_0:def_4;
then proj1 . pp in E0 by FUNCT_2:35;
then A34: pp `1 in E0 by PSCOMP_1:def_5;
E0 is real-bounded by RCOMP_1:10;
then E0 is bounded_above by XXREAL_2:def_11;
then |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 >= pp `1 by A15, A22, A34, SEQ_4:def_1;
then A35: pp `1 = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 by A33, XXREAL_0:1;
pp `2 = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `2 by A24, A21, A22, A32, GOBOARD7:6;
hence x = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| by A35, TOPREAL3:6; ::_thesis: verum
end;
assume A36: x = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| ; ::_thesis: x in (LSeg (((Gauge (C,n)) * (k,i)),|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))
then x in LSeg (((Gauge (C,n)) * (k,i)),|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|) by RLTOPSP1:68;
hence x in (LSeg (((Gauge (C,n)) * (k,i)),|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n))) by A27, A36, XBOOLE_0:def_4; ::_thesis: verum
end;
hence (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))} by A22, TARSKI:def_1; ::_thesis: verum
end;
theorem :: JORDAN15:19
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i, j, k being Element of NAT st 1 <= j & j <= k & k <= len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (k,i) in L~ (Lower_Seq (C,n)) holds
ex k1 being Element of NAT st
( j <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (k1,i))} )
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i, j, k being Element of NAT st 1 <= j & j <= k & k <= len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (k,i) in L~ (Lower_Seq (C,n)) holds
ex k1 being Element of NAT st
( j <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (k1,i))} )
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for i, j, k being Element of NAT st 1 <= j & j <= k & k <= len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (k,i) in L~ (Lower_Seq (C,n)) holds
ex k1 being Element of NAT st
( j <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (k1,i))} )
let i, j, k be Element of NAT ; ::_thesis: ( 1 <= j & j <= k & k <= len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (k,i) in L~ (Lower_Seq (C,n)) implies ex k1 being Element of NAT st
( j <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (k1,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: (Gauge (C,n)) * (k,i) in L~ (Lower_Seq (C,n)) ; ::_thesis: ex k1 being Element of NAT st
( j <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (k1,i))} )
set G = Gauge (C,n);
A7: k >= 1 by A1, A2, XXREAL_0:2;
then A8: [k,i] in Indices (Gauge (C,n)) by A3, A4, A5, MATRIX_1:36;
set X = (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)));
A9: (Gauge (C,n)) * (k,i) in LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) by RLTOPSP1:68;
then reconsider X1 = (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))) as non empty compact Subset of (TOP-REAL 2) by A6, XBOOLE_0:def_4;
A10: LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets L~ (Lower_Seq (C,n)) by A6, A9, XBOOLE_0:3;
set s = ((Gauge (C,n)) * (1,i)) `2 ;
set e = (Gauge (C,n)) * (k,i);
set f = (Gauge (C,n)) * (j,i);
set w1 = lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))));
A11: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
then A12: j <= width (Gauge (C,n)) by A2, A3, XXREAL_0:2;
then [j,i] in Indices (Gauge (C,n)) by A1, A4, A5, A11, MATRIX_1:36;
then consider k1 being Element of NAT such that
A13: j <= k1 and
A14: k1 <= k and
A15: ((Gauge (C,n)) * (k1,i)) `1 = lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))))) by A2, A10, A8, JORDAN1F:3, JORDAN1G:5;
set p = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|;
A16: k1 <= width (Gauge (C,n)) by A3, A11, A14, XXREAL_0:2;
((Gauge (C,n)) * (j,i)) `2 = ((Gauge (C,n)) * (1,i)) `2 by A1, A4, A5, A11, A12, GOBOARD5:1
.= ((Gauge (C,n)) * (k,i)) `2 by A3, A4, A5, A7, GOBOARD5:1 ;
then A17: LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) is horizontal by SPPOL_1:15;
take k1 ; ::_thesis: ( j <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (k1,i))} )
thus ( j <= k1 & k1 <= k ) by A13, A14; ::_thesis: (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (k1,i))}
consider pp being set such that
A18: pp in W-most X1 by XBOOLE_0:def_1;
A19: 1 <= k1 by A1, A13, XXREAL_0:2;
then A20: ((Gauge (C,n)) * (k1,i)) `2 = ((Gauge (C,n)) * (1,i)) `2 by A4, A5, A11, A16, GOBOARD5:1;
then A21: |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| = (Gauge (C,n)) * (k1,i) by A15, EUCLID:53;
then A22: ((Gauge (C,n)) * (j,i)) `1 <= |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 by A1, A4, A5, A11, A13, A16, SPRECT_3:13;
A23: ((Gauge (C,n)) * (j,i)) `2 = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `2 by A1, A4, A5, A11, A12, A20, A21, GOBOARD5:1;
reconsider pp = pp as Point of (TOP-REAL 2) by A18;
A24: pp in (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))) by A18, XBOOLE_0:def_4;
then A25: pp in L~ (Lower_Seq (C,n)) by XBOOLE_0:def_4;
A26: |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 = W-bound ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))) by A15, A21, SPRECT_1:43
.= (W-min ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))))) `1 by EUCLID:52
.= pp `1 by A18, PSCOMP_1:31 ;
pp in LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) by A24, XBOOLE_0:def_4;
then pp `2 = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `2 by A23, A17, SPPOL_1:40;
then A27: |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| in L~ (Lower_Seq (C,n)) by A25, A26, TOPREAL3:6;
for x being set holds
( x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,((Gauge (C,n)) * (j,i)))) /\ (L~ (Lower_Seq (C,n))) iff x = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| )
proof
let x be set ; ::_thesis: ( x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,((Gauge (C,n)) * (j,i)))) /\ (L~ (Lower_Seq (C,n))) iff x = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| )
thus ( x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,((Gauge (C,n)) * (j,i)))) /\ (L~ (Lower_Seq (C,n))) implies x = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| ) ::_thesis: ( x = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| implies x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,((Gauge (C,n)) * (j,i)))) /\ (L~ (Lower_Seq (C,n))) )
proof
reconsider EE = (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))) as compact Subset of (TOP-REAL 2) ;
assume A28: x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,((Gauge (C,n)) * (j,i)))) /\ (L~ (Lower_Seq (C,n))) ; ::_thesis: x = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|
then reconsider pp = x as Point of (TOP-REAL 2) ;
A29: pp in LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,((Gauge (C,n)) * (j,i))) by A28, XBOOLE_0:def_4;
then A30: pp `1 <= |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 by A22, TOPREAL1:3;
A31: |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 <= ((Gauge (C,n)) * (k,i)) `1 by A3, A4, A5, A14, A19, A21, SPRECT_3:13;
A32: ((Gauge (C,n)) * (j,i)) `1 <= |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 by A1, A4, A5, A11, A13, A16, A21, SPRECT_3:13;
A33: ((Gauge (C,n)) * (k,i)) `2 = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `2 by A3, A4, A5, A7, A20, A21, GOBOARD5:1;
reconsider E0 = proj1 .: EE as compact Subset of REAL by Th4;
A34: (Gauge (C,n)) * (j,i) in LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) by RLTOPSP1:68;
((Gauge (C,n)) * (j,i)) `2 = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `2 by A1, A4, A5, A11, A12, A20, A21, GOBOARD5:1;
then |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| in LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) by A33, A32, A31, GOBOARD7:8;
then A35: LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,((Gauge (C,n)) * (j,i))) c= LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) by A34, TOPREAL1:6;
pp in L~ (Lower_Seq (C,n)) by A28, XBOOLE_0:def_4;
then pp in EE by A29, A35, XBOOLE_0:def_4;
then proj1 . pp in E0 by FUNCT_2:35;
then A36: pp `1 in E0 by PSCOMP_1:def_5;
E0 is real-bounded by RCOMP_1:10;
then E0 is bounded_below by XXREAL_2:def_11;
then |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 <= pp `1 by A15, A21, A36, SEQ_4:def_2;
then A37: pp `1 = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 by A30, XXREAL_0:1;
pp `2 = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `2 by A23, A29, GOBOARD7:6;
hence x = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| by A37, TOPREAL3:6; ::_thesis: verum
end;
assume A38: x = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| ; ::_thesis: x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,((Gauge (C,n)) * (j,i)))) /\ (L~ (Lower_Seq (C,n)))
then x in LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,((Gauge (C,n)) * (j,i))) by RLTOPSP1:68;
hence x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,((Gauge (C,n)) * (j,i)))) /\ (L~ (Lower_Seq (C,n))) by A27, A38, XBOOLE_0:def_4; ::_thesis: verum
end;
hence (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (k1,i))} by A21, TARSKI:def_1; ::_thesis: verum
end;
theorem Th20: :: JORDAN15:20
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i, j, k being Element of NAT st 1 <= j & j <= k & k <= len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (k,i) in L~ (Lower_Seq (C,n)) holds
ex j1, k1 being Element of NAT st
( j <= j1 & j1 <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))} & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (k1,i))} )
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i, j, k being Element of NAT st 1 <= j & j <= k & k <= len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (k,i) in L~ (Lower_Seq (C,n)) holds
ex j1, k1 being Element of NAT st
( j <= j1 & j1 <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))} & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (k1,i))} )
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for i, j, k being Element of NAT st 1 <= j & j <= k & k <= len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (k,i) in L~ (Lower_Seq (C,n)) holds
ex j1, k1 being Element of NAT st
( j <= j1 & j1 <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))} & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (k1,i))} )
let i, j, k be Element of NAT ; ::_thesis: ( 1 <= j & j <= k & k <= len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (k,i) in L~ (Lower_Seq (C,n)) implies ex j1, k1 being Element of NAT st
( j <= j1 & j1 <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))} & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (k1,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: (Gauge (C,n)) * (j,i) in L~ (Upper_Seq (C,n)) and
A7: (Gauge (C,n)) * (k,i) in L~ (Lower_Seq (C,n)) ; ::_thesis: ex j1, k1 being Element of NAT st
( j <= j1 & j1 <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))} & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (k1,i))} )
set G = Gauge (C,n);
A8: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
then A9: j <= width (Gauge (C,n)) by A2, A3, XXREAL_0:2;
then A10: [j,i] in Indices (Gauge (C,n)) by A1, A4, A5, A8, MATRIX_1:36;
set s = ((Gauge (C,n)) * (1,i)) `2 ;
set e = (Gauge (C,n)) * (k,i);
set f = (Gauge (C,n)) * (j,i);
set w1 = lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))));
A11: (Gauge (C,n)) * (k,i) in LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) by RLTOPSP1:68;
then A12: LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets L~ (Lower_Seq (C,n)) by A7, XBOOLE_0:3;
A13: k >= 1 by A1, A2, XXREAL_0:2;
then [k,i] in Indices (Gauge (C,n)) by A3, A4, A5, MATRIX_1:36;
then consider k1 being Element of NAT such that
A14: j <= k1 and
A15: k1 <= k and
A16: ((Gauge (C,n)) * (k1,i)) `1 = lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))))) by A2, A12, A10, JORDAN1F:3, JORDAN1G:5;
A17: k1 <= width (Gauge (C,n)) by A3, A8, A15, XXREAL_0:2;
set p = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|;
set w2 = upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))));
set q = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|;
A18: (Gauge (C,n)) * (j,i) in LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k1,i))) by RLTOPSP1:68;
then A19: LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k1,i))) meets L~ (Upper_Seq (C,n)) by A6, XBOOLE_0:3;
A20: 1 <= k1 by A1, A14, XXREAL_0:2;
then A21: ((Gauge (C,n)) * (k1,i)) `2 = ((Gauge (C,n)) * (1,i)) `2 by A4, A5, A8, A17, GOBOARD5:1;
then A22: |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| = (Gauge (C,n)) * (k1,i) by A16, EUCLID:53;
((Gauge (C,n)) * (j,i)) `2 = ((Gauge (C,n)) * (1,i)) `2 by A1, A4, A5, A8, A9, GOBOARD5:1
.= ((Gauge (C,n)) * (k,i)) `2 by A3, A4, A5, A13, GOBOARD5:1 ;
then A23: LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) is horizontal by SPPOL_1:15;
set X = (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Upper_Seq (C,n)));
reconsider X1 = (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Upper_Seq (C,n))) as non empty compact Subset of (TOP-REAL 2) by A6, A18, XBOOLE_0:def_4;
consider pp being set such that
A24: pp in E-most X1 by XBOOLE_0:def_1;
[k1,i] in Indices (Gauge (C,n)) by A4, A5, A8, A20, A17, MATRIX_1:36;
then consider j1 being Element of NAT such that
A25: j <= j1 and
A26: j1 <= k1 and
A27: ((Gauge (C,n)) * (j1,i)) `1 = upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n))))) by A10, A14, A22, A19, JORDAN1F:4, JORDAN1G:4;
A28: j1 <= width (Gauge (C,n)) by A17, A26, XXREAL_0:2;
reconsider pp = pp as Point of (TOP-REAL 2) by A24;
A29: pp in (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Upper_Seq (C,n))) by A24, XBOOLE_0:def_4;
then A30: pp in L~ (Upper_Seq (C,n)) by XBOOLE_0:def_4;
take j1 ; ::_thesis: ex k1 being Element of NAT st
( j <= j1 & j1 <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))} & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (k1,i))} )
take k1 ; ::_thesis: ( j <= j1 & j1 <= k1 & k1 <= k & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))} & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (k1,i))} )
thus ( j <= j1 & j1 <= k1 & k1 <= k ) by A15, A25, A26; ::_thesis: ( (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))} & (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (k1,i))} )
A31: pp in LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k1,i))) by A29, XBOOLE_0:def_4;
A32: 1 <= j1 by A1, A25, XXREAL_0:2;
then A33: ((Gauge (C,n)) * (j1,i)) `2 = ((Gauge (C,n)) * (1,i)) `2 by A4, A5, A8, A28, GOBOARD5:1;
then A34: |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| = (Gauge (C,n)) * (j1,i) by A27, EUCLID:53;
then A35: |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 <= |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 by A4, A5, A8, A17, A22, A26, A32, SPRECT_3:13;
A36: |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 = E-bound ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Upper_Seq (C,n)))) by A22, A27, A34, SPRECT_1:46
.= (E-min ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Upper_Seq (C,n))))) `1 by EUCLID:52
.= pp `1 by A24, PSCOMP_1:47 ;
A37: ((Gauge (C,n)) * (j,i)) `2 = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `2 by A1, A4, A5, A8, A9, A21, A22, GOBOARD5:1;
then LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|) is horizontal by SPPOL_1:15;
then pp `2 = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `2 by A21, A22, A33, A34, A31, SPPOL_1:40;
then A38: |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| in L~ (Upper_Seq (C,n)) by A30, A36, TOPREAL3:6;
for x being set holds
( x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n))) iff x = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| )
proof
let x be set ; ::_thesis: ( x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n))) iff x = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| )
thus ( x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n))) implies x = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| ) ::_thesis: ( x = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| implies x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n))) )
proof
reconsider EE = (LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n))) as compact Subset of (TOP-REAL 2) ;
assume A39: x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n))) ; ::_thesis: x = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|
then reconsider pp = x as Point of (TOP-REAL 2) ;
A40: pp in LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|) by A39, XBOOLE_0:def_4;
then A41: pp `1 >= |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 by A35, TOPREAL1:3;
A42: ((Gauge (C,n)) * (j,i)) `1 <= |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 by A1, A4, A5, A8, A25, A28, A34, SPRECT_3:13;
reconsider E0 = proj1 .: EE as compact Subset of REAL by Th4;
A43: |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| in LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|) by RLTOPSP1:68;
((Gauge (C,n)) * (j,i)) `2 = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `2 by A1, A4, A5, A8, A9, A33, A34, GOBOARD5:1;
then |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| in LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,((Gauge (C,n)) * (j,i))) by A21, A22, A33, A34, A35, A42, GOBOARD7:8;
then A44: LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|) c= LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|) by A43, TOPREAL1:6;
pp in L~ (Upper_Seq (C,n)) by A39, XBOOLE_0:def_4;
then pp in EE by A40, A44, XBOOLE_0:def_4;
then proj1 . pp in E0 by FUNCT_2:35;
then A45: pp `1 in E0 by PSCOMP_1:def_5;
E0 is real-bounded by RCOMP_1:10;
then E0 is bounded_above by XXREAL_2:def_11;
then |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 >= pp `1 by A27, A34, A45, SEQ_4:def_1;
then A46: pp `1 = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 by A41, XXREAL_0:1;
pp `2 = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `2 by A21, A22, A33, A34, A40, GOBOARD7:6;
hence x = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| by A46, TOPREAL3:6; ::_thesis: verum
end;
assume A47: x = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| ; ::_thesis: x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))
then x in LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|) by RLTOPSP1:68;
hence x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n))) by A38, A47, XBOOLE_0:def_4; ::_thesis: verum
end;
hence (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))} by A22, A34, TARSKI:def_1; ::_thesis: (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (k1,i))}
set X = (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)));
reconsider X1 = (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))) as non empty compact Subset of (TOP-REAL 2) by A7, A11, XBOOLE_0:def_4;
consider pp being set such that
A48: pp in W-most X1 by XBOOLE_0:def_1;
reconsider pp = pp as Point of (TOP-REAL 2) by A48;
A49: pp in (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))) by A48, XBOOLE_0:def_4;
then A50: pp in L~ (Lower_Seq (C,n)) by XBOOLE_0:def_4;
pp in LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) by A49, XBOOLE_0:def_4;
then A51: pp `2 = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `2 by A37, A23, SPPOL_1:40;
|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 = W-bound ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))) by A16, A22, SPRECT_1:43
.= (W-min ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))))) `1 by EUCLID:52
.= pp `1 by A48, PSCOMP_1:31 ;
then A52: |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| in L~ (Lower_Seq (C,n)) by A50, A51, TOPREAL3:6;
for x being set holds
( x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n))) iff x = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| )
proof
let x be set ; ::_thesis: ( x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n))) iff x = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| )
thus ( x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n))) implies x = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| ) ::_thesis: ( x = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| implies x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n))) )
proof
j1 <= k by A15, A26, XXREAL_0:2;
then A53: |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 <= ((Gauge (C,n)) * (k,i)) `1 by A3, A4, A5, A32, A34, SPRECT_3:13;
A54: ((Gauge (C,n)) * (k,i)) `2 = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `2 by A3, A4, A5, A13, A21, A22, GOBOARD5:1;
A55: ((Gauge (C,n)) * (j,i)) `1 <= |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 by A1, A4, A5, A8, A14, A17, A22, SPRECT_3:13;
A56: ((Gauge (C,n)) * (j,i)) `1 <= |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 by A1, A4, A5, A8, A25, A28, A34, SPRECT_3:13;
A57: |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 <= ((Gauge (C,n)) * (k,i)) `1 by A3, A4, A5, A15, A20, A22, SPRECT_3:13;
((Gauge (C,n)) * (j,i)) `2 = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `2 by A1, A4, A5, A8, A9, A21, A22, GOBOARD5:1;
then A58: |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| in LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) by A54, A55, A57, GOBOARD7:8;
A59: ((Gauge (C,n)) * (k,i)) `2 = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `2 by A3, A4, A5, A13, A33, A34, GOBOARD5:1;
((Gauge (C,n)) * (j,i)) `2 = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `2 by A1, A4, A5, A8, A9, A33, A34, GOBOARD5:1;
then |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| in LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) by A59, A56, A53, GOBOARD7:8;
then A60: LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|) c= LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) by A58, TOPREAL1:6;
reconsider EE = (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))) as compact Subset of (TOP-REAL 2) ;
reconsider E0 = proj1 .: EE as compact Subset of REAL by Th4;
assume A61: x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n))) ; ::_thesis: x = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|
then reconsider pp = x as Point of (TOP-REAL 2) ;
A62: pp in LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|) by A61, XBOOLE_0:def_4;
then A63: pp `1 <= |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 by A35, TOPREAL1:3;
pp in L~ (Lower_Seq (C,n)) by A61, XBOOLE_0:def_4;
then pp in EE by A62, A60, XBOOLE_0:def_4;
then proj1 . pp in E0 by FUNCT_2:35;
then A64: pp `1 in E0 by PSCOMP_1:def_5;
E0 is real-bounded by RCOMP_1:10;
then E0 is bounded_below by XXREAL_2:def_11;
then |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 <= pp `1 by A16, A22, A64, SEQ_4:def_2;
then A65: pp `1 = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `1 by A63, XXREAL_0:1;
pp `2 = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| `2 by A21, A22, A33, A34, A62, GOBOARD7:6;
hence x = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| by A65, TOPREAL3:6; ::_thesis: verum
end;
assume A66: x = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]| ; ::_thesis: x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n)))
then x in LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|) by RLTOPSP1:68;
hence x in (LSeg (|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|,|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Upper_Seq (C,n)))))),(((Gauge (C,n)) * (1,i)) `2)]|)) /\ (L~ (Lower_Seq (C,n))) by A52, A66, XBOOLE_0:def_4; ::_thesis: verum
end;
hence (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (k1,i))} by A22, A34, TARSKI:def_1; ::_thesis: verum
end;
theorem Th21: :: JORDAN15:21
for n being Element of NAT
for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C
let C be Simple_closed_curve; ::_thesis: for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C
let i, j, k be Element of NAT ; ::_thesis: ( 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) implies LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C )
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: (Gauge (C,n)) * (i,k) in L~ (Upper_Seq (C,n)) and
A7: (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) ; ::_thesis: LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C
consider j1, k1 being Element of NAT such that
A8: j <= j1 and
A9: j1 <= k1 and
A10: k1 <= k and
A11: (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j1))} and
A12: (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))} by A1, A2, A3, A4, A5, A6, A7, Th11;
A13: k1 <= width (Gauge (C,n)) by A5, A10, XXREAL_0:2;
1 <= j1 by A3, A8, XXREAL_0:2;
then LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1))) meets Lower_Arc C by A1, A2, A9, A11, A12, A13, JORDAN1J:58;
hence LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C by A1, A2, A3, A5, A8, A9, A10, Th5, XBOOLE_1:63; ::_thesis: verum
end;
theorem Th22: :: JORDAN15:22
for n being Element of NAT
for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C
let C be Simple_closed_curve; ::_thesis: for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C
let i, j, k be Element of NAT ; ::_thesis: ( 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) implies LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C )
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: (Gauge (C,n)) * (i,k) in L~ (Upper_Seq (C,n)) and
A7: (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) ; ::_thesis: LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C
consider j1, k1 being Element of NAT such that
A8: j <= j1 and
A9: j1 <= k1 and
A10: k1 <= k and
A11: (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j1))} and
A12: (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))} by A1, A2, A3, A4, A5, A6, A7, Th11;
A13: k1 <= width (Gauge (C,n)) by A5, A10, XXREAL_0:2;
1 <= j1 by A3, A8, XXREAL_0:2;
then LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1))) meets Upper_Arc C by A1, A2, A9, A11, A12, A13, JORDAN1J:59;
hence LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C by A1, A2, A3, A5, A8, A9, A10, Th5, XBOOLE_1:63; ::_thesis: verum
end;
theorem Th23: :: JORDAN15:23
for n being Element of NAT
for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (Gauge (C,n)) * (i,k) in Upper_Arc (L~ (Cage (C,n))) & (Gauge (C,n)) * (i,j) in Lower_Arc (L~ (Cage (C,n))) holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (Gauge (C,n)) * (i,k) in Upper_Arc (L~ (Cage (C,n))) & (Gauge (C,n)) * (i,j) in Lower_Arc (L~ (Cage (C,n))) holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C
let C be Simple_closed_curve; ::_thesis: for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (Gauge (C,n)) * (i,k) in Upper_Arc (L~ (Cage (C,n))) & (Gauge (C,n)) * (i,j) in Lower_Arc (L~ (Cage (C,n))) holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C
let i, j, k be Element of NAT ; ::_thesis: ( 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (Gauge (C,n)) * (i,k) in Upper_Arc (L~ (Cage (C,n))) & (Gauge (C,n)) * (i,j) in Lower_Arc (L~ (Cage (C,n))) implies LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C )
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: n > 0 and
A7: (Gauge (C,n)) * (i,k) in Upper_Arc (L~ (Cage (C,n))) and
A8: (Gauge (C,n)) * (i,j) in Lower_Arc (L~ (Cage (C,n))) ; ::_thesis: LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C
A9: L~ (Lower_Seq (C,n)) = Lower_Arc (L~ (Cage (C,n))) by A6, JORDAN1G:56;
L~ (Upper_Seq (C,n)) = Upper_Arc (L~ (Cage (C,n))) by A6, JORDAN1G:55;
hence LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C by A1, A2, A3, A4, A5, A7, A8, A9, Th21; ::_thesis: verum
end;
theorem Th24: :: JORDAN15:24
for n being Element of NAT
for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (Gauge (C,n)) * (i,k) in Upper_Arc (L~ (Cage (C,n))) & (Gauge (C,n)) * (i,j) in Lower_Arc (L~ (Cage (C,n))) holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (Gauge (C,n)) * (i,k) in Upper_Arc (L~ (Cage (C,n))) & (Gauge (C,n)) * (i,j) in Lower_Arc (L~ (Cage (C,n))) holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C
let C be Simple_closed_curve; ::_thesis: for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (Gauge (C,n)) * (i,k) in Upper_Arc (L~ (Cage (C,n))) & (Gauge (C,n)) * (i,j) in Lower_Arc (L~ (Cage (C,n))) holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C
let i, j, k be Element of NAT ; ::_thesis: ( 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (Gauge (C,n)) * (i,k) in Upper_Arc (L~ (Cage (C,n))) & (Gauge (C,n)) * (i,j) in Lower_Arc (L~ (Cage (C,n))) implies LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C )
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: n > 0 and
A7: (Gauge (C,n)) * (i,k) in Upper_Arc (L~ (Cage (C,n))) and
A8: (Gauge (C,n)) * (i,j) in Lower_Arc (L~ (Cage (C,n))) ; ::_thesis: LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C
A9: L~ (Lower_Seq (C,n)) = Lower_Arc (L~ (Cage (C,n))) by A6, JORDAN1G:56;
L~ (Upper_Seq (C,n)) = Upper_Arc (L~ (Cage (C,n))) by A6, JORDAN1G:55;
hence LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C by A1, A2, A3, A4, A5, A7, A8, A9, Th22; ::_thesis: verum
end;
theorem :: JORDAN15:25
for n being Element of NAT
for C being Simple_closed_curve
for j, k being Element of NAT st 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j) in Lower_Arc (L~ (Cage (C,(n + 1)))) holds
LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))) meets Lower_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for j, k being Element of NAT st 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j) in Lower_Arc (L~ (Cage (C,(n + 1)))) holds
LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))) meets Lower_Arc C
let C be Simple_closed_curve; ::_thesis: for j, k being Element of NAT st 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j) in Lower_Arc (L~ (Cage (C,(n + 1)))) holds
LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))) meets Lower_Arc C
let j, k be Element of NAT ; ::_thesis: ( 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j) in Lower_Arc (L~ (Cage (C,(n + 1)))) implies LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))) meets Lower_Arc C )
assume that
A1: 1 <= j and
A2: j <= k and
A3: k <= width (Gauge (C,(n + 1))) and
A4: (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k) in Upper_Arc (L~ (Cage (C,(n + 1)))) and
A5: (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j) in Lower_Arc (L~ (Cage (C,(n + 1)))) ; ::_thesis: LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))) meets Lower_Arc C
A6: len (Gauge (C,(n + 1))) >= 4 by JORDAN8:10;
then len (Gauge (C,(n + 1))) >= 2 by XXREAL_0:2;
then A7: 1 < Center (Gauge (C,(n + 1))) by JORDAN1B:14;
len (Gauge (C,(n + 1))) >= 3 by A6, XXREAL_0:2;
hence LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))) meets Lower_Arc C by A1, A2, A3, A4, A5, A7, Th23, JORDAN1B:15; ::_thesis: verum
end;
theorem :: JORDAN15:26
for n being Element of NAT
for C being Simple_closed_curve
for j, k being Element of NAT st 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j) in Lower_Arc (L~ (Cage (C,(n + 1)))) holds
LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))) meets Upper_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for j, k being Element of NAT st 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j) in Lower_Arc (L~ (Cage (C,(n + 1)))) holds
LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))) meets Upper_Arc C
let C be Simple_closed_curve; ::_thesis: for j, k being Element of NAT st 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j) in Lower_Arc (L~ (Cage (C,(n + 1)))) holds
LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))) meets Upper_Arc C
let j, k be Element of NAT ; ::_thesis: ( 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j) in Lower_Arc (L~ (Cage (C,(n + 1)))) implies LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))) meets Upper_Arc C )
assume that
A1: 1 <= j and
A2: j <= k and
A3: k <= width (Gauge (C,(n + 1))) and
A4: (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k) in Upper_Arc (L~ (Cage (C,(n + 1)))) and
A5: (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j) in Lower_Arc (L~ (Cage (C,(n + 1)))) ; ::_thesis: LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))) meets Upper_Arc C
A6: len (Gauge (C,(n + 1))) >= 4 by JORDAN8:10;
then len (Gauge (C,(n + 1))) >= 2 by XXREAL_0:2;
then A7: 1 < Center (Gauge (C,(n + 1))) by JORDAN1B:14;
len (Gauge (C,(n + 1))) >= 3 by A6, XXREAL_0:2;
hence LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))) meets Upper_Arc C by A1, A2, A3, A4, A5, A7, Th24, JORDAN1B:15; ::_thesis: verum
end;
theorem Th27: :: JORDAN15:27
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i, j, k being Element of NAT st 1 < j & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (k,i) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Lower_Seq (C,n)) holds
j <> k
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i, j, k being Element of NAT st 1 < j & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (k,i) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Lower_Seq (C,n)) holds
j <> k
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for i, j, k being Element of NAT st 1 < j & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (k,i) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Lower_Seq (C,n)) holds
j <> k
let i, j, k be Element of NAT ; ::_thesis: ( 1 < j & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (k,i) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Lower_Seq (C,n)) implies j <> k )
assume that
A1: 1 < j and
A2: k < len (Gauge (C,n)) and
A3: 1 <= i and
A4: i <= width (Gauge (C,n)) and
A5: (Gauge (C,n)) * (k,i) in L~ (Upper_Seq (C,n)) and
A6: (Gauge (C,n)) * (j,i) in L~ (Lower_Seq (C,n)) and
A7: j = k ; ::_thesis: contradiction
A8: [j,i] in Indices (Gauge (C,n)) by A1, A2, A3, A4, A7, MATRIX_1:36;
(Gauge (C,n)) * (k,i) in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A5, A6, A7, XBOOLE_0:def_4;
then A9: (Gauge (C,n)) * (k,i) in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16;
A10: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
len (Gauge (C,n)) >= 4 by JORDAN8:10;
then A11: len (Gauge (C,n)) >= 1 by XXREAL_0:2;
then A12: [(len (Gauge (C,n))),i] in Indices (Gauge (C,n)) by A3, A4, MATRIX_1:36;
A13: [1,i] in Indices (Gauge (C,n)) by A3, A4, A11, MATRIX_1:36;
percases ( (Gauge (C,n)) * (k,i) = W-min (L~ (Cage (C,n))) or (Gauge (C,n)) * (k,i) = E-max (L~ (Cage (C,n))) ) by A9, TARSKI:def_2;
supposeA14: (Gauge (C,n)) * (k,i) = W-min (L~ (Cage (C,n))) ; ::_thesis: contradiction
((Gauge (C,n)) * (1,i)) `1 = W-bound (L~ (Cage (C,n))) by A3, A4, A10, JORDAN1A:73;
then (W-min (L~ (Cage (C,n)))) `1 <> W-bound (L~ (Cage (C,n))) by A1, A7, A8, A13, A14, JORDAN1G:7;
hence contradiction by EUCLID:52; ::_thesis: verum
end;
supposeA15: (Gauge (C,n)) * (k,i) = E-max (L~ (Cage (C,n))) ; ::_thesis: contradiction
((Gauge (C,n)) * ((len (Gauge (C,n))),i)) `1 = E-bound (L~ (Cage (C,n))) by A3, A4, A10, JORDAN1A:71;
then (E-max (L~ (Cage (C,n)))) `1 <> E-bound (L~ (Cage (C,n))) by A2, A7, A8, A12, A15, JORDAN1G:7;
hence contradiction by EUCLID:52; ::_thesis: verum
end;
end;
end;
theorem Th28: :: JORDAN15:28
for n being Element of NAT
for C being Simple_closed_curve
for i, j, k being Element of 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)) * (k,i))} & (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (j,i))} holds
LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Lower_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i, j, k being Element of 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)) * (k,i))} & (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (j,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 Element of 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)) * (k,i))} & (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (j,i))} holds
LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Lower_Arc C
let i, j, k be Element of 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)) * (k,i))} & (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (j,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)) * (k,i))} and
A7: (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (j,i))} and
A8: LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) misses Lower_Arc C ; ::_thesis: contradiction
(Gauge (C,n)) * (j,i) in {((Gauge (C,n)) * (j,i))} by TARSKI:def_1;
then A9: (Gauge (C,n)) * (j,i) in L~ (Lower_Seq (C,n)) by A7, XBOOLE_0:def_4;
(Gauge (C,n)) * (k,i) in {((Gauge (C,n)) * (k,i))} by TARSKI:def_1;
then A10: (Gauge (C,n)) * (k,i) in L~ (Upper_Seq (C,n)) by A6, XBOOLE_0:def_4;
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 A2, A3, A11, XXREAL_0:2;
A14: 1 <= k by A1, A2, XXREAL_0:2;
A15: k <= width (Gauge (C,n)) by A3, JORDAN8:def_1;
A16: [j,i] in Indices (Gauge (C,n)) by A1, A4, A5, A11, A13, MATRIX_1:36;
A17: [k,i] in Indices (Gauge (C,n)) by A3, A4, A5, A14, MATRIX_1:36;
set go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (k,i)));
set do = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (j,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,k)) `1 by A3, A14, JORDAN1A:73 ;
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 A14, A15, MATRIX_1:36;
then A23: (Gauge (C,n)) * (k,i) <> (Upper_Seq (C,n)) . 1 by A1, A2, A17, A19, A20, JORDAN1G:7;
then reconsider go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (k,i))) as being_S-Seq FinSequence of (TOP-REAL 2) by A10, JORDAN3:35;
A24: [1,j] in Indices (Gauge (C,n)) by A1, A13, A21, MATRIX_1:36;
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 A26, FINSEQ_3:25;
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 ;
A29: (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * (1,k)) `1 by A3, A14, JORDAN1A:73 ;
A30: [j,i] in Indices (Gauge (C,n)) by A1, A4, A5, A11, A13, MATRIX_1:36;
then A31: (Gauge (C,n)) * (j,i) <> (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) by A1, A22, A28, A29, JORDAN1G:7;
then reconsider do = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (j,i))) as being_S-Seq FinSequence of (TOP-REAL 2) by A9, JORDAN3:34;
A32: [(len (Gauge (C,n))),k] in Indices (Gauge (C,n)) by A14, A15, A21, MATRIX_1:36;
A33: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by A27, 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 A3, A14, JORDAN1A:71 ;
then A34: (Gauge (C,n)) * (j,i) <> (Lower_Seq (C,n)) . 1 by A2, A3, A30, A32, A33, JORDAN1G:7;
A35: len go >= 1 + 1 by TOPREAL1:def_8;
A36: (Gauge (C,n)) * (k,i) in rng (Upper_Seq (C,n)) by A4, A5, A10, A11, A14, A15, JORDAN1G:4, JORDAN1J:40;
then A37: go is_sequence_on Gauge (C,n) by JORDAN1G:4, JORDAN1J:38;
A38: len do >= 1 + 1 by TOPREAL1:def_8;
A39: (Gauge (C,n)) * (j,i) in rng (Lower_Seq (C,n)) by A1, A4, A5, A9, A11, A13, JORDAN1G:5, JORDAN1J:40;
then A40: do is_sequence_on Gauge (C,n) by JORDAN1G:5, JORDAN1J:39;
reconsider go = go as non constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A35, A37, JGRAPH_1:12, JORDAN8:5;
reconsider do = do as non constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A38, A40, JGRAPH_1:12, JORDAN8:5;
A41: len go > 1 by A35, NAT_1:13;
then A42: len go in dom go by FINSEQ_3:25;
then A43: go /. (len go) = go . (len go) by PARTFUN1:def_6
.= (Gauge (C,n)) * (k,i) by A10, JORDAN3:24 ;
len do >= 1 by A38, XXREAL_0:2;
then 1 in dom do by FINSEQ_3:25;
then A44: do /. 1 = do . 1 by PARTFUN1:def_6
.= (Gauge (C,n)) * (j,i) by A9, JORDAN3:23 ;
reconsider m = (len go) - 1 as Element of NAT by A42, FINSEQ_3:26;
A45: m + 1 = len go ;
then A46: (len go) -' 1 = m by NAT_D:34;
A47: LSeg (go,m) c= L~ go by TOPREAL3:19;
A48: L~ go c= L~ (Upper_Seq (C,n)) by A10, JORDAN3:41;
then LSeg (go,m) c= L~ (Upper_Seq (C,n)) by A47, XBOOLE_1:1;
then A49: (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i)))) c= {((Gauge (C,n)) * (k,i))} by A6, XBOOLE_1:26;
m >= 1 by A35, XREAL_1:19;
then A50: LSeg (go,m) = LSeg ((go /. m),((Gauge (C,n)) * (k,i))) by A43, A45, TOPREAL1:def_3;
{((Gauge (C,n)) * (k,i))} c= (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (k,i))} or x in (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i)))) )
A51: (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 (go,m)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))))
then A52: x = (Gauge (C,n)) * (k,i) by TARSKI:def_1;
(Gauge (C,n)) * (k,i) in LSeg (go,m) by A50, RLTOPSP1:68;
hence x in (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i)))) by A52, A51, XBOOLE_0:def_4; ::_thesis: verum
end;
then A53: (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i)))) = {((Gauge (C,n)) * (k,i))} by A49, XBOOLE_0:def_10;
A54: LSeg (do,1) c= L~ do by TOPREAL3:19;
A55: L~ do c= L~ (Lower_Seq (C,n)) by A9, JORDAN3:42;
then LSeg (do,1) c= L~ (Lower_Seq (C,n)) by A54, XBOOLE_1:1;
then A56: (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i)))) c= {((Gauge (C,n)) * (j,i))} by A7, XBOOLE_1:26;
A57: LSeg (do,1) = LSeg (((Gauge (C,n)) * (j,i)),(do /. (1 + 1))) by A38, A44, TOPREAL1:def_3;
{((Gauge (C,n)) * (j,i))} c= (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (j,i))} or x in (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i)))) )
A58: (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 (do,1)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))))
then A59: x = (Gauge (C,n)) * (j,i) by TARSKI:def_1;
(Gauge (C,n)) * (j,i) in LSeg (do,1) by A57, RLTOPSP1:68;
hence x in (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i)))) by A59, A58, XBOOLE_0:def_4; ::_thesis: verum
end;
then A60: (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i)))) /\ (LSeg (do,1)) = {((Gauge (C,n)) * (j,i))} by A56, XBOOLE_0:def_10;
A61: go /. 1 = (Upper_Seq (C,n)) /. 1 by A10, SPRECT_3:22
.= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ;
then A62: go /. 1 = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by JORDAN1F:8
.= do /. (len do) by A9, JORDAN1J:35 ;
A63: rng go c= L~ go by A35, SPPOL_2:18;
A64: rng do c= L~ do by A38, SPPOL_2:18;
A65: {(go /. 1)} c= (L~ go) /\ (L~ do)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(go /. 1)} or x in (L~ go) /\ (L~ do) )
assume x in {(go /. 1)} ; ::_thesis: x in (L~ go) /\ (L~ do)
then A66: x = go /. 1 by TARSKI:def_1;
then A67: x in rng go by FINSEQ_6:42;
x in rng do by A62, A66, REVROT_1:3;
hence x in (L~ go) /\ (L~ do) by A63, A64, A67, XBOOLE_0:def_4; ::_thesis: verum
end;
A68: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by A27, PARTFUN1:def_6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;
A69: [(len (Gauge (C,n))),j] in Indices (Gauge (C,n)) by A1, A13, A21, MATRIX_1:36;
(L~ go) /\ (L~ do) c= {(go /. 1)}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ go) /\ (L~ do) or x in {(go /. 1)} )
assume A70: x in (L~ go) /\ (L~ do) ; ::_thesis: x in {(go /. 1)}
then A71: x in L~ do by XBOOLE_0:def_4;
A72: now__::_thesis:_not_x_=_E-max_(L~_(Cage_(C,n)))
assume x = E-max (L~ (Cage (C,n))) ; ::_thesis: contradiction
then A73: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (j,i) by A9, A68, A71, JORDAN1E:7;
((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 = E-bound (L~ (Cage (C,n))) by A1, A11, A13, JORDAN1A:71;
then (E-max (L~ (Cage (C,n)))) `1 <> E-bound (L~ (Cage (C,n))) by A2, A3, A16, A69, A73, JORDAN1G:7;
hence contradiction by EUCLID:52; ::_thesis: verum
end;
x in L~ go by A70, XBOOLE_0:def_4;
then x in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A48, A55, A71, 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 A61, A72, TARSKI:def_1; ::_thesis: verum
end;
then A74: (L~ go) /\ (L~ do) = {(go /. 1)} by A65, XBOOLE_0:def_10;
set W2 = go /. 2;
A75: 2 in dom go by A35, FINSEQ_3:25;
A76: 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,j)) `1 = ((Gauge (C,n)) * (j,i)) `1 by A1, A11, A13, JORDAN1A:73;
hence contradiction by A1, A16, A24, JORDAN1G:7; ::_thesis: verum
end;
go = mid ((Upper_Seq (C,n)),1,(((Gauge (C,n)) * (k,i)) .. (Upper_Seq (C,n)))) by A36, JORDAN1G:49
.= (Upper_Seq (C,n)) | (((Gauge (C,n)) * (k,i)) .. (Upper_Seq (C,n))) by A36, FINSEQ_4:21, FINSEQ_6:116 ;
then A77: go /. 2 = (Upper_Seq (C,n)) /. 2 by A75, FINSEQ_4:70;
A78: W-min (L~ (Cage (C,n))) in rng go by A61, FINSEQ_6:42;
set pion = <*((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))*>;
A79: now__::_thesis:_for_n_being_Element_of_NAT_st_n_in_dom_<*((Gauge_(C,n))_*_(k,i)),((Gauge_(C,n))_*_(j,i))*>_holds_
ex_j,_i_being_Element_of_NAT_st_
(_[j,i]_in_Indices_(Gauge_(C,n))_&_<*((Gauge_(C,n))_*_(k,i)),((Gauge_(C,n))_*_(j,i))*>_/._n_=_(Gauge_(C,n))_*_(j,i)_)
let n be Element of NAT ; ::_thesis: ( n in dom <*((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))*> implies ex j, i being Element of NAT st
( [j,i] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))*> /. n = (Gauge (C,n)) * (j,i) ) )
assume n in dom <*((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))*> ; ::_thesis: ex j, i being Element of NAT st
( [j,i] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))*> /. n = (Gauge (C,n)) * (j,i) )
then n in {1,2} by FINSEQ_1:2, FINSEQ_1:89;
then ( n = 1 or n = 2 ) by TARSKI:def_2;
hence ex j, i being Element of NAT st
( [j,i] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))*> /. n = (Gauge (C,n)) * (j,i) ) by A16, A17, FINSEQ_4:17; ::_thesis: verum
end;
A80: (Gauge (C,n)) * (k,i) <> (Gauge (C,n)) * (j,i) by A12, A16, A17, GOBOARD1:5;
((Gauge (C,n)) * (k,i)) `2 = ((Gauge (C,n)) * (1,i)) `2 by A3, A4, A5, A14, GOBOARD5:1
.= ((Gauge (C,n)) * (j,i)) `2 by A1, A4, A5, A11, A13, GOBOARD5:1 ;
then LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))) is horizontal by SPPOL_1:15;
then <*((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))*> is being_S-Seq by A80, JORDAN1B:8;
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)) * (k,i)),((Gauge (C,n)) * (j,i))*> = L~ pion1 and
A84: <*((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))*> /. 1 = pion1 /. 1 and
A85: <*((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))*> /. (len <*((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))*>) = pion1 /. (len pion1) and
A86: len <*((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))*> <= len pion1 by A79, GOBOARD3:2;
reconsider pion1 = pion1 as being_S-Seq FinSequence of (TOP-REAL 2) by A82;
set godo = (go ^' pion1) ^' do;
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 A35, XXREAL_0:2;
then A90: len (go ^' pion1) > 1 + 0 by NAT_1:13;
A91: len ((go ^' pion1) ^' do) >= len (go ^' pion1) by TOPREAL8:7;
then A92: 1 + 1 <= len ((go ^' pion1) ^' do) 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 A43, A84, FINSEQ_4:17;
then A95: go ^' pion1 is_sequence_on Gauge (C,n) by A37, A81, TOPREAL8:12;
A96: (go ^' pion1) /. (len (go ^' pion1)) = <*((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))*> /. (len <*((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))*>) by A85, GRAPH_2:54
.= <*((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))*> /. 2 by FINSEQ_1:44
.= do /. 1 by A44, FINSEQ_4:17 ;
then A97: (go ^' pion1) ^' do is_sequence_on Gauge (C,n) by A40, A95, TOPREAL8:12;
LSeg (pion1,1) c= L~ <*((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))*> by A83, TOPREAL3:19;
then LSeg (pion1,1) c= LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))) by SPPOL_2:21;
then A98: (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) c= {((Gauge (C,n)) * (k,i))} by A46, A53, XBOOLE_1:27;
A99: len pion1 >= 1 + 1 by A86, FINSEQ_1:44;
{((Gauge (C,n)) * (k,i))} c= (LSeg (go,m)) /\ (LSeg (pion1,1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (k,i))} or x in (LSeg (go,m)) /\ (LSeg (pion1,1)) )
assume x in {((Gauge (C,n)) * (k,i))} ; ::_thesis: x in (LSeg (go,m)) /\ (LSeg (pion1,1))
then A100: x = (Gauge (C,n)) * (k,i) by TARSKI:def_1;
A101: (Gauge (C,n)) * (k,i) in LSeg (go,m) by A50, RLTOPSP1:68;
(Gauge (C,n)) * (k,i) in LSeg (pion1,1) by A43, A94, A99, TOPREAL1:21;
hence x in (LSeg (go,m)) /\ (LSeg (pion1,1)) by A100, A101, XBOOLE_0:def_4; ::_thesis: verum
end;
then (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) = {(go /. (len go))} by A43, A46, A98, XBOOLE_0:def_10;
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 GRAPH_2:13;
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)) * (k,i)),((Gauge (C,n)) * (j,i))*> by A83, TOPREAL3:19;
then LSeg (pion1,((len pion1) -' 1)) c= LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))) by SPPOL_2:21;
then A109: (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) c= {((Gauge (C,n)) * (j,i))} by A60, XBOOLE_1:27;
{((Gauge (C,n)) * (j,i))} c= (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (j,i))} or x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) )
assume x in {((Gauge (C,n)) * (j,i))} ; ::_thesis: x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1))
then A110: x = (Gauge (C,n)) * (j,i) by TARSKI:def_1;
pion1 /. (((len pion1) -' 1) + 1) = <*((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))*> /. 2 by A85, A106, FINSEQ_1:44
.= (Gauge (C,n)) * (j,i) by FINSEQ_4:17 ;
then A111: (Gauge (C,n)) * (j,i) in LSeg (pion1,((len pion1) -' 1)) by A105, A106, TOPREAL1:21;
(Gauge (C,n)) * (j,i) in LSeg (do,1) by A57, RLTOPSP1:68;
hence x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) by A110, A111, XBOOLE_0:def_4; ::_thesis: verum
end;
then (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) = {((Gauge (C,n)) * (j,i))} by A109, XBOOLE_0:def_10;
then A112: (LSeg ((go ^' pion1),((len go) + ((len pion1) -' 2)))) /\ (LSeg (do,1)) = {((go ^' pion1) /. (len (go ^' pion1)))} by A44, 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 set ; :: 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 A116: x = pion1 /. 1 by TARSKI:def_1;
then A117: x in rng pion1 by FINSEQ_6:42;
x in rng go by A94, A116, REVROT_1:3;
hence x in (L~ go) /\ (L~ pion1) by A63, A114, A117, XBOOLE_0:def_4; ::_thesis: verum
end;
(L~ go) /\ (L~ pion1) c= {(pion1 /. 1)}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ go) /\ (L~ pion1) or x in {(pion1 /. 1)} )
assume A118: x in (L~ go) /\ (L~ pion1) ; ::_thesis: x in {(pion1 /. 1)}
then A119: x in L~ pion1 by XBOOLE_0:def_4;
x in L~ go by A118, XBOOLE_0:def_4;
then x in (L~ pion1) /\ (L~ (Upper_Seq (C,n))) by A48, A119, XBOOLE_0:def_4;
hence x in {(pion1 /. 1)} by A6, A43, A83, A94, SPPOL_2:21; ::_thesis: verum
end;
then A120: (L~ go) /\ (L~ pion1) = {(pion1 /. 1)} by A115, XBOOLE_0:def_10;
then A121: go ^' pion1 is s.n.c. by A94, JORDAN1J:54;
(rng go) /\ (rng pion1) c= {(pion1 /. 1)} by A63, A114, A120, XBOOLE_1:27;
then A122: go ^' pion1 is one-to-one by JORDAN1J:55;
A123: <*((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))*> /. (len <*((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))*>) = <*((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))*> /. 2 by FINSEQ_1:44
.= do /. 1 by A44, FINSEQ_4:17 ;
A124: {(pion1 /. (len pion1))} c= (L~ do) /\ (L~ pion1)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(pion1 /. (len pion1))} or x in (L~ do) /\ (L~ pion1) )
assume x in {(pion1 /. (len pion1))} ; ::_thesis: x in (L~ do) /\ (L~ pion1)
then A125: x = pion1 /. (len pion1) by TARSKI:def_1;
then A126: x in rng pion1 by REVROT_1:3;
x in rng do by A85, A123, A125, FINSEQ_6:42;
hence x in (L~ do) /\ (L~ pion1) by A64, A114, A126, XBOOLE_0:def_4; ::_thesis: verum
end;
(L~ do) /\ (L~ pion1) c= {(pion1 /. (len pion1))}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ do) /\ (L~ pion1) or x in {(pion1 /. (len pion1))} )
assume A127: x in (L~ do) /\ (L~ pion1) ; ::_thesis: x in {(pion1 /. (len pion1))}
then A128: x in L~ pion1 by XBOOLE_0:def_4;
x in L~ do by A127, XBOOLE_0:def_4;
then x in (L~ pion1) /\ (L~ (Lower_Seq (C,n))) by A55, A128, XBOOLE_0:def_4;
hence x in {(pion1 /. (len pion1))} by A7, A44, A83, A85, A123, SPPOL_2:21; ::_thesis: verum
end;
then A129: (L~ do) /\ (L~ pion1) = {(pion1 /. (len pion1))} by A124, XBOOLE_0:def_10;
A130: (L~ (go ^' pion1)) /\ (L~ do) = ((L~ go) \/ (L~ pion1)) /\ (L~ do) by A94, TOPREAL8:35
.= {(go /. 1)} \/ {(do /. 1)} by A74, A85, A123, A129, XBOOLE_1:23
.= {((go ^' pion1) /. 1)} \/ {(do /. 1)} by GRAPH_2:53
.= {((go ^' pion1) /. 1),(do /. 1)} by ENUMSET1:1 ;
do /. (len do) = (go ^' pion1) /. 1 by A62, GRAPH_2:53;
then reconsider godo = (go ^' pion1) ^' do as non constant standard special_circular_sequence by A92, A96, A97, A102, A104, A112, A113, A121, A122, A130, JORDAN8:4, JORDAN8:5, TOPREAL8:11, TOPREAL8:33, TOPREAL8:34;
A131: Lower_Arc C is_an_arc_of E-max C, W-min C by JORDAN6:def_9;
then A132: Lower_Arc C is connected by JORDAN6:10;
A133: W-min C in Lower_Arc C by A131, TOPREAL1:1;
A134: E-max C in Lower_Arc C by A131, 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 A135: (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 = W-min (L~ (Cage (C,n))) by FINSEQ_6:92;
A136: 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 A135, 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 A135, A136, 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 A135, A136, SPRECT_5:24, XXREAL_0:2;
then A137: (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A135, A136, SPRECT_5:25, XXREAL_0:2;
A138: now__::_thesis:_not_((Gauge_(C,n))_*_(k,i))_.._(Upper_Seq_(C,n))_<=_1
assume A139: ((Gauge (C,n)) * (k,i)) .. (Upper_Seq (C,n)) <= 1 ; ::_thesis: contradiction
((Gauge (C,n)) * (k,i)) .. (Upper_Seq (C,n)) >= 1 by A36, FINSEQ_4:21;
then ((Gauge (C,n)) * (k,i)) .. (Upper_Seq (C,n)) = 1 by A139, XXREAL_0:1;
then (Gauge (C,n)) * (k,i) = (Upper_Seq (C,n)) /. 1 by A36, FINSEQ_5:38;
hence contradiction by A19, A23, JORDAN1F:5; ::_thesis: verum
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~ do) by A96, TOPREAL8:35
.= ((L~ go) \/ (L~ pion1)) \/ (L~ do) 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 A48, A145, XBOOLE_1:1;
A148: L~ do c= L~ (Cage (C,n)) by A55, A146, XBOOLE_1:1;
A149: W-min C in C by SPRECT_1:13;
A150: L~ <*((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))*> = LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))) by SPPOL_2:21;
A151: now__::_thesis:_not_W-min_C_in_L~_godo
assume W-min C in L~ godo ; ::_thesis: contradiction
then A152: ( W-min C in (L~ go) \/ (L~ pion1) or W-min C in L~ do ) by A143, XBOOLE_0:def_3;
percases ( W-min C in L~ go or W-min C in L~ pion1 or W-min C in L~ do ) by A152, XBOOLE_0:def_3;
suppose W-min C in L~ go ; ::_thesis: contradiction
then C meets L~ (Cage (C,n)) by A147, A149, XBOOLE_0:3;
hence contradiction by JORDAN10:5; ::_thesis: verum
end;
suppose W-min C in L~ pion1 ; ::_thesis: contradiction
hence contradiction by A8, A83, A133, A150, XBOOLE_0:3; ::_thesis: verum
end;
suppose W-min C in L~ do ; ::_thesis: contradiction
then C meets L~ (Cage (C,n)) by A148, A149, XBOOLE_0:3;
hence contradiction by JORDAN10:5; ::_thesis: verum
end;
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 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)) * (k,i)))),1,(Gauge (C,n))) by A36, A93, A138, JORDAN1J:52
.= right_cell ((go ^' pion1),1,(Gauge (C,n))) by A41, 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 GRAPH_2:53
.= W-min (L~ (Cage (C,n))) by A61, GRAPH_2:53 ;
A155: len (Upper_Seq (C,n)) >= 2 by A18, XXREAL_0:2;
A156: godo /. 2 = (go ^' pion1) /. 2 by A89, GRAPH_2:57
.= (Upper_Seq (C,n)) /. 2 by A35, A77, GRAPH_2:57
.= ((Upper_Seq (C,n)) ^' (Lower_Seq (C,n))) /. 2 by A155, GRAPH_2:57
.= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 2 by JORDAN1E:11 ;
A157: (L~ go) \/ (L~ do) is compact by COMPTS_1:10;
W-min (L~ (Cage (C,n))) in (L~ go) \/ (L~ do) by A63, A78, XBOOLE_0:def_3;
then A158: W-min ((L~ go) \/ (L~ do)) = W-min (L~ (Cage (C,n))) by A147, A148, A157, JORDAN1J:21, XBOOLE_1:8;
A159: (W-min ((L~ go) \/ (L~ do))) `1 = W-bound ((L~ go) \/ (L~ do)) by EUCLID:52;
A160: (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52;
A161: ((Gauge (C,n)) * (j,i)) `1 <= ((Gauge (C,n)) * (k,i)) `1 by A1, A2, A3, A4, A5, SPRECT_3:13;
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 A162: W-bound (L~ pion1) = ((Gauge (C,n)) * (j,i)) `1 by A83, SPPOL_2:21;
((Gauge (C,n)) * (j,i)) `1 >= W-bound (L~ (Cage (C,n))) by A9, A146, PSCOMP_1:24;
then ((Gauge (C,n)) * (j,i)) `1 > W-bound (L~ (Cage (C,n))) by A76, XXREAL_0:1;
then W-min (((L~ go) \/ (L~ do)) \/ (L~ pion1)) = W-min ((L~ go) \/ (L~ do)) by A157, A158, A159, A160, A162, JORDAN1J:33;
then A163: W-min (L~ godo) = W-min (L~ (Cage (C,n))) by A143, A158, XBOOLE_1:4;
A164: rng godo c= L~ godo by A89, A91, SPPOL_2:18, XXREAL_0:2;
2 in dom godo by A92, FINSEQ_3:25;
then A165: 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 A163, PSCOMP_1:31
.= W-bound (L~ godo) by EUCLID:52 ;
then godo /. 2 in W-most (L~ godo) by A164, A165, SPRECT_2:12;
then (Rotate (godo,(W-min (L~ godo)))) /. 2 in W-most (L~ godo) by A154, A163, FINSEQ_6:89;
then reconsider godo = godo as non 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 A166: (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 ;
A167: east_halfline (E-max C) misses L~ go
proof
assume east_halfline (E-max C) meets L~ go ; ::_thesis: contradiction
then consider p being set such that
A168: p in east_halfline (E-max C) and
A169: p in L~ go by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A168;
p in L~ (Upper_Seq (C,n)) by A48, A169;
then p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A145, A168, XBOOLE_0:def_4;
then A170: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;
then A171: p = E-max (L~ (Cage (C,n))) by A48, A169, JORDAN1J:46;
then E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (k,i) by A10, A166, A169, JORDAN1J:43;
then ((Gauge (C,n)) * (k,i)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by A3, A14, A170, A171, JORDAN1A:71;
hence contradiction by A3, A17, A32, JORDAN1G:7; ::_thesis: verum
end;
now__::_thesis:_not_east_halfline_(E-max_C)_meets_L~_godo
assume east_halfline (E-max C) meets L~ godo ; ::_thesis: contradiction
then A172: ( east_halfline (E-max C) meets (L~ go) \/ (L~ pion1) or east_halfline (E-max C) meets L~ do ) by A143, XBOOLE_1:70;
percases ( 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~ do ) by A172, XBOOLE_1:70;
suppose east_halfline (E-max C) meets L~ go ; ::_thesis: contradiction
hence contradiction by A167; ::_thesis: verum
end;
suppose east_halfline (E-max C) meets L~ pion1 ; ::_thesis: contradiction
then consider p being set such that
A173: p in east_halfline (E-max C) and
A174: p in L~ pion1 by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A173;
A175: p `2 = (E-max C) `2 by A173, TOPREAL1:def_11;
k + 1 <= len (Gauge (C,n)) by A3, NAT_1:13;
then (k + 1) - 1 <= (len (Gauge (C,n))) - 1 by XREAL_1:9;
then A176: 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 A177: ((Gauge (C,n)) * (k,i)) `1 <= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),1)) `1 by A4, A5, A11, A14, A21, A176, JORDAN1A:18;
p `1 <= ((Gauge (C,n)) * (k,i)) `1 by A83, A150, A161, A174, TOPREAL1:3;
then p `1 <= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),1)) `1 by A177, XXREAL_0:2;
then p `1 <= E-bound C by A21, JORDAN8:12;
then A178: p `1 <= (E-max C) `1 by EUCLID:52;
p `1 >= (E-max C) `1 by A173, TOPREAL1:def_11;
then p `1 = (E-max C) `1 by A178, XXREAL_0:1;
then p = E-max C by A175, TOPREAL3:6;
hence contradiction by A8, A83, A134, A150, A174, XBOOLE_0:3; ::_thesis: verum
end;
suppose east_halfline (E-max C) meets L~ do ; ::_thesis: contradiction
then consider p being set such that
A179: p in east_halfline (E-max C) and
A180: p in L~ do by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A179;
A181: p in LSeg (do,(Index (p,do))) by A180, JORDAN3:9;
consider t being Nat such that
A182: t in dom (Lower_Seq (C,n)) and
A183: (Lower_Seq (C,n)) . t = (Gauge (C,n)) * (j,i) by A39, FINSEQ_2:10;
1 <= t by A182, FINSEQ_3:25;
then A184: 1 < t by A34, A183, XXREAL_0:1;
t <= len (Lower_Seq (C,n)) by A182, FINSEQ_3:25;
then (Index (((Gauge (C,n)) * (j,i)),(Lower_Seq (C,n)))) + 1 = t by A183, A184, JORDAN3:12;
then A185: len (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (j,i)))) = (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (j,i)),(Lower_Seq (C,n)))) by A9, A183, JORDAN3:26;
Index (p,do) < len do by A180, JORDAN3:8;
then Index (p,do) < (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (j,i)),(Lower_Seq (C,n)))) by A185, XREAL_0:def_2;
then (Index (p,do)) + 1 <= (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (j,i)),(Lower_Seq (C,n)))) by NAT_1:13;
then A186: Index (p,do) <= ((len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (j,i)),(Lower_Seq (C,n))))) - 1 by XREAL_1:19;
A187: do = mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n)))) by A39, JORDAN1J:37;
p in L~ (Lower_Seq (C,n)) by A55, A180;
then p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A146, A179, XBOOLE_0:def_4;
then A188: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;
A189: (Index (((Gauge (C,n)) * (j,i)),(Lower_Seq (C,n)))) + 1 = ((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n)) by A34, A39, JORDAN1J:56;
0 + (Index (((Gauge (C,n)) * (j,i)),(Lower_Seq (C,n)))) < len (Lower_Seq (C,n)) by A9, JORDAN3:8;
then (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (j,i)),(Lower_Seq (C,n)))) > 0 by XREAL_1:20;
then Index (p,do) <= ((len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (j,i)),(Lower_Seq (C,n))))) - 1 by A186, XREAL_0:def_2;
then Index (p,do) <= (len (Lower_Seq (C,n))) - (((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n))) by A189;
then Index (p,do) <= (len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n))) by XREAL_0:def_2;
then A190: Index (p,do) < ((len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n)))) + 1 by NAT_1:13;
A191: 1 <= Index (p,do) by A180, JORDAN3:8;
A192: ((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n)) <= len (Lower_Seq (C,n)) by A39, FINSEQ_4:21;
((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n)) <> len (Lower_Seq (C,n)) by A31, A39, FINSEQ_4:19;
then A193: ((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n)) < len (Lower_Seq (C,n)) by A192, XXREAL_0:1;
A194: 1 + 1 <= len (Lower_Seq (C,n)) by A25, XXREAL_0:2;
then A195: 2 in dom (Lower_Seq (C,n)) by FINSEQ_3:25;
set tt = ((Index (p,do)) + (((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n)))) -' 1;
set RC = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))));
A196: E-max C in right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by JORDAN1I:7;
A197: 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 ;
A198: L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33;
consider jj2 being Element of NAT such that
A199: 1 <= jj2 and
A200: jj2 <= width (Gauge (C,n)) and
A201: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * ((len (Gauge (C,n))),jj2) by JORDAN1D:25;
A202: len (Gauge (C,n)) >= 4 by JORDAN8:10;
then len (Gauge (C,n)) >= 1 by XXREAL_0:2;
then A203: [(len (Gauge (C,n))),jj2] in Indices (Gauge (C,n)) by A199, A200, MATRIX_1:36;
A204: len (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = len (Cage (C,n)) by REVROT_1:14;
Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) by JORDAN1G:18;
then A205: LSeg ((Lower_Seq (C,n)),1) = LSeg ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by A194, SPPOL_2:9;
A206: 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 A140, REVROT_1:34;
then consider ii, jj being Element of NAT such that
A207: [ii,(jj + 1)] in Indices (Gauge (C,n)) and
A208: [ii,jj] in Indices (Gauge (C,n)) and
A209: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = (Gauge (C,n)) * (ii,(jj + 1)) and
A210: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1) = (Gauge (C,n)) * (ii,jj) by A87, A198, A204, A206, FINSEQ_6:92, JORDAN1I:23;
A211: (jj + 1) + 1 <> jj ;
A212: 1 <= jj by A208, MATRIX_1:38;
(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 A198, A206, FINSEQ_6:92;
then A213: ii = len (Gauge (C,n)) by A198, A207, A209, A201, A203, GOBOARD1:5;
then ii - 1 >= 4 - 1 by A202, XREAL_1:9;
then A214: ii - 1 >= 1 by XXREAL_0:2;
then A215: 1 <= ii -' 1 by XREAL_0:def_2;
A216: jj <= width (Gauge (C,n)) by A208, MATRIX_1:38;
then A217: ((Gauge (C,n)) * ((len (Gauge (C,n))),jj)) `1 = E-bound (L~ (Cage (C,n))) by A11, A212, JORDAN1A:71;
A218: jj + 1 <= width (Gauge (C,n)) by A207, MATRIX_1:38;
ii + 1 <> ii ;
then A219: right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) = cell ((Gauge (C,n)),(ii -' 1),jj) by A87, A204, A197, A207, A208, A209, A210, A211, GOBOARD5:def_6;
A220: ii <= len (Gauge (C,n)) by A208, MATRIX_1:38;
A221: 1 <= ii by A208, MATRIX_1:38;
A222: ii <= len (Gauge (C,n)) by A207, MATRIX_1:38;
A223: 1 <= jj + 1 by A207, MATRIX_1:38;
then A224: E-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * ((len (Gauge (C,n))),(jj + 1))) `1 by A11, A218, JORDAN1A:71;
A225: 1 <= ii by A207, MATRIX_1:38;
then A226: (ii -' 1) + 1 = ii by XREAL_1:235;
then A227: ii -' 1 < len (Gauge (C,n)) by A222, NAT_1:13;
then A228: ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 = ((Gauge (C,n)) * (1,(jj + 1))) `2 by A223, A218, A215, GOBOARD5:1
.= ((Gauge (C,n)) * (ii,(jj + 1))) `2 by A225, A222, A223, A218, GOBOARD5:1 ;
A229: (E-max C) `2 = p `2 by A179, TOPREAL1:def_11;
then A230: p `2 <= ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 by A196, A222, A218, A212, A219, A226, A214, JORDAN9:17;
A231: ((Gauge (C,n)) * ((ii -' 1),jj)) `2 = ((Gauge (C,n)) * (1,jj)) `2 by A212, A216, A215, A227, GOBOARD5:1
.= ((Gauge (C,n)) * (ii,jj)) `2 by A221, A220, A212, A216, GOBOARD5:1 ;
((Gauge (C,n)) * ((ii -' 1),jj)) `2 <= p `2 by A229, A196, A222, A218, A212, A219, A226, A214, JORDAN9:17;
then p in LSeg (((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1),((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1))) by A188, A209, A210, A213, A230, A231, A228, A217, A224, GOBOARD7:7;
then A232: p in LSeg ((Lower_Seq (C,n)),1) by A87, A205, A204, TOPREAL1:def_3;
1 <= ((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n)) by A39, FINSEQ_4:21;
then A233: LSeg ((mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n))))),(Index (p,do))) = LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n)))) -' 1)) by A193, A191, A190, JORDAN4:19;
1 <= Index (((Gauge (C,n)) * (j,i)),(Lower_Seq (C,n))) by A9, JORDAN3:8;
then A234: 1 + 1 <= ((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n)) by A189, XREAL_1:7;
then (Index (p,do)) + (((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n))) >= (1 + 1) + 1 by A191, XREAL_1:7;
then ((Index (p,do)) + (((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n)))) - 1 >= ((1 + 1) + 1) - 1 by XREAL_1:9;
then A235: ((Index (p,do)) + (((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n)))) -' 1 >= 1 + 1 by XREAL_0:def_2;
now__::_thesis:_contradiction
percases ( ((Index (p,do)) + (((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n)))) -' 1 > 1 + 1 or ((Index (p,do)) + (((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ) by A235, XXREAL_0:1;
suppose ((Index (p,do)) + (((Gauge (C,n)) * (j,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,do)) + (((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n)))) -' 1)) by TOPREAL1:def_7;
hence contradiction by A232, A181, A187, A233, XBOOLE_0:3; ::_thesis: verum
end;
supposeA236: ((Index (p,do)) + (((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ; ::_thesis: contradiction
then 1 + 1 = ((Index (p,do)) + (((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n)))) - 1 by XREAL_0:def_2;
then (1 + 1) + 1 = (Index (p,do)) + (((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n))) ;
then A237: ((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n)) = 2 by A191, A234, JORDAN1E:6;
(LSeg ((Lower_Seq (C,n)),1)) /\ (LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n)))) -' 1))) = {((Lower_Seq (C,n)) /. 2)} by A25, A236, TOPREAL1:def_6;
then p in {((Lower_Seq (C,n)) /. 2)} by A232, A181, A187, A233, XBOOLE_0:def_4;
then A238: p = (Lower_Seq (C,n)) /. 2 by TARSKI:def_1;
then A239: p in rng (Lower_Seq (C,n)) by A195, PARTFUN2:2;
p .. (Lower_Seq (C,n)) = 2 by A195, A238, FINSEQ_5:41;
then p = (Gauge (C,n)) * (j,i) by A39, A237, A239, FINSEQ_5:9;
then ((Gauge (C,n)) * (j,i)) `1 = E-bound (L~ (Cage (C,n))) by A238, JORDAN1G:32;
then ((Gauge (C,n)) * (j,i)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 by A1, A11, A13, JORDAN1A:71;
hence contradiction by A2, A3, A16, A69, JORDAN1G:7; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
end;
end;
then east_halfline (E-max C) c= (L~ godo) ` by SUBSET_1:23;
then consider W being Subset of (TOP-REAL 2) such that
A240: W is_a_component_of (L~ godo) ` and
A241: east_halfline (E-max C) c= W by GOBOARD9:3;
not W is bounded by A241, JORDAN2C:121, RLTOPSP1:42;
then W is_outside_component_of L~ godo by A240, JORDAN2C:def_3;
then W c= UBD (L~ godo) by JORDAN2C:23;
then A242: east_halfline (E-max C) c= UBD (L~ godo) by A241, XBOOLE_1:1;
E-max C in east_halfline (E-max C) by TOPREAL1:38;
then E-max C in UBD (L~ godo) by A242;
then E-max C in LeftComp godo by GOBRD14:36;
then Lower_Arc C meets L~ godo by A132, A133, A134, A142, A153, JORDAN1J:36;
then A243: ( Lower_Arc C meets (L~ go) \/ (L~ pion1) or Lower_Arc C meets L~ do ) by A143, XBOOLE_1:70;
A244: Lower_Arc C c= C by JORDAN6:61;
percases ( Lower_Arc C meets L~ go or Lower_Arc C meets L~ pion1 or Lower_Arc C meets L~ do ) by A243, XBOOLE_1:70;
suppose Lower_Arc C meets L~ go ; ::_thesis: contradiction
then Lower_Arc C meets L~ (Cage (C,n)) by A48, A145, XBOOLE_1:1, XBOOLE_1:63;
hence contradiction by A244, JORDAN10:5, XBOOLE_1:63; ::_thesis: verum
end;
suppose Lower_Arc C meets L~ pion1 ; ::_thesis: contradiction
hence contradiction by A8, A83, A150; ::_thesis: verum
end;
suppose Lower_Arc C meets L~ do ; ::_thesis: contradiction
then Lower_Arc C meets L~ (Cage (C,n)) by A55, A146, XBOOLE_1:1, XBOOLE_1:63;
hence contradiction by A244, JORDAN10:5, XBOOLE_1:63; ::_thesis: verum
end;
end;
end;
theorem Th29: :: JORDAN15:29
for n being Element of NAT
for C being Simple_closed_curve
for i, j, k being Element of 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)) * (k,i))} & (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (j,i))} holds
LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Upper_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i, j, k being Element of 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)) * (k,i))} & (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (j,i))} holds
LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Upper_Arc C
let C be Simple_closed_curve; ::_thesis: for i, j, k being Element of 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)) * (k,i))} & (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (j,i))} holds
LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Upper_Arc C
let i, j, k be Element of 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)) * (k,i))} & (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (j,i))} implies LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) 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)) * (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)) * (k,i))} and
A7: (LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (j,i))} and
A8: LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) misses Upper_Arc C ; ::_thesis: contradiction
(Gauge (C,n)) * (j,i) in {((Gauge (C,n)) * (j,i))} by TARSKI:def_1;
then A9: (Gauge (C,n)) * (j,i) in L~ (Lower_Seq (C,n)) by A7, XBOOLE_0:def_4;
(Gauge (C,n)) * (k,i) in {((Gauge (C,n)) * (k,i))} by TARSKI:def_1;
then A10: (Gauge (C,n)) * (k,i) in L~ (Upper_Seq (C,n)) by A6, XBOOLE_0:def_4;
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 A2, A3, A11, XXREAL_0:2;
A14: 1 <= k by A1, A2, XXREAL_0:2;
A15: k <= width (Gauge (C,n)) by A3, JORDAN8:def_1;
A16: [j,i] in Indices (Gauge (C,n)) by A1, A4, A5, A11, A13, MATRIX_1:36;
A17: [k,i] in Indices (Gauge (C,n)) by A3, A4, A5, A14, MATRIX_1:36;
set go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (k,i)));
set do = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (j,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,k)) `1 by A3, A14, JORDAN1A:73 ;
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 A14, A15, MATRIX_1:36;
then A23: (Gauge (C,n)) * (k,i) <> (Upper_Seq (C,n)) . 1 by A1, A2, A17, A19, A20, JORDAN1G:7;
then reconsider go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (k,i))) as being_S-Seq FinSequence of (TOP-REAL 2) by A10, JORDAN3:35;
A24: [1,j] in Indices (Gauge (C,n)) by A1, A13, A21, MATRIX_1:36;
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 A26, FINSEQ_3:25;
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 ;
A29: (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * (1,k)) `1 by A3, A14, JORDAN1A:73 ;
A30: [j,i] in Indices (Gauge (C,n)) by A1, A4, A5, A11, A13, MATRIX_1:36;
then A31: (Gauge (C,n)) * (j,i) <> (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) by A1, A22, A28, A29, JORDAN1G:7;
then reconsider do = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (j,i))) as being_S-Seq FinSequence of (TOP-REAL 2) by A9, JORDAN3:34;
A32: [(len (Gauge (C,n))),k] in Indices (Gauge (C,n)) by A14, A15, A21, MATRIX_1:36;
A33: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by A27, 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 A3, A14, JORDAN1A:71 ;
then A34: (Gauge (C,n)) * (j,i) <> (Lower_Seq (C,n)) . 1 by A2, A3, A30, A32, A33, JORDAN1G:7;
A35: len go >= 1 + 1 by TOPREAL1:def_8;
A36: (Gauge (C,n)) * (k,i) in rng (Upper_Seq (C,n)) by A4, A5, A10, A11, A14, A15, JORDAN1G:4, JORDAN1J:40;
then A37: go is_sequence_on Gauge (C,n) by JORDAN1G:4, JORDAN1J:38;
A38: len do >= 1 + 1 by TOPREAL1:def_8;
A39: (Gauge (C,n)) * (j,i) in rng (Lower_Seq (C,n)) by A1, A4, A5, A9, A11, A13, JORDAN1G:5, JORDAN1J:40;
then A40: do is_sequence_on Gauge (C,n) by JORDAN1G:5, JORDAN1J:39;
reconsider go = go as non constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A35, A37, JGRAPH_1:12, JORDAN8:5;
reconsider do = do as non constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A38, A40, JGRAPH_1:12, JORDAN8:5;
A41: len go > 1 by A35, NAT_1:13;
then A42: len go in dom go by FINSEQ_3:25;
then A43: go /. (len go) = go . (len go) by PARTFUN1:def_6
.= (Gauge (C,n)) * (k,i) by A10, JORDAN3:24 ;
len do >= 1 by A38, XXREAL_0:2;
then 1 in dom do by FINSEQ_3:25;
then A44: do /. 1 = do . 1 by PARTFUN1:def_6
.= (Gauge (C,n)) * (j,i) by A9, JORDAN3:23 ;
reconsider m = (len go) - 1 as Element of NAT by A42, FINSEQ_3:26;
A45: m + 1 = len go ;
then A46: (len go) -' 1 = m by NAT_D:34;
A47: LSeg (go,m) c= L~ go by TOPREAL3:19;
A48: L~ go c= L~ (Upper_Seq (C,n)) by A10, JORDAN3:41;
then LSeg (go,m) c= L~ (Upper_Seq (C,n)) by A47, XBOOLE_1:1;
then A49: (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i)))) c= {((Gauge (C,n)) * (k,i))} by A6, XBOOLE_1:26;
m >= 1 by A35, XREAL_1:19;
then A50: LSeg (go,m) = LSeg ((go /. m),((Gauge (C,n)) * (k,i))) by A43, A45, TOPREAL1:def_3;
{((Gauge (C,n)) * (k,i))} c= (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (k,i))} or x in (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i)))) )
A51: (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 (go,m)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))))
then A52: x = (Gauge (C,n)) * (k,i) by TARSKI:def_1;
(Gauge (C,n)) * (k,i) in LSeg (go,m) by A50, RLTOPSP1:68;
hence x in (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i)))) by A52, A51, XBOOLE_0:def_4; ::_thesis: verum
end;
then A53: (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i)))) = {((Gauge (C,n)) * (k,i))} by A49, XBOOLE_0:def_10;
A54: LSeg (do,1) c= L~ do by TOPREAL3:19;
A55: L~ do c= L~ (Lower_Seq (C,n)) by A9, JORDAN3:42;
then LSeg (do,1) c= L~ (Lower_Seq (C,n)) by A54, XBOOLE_1:1;
then A56: (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i)))) c= {((Gauge (C,n)) * (j,i))} by A7, XBOOLE_1:26;
A57: LSeg (do,1) = LSeg (((Gauge (C,n)) * (j,i)),(do /. (1 + 1))) by A38, A44, TOPREAL1:def_3;
{((Gauge (C,n)) * (j,i))} c= (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (j,i))} or x in (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i)))) )
A58: (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 (do,1)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))))
then A59: x = (Gauge (C,n)) * (j,i) by TARSKI:def_1;
(Gauge (C,n)) * (j,i) in LSeg (do,1) by A57, RLTOPSP1:68;
hence x in (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i)))) by A59, A58, XBOOLE_0:def_4; ::_thesis: verum
end;
then A60: (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i)))) /\ (LSeg (do,1)) = {((Gauge (C,n)) * (j,i))} by A56, XBOOLE_0:def_10;
A61: go /. 1 = (Upper_Seq (C,n)) /. 1 by A10, SPRECT_3:22
.= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ;
then A62: go /. 1 = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by JORDAN1F:8
.= do /. (len do) by A9, JORDAN1J:35 ;
A63: rng go c= L~ go by A35, SPPOL_2:18;
A64: rng do c= L~ do by A38, SPPOL_2:18;
A65: {(go /. 1)} c= (L~ go) /\ (L~ do)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(go /. 1)} or x in (L~ go) /\ (L~ do) )
assume x in {(go /. 1)} ; ::_thesis: x in (L~ go) /\ (L~ do)
then A66: x = go /. 1 by TARSKI:def_1;
then A67: x in rng go by FINSEQ_6:42;
x in rng do by A62, A66, REVROT_1:3;
hence x in (L~ go) /\ (L~ do) by A63, A64, A67, XBOOLE_0:def_4; ::_thesis: verum
end;
A68: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by A27, PARTFUN1:def_6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;
A69: [(len (Gauge (C,n))),j] in Indices (Gauge (C,n)) by A1, A13, A21, MATRIX_1:36;
(L~ go) /\ (L~ do) c= {(go /. 1)}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ go) /\ (L~ do) or x in {(go /. 1)} )
assume A70: x in (L~ go) /\ (L~ do) ; ::_thesis: x in {(go /. 1)}
then A71: x in L~ do by XBOOLE_0:def_4;
A72: now__::_thesis:_not_x_=_E-max_(L~_(Cage_(C,n)))
assume x = E-max (L~ (Cage (C,n))) ; ::_thesis: contradiction
then A73: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (j,i) by A9, A68, A71, JORDAN1E:7;
((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 = E-bound (L~ (Cage (C,n))) by A1, A11, A13, JORDAN1A:71;
then (E-max (L~ (Cage (C,n)))) `1 <> E-bound (L~ (Cage (C,n))) by A2, A3, A16, A69, A73, JORDAN1G:7;
hence contradiction by EUCLID:52; ::_thesis: verum
end;
x in L~ go by A70, XBOOLE_0:def_4;
then x in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A48, A55, A71, 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 A61, A72, TARSKI:def_1; ::_thesis: verum
end;
then A74: (L~ go) /\ (L~ do) = {(go /. 1)} by A65, XBOOLE_0:def_10;
set W2 = go /. 2;
A75: 2 in dom go by A35, FINSEQ_3:25;
A76: 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,j)) `1 = ((Gauge (C,n)) * (j,i)) `1 by A1, A11, A13, JORDAN1A:73;
hence contradiction by A1, A16, A24, JORDAN1G:7; ::_thesis: verum
end;
go = mid ((Upper_Seq (C,n)),1,(((Gauge (C,n)) * (k,i)) .. (Upper_Seq (C,n)))) by A36, JORDAN1G:49
.= (Upper_Seq (C,n)) | (((Gauge (C,n)) * (k,i)) .. (Upper_Seq (C,n))) by A36, FINSEQ_4:21, FINSEQ_6:116 ;
then A77: go /. 2 = (Upper_Seq (C,n)) /. 2 by A75, FINSEQ_4:70;
A78: W-min (L~ (Cage (C,n))) in rng go by A61, FINSEQ_6:42;
set pion = <*((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))*>;
A79: now__::_thesis:_for_n_being_Element_of_NAT_st_n_in_dom_<*((Gauge_(C,n))_*_(k,i)),((Gauge_(C,n))_*_(j,i))*>_holds_
ex_j,_i_being_Element_of_NAT_st_
(_[j,i]_in_Indices_(Gauge_(C,n))_&_<*((Gauge_(C,n))_*_(k,i)),((Gauge_(C,n))_*_(j,i))*>_/._n_=_(Gauge_(C,n))_*_(j,i)_)
let n be Element of NAT ; ::_thesis: ( n in dom <*((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))*> implies ex j, i being Element of NAT st
( [j,i] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))*> /. n = (Gauge (C,n)) * (j,i) ) )
assume n in dom <*((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))*> ; ::_thesis: ex j, i being Element of NAT st
( [j,i] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))*> /. n = (Gauge (C,n)) * (j,i) )
then n in {1,2} by FINSEQ_1:2, FINSEQ_1:89;
then ( n = 1 or n = 2 ) by TARSKI:def_2;
hence ex j, i being Element of NAT st
( [j,i] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))*> /. n = (Gauge (C,n)) * (j,i) ) by A16, A17, FINSEQ_4:17; ::_thesis: verum
end;
A80: (Gauge (C,n)) * (k,i) <> (Gauge (C,n)) * (j,i) by A12, A16, A17, GOBOARD1:5;
((Gauge (C,n)) * (k,i)) `2 = ((Gauge (C,n)) * (1,i)) `2 by A3, A4, A5, A14, GOBOARD5:1
.= ((Gauge (C,n)) * (j,i)) `2 by A1, A4, A5, A11, A13, GOBOARD5:1 ;
then LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))) is horizontal by SPPOL_1:15;
then <*((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))*> is being_S-Seq by A80, JORDAN1B:8;
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)) * (k,i)),((Gauge (C,n)) * (j,i))*> = L~ pion1 and
A84: <*((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))*> /. 1 = pion1 /. 1 and
A85: <*((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))*> /. (len <*((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))*>) = pion1 /. (len pion1) and
A86: len <*((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))*> <= len pion1 by A79, GOBOARD3:2;
reconsider pion1 = pion1 as being_S-Seq FinSequence of (TOP-REAL 2) by A82;
set godo = (go ^' pion1) ^' do;
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 A35, XXREAL_0:2;
then A90: len (go ^' pion1) > 1 + 0 by NAT_1:13;
A91: len ((go ^' pion1) ^' do) >= len (go ^' pion1) by TOPREAL8:7;
then A92: 1 + 1 <= len ((go ^' pion1) ^' do) 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 A43, A84, FINSEQ_4:17;
then A95: go ^' pion1 is_sequence_on Gauge (C,n) by A37, A81, TOPREAL8:12;
A96: (go ^' pion1) /. (len (go ^' pion1)) = <*((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))*> /. (len <*((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))*>) by A85, GRAPH_2:54
.= <*((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))*> /. 2 by FINSEQ_1:44
.= do /. 1 by A44, FINSEQ_4:17 ;
then A97: (go ^' pion1) ^' do is_sequence_on Gauge (C,n) by A40, A95, TOPREAL8:12;
LSeg (pion1,1) c= L~ <*((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))*> by A83, TOPREAL3:19;
then LSeg (pion1,1) c= LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))) by SPPOL_2:21;
then A98: (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) c= {((Gauge (C,n)) * (k,i))} by A46, A53, XBOOLE_1:27;
A99: len pion1 >= 1 + 1 by A86, FINSEQ_1:44;
{((Gauge (C,n)) * (k,i))} c= (LSeg (go,m)) /\ (LSeg (pion1,1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (k,i))} or x in (LSeg (go,m)) /\ (LSeg (pion1,1)) )
assume x in {((Gauge (C,n)) * (k,i))} ; ::_thesis: x in (LSeg (go,m)) /\ (LSeg (pion1,1))
then A100: x = (Gauge (C,n)) * (k,i) by TARSKI:def_1;
A101: (Gauge (C,n)) * (k,i) in LSeg (go,m) by A50, RLTOPSP1:68;
(Gauge (C,n)) * (k,i) in LSeg (pion1,1) by A43, A94, A99, TOPREAL1:21;
hence x in (LSeg (go,m)) /\ (LSeg (pion1,1)) by A100, A101, XBOOLE_0:def_4; ::_thesis: verum
end;
then (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) = {(go /. (len go))} by A43, A46, A98, XBOOLE_0:def_10;
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 GRAPH_2:13;
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)) * (k,i)),((Gauge (C,n)) * (j,i))*> by A83, TOPREAL3:19;
then LSeg (pion1,((len pion1) -' 1)) c= LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))) by SPPOL_2:21;
then A109: (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) c= {((Gauge (C,n)) * (j,i))} by A60, XBOOLE_1:27;
{((Gauge (C,n)) * (j,i))} c= (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (j,i))} or x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) )
assume x in {((Gauge (C,n)) * (j,i))} ; ::_thesis: x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1))
then A110: x = (Gauge (C,n)) * (j,i) by TARSKI:def_1;
pion1 /. (((len pion1) -' 1) + 1) = <*((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))*> /. 2 by A85, A106, FINSEQ_1:44
.= (Gauge (C,n)) * (j,i) by FINSEQ_4:17 ;
then A111: (Gauge (C,n)) * (j,i) in LSeg (pion1,((len pion1) -' 1)) by A105, A106, TOPREAL1:21;
(Gauge (C,n)) * (j,i) in LSeg (do,1) by A57, RLTOPSP1:68;
hence x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) by A110, A111, XBOOLE_0:def_4; ::_thesis: verum
end;
then (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) = {((Gauge (C,n)) * (j,i))} by A109, XBOOLE_0:def_10;
then A112: (LSeg ((go ^' pion1),((len go) + ((len pion1) -' 2)))) /\ (LSeg (do,1)) = {((go ^' pion1) /. (len (go ^' pion1)))} by A44, 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 set ; :: 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 A116: x = pion1 /. 1 by TARSKI:def_1;
then A117: x in rng pion1 by FINSEQ_6:42;
x in rng go by A94, A116, REVROT_1:3;
hence x in (L~ go) /\ (L~ pion1) by A63, A114, A117, XBOOLE_0:def_4; ::_thesis: verum
end;
(L~ go) /\ (L~ pion1) c= {(pion1 /. 1)}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ go) /\ (L~ pion1) or x in {(pion1 /. 1)} )
assume A118: x in (L~ go) /\ (L~ pion1) ; ::_thesis: x in {(pion1 /. 1)}
then A119: x in L~ pion1 by XBOOLE_0:def_4;
x in L~ go by A118, XBOOLE_0:def_4;
then x in (L~ pion1) /\ (L~ (Upper_Seq (C,n))) by A48, A119, XBOOLE_0:def_4;
hence x in {(pion1 /. 1)} by A6, A43, A83, A94, SPPOL_2:21; ::_thesis: verum
end;
then A120: (L~ go) /\ (L~ pion1) = {(pion1 /. 1)} by A115, XBOOLE_0:def_10;
then A121: go ^' pion1 is s.n.c. by A94, JORDAN1J:54;
(rng go) /\ (rng pion1) c= {(pion1 /. 1)} by A63, A114, A120, XBOOLE_1:27;
then A122: go ^' pion1 is one-to-one by JORDAN1J:55;
A123: <*((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))*> /. (len <*((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))*>) = <*((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))*> /. 2 by FINSEQ_1:44
.= do /. 1 by A44, FINSEQ_4:17 ;
A124: {(pion1 /. (len pion1))} c= (L~ do) /\ (L~ pion1)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(pion1 /. (len pion1))} or x in (L~ do) /\ (L~ pion1) )
assume x in {(pion1 /. (len pion1))} ; ::_thesis: x in (L~ do) /\ (L~ pion1)
then A125: x = pion1 /. (len pion1) by TARSKI:def_1;
then A126: x in rng pion1 by REVROT_1:3;
x in rng do by A85, A123, A125, FINSEQ_6:42;
hence x in (L~ do) /\ (L~ pion1) by A64, A114, A126, XBOOLE_0:def_4; ::_thesis: verum
end;
(L~ do) /\ (L~ pion1) c= {(pion1 /. (len pion1))}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ do) /\ (L~ pion1) or x in {(pion1 /. (len pion1))} )
assume A127: x in (L~ do) /\ (L~ pion1) ; ::_thesis: x in {(pion1 /. (len pion1))}
then A128: x in L~ pion1 by XBOOLE_0:def_4;
x in L~ do by A127, XBOOLE_0:def_4;
then x in (L~ pion1) /\ (L~ (Lower_Seq (C,n))) by A55, A128, XBOOLE_0:def_4;
hence x in {(pion1 /. (len pion1))} by A7, A44, A83, A85, A123, SPPOL_2:21; ::_thesis: verum
end;
then A129: (L~ do) /\ (L~ pion1) = {(pion1 /. (len pion1))} by A124, XBOOLE_0:def_10;
A130: (L~ (go ^' pion1)) /\ (L~ do) = ((L~ go) \/ (L~ pion1)) /\ (L~ do) by A94, TOPREAL8:35
.= {(go /. 1)} \/ {(do /. 1)} by A74, A85, A123, A129, XBOOLE_1:23
.= {((go ^' pion1) /. 1)} \/ {(do /. 1)} by GRAPH_2:53
.= {((go ^' pion1) /. 1),(do /. 1)} by ENUMSET1:1 ;
do /. (len do) = (go ^' pion1) /. 1 by A62, GRAPH_2:53;
then reconsider godo = (go ^' pion1) ^' do as non constant standard special_circular_sequence by A92, A96, A97, A102, A104, A112, A113, A121, A122, A130, JORDAN8:4, JORDAN8:5, TOPREAL8:11, TOPREAL8:33, TOPREAL8:34;
A131: Upper_Arc C is_an_arc_of W-min C, E-max C by JORDAN6:def_8;
then A132: Upper_Arc C is connected by JORDAN6:10;
A133: W-min C in Upper_Arc C by A131, TOPREAL1:1;
A134: E-max C in Upper_Arc C by A131, 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 A135: (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 = W-min (L~ (Cage (C,n))) by FINSEQ_6:92;
A136: 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 A135, 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 A135, A136, 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 A135, A136, SPRECT_5:24, XXREAL_0:2;
then A137: (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A135, A136, SPRECT_5:25, XXREAL_0:2;
A138: now__::_thesis:_not_((Gauge_(C,n))_*_(k,i))_.._(Upper_Seq_(C,n))_<=_1
assume A139: ((Gauge (C,n)) * (k,i)) .. (Upper_Seq (C,n)) <= 1 ; ::_thesis: contradiction
((Gauge (C,n)) * (k,i)) .. (Upper_Seq (C,n)) >= 1 by A36, FINSEQ_4:21;
then ((Gauge (C,n)) * (k,i)) .. (Upper_Seq (C,n)) = 1 by A139, XXREAL_0:1;
then (Gauge (C,n)) * (k,i) = (Upper_Seq (C,n)) /. 1 by A36, FINSEQ_5:38;
hence contradiction by A19, A23, JORDAN1F:5; ::_thesis: verum
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~ do) by A96, TOPREAL8:35
.= ((L~ go) \/ (L~ pion1)) \/ (L~ do) 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 A48, A145, XBOOLE_1:1;
A148: L~ do c= L~ (Cage (C,n)) by A55, A146, XBOOLE_1:1;
A149: W-min C in C by SPRECT_1:13;
A150: L~ <*((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))*> = LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))) by SPPOL_2:21;
A151: now__::_thesis:_not_W-min_C_in_L~_godo
assume W-min C in L~ godo ; ::_thesis: contradiction
then A152: ( W-min C in (L~ go) \/ (L~ pion1) or W-min C in L~ do ) by A143, XBOOLE_0:def_3;
percases ( W-min C in L~ go or W-min C in L~ pion1 or W-min C in L~ do ) by A152, XBOOLE_0:def_3;
suppose W-min C in L~ go ; ::_thesis: contradiction
then C meets L~ (Cage (C,n)) by A147, A149, XBOOLE_0:3;
hence contradiction by JORDAN10:5; ::_thesis: verum
end;
suppose W-min C in L~ pion1 ; ::_thesis: contradiction
hence contradiction by A8, A83, A133, A150, XBOOLE_0:3; ::_thesis: verum
end;
suppose W-min C in L~ do ; ::_thesis: contradiction
then C meets L~ (Cage (C,n)) by A148, A149, XBOOLE_0:3;
hence contradiction by JORDAN10:5; ::_thesis: verum
end;
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 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)) * (k,i)))),1,(Gauge (C,n))) by A36, A93, A138, JORDAN1J:52
.= right_cell ((go ^' pion1),1,(Gauge (C,n))) by A41, 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 GRAPH_2:53
.= W-min (L~ (Cage (C,n))) by A61, GRAPH_2:53 ;
A155: len (Upper_Seq (C,n)) >= 2 by A18, XXREAL_0:2;
A156: godo /. 2 = (go ^' pion1) /. 2 by A89, GRAPH_2:57
.= (Upper_Seq (C,n)) /. 2 by A35, A77, GRAPH_2:57
.= ((Upper_Seq (C,n)) ^' (Lower_Seq (C,n))) /. 2 by A155, GRAPH_2:57
.= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 2 by JORDAN1E:11 ;
A157: (L~ go) \/ (L~ do) is compact by COMPTS_1:10;
W-min (L~ (Cage (C,n))) in (L~ go) \/ (L~ do) by A63, A78, XBOOLE_0:def_3;
then A158: W-min ((L~ go) \/ (L~ do)) = W-min (L~ (Cage (C,n))) by A147, A148, A157, JORDAN1J:21, XBOOLE_1:8;
A159: (W-min ((L~ go) \/ (L~ do))) `1 = W-bound ((L~ go) \/ (L~ do)) by EUCLID:52;
A160: (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52;
A161: ((Gauge (C,n)) * (j,i)) `1 <= ((Gauge (C,n)) * (k,i)) `1 by A1, A2, A3, A4, A5, SPRECT_3:13;
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 A162: W-bound (L~ pion1) = ((Gauge (C,n)) * (j,i)) `1 by A83, SPPOL_2:21;
((Gauge (C,n)) * (j,i)) `1 >= W-bound (L~ (Cage (C,n))) by A9, A146, PSCOMP_1:24;
then ((Gauge (C,n)) * (j,i)) `1 > W-bound (L~ (Cage (C,n))) by A76, XXREAL_0:1;
then W-min (((L~ go) \/ (L~ do)) \/ (L~ pion1)) = W-min ((L~ go) \/ (L~ do)) by A157, A158, A159, A160, A162, JORDAN1J:33;
then A163: W-min (L~ godo) = W-min (L~ (Cage (C,n))) by A143, A158, XBOOLE_1:4;
A164: rng godo c= L~ godo by A89, A91, SPPOL_2:18, XXREAL_0:2;
2 in dom godo by A92, FINSEQ_3:25;
then A165: 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 A163, PSCOMP_1:31
.= W-bound (L~ godo) by EUCLID:52 ;
then godo /. 2 in W-most (L~ godo) by A164, A165, SPRECT_2:12;
then (Rotate (godo,(W-min (L~ godo)))) /. 2 in W-most (L~ godo) by A154, A163, FINSEQ_6:89;
then reconsider godo = godo as non 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 A166: (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 ;
A167: east_halfline (E-max C) misses L~ go
proof
assume east_halfline (E-max C) meets L~ go ; ::_thesis: contradiction
then consider p being set such that
A168: p in east_halfline (E-max C) and
A169: p in L~ go by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A168;
p in L~ (Upper_Seq (C,n)) by A48, A169;
then p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A145, A168, XBOOLE_0:def_4;
then A170: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;
then A171: p = E-max (L~ (Cage (C,n))) by A48, A169, JORDAN1J:46;
then E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (k,i) by A10, A166, A169, JORDAN1J:43;
then ((Gauge (C,n)) * (k,i)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by A3, A14, A170, A171, JORDAN1A:71;
hence contradiction by A3, A17, A32, JORDAN1G:7; ::_thesis: verum
end;
now__::_thesis:_not_east_halfline_(E-max_C)_meets_L~_godo
assume east_halfline (E-max C) meets L~ godo ; ::_thesis: contradiction
then A172: ( east_halfline (E-max C) meets (L~ go) \/ (L~ pion1) or east_halfline (E-max C) meets L~ do ) by A143, XBOOLE_1:70;
percases ( 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~ do ) by A172, XBOOLE_1:70;
suppose east_halfline (E-max C) meets L~ go ; ::_thesis: contradiction
hence contradiction by A167; ::_thesis: verum
end;
suppose east_halfline (E-max C) meets L~ pion1 ; ::_thesis: contradiction
then consider p being set such that
A173: p in east_halfline (E-max C) and
A174: p in L~ pion1 by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A173;
A175: p `2 = (E-max C) `2 by A173, TOPREAL1:def_11;
k + 1 <= len (Gauge (C,n)) by A3, NAT_1:13;
then (k + 1) - 1 <= (len (Gauge (C,n))) - 1 by XREAL_1:9;
then A176: 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 A177: ((Gauge (C,n)) * (k,i)) `1 <= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),1)) `1 by A4, A5, A11, A14, A21, A176, JORDAN1A:18;
p `1 <= ((Gauge (C,n)) * (k,i)) `1 by A83, A150, A161, A174, TOPREAL1:3;
then p `1 <= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),1)) `1 by A177, XXREAL_0:2;
then p `1 <= E-bound C by A21, JORDAN8:12;
then A178: p `1 <= (E-max C) `1 by EUCLID:52;
p `1 >= (E-max C) `1 by A173, TOPREAL1:def_11;
then p `1 = (E-max C) `1 by A178, XXREAL_0:1;
then p = E-max C by A175, TOPREAL3:6;
hence contradiction by A8, A83, A134, A150, A174, XBOOLE_0:3; ::_thesis: verum
end;
suppose east_halfline (E-max C) meets L~ do ; ::_thesis: contradiction
then consider p being set such that
A179: p in east_halfline (E-max C) and
A180: p in L~ do by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A179;
A181: p in LSeg (do,(Index (p,do))) by A180, JORDAN3:9;
consider t being Nat such that
A182: t in dom (Lower_Seq (C,n)) and
A183: (Lower_Seq (C,n)) . t = (Gauge (C,n)) * (j,i) by A39, FINSEQ_2:10;
1 <= t by A182, FINSEQ_3:25;
then A184: 1 < t by A34, A183, XXREAL_0:1;
t <= len (Lower_Seq (C,n)) by A182, FINSEQ_3:25;
then (Index (((Gauge (C,n)) * (j,i)),(Lower_Seq (C,n)))) + 1 = t by A183, A184, JORDAN3:12;
then A185: len (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (j,i)))) = (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (j,i)),(Lower_Seq (C,n)))) by A9, A183, JORDAN3:26;
Index (p,do) < len do by A180, JORDAN3:8;
then Index (p,do) < (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (j,i)),(Lower_Seq (C,n)))) by A185, XREAL_0:def_2;
then (Index (p,do)) + 1 <= (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (j,i)),(Lower_Seq (C,n)))) by NAT_1:13;
then A186: Index (p,do) <= ((len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (j,i)),(Lower_Seq (C,n))))) - 1 by XREAL_1:19;
A187: do = mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n)))) by A39, JORDAN1J:37;
p in L~ (Lower_Seq (C,n)) by A55, A180;
then p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A146, A179, XBOOLE_0:def_4;
then A188: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;
A189: (Index (((Gauge (C,n)) * (j,i)),(Lower_Seq (C,n)))) + 1 = ((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n)) by A34, A39, JORDAN1J:56;
0 + (Index (((Gauge (C,n)) * (j,i)),(Lower_Seq (C,n)))) < len (Lower_Seq (C,n)) by A9, JORDAN3:8;
then (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (j,i)),(Lower_Seq (C,n)))) > 0 by XREAL_1:20;
then Index (p,do) <= ((len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (j,i)),(Lower_Seq (C,n))))) - 1 by A186, XREAL_0:def_2;
then Index (p,do) <= (len (Lower_Seq (C,n))) - (((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n))) by A189;
then Index (p,do) <= (len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n))) by XREAL_0:def_2;
then A190: Index (p,do) < ((len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n)))) + 1 by NAT_1:13;
A191: 1 <= Index (p,do) by A180, JORDAN3:8;
A192: ((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n)) <= len (Lower_Seq (C,n)) by A39, FINSEQ_4:21;
((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n)) <> len (Lower_Seq (C,n)) by A31, A39, FINSEQ_4:19;
then A193: ((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n)) < len (Lower_Seq (C,n)) by A192, XXREAL_0:1;
A194: 1 + 1 <= len (Lower_Seq (C,n)) by A25, XXREAL_0:2;
then A195: 2 in dom (Lower_Seq (C,n)) by FINSEQ_3:25;
set tt = ((Index (p,do)) + (((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n)))) -' 1;
set RC = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))));
A196: E-max C in right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by JORDAN1I:7;
A197: 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 ;
A198: L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33;
consider jj2 being Element of NAT such that
A199: 1 <= jj2 and
A200: jj2 <= width (Gauge (C,n)) and
A201: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * ((len (Gauge (C,n))),jj2) by JORDAN1D:25;
A202: len (Gauge (C,n)) >= 4 by JORDAN8:10;
then len (Gauge (C,n)) >= 1 by XXREAL_0:2;
then A203: [(len (Gauge (C,n))),jj2] in Indices (Gauge (C,n)) by A199, A200, MATRIX_1:36;
A204: len (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = len (Cage (C,n)) by REVROT_1:14;
Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) by JORDAN1G:18;
then A205: LSeg ((Lower_Seq (C,n)),1) = LSeg ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by A194, SPPOL_2:9;
A206: 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 A140, REVROT_1:34;
then consider ii, jj being Element of NAT such that
A207: [ii,(jj + 1)] in Indices (Gauge (C,n)) and
A208: [ii,jj] in Indices (Gauge (C,n)) and
A209: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = (Gauge (C,n)) * (ii,(jj + 1)) and
A210: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1) = (Gauge (C,n)) * (ii,jj) by A87, A198, A204, A206, FINSEQ_6:92, JORDAN1I:23;
A211: (jj + 1) + 1 <> jj ;
A212: 1 <= jj by A208, MATRIX_1:38;
(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 A198, A206, FINSEQ_6:92;
then A213: ii = len (Gauge (C,n)) by A198, A207, A209, A201, A203, GOBOARD1:5;
then ii - 1 >= 4 - 1 by A202, XREAL_1:9;
then A214: ii - 1 >= 1 by XXREAL_0:2;
then A215: 1 <= ii -' 1 by XREAL_0:def_2;
A216: jj <= width (Gauge (C,n)) by A208, MATRIX_1:38;
then A217: ((Gauge (C,n)) * ((len (Gauge (C,n))),jj)) `1 = E-bound (L~ (Cage (C,n))) by A11, A212, JORDAN1A:71;
A218: jj + 1 <= width (Gauge (C,n)) by A207, MATRIX_1:38;
ii + 1 <> ii ;
then A219: right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) = cell ((Gauge (C,n)),(ii -' 1),jj) by A87, A204, A197, A207, A208, A209, A210, A211, GOBOARD5:def_6;
A220: ii <= len (Gauge (C,n)) by A208, MATRIX_1:38;
A221: 1 <= ii by A208, MATRIX_1:38;
A222: ii <= len (Gauge (C,n)) by A207, MATRIX_1:38;
A223: 1 <= jj + 1 by A207, MATRIX_1:38;
then A224: E-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * ((len (Gauge (C,n))),(jj + 1))) `1 by A11, A218, JORDAN1A:71;
A225: 1 <= ii by A207, MATRIX_1:38;
then A226: (ii -' 1) + 1 = ii by XREAL_1:235;
then A227: ii -' 1 < len (Gauge (C,n)) by A222, NAT_1:13;
then A228: ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 = ((Gauge (C,n)) * (1,(jj + 1))) `2 by A223, A218, A215, GOBOARD5:1
.= ((Gauge (C,n)) * (ii,(jj + 1))) `2 by A225, A222, A223, A218, GOBOARD5:1 ;
A229: (E-max C) `2 = p `2 by A179, TOPREAL1:def_11;
then A230: p `2 <= ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 by A196, A222, A218, A212, A219, A226, A214, JORDAN9:17;
A231: ((Gauge (C,n)) * ((ii -' 1),jj)) `2 = ((Gauge (C,n)) * (1,jj)) `2 by A212, A216, A215, A227, GOBOARD5:1
.= ((Gauge (C,n)) * (ii,jj)) `2 by A221, A220, A212, A216, GOBOARD5:1 ;
((Gauge (C,n)) * ((ii -' 1),jj)) `2 <= p `2 by A229, A196, A222, A218, A212, A219, A226, A214, JORDAN9:17;
then p in LSeg (((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1),((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1))) by A188, A209, A210, A213, A230, A231, A228, A217, A224, GOBOARD7:7;
then A232: p in LSeg ((Lower_Seq (C,n)),1) by A87, A205, A204, TOPREAL1:def_3;
1 <= ((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n)) by A39, FINSEQ_4:21;
then A233: LSeg ((mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n))))),(Index (p,do))) = LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n)))) -' 1)) by A193, A191, A190, JORDAN4:19;
1 <= Index (((Gauge (C,n)) * (j,i)),(Lower_Seq (C,n))) by A9, JORDAN3:8;
then A234: 1 + 1 <= ((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n)) by A189, XREAL_1:7;
then (Index (p,do)) + (((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n))) >= (1 + 1) + 1 by A191, XREAL_1:7;
then ((Index (p,do)) + (((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n)))) - 1 >= ((1 + 1) + 1) - 1 by XREAL_1:9;
then A235: ((Index (p,do)) + (((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n)))) -' 1 >= 1 + 1 by XREAL_0:def_2;
now__::_thesis:_contradiction
percases ( ((Index (p,do)) + (((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n)))) -' 1 > 1 + 1 or ((Index (p,do)) + (((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ) by A235, XXREAL_0:1;
suppose ((Index (p,do)) + (((Gauge (C,n)) * (j,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,do)) + (((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n)))) -' 1)) by TOPREAL1:def_7;
hence contradiction by A232, A181, A187, A233, XBOOLE_0:3; ::_thesis: verum
end;
supposeA236: ((Index (p,do)) + (((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ; ::_thesis: contradiction
then 1 + 1 = ((Index (p,do)) + (((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n)))) - 1 by XREAL_0:def_2;
then (1 + 1) + 1 = (Index (p,do)) + (((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n))) ;
then A237: ((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n)) = 2 by A191, A234, JORDAN1E:6;
(LSeg ((Lower_Seq (C,n)),1)) /\ (LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (j,i)) .. (Lower_Seq (C,n)))) -' 1))) = {((Lower_Seq (C,n)) /. 2)} by A25, A236, TOPREAL1:def_6;
then p in {((Lower_Seq (C,n)) /. 2)} by A232, A181, A187, A233, XBOOLE_0:def_4;
then A238: p = (Lower_Seq (C,n)) /. 2 by TARSKI:def_1;
then A239: p in rng (Lower_Seq (C,n)) by A195, PARTFUN2:2;
p .. (Lower_Seq (C,n)) = 2 by A195, A238, FINSEQ_5:41;
then p = (Gauge (C,n)) * (j,i) by A39, A237, A239, FINSEQ_5:9;
then ((Gauge (C,n)) * (j,i)) `1 = E-bound (L~ (Cage (C,n))) by A238, JORDAN1G:32;
then ((Gauge (C,n)) * (j,i)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 by A1, A11, A13, JORDAN1A:71;
hence contradiction by A2, A3, A16, A69, JORDAN1G:7; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
end;
end;
then east_halfline (E-max C) c= (L~ godo) ` by SUBSET_1:23;
then consider W being Subset of (TOP-REAL 2) such that
A240: W is_a_component_of (L~ godo) ` and
A241: east_halfline (E-max C) c= W by GOBOARD9:3;
not W is bounded by A241, JORDAN2C:121, RLTOPSP1:42;
then W is_outside_component_of L~ godo by A240, JORDAN2C:def_3;
then W c= UBD (L~ godo) by JORDAN2C:23;
then A242: east_halfline (E-max C) c= UBD (L~ godo) by A241, XBOOLE_1:1;
E-max C in east_halfline (E-max C) by TOPREAL1:38;
then E-max C in UBD (L~ godo) by A242;
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 A243: ( Upper_Arc C meets (L~ go) \/ (L~ pion1) or Upper_Arc C meets L~ do ) by A143, XBOOLE_1:70;
A244: Upper_Arc C c= C by JORDAN6:61;
percases ( Upper_Arc C meets L~ go or Upper_Arc C meets L~ pion1 or Upper_Arc C meets L~ do ) by A243, XBOOLE_1:70;
suppose Upper_Arc C meets L~ go ; ::_thesis: contradiction
then Upper_Arc C meets L~ (Cage (C,n)) by A48, A145, XBOOLE_1:1, XBOOLE_1:63;
hence contradiction by A244, JORDAN10:5, XBOOLE_1:63; ::_thesis: verum
end;
suppose Upper_Arc C meets L~ pion1 ; ::_thesis: contradiction
hence contradiction by A8, A83, A150; ::_thesis: verum
end;
suppose Upper_Arc C meets L~ do ; ::_thesis: contradiction
then Upper_Arc C meets L~ (Cage (C,n)) by A55, A146, XBOOLE_1:1, XBOOLE_1:63;
hence contradiction by A244, JORDAN10:5, XBOOLE_1:63; ::_thesis: verum
end;
end;
end;
theorem Th30: :: JORDAN15:30
for n being Element of NAT
for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (k,i) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Lower_Seq (C,n)) holds
LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Lower_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (k,i) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Lower_Seq (C,n)) 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 Element of NAT st 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (k,i) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Lower_Seq (C,n)) holds
LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Lower_Arc C
let i, j, k be Element of NAT ; ::_thesis: ( 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (k,i) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Lower_Seq (C,n)) implies LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Lower_Arc C )
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: (Gauge (C,n)) * (k,i) in L~ (Upper_Seq (C,n)) and
A7: (Gauge (C,n)) * (j,i) in L~ (Lower_Seq (C,n)) ; ::_thesis: LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Lower_Arc C
consider j1, k1 being Element of NAT such that
A8: j <= j1 and
A9: j1 <= k1 and
A10: k1 <= k and
A11: (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))} and
A12: (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (k1,i))} by A1, A2, A3, A4, A5, A6, A7, Th14;
A13: k1 < len (Gauge (C,n)) by A3, A10, XXREAL_0:2;
1 < j1 by A1, A8, XXREAL_0:2;
then LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i))) meets Lower_Arc C by A4, A5, A9, A11, A12, A13, Th28;
hence LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Lower_Arc C by A1, A3, A4, A5, A8, A9, A10, Th6, XBOOLE_1:63; ::_thesis: verum
end;
theorem Th31: :: JORDAN15:31
for n being Element of NAT
for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (k,i) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Lower_Seq (C,n)) holds
LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Upper_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (k,i) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Lower_Seq (C,n)) holds
LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Upper_Arc C
let C be Simple_closed_curve; ::_thesis: for i, j, k being Element of NAT st 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (k,i) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Lower_Seq (C,n)) holds
LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Upper_Arc C
let i, j, k be Element of NAT ; ::_thesis: ( 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (k,i) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Lower_Seq (C,n)) implies LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Upper_Arc C )
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: (Gauge (C,n)) * (k,i) in L~ (Upper_Seq (C,n)) and
A7: (Gauge (C,n)) * (j,i) in L~ (Lower_Seq (C,n)) ; ::_thesis: LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Upper_Arc C
consider j1, k1 being Element of NAT such that
A8: j <= j1 and
A9: j1 <= k1 and
A10: k1 <= k and
A11: (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))} and
A12: (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (k1,i))} by A1, A2, A3, A4, A5, A6, A7, Th14;
A13: k1 < len (Gauge (C,n)) by A3, A10, XXREAL_0:2;
1 < j1 by A1, A8, XXREAL_0:2;
then LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i))) meets Upper_Arc C by A4, A5, A9, A11, A12, A13, Th29;
hence LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Upper_Arc C by A1, A3, A4, A5, A8, A9, A10, Th6, XBOOLE_1:63; ::_thesis: verum
end;
theorem Th32: :: JORDAN15:32
for n being Element of NAT
for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & n > 0 & (Gauge (C,n)) * (k,i) in Upper_Arc (L~ (Cage (C,n))) & (Gauge (C,n)) * (j,i) in Lower_Arc (L~ (Cage (C,n))) holds
LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Lower_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & n > 0 & (Gauge (C,n)) * (k,i) in Upper_Arc (L~ (Cage (C,n))) & (Gauge (C,n)) * (j,i) in Lower_Arc (L~ (Cage (C,n))) 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 Element of NAT st 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & n > 0 & (Gauge (C,n)) * (k,i) in Upper_Arc (L~ (Cage (C,n))) & (Gauge (C,n)) * (j,i) in Lower_Arc (L~ (Cage (C,n))) holds
LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Lower_Arc C
let i, j, k be Element of NAT ; ::_thesis: ( 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & n > 0 & (Gauge (C,n)) * (k,i) in Upper_Arc (L~ (Cage (C,n))) & (Gauge (C,n)) * (j,i) in Lower_Arc (L~ (Cage (C,n))) implies LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Lower_Arc C )
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: n > 0 and
A7: (Gauge (C,n)) * (k,i) in Upper_Arc (L~ (Cage (C,n))) and
A8: (Gauge (C,n)) * (j,i) in Lower_Arc (L~ (Cage (C,n))) ; ::_thesis: LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Lower_Arc C
A9: L~ (Lower_Seq (C,n)) = Lower_Arc (L~ (Cage (C,n))) by A6, JORDAN1G:56;
L~ (Upper_Seq (C,n)) = Upper_Arc (L~ (Cage (C,n))) by A6, JORDAN1G:55;
hence LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Lower_Arc C by A1, A2, A3, A4, A5, A7, A8, A9, Th30; ::_thesis: verum
end;
theorem Th33: :: JORDAN15:33
for n being Element of NAT
for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & n > 0 & (Gauge (C,n)) * (k,i) in Upper_Arc (L~ (Cage (C,n))) & (Gauge (C,n)) * (j,i) in Lower_Arc (L~ (Cage (C,n))) holds
LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Upper_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & n > 0 & (Gauge (C,n)) * (k,i) in Upper_Arc (L~ (Cage (C,n))) & (Gauge (C,n)) * (j,i) in Lower_Arc (L~ (Cage (C,n))) holds
LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Upper_Arc C
let C be Simple_closed_curve; ::_thesis: for i, j, k being Element of NAT st 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & n > 0 & (Gauge (C,n)) * (k,i) in Upper_Arc (L~ (Cage (C,n))) & (Gauge (C,n)) * (j,i) in Lower_Arc (L~ (Cage (C,n))) holds
LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Upper_Arc C
let i, j, k be Element of NAT ; ::_thesis: ( 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & n > 0 & (Gauge (C,n)) * (k,i) in Upper_Arc (L~ (Cage (C,n))) & (Gauge (C,n)) * (j,i) in Lower_Arc (L~ (Cage (C,n))) implies LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Upper_Arc C )
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: n > 0 and
A7: (Gauge (C,n)) * (k,i) in Upper_Arc (L~ (Cage (C,n))) and
A8: (Gauge (C,n)) * (j,i) in Lower_Arc (L~ (Cage (C,n))) ; ::_thesis: LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Upper_Arc C
A9: L~ (Lower_Seq (C,n)) = Lower_Arc (L~ (Cage (C,n))) by A6, JORDAN1G:56;
L~ (Upper_Seq (C,n)) = Upper_Arc (L~ (Cage (C,n))) by A6, JORDAN1G:55;
hence LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Upper_Arc C by A1, A2, A3, A4, A5, A7, A8, A9, Th31; ::_thesis: verum
end;
theorem :: JORDAN15:34
for n being Element of NAT
for C being Simple_closed_curve
for j, k being Element of NAT st 1 < j & j <= k & k < len (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1))))) in Lower_Arc (L~ (Cage (C,(n + 1)))) holds
LSeg (((Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1)))))),((Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))))) meets Lower_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for j, k being Element of NAT st 1 < j & j <= k & k < len (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1))))) in Lower_Arc (L~ (Cage (C,(n + 1)))) holds
LSeg (((Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1)))))),((Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))))) meets Lower_Arc C
let C be Simple_closed_curve; ::_thesis: for j, k being Element of NAT st 1 < j & j <= k & k < len (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1))))) in Lower_Arc (L~ (Cage (C,(n + 1)))) holds
LSeg (((Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1)))))),((Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))))) meets Lower_Arc C
let j, k be Element of NAT ; ::_thesis: ( 1 < j & j <= k & k < len (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1))))) in Lower_Arc (L~ (Cage (C,(n + 1)))) implies LSeg (((Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1)))))),((Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))))) meets Lower_Arc C )
assume that
A1: 1 < j and
A2: j <= k and
A3: k < len (Gauge (C,(n + 1))) and
A4: (Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))) in Upper_Arc (L~ (Cage (C,(n + 1)))) and
A5: (Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1))))) in Lower_Arc (L~ (Cage (C,(n + 1)))) ; ::_thesis: LSeg (((Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1)))))),((Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))))) meets Lower_Arc C
A6: len (Gauge (C,(n + 1))) >= 4 by JORDAN8:10;
then len (Gauge (C,(n + 1))) >= 3 by XXREAL_0:2;
then Center (Gauge (C,(n + 1))) < len (Gauge (C,(n + 1))) by JORDAN1B:15;
then A7: Center (Gauge (C,(n + 1))) < width (Gauge (C,(n + 1))) by JORDAN8:def_1;
len (Gauge (C,(n + 1))) >= 2 by A6, XXREAL_0:2;
then 1 < Center (Gauge (C,(n + 1))) by JORDAN1B:14;
hence LSeg (((Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1)))))),((Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))))) meets Lower_Arc C by A1, A2, A3, A4, A5, A7, Th32; ::_thesis: verum
end;
theorem :: JORDAN15:35
for n being Element of NAT
for C being Simple_closed_curve
for j, k being Element of NAT st 1 < j & j <= k & k < len (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1))))) in Lower_Arc (L~ (Cage (C,(n + 1)))) holds
LSeg (((Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1)))))),((Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))))) meets Upper_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for j, k being Element of NAT st 1 < j & j <= k & k < len (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1))))) in Lower_Arc (L~ (Cage (C,(n + 1)))) holds
LSeg (((Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1)))))),((Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))))) meets Upper_Arc C
let C be Simple_closed_curve; ::_thesis: for j, k being Element of NAT st 1 < j & j <= k & k < len (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1))))) in Lower_Arc (L~ (Cage (C,(n + 1)))) holds
LSeg (((Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1)))))),((Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))))) meets Upper_Arc C
let j, k be Element of NAT ; ::_thesis: ( 1 < j & j <= k & k < len (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1))))) in Lower_Arc (L~ (Cage (C,(n + 1)))) implies LSeg (((Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1)))))),((Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))))) meets Upper_Arc C )
assume that
A1: 1 < j and
A2: j <= k and
A3: k < len (Gauge (C,(n + 1))) and
A4: (Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))) in Upper_Arc (L~ (Cage (C,(n + 1)))) and
A5: (Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1))))) in Lower_Arc (L~ (Cage (C,(n + 1)))) ; ::_thesis: LSeg (((Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1)))))),((Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))))) meets Upper_Arc C
A6: len (Gauge (C,(n + 1))) >= 4 by JORDAN8:10;
then len (Gauge (C,(n + 1))) >= 3 by XXREAL_0:2;
then Center (Gauge (C,(n + 1))) < len (Gauge (C,(n + 1))) by JORDAN1B:15;
then A7: Center (Gauge (C,(n + 1))) < width (Gauge (C,(n + 1))) by JORDAN8:def_1;
len (Gauge (C,(n + 1))) >= 2 by A6, XXREAL_0:2;
then 1 < Center (Gauge (C,(n + 1))) by JORDAN1B:14;
hence LSeg (((Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1)))))),((Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))))) meets Upper_Arc C by A1, A2, A3, A4, A5, A7, Th33; ::_thesis: verum
end;
theorem Th36: :: JORDAN15:36
for n being Element of NAT
for C being Simple_closed_curve
for i, j, k being Element of 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
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i, j, k being Element of 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 Element of 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 Element of 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 A7, XBOOLE_0:def_4;
(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 A6, XBOOLE_0:def_4;
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 A2, A3, A11, XXREAL_0:2;
A14: 1 < k by A1, A2, XXREAL_0:2;
A15: k < width (Gauge (C,n)) by A3, JORDAN8:def_1;
A16: [j,i] in Indices (Gauge (C,n)) by A1, A4, A5, A11, A13, MATRIX_1:36;
A17: [k,i] in Indices (Gauge (C,n)) by A3, A4, A5, A14, MATRIX_1:36;
set go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (j,i)));
set do = 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 A4, A5, A11, JORDAN1A:73 ;
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 A14, A15, MATRIX_1:36;
A23: [1,i] in Indices (Gauge (C,n)) by A4, A5, A21, MATRIX_1:36;
then A24: (Gauge (C,n)) * (j,i) <> (Upper_Seq (C,n)) . 1 by A1, A16, A19, A20, JORDAN1G:7;
then reconsider go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (j,i))) as being_S-Seq FinSequence of (TOP-REAL 2) by A10, JORDAN3:35;
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 A26, FINSEQ_3:25;
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 A4, A5, A11, JORDAN1A:73 ;
then A29: (Gauge (C,n)) * (k,i) <> (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) by A1, A2, A17, A23, A28, JORDAN1G:7;
then reconsider do = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (k,i))) 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 A14, A15, A21, MATRIX_1:36;
A31: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by A27, 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 A3, A14, JORDAN1A:71 ;
then A32: (Gauge (C,n)) * (k,i) <> (Lower_Seq (C,n)) . 1 by A3, A17, A30, A31, JORDAN1G:7;
A33: len go >= 1 + 1 by TOPREAL1:def_8;
A34: (Gauge (C,n)) * (j,i) in rng (Upper_Seq (C,n)) by A1, A4, A5, A10, A11, A13, JORDAN1G:4, JORDAN1J:40;
then A35: go is_sequence_on Gauge (C,n) by JORDAN1G:4, JORDAN1J:38;
A36: len do >= 1 + 1 by TOPREAL1:def_8;
A37: (Gauge (C,n)) * (k,i) in rng (Lower_Seq (C,n)) by A4, A5, A9, A11, A14, A15, JORDAN1G:5, JORDAN1J:40;
then A38: do is_sequence_on Gauge (C,n) by JORDAN1G:5, JORDAN1J:39;
reconsider go = go as non constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A33, A35, JGRAPH_1:12, JORDAN8:5;
reconsider do = do as non 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)) * (j,i) by A10, JORDAN3:24 ;
len do >= 1 by A36, XXREAL_0:2;
then 1 in dom do by FINSEQ_3:25;
then A42: do /. 1 = do . 1 by PARTFUN1:def_6
.= (Gauge (C,n)) * (k,i) by A9, JORDAN3:23 ;
reconsider m = (len go) - 1 as Element of 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, XBOOLE_1:1;
then A47: (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i)))) c= {((Gauge (C,n)) * (j,i))} by A6, XBOOLE_1:26;
m >= 1 by A33, XREAL_1:19;
then A48: LSeg (go,m) = LSeg ((go /. m),((Gauge (C,n)) * (j,i))) by A41, A43, TOPREAL1:def_3;
{((Gauge (C,n)) * (j,i))} c= (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))))
proof
let x be set ; :: 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 A48, RLTOPSP1:68;
hence x in (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i)))) by A50, A49, XBOOLE_0:def_4; ::_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, XBOOLE_0:def_10;
A52: LSeg (do,1) c= L~ do by TOPREAL3:19;
A53: L~ do c= L~ (Lower_Seq (C,n)) by A9, JORDAN3:42;
then LSeg (do,1) c= L~ (Lower_Seq (C,n)) by A52, XBOOLE_1:1;
then A54: (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i)))) c= {((Gauge (C,n)) * (k,i))} by A7, XBOOLE_1:26;
A55: LSeg (do,1) = LSeg (((Gauge (C,n)) * (k,i)),(do /. (1 + 1))) by A36, A42, TOPREAL1:def_3;
{((Gauge (C,n)) * (k,i))} c= (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (k,i))} or x in (LSeg (do,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 (do,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 (do,1) by A55, RLTOPSP1:68;
hence x in (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i)))) by A57, A56, XBOOLE_0:def_4; ::_thesis: verum
end;
then A58: (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i)))) /\ (LSeg (do,1)) = {((Gauge (C,n)) * (k,i))} by A54, XBOOLE_0:def_10;
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
.= do /. (len do) by A9, JORDAN1J:35 ;
A61: rng go c= L~ go by A33, SPPOL_2:18;
A62: rng do c= L~ do by A36, SPPOL_2:18;
A63: {(go /. 1)} c= (L~ go) /\ (L~ do)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(go /. 1)} or x in (L~ go) /\ (L~ do) )
assume x in {(go /. 1)} ; ::_thesis: x in (L~ go) /\ (L~ do)
then A64: x = go /. 1 by TARSKI:def_1;
then A65: x in rng go by FINSEQ_6:42;
x in rng do by A60, A64, REVROT_1:3;
hence x in (L~ go) /\ (L~ do) by A61, A62, A65, XBOOLE_0:def_4; ::_thesis: verum
end;
A66: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by A27, PARTFUN1:def_6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;
A67: [(len (Gauge (C,n))),j] in Indices (Gauge (C,n)) by A1, A13, A21, MATRIX_1:36;
(L~ go) /\ (L~ do) c= {(go /. 1)}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ go) /\ (L~ do) or x in {(go /. 1)} )
assume A68: x in (L~ go) /\ (L~ do) ; ::_thesis: x in {(go /. 1)}
then A69: x in L~ do 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 A9, A66, A69, JORDAN1E:7;
((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 = E-bound (L~ (Cage (C,n))) by A1, A11, A13, JORDAN1A:71;
then (E-max (L~ (Cage (C,n)))) `1 <> E-bound (L~ (Cage (C,n))) by A3, A17, A67, A71, JORDAN1G:7;
hence contradiction by EUCLID:52; ::_thesis: 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; ::_thesis: verum
end;
then A72: (L~ go) /\ (L~ do) = {(go /. 1)} by A63, XBOOLE_0:def_10;
set W2 = go /. 2;
A73: 2 in dom go by A33, FINSEQ_3:25;
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 A3, A14, JORDAN1A:73;
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 A34, JORDAN1G:49
.= (Upper_Seq (C,n)) | (((Gauge (C,n)) * (j,i)) .. (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;
A76: W-min (L~ (Cage (C,n))) in rng go by A59, FINSEQ_6:42;
set pion = <*((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))*>;
A77: now__::_thesis:_for_n_being_Element_of_NAT_st_n_in_dom_<*((Gauge_(C,n))_*_(j,i)),((Gauge_(C,n))_*_(k,i))*>_holds_
ex_j,_i_being_Element_of_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 Element of NAT ; ::_thesis: ( n in dom <*((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))*> implies ex j, i being Element of 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 Element of 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 FINSEQ_1:2, FINSEQ_1:89;
then ( n = 1 or n = 2 ) by TARSKI:def_2;
hence ex j, i being Element of 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 A16, A17, FINSEQ_4:17; ::_thesis: verum
end;
A78: (Gauge (C,n)) * (k,i) <> (Gauge (C,n)) * (j,i) by A12, A16, A17, GOBOARD1:5;
((Gauge (C,n)) * (k,i)) `2 = ((Gauge (C,n)) * (1,i)) `2 by A3, A4, A5, A14, GOBOARD5:1
.= ((Gauge (C,n)) * (j,i)) `2 by A1, A4, A5, A11, A13, GOBOARD5:1 ;
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 A78, JORDAN1B:8;
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)) * (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 A77, GOBOARD3:2;
reconsider pion1 = pion1 as being_S-Seq FinSequence of (TOP-REAL 2) by A80;
set godo = (go ^' pion1) ^' do;
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) ^' do) >= len (go ^' pion1) by TOPREAL8:7;
then A90: 1 + 1 <= len ((go ^' pion1) ^' do) 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)) * (j,i)),((Gauge (C,n)) * (k,i))*> /. (len <*((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))*>) by A83, GRAPH_2:54
.= <*((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))*> /. 2 by FINSEQ_1:44
.= do /. 1 by A42, FINSEQ_4:17 ;
then A95: (go ^' pion1) ^' do is_sequence_on Gauge (C,n) by A38, A93, TOPREAL8:12;
LSeg (pion1,1) c= L~ <*((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))*> by A81, TOPREAL3:19;
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 A44, A51, XBOOLE_1:27;
A97: len pion1 >= 1 + 1 by A84, FINSEQ_1:44;
{((Gauge (C,n)) * (j,i))} c= (LSeg (go,m)) /\ (LSeg (pion1,1))
proof
let x be set ; :: 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 A48, RLTOPSP1:68;
(Gauge (C,n)) * (j,i) 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; ::_thesis: verum
end;
then (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) = {(go /. (len go))} by A41, A44, A96, XBOOLE_0:def_10;
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 GRAPH_2:13;
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)) * (j,i)),((Gauge (C,n)) * (k,i))*> by A81, TOPREAL3:19;
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 (do,1)) c= {((Gauge (C,n)) * (k,i))} by A58, XBOOLE_1:27;
{((Gauge (C,n)) * (k,i))} c= (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (k,i))} or x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) )
assume x in {((Gauge (C,n)) * (k,i))} ; ::_thesis: x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,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 A83, A104, FINSEQ_1:44
.= (Gauge (C,n)) * (k,i) by FINSEQ_4:17 ;
then A109: (Gauge (C,n)) * (k,i) in LSeg (pion1,((len pion1) -' 1)) by A103, A104, TOPREAL1:21;
(Gauge (C,n)) * (k,i) in LSeg (do,1) by A55, RLTOPSP1:68;
hence x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) by A108, A109, XBOOLE_0:def_4; ::_thesis: verum
end;
then (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) = {((Gauge (C,n)) * (k,i))} by A107, XBOOLE_0:def_10;
then A110: (LSeg ((go ^' pion1),((len go) + ((len pion1) -' 2)))) /\ (LSeg (do,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)
proof
let x be set ; :: 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 A92, A114, REVROT_1:3;
hence x in (L~ go) /\ (L~ pion1) by A61, A112, A115, XBOOLE_0:def_4; ::_thesis: verum
end;
(L~ go) /\ (L~ pion1) c= {(pion1 /. 1)}
proof
let x be set ; :: 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 A116, XBOOLE_0:def_4;
then x in (L~ pion1) /\ (L~ (Upper_Seq (C,n))) by A46, A117, XBOOLE_0:def_4;
hence x in {(pion1 /. 1)} by A6, A41, A81, A92, SPPOL_2:21; ::_thesis: verum
end;
then A118: (L~ go) /\ (L~ pion1) = {(pion1 /. 1)} by A113, XBOOLE_0:def_10;
then A119: go ^' pion1 is s.n.c. by A92, JORDAN1J:54;
(rng go) /\ (rng pion1) c= {(pion1 /. 1)} by A61, A112, A118, XBOOLE_1:27;
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
.= do /. 1 by A42, FINSEQ_4:17 ;
A122: {(pion1 /. (len pion1))} c= (L~ do) /\ (L~ pion1)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(pion1 /. (len pion1))} or x in (L~ do) /\ (L~ pion1) )
assume x in {(pion1 /. (len pion1))} ; ::_thesis: x in (L~ do) /\ (L~ pion1)
then A123: x = pion1 /. (len pion1) by TARSKI:def_1;
then A124: x in rng pion1 by REVROT_1:3;
x in rng do by A83, A121, A123, FINSEQ_6:42;
hence x in (L~ do) /\ (L~ pion1) by A62, A112, A124, XBOOLE_0:def_4; ::_thesis: verum
end;
(L~ do) /\ (L~ pion1) c= {(pion1 /. (len pion1))}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ do) /\ (L~ pion1) or x in {(pion1 /. (len pion1))} )
assume A125: x in (L~ do) /\ (L~ pion1) ; ::_thesis: x in {(pion1 /. (len pion1))}
then A126: x in L~ pion1 by XBOOLE_0:def_4;
x in L~ do by A125, XBOOLE_0:def_4;
then x in (L~ pion1) /\ (L~ (Lower_Seq (C,n))) by A53, A126, XBOOLE_0:def_4;
hence x in {(pion1 /. (len pion1))} by A7, A42, A81, A83, A121, SPPOL_2:21; ::_thesis: verum
end;
then A127: (L~ do) /\ (L~ pion1) = {(pion1 /. (len pion1))} by A122, XBOOLE_0:def_10;
A128: (L~ (go ^' pion1)) /\ (L~ do) = ((L~ go) \/ (L~ pion1)) /\ (L~ do) by A92, TOPREAL8:35
.= {(go /. 1)} \/ {(do /. 1)} by A72, A83, A121, A127, XBOOLE_1:23
.= {((go ^' pion1) /. 1)} \/ {(do /. 1)} by GRAPH_2:53
.= {((go ^' pion1) /. 1),(do /. 1)} by ENUMSET1:1 ;
do /. (len do) = (go ^' pion1) /. 1 by A60, GRAPH_2:53;
then reconsider godo = (go ^' pion1) ^' do as non constant 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: 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 A129, TOPREAL1:1;
A132: E-max C in Lower_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__::_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 A34, FINSEQ_4:21;
then ((Gauge (C,n)) * (j,i)) .. (Upper_Seq (C,n)) = 1 by A137, XXREAL_0:1;
then (Gauge (C,n)) * (j,i) = (Upper_Seq (C,n)) /. 1 by A34, FINSEQ_5:38;
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 A90, A95, JORDAN9:27;
A141: L~ godo = (L~ (go ^' pion1)) \/ (L~ do) by A94, TOPREAL8:35
.= ((L~ go) \/ (L~ pion1)) \/ (L~ do) 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, XBOOLE_1:1;
A146: L~ do c= L~ (Cage (C,n)) by A53, A144, XBOOLE_1:1;
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~ do ) by A141, XBOOLE_0:def_3;
percases ( W-min C in L~ go or W-min C in L~ pion1 or W-min C in L~ do ) by A150, XBOOLE_0:def_3;
suppose W-min C in L~ go ; ::_thesis: contradiction
then C meets L~ (Cage (C,n)) by A145, A147, XBOOLE_0:3;
hence contradiction by JORDAN10:5; ::_thesis: verum
end;
suppose W-min C in L~ pion1 ; ::_thesis: contradiction
hence contradiction by A8, A81, A131, A148, XBOOLE_0:3; ::_thesis: verum
end;
suppose W-min C in L~ do ; ::_thesis: contradiction
then C meets L~ (Cage (C,n)) by A146, A147, XBOOLE_0:3;
hence contradiction by JORDAN10:5; ::_thesis: verum
end;
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 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, 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)) * (j,i)))),1,(Gauge (C,n))) by A34, A91, A136, JORDAN1J:52
.= right_cell ((go ^' pion1),1,(Gauge (C,n))) by A39, A93, JORDAN1J:51
.= right_cell (godo,1,(Gauge (C,n))) by A88, A95, JORDAN1J:51 ;
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 GRAPH_2:53
.= W-min (L~ (Cage (C,n))) by A59, GRAPH_2:53 ;
A153: len (Upper_Seq (C,n)) >= 2 by A18, XXREAL_0:2;
A154: godo /. 2 = (go ^' pion1) /. 2 by A87, GRAPH_2:57
.= (Upper_Seq (C,n)) /. 2 by A33, A75, GRAPH_2:57
.= ((Upper_Seq (C,n)) ^' (Lower_Seq (C,n))) /. 2 by A153, GRAPH_2:57
.= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 2 by JORDAN1E:11 ;
A155: (L~ go) \/ (L~ do) is compact by COMPTS_1:10;
W-min (L~ (Cage (C,n))) in (L~ go) \/ (L~ do) by A61, A76, XBOOLE_0:def_3;
then A156: W-min ((L~ go) \/ (L~ do)) = W-min (L~ (Cage (C,n))) by A145, A146, A155, JORDAN1J:21, XBOOLE_1:8;
A157: (W-min ((L~ go) \/ (L~ do))) `1 = W-bound ((L~ go) \/ (L~ do)) 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 A1, A2, A3, A4, A5, SPRECT_3:13;
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 A81, SPPOL_2:21;
((Gauge (C,n)) * (j,i)) `1 >= W-bound (L~ (Cage (C,n))) by A10, A143, PSCOMP_1:24;
then ((Gauge (C,n)) * (j,i)) `1 > W-bound (L~ (Cage (C,n))) by A74, XXREAL_0:1;
then W-min (((L~ go) \/ (L~ do)) \/ (L~ pion1)) = W-min ((L~ go) \/ (L~ do)) by A155, A156, A157, A158, A160, JORDAN1J:33;
then A161: W-min (L~ godo) = W-min (L~ (Cage (C,n))) by A141, A156, XBOOLE_1:4;
A162: rng godo c= L~ godo by A87, A89, SPPOL_2:18, XXREAL_0:2;
2 in dom godo by A90, FINSEQ_3:25;
then A163: 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 A161, PSCOMP_1:31
.= W-bound (L~ godo) by EUCLID:52 ;
then godo /. 2 in W-most (L~ godo) by A162, A163, SPRECT_2:12;
then (Rotate (godo,(W-min (L~ godo)))) /. 2 in W-most (L~ godo) by A152, A161, FINSEQ_6:89;
then reconsider godo = godo as non 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 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 (E-max C) misses L~ go
proof
assume east_halfline (E-max C) meets L~ go ; ::_thesis: contradiction
then consider p being set such that
A166: p in east_halfline (E-max C) and
A167: p in L~ go by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A166;
p in L~ (Upper_Seq (C,n)) by A46, A167;
then p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A143, A166, XBOOLE_0:def_4;
then A168: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;
then A169: p = E-max (L~ (Cage (C,n))) by A46, A167, JORDAN1J:46;
then E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (j,i) by A10, A164, A167, JORDAN1J:43;
then ((Gauge (C,n)) * (j,i)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by A3, A14, A168, A169, JORDAN1A:71;
hence contradiction by A2, A3, A16, A30, JORDAN1G:7; ::_thesis: verum
end;
now__::_thesis:_not_east_halfline_(E-max_C)_meets_L~_godo
assume east_halfline (E-max C) meets L~ godo ; ::_thesis: contradiction
then A170: ( east_halfline (E-max C) meets (L~ go) \/ (L~ pion1) or east_halfline (E-max C) meets L~ do ) by A141, XBOOLE_1:70;
percases ( 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~ do ) by A170, XBOOLE_1:70;
suppose east_halfline (E-max C) meets L~ go ; ::_thesis: contradiction
hence contradiction by A165; ::_thesis: verum
end;
suppose east_halfline (E-max C) meets L~ pion1 ; ::_thesis: contradiction
then consider p being set such that
A171: p in east_halfline (E-max C) and
A172: p in L~ pion1 by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A171;
A173: p `2 = (E-max C) `2 by A171, TOPREAL1:def_11;
k + 1 <= len (Gauge (C,n)) by A3, NAT_1:13;
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 A4, A5, A11, A14, A21, A174, JORDAN1A:18;
p `1 <= ((Gauge (C,n)) * (k,i)) `1 by A81, A148, A159, A172, TOPREAL1:3;
then p `1 <= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),1)) `1 by A175, XXREAL_0:2;
then p `1 <= E-bound C by A21, JORDAN8:12;
then A176: p `1 <= (E-max C) `1 by EUCLID:52;
p `1 >= (E-max C) `1 by A171, TOPREAL1:def_11;
then p `1 = (E-max C) `1 by A176, XXREAL_0:1;
then p = E-max C by A173, TOPREAL3:6;
hence contradiction by A8, A81, A132, A148, A172, XBOOLE_0:3; ::_thesis: verum
end;
suppose east_halfline (E-max C) meets L~ do ; ::_thesis: contradiction
then consider p being set such that
A177: p in east_halfline (E-max C) and
A178: p in L~ do by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A177;
A179: p in LSeg (do,(Index (p,do))) by A178, JORDAN3:9;
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 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)) * (k,i)),(Lower_Seq (C,n)))) + 1 = t by A181, A182, JORDAN3:12;
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 A9, A181, JORDAN3:26;
Index (p,do) < len do by A178, JORDAN3:8;
then Index (p,do) < (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (k,i)),(Lower_Seq (C,n)))) by A183, XREAL_0:def_2;
then (Index (p,do)) + 1 <= (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (k,i)),(Lower_Seq (C,n)))) by NAT_1:13;
then A184: Index (p,do) <= ((len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (k,i)),(Lower_Seq (C,n))))) - 1 by XREAL_1:19;
A185: do = mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n)))) by A37, JORDAN1J:37;
p in L~ (Lower_Seq (C,n)) by A53, A178;
then p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A144, A177, XBOOLE_0:def_4;
then A186: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;
A187: (Index (((Gauge (C,n)) * (k,i)),(Lower_Seq (C,n)))) + 1 = ((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)) by A32, A37, JORDAN1J:56;
0 + (Index (((Gauge (C,n)) * (k,i)),(Lower_Seq (C,n)))) < len (Lower_Seq (C,n)) by A9, JORDAN3:8;
then (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (k,i)),(Lower_Seq (C,n)))) > 0 by XREAL_1:20;
then Index (p,do) <= ((len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (k,i)),(Lower_Seq (C,n))))) - 1 by A184, XREAL_0:def_2;
then Index (p,do) <= (len (Lower_Seq (C,n))) - (((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n))) by A187;
then Index (p,do) <= (len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n))) by XREAL_0:def_2;
then A188: Index (p,do) < ((len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)))) + 1 by NAT_1:13;
A189: 1 <= Index (p,do) by A178, JORDAN3:8;
A190: ((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)) <= len (Lower_Seq (C,n)) by A37, FINSEQ_4:21;
((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)) <> len (Lower_Seq (C,n)) by A29, A37, FINSEQ_4:19;
then A191: ((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)) < len (Lower_Seq (C,n)) by A190, XXREAL_0:1;
A192: 1 + 1 <= len (Lower_Seq (C,n)) by A25, XXREAL_0:2;
then A193: 2 in dom (Lower_Seq (C,n)) by FINSEQ_3:25;
set tt = ((Index (p,do)) + (((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 Element of 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 A197, A198, MATRIX_1:36;
A202: len (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = len (Cage (C,n)) by REVROT_1:14;
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 A192, SPPOL_2:9;
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 A138, REVROT_1:34;
then consider ii, g being Element of 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 A85, A196, A202, A204, FINSEQ_6:92, JORDAN1I:23;
A209: (g + 1) + 1 <> g ;
A210: 1 <= g by A206, MATRIX_1:38;
(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 A196, A204, FINSEQ_6:92;
then A211: ii = len (Gauge (C,n)) by A196, A205, A207, A199, A201, GOBOARD1:5;
then ii - 1 >= 4 - 1 by A200, XREAL_1:9;
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 A206, MATRIX_1:38;
then A215: ((Gauge (C,n)) * ((len (Gauge (C,n))),g)) `1 = E-bound (L~ (Cage (C,n))) by A11, A210, JORDAN1A:71;
A216: g + 1 <= width (Gauge (C,n)) by A205, MATRIX_1:38;
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 A85, A202, A195, A205, A206, A207, A208, A209, GOBOARD5:def_6;
A218: ii <= len (Gauge (C,n)) by A206, MATRIX_1:38;
A219: 1 <= ii by A206, MATRIX_1:38;
A220: ii <= len (Gauge (C,n)) by A205, MATRIX_1:38;
A221: 1 <= g + 1 by A205, MATRIX_1:38;
then A222: E-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * ((len (Gauge (C,n))),(g + 1))) `1 by A11, A216, JORDAN1A:71;
A223: 1 <= ii by A205, MATRIX_1:38;
then A224: (ii -' 1) + 1 = ii by XREAL_1:235;
then A225: ii -' 1 < len (Gauge (C,n)) by A220, NAT_1:13;
then A226: ((Gauge (C,n)) * ((ii -' 1),(g + 1))) `2 = ((Gauge (C,n)) * (1,(g + 1))) `2 by A221, A216, A213, GOBOARD5:1
.= ((Gauge (C,n)) * (ii,(g + 1))) `2 by A223, A220, A221, A216, GOBOARD5:1 ;
A227: (E-max C) `2 = p `2 by A177, TOPREAL1:def_11;
then A228: p `2 <= ((Gauge (C,n)) * ((ii -' 1),(g + 1))) `2 by A194, A220, A216, A210, A217, A224, A212, JORDAN9:17;
A229: ((Gauge (C,n)) * ((ii -' 1),g)) `2 = ((Gauge (C,n)) * (1,g)) `2 by A210, A214, A213, A225, GOBOARD5:1
.= ((Gauge (C,n)) * (ii,g)) `2 by A219, A218, A210, A214, GOBOARD5:1 ;
((Gauge (C,n)) * ((ii -' 1),g)) `2 <= p `2 by A227, A194, A220, A216, A210, A217, A224, A212, JORDAN9:17;
then p in LSeg (((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1),((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1))) by A186, A207, A208, A211, A228, A229, A226, A215, A222, GOBOARD7:7;
then A230: p in LSeg ((Lower_Seq (C,n)),1) by A85, A203, A202, TOPREAL1:def_3;
1 <= ((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)) by A37, FINSEQ_4:21;
then A231: LSeg ((mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n))))),(Index (p,do))) = LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)))) -' 1)) by A191, A189, A188, JORDAN4:19;
1 <= Index (((Gauge (C,n)) * (k,i)),(Lower_Seq (C,n))) by A9, JORDAN3:8;
then A232: 1 + 1 <= ((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)) by A187, XREAL_1:7;
then (Index (p,do)) + (((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n))) >= (1 + 1) + 1 by A189, XREAL_1:7;
then ((Index (p,do)) + (((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)))) - 1 >= ((1 + 1) + 1) - 1 by XREAL_1:9;
then A233: ((Index (p,do)) + (((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)))) -' 1 >= 1 + 1 by XREAL_0:def_2;
now__::_thesis:_contradiction
percases ( ((Index (p,do)) + (((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)))) -' 1 > 1 + 1 or ((Index (p,do)) + (((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ) by A233, XXREAL_0:1;
suppose ((Index (p,do)) + (((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,do)) + (((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;
supposeA234: ((Index (p,do)) + (((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ; ::_thesis: contradiction
then 1 + 1 = ((Index (p,do)) + (((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)))) - 1 by XREAL_0:def_2;
then (1 + 1) + 1 = (Index (p,do)) + (((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n))) ;
then A235: ((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)) = 2 by A189, A232, JORDAN1E:6;
(LSeg ((Lower_Seq (C,n)),1)) /\ (LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)))) -' 1))) = {((Lower_Seq (C,n)) /. 2)} by A25, A234, TOPREAL1:def_6;
then p in {((Lower_Seq (C,n)) /. 2)} by A230, A179, A185, A231, XBOOLE_0:def_4;
then A236: p = (Lower_Seq (C,n)) /. 2 by TARSKI:def_1;
then A237: p in rng (Lower_Seq (C,n)) by A193, PARTFUN2:2;
p .. (Lower_Seq (C,n)) = 2 by A193, A236, FINSEQ_5:41;
then p = (Gauge (C,n)) * (k,i) by A37, A235, A237, FINSEQ_5:9;
then ((Gauge (C,n)) * (k,i)) `1 = E-bound (L~ (Cage (C,n))) by A236, JORDAN1G:32;
then ((Gauge (C,n)) * (k,i)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 by A1, A11, A13, JORDAN1A:71;
hence contradiction by A3, A17, A67, JORDAN1G:7; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
end;
end;
then east_halfline (E-max C) c= (L~ godo) ` by SUBSET_1:23;
then consider W being Subset of (TOP-REAL 2) such that
A238: W is_a_component_of (L~ godo) ` and
A239: east_halfline (E-max C) c= W by GOBOARD9:3;
not W is bounded by A239, JORDAN2C:121, RLTOPSP1:42;
then W is_outside_component_of L~ godo by A238, JORDAN2C:def_3;
then W c= UBD (L~ godo) by JORDAN2C:23;
then A240: east_halfline (E-max C) c= UBD (L~ godo) by A239, XBOOLE_1:1;
E-max C in east_halfline (E-max C) 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 A130, A131, A132, A140, A151, JORDAN1J:36;
then A241: ( Lower_Arc C meets (L~ go) \/ (L~ pion1) or Lower_Arc C meets L~ do ) by A141, XBOOLE_1:70;
A242: Lower_Arc C c= C by JORDAN6:61;
percases ( Lower_Arc C meets L~ go or Lower_Arc C meets L~ pion1 or Lower_Arc C meets L~ do ) by A241, XBOOLE_1:70;
suppose Lower_Arc C meets L~ go ; ::_thesis: contradiction
then Lower_Arc C meets L~ (Cage (C,n)) by A46, A143, XBOOLE_1:1, XBOOLE_1:63;
hence contradiction by A242, JORDAN10:5, XBOOLE_1:63; ::_thesis: verum
end;
suppose Lower_Arc C meets L~ pion1 ; ::_thesis: contradiction
hence contradiction by A8, A81, A148; ::_thesis: verum
end;
suppose Lower_Arc C meets L~ do ; ::_thesis: contradiction
then Lower_Arc C meets L~ (Cage (C,n)) by A53, A144, XBOOLE_1:1, XBOOLE_1:63;
hence contradiction by A242, JORDAN10:5, XBOOLE_1:63; ::_thesis: verum
end;
end;
end;
theorem Th37: :: JORDAN15:37
for n being Element of NAT
for C being Simple_closed_curve
for i, j, k being Element of 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 Upper_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i, j, k being Element of 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 Upper_Arc C
let C be Simple_closed_curve; ::_thesis: for i, j, k being Element of 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 Upper_Arc C
let i, j, k be Element of 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 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)) * (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 Upper_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 A7, XBOOLE_0:def_4;
(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 A6, XBOOLE_0:def_4;
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 A2, A3, A11, XXREAL_0:2;
A14: 1 < k by A1, A2, XXREAL_0:2;
A15: k < width (Gauge (C,n)) by A3, JORDAN8:def_1;
A16: [j,i] in Indices (Gauge (C,n)) by A1, A4, A5, A11, A13, MATRIX_1:36;
A17: [k,i] in Indices (Gauge (C,n)) by A3, A4, A5, A14, MATRIX_1:36;
set go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (j,i)));
set do = 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 A4, A5, A11, JORDAN1A:73 ;
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 A14, A15, MATRIX_1:36;
A23: [1,i] in Indices (Gauge (C,n)) by A4, A5, A21, MATRIX_1:36;
then A24: (Gauge (C,n)) * (j,i) <> (Upper_Seq (C,n)) . 1 by A1, A16, A19, A20, JORDAN1G:7;
then reconsider go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (j,i))) as being_S-Seq FinSequence of (TOP-REAL 2) by A10, JORDAN3:35;
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 A26, FINSEQ_3:25;
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 A4, A5, A11, JORDAN1A:73 ;
then A29: (Gauge (C,n)) * (k,i) <> (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) by A1, A2, A17, A23, A28, JORDAN1G:7;
then reconsider do = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (k,i))) 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 A14, A15, A21, MATRIX_1:36;
A31: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by A27, 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 A3, A14, JORDAN1A:71 ;
then A32: (Gauge (C,n)) * (k,i) <> (Lower_Seq (C,n)) . 1 by A3, A17, A30, A31, JORDAN1G:7;
A33: len go >= 1 + 1 by TOPREAL1:def_8;
A34: (Gauge (C,n)) * (j,i) in rng (Upper_Seq (C,n)) by A1, A4, A5, A10, A11, A13, JORDAN1G:4, JORDAN1J:40;
then A35: go is_sequence_on Gauge (C,n) by JORDAN1G:4, JORDAN1J:38;
A36: len do >= 1 + 1 by TOPREAL1:def_8;
A37: (Gauge (C,n)) * (k,i) in rng (Lower_Seq (C,n)) by A4, A5, A9, A11, A14, A15, JORDAN1G:5, JORDAN1J:40;
then A38: do is_sequence_on Gauge (C,n) by JORDAN1G:5, JORDAN1J:39;
reconsider go = go as non constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A33, A35, JGRAPH_1:12, JORDAN8:5;
reconsider do = do as non 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)) * (j,i) by A10, JORDAN3:24 ;
len do >= 1 by A36, XXREAL_0:2;
then 1 in dom do by FINSEQ_3:25;
then A42: do /. 1 = do . 1 by PARTFUN1:def_6
.= (Gauge (C,n)) * (k,i) by A9, JORDAN3:23 ;
reconsider m = (len go) - 1 as Element of 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, XBOOLE_1:1;
then A47: (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i)))) c= {((Gauge (C,n)) * (j,i))} by A6, XBOOLE_1:26;
m >= 1 by A33, XREAL_1:19;
then A48: LSeg (go,m) = LSeg ((go /. m),((Gauge (C,n)) * (j,i))) by A41, A43, TOPREAL1:def_3;
{((Gauge (C,n)) * (j,i))} c= (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))))
proof
let x be set ; :: 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 A48, RLTOPSP1:68;
hence x in (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i)))) by A50, A49, XBOOLE_0:def_4; ::_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, XBOOLE_0:def_10;
A52: LSeg (do,1) c= L~ do by TOPREAL3:19;
A53: L~ do c= L~ (Lower_Seq (C,n)) by A9, JORDAN3:42;
then LSeg (do,1) c= L~ (Lower_Seq (C,n)) by A52, XBOOLE_1:1;
then A54: (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i)))) c= {((Gauge (C,n)) * (k,i))} by A7, XBOOLE_1:26;
A55: LSeg (do,1) = LSeg (((Gauge (C,n)) * (k,i)),(do /. (1 + 1))) by A36, A42, TOPREAL1:def_3;
{((Gauge (C,n)) * (k,i))} c= (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (k,i))} or x in (LSeg (do,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 (do,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 (do,1) by A55, RLTOPSP1:68;
hence x in (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i)))) by A57, A56, XBOOLE_0:def_4; ::_thesis: verum
end;
then A58: (LSeg (((Gauge (C,n)) * (k,i)),((Gauge (C,n)) * (j,i)))) /\ (LSeg (do,1)) = {((Gauge (C,n)) * (k,i))} by A54, XBOOLE_0:def_10;
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
.= do /. (len do) by A9, JORDAN1J:35 ;
A61: rng go c= L~ go by A33, SPPOL_2:18;
A62: rng do c= L~ do by A36, SPPOL_2:18;
A63: {(go /. 1)} c= (L~ go) /\ (L~ do)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(go /. 1)} or x in (L~ go) /\ (L~ do) )
assume x in {(go /. 1)} ; ::_thesis: x in (L~ go) /\ (L~ do)
then A64: x = go /. 1 by TARSKI:def_1;
then A65: x in rng go by FINSEQ_6:42;
x in rng do by A60, A64, REVROT_1:3;
hence x in (L~ go) /\ (L~ do) by A61, A62, A65, XBOOLE_0:def_4; ::_thesis: verum
end;
A66: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by A27, PARTFUN1:def_6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;
A67: [(len (Gauge (C,n))),j] in Indices (Gauge (C,n)) by A1, A13, A21, MATRIX_1:36;
(L~ go) /\ (L~ do) c= {(go /. 1)}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ go) /\ (L~ do) or x in {(go /. 1)} )
assume A68: x in (L~ go) /\ (L~ do) ; ::_thesis: x in {(go /. 1)}
then A69: x in L~ do 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 A9, A66, A69, JORDAN1E:7;
((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 = E-bound (L~ (Cage (C,n))) by A1, A11, A13, JORDAN1A:71;
then (E-max (L~ (Cage (C,n)))) `1 <> E-bound (L~ (Cage (C,n))) by A3, A17, A67, A71, JORDAN1G:7;
hence contradiction by EUCLID:52; ::_thesis: 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; ::_thesis: verum
end;
then A72: (L~ go) /\ (L~ do) = {(go /. 1)} by A63, XBOOLE_0:def_10;
set W2 = go /. 2;
A73: 2 in dom go by A33, FINSEQ_3:25;
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 A3, A14, JORDAN1A:73;
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 A34, JORDAN1G:49
.= (Upper_Seq (C,n)) | (((Gauge (C,n)) * (j,i)) .. (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;
A76: W-min (L~ (Cage (C,n))) in rng go by A59, FINSEQ_6:42;
set pion = <*((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))*>;
A77: now__::_thesis:_for_n_being_Element_of_NAT_st_n_in_dom_<*((Gauge_(C,n))_*_(j,i)),((Gauge_(C,n))_*_(k,i))*>_holds_
ex_j,_i_being_Element_of_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 Element of NAT ; ::_thesis: ( n in dom <*((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))*> implies ex j, i being Element of 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 Element of 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 FINSEQ_1:2, FINSEQ_1:89;
then ( n = 1 or n = 2 ) by TARSKI:def_2;
hence ex j, i being Element of 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 A16, A17, FINSEQ_4:17; ::_thesis: verum
end;
A78: (Gauge (C,n)) * (k,i) <> (Gauge (C,n)) * (j,i) by A12, A16, A17, GOBOARD1:5;
((Gauge (C,n)) * (k,i)) `2 = ((Gauge (C,n)) * (1,i)) `2 by A3, A4, A5, A14, GOBOARD5:1
.= ((Gauge (C,n)) * (j,i)) `2 by A1, A4, A5, A11, A13, GOBOARD5:1 ;
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 A78, JORDAN1B:8;
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)) * (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 A77, GOBOARD3:2;
reconsider pion1 = pion1 as being_S-Seq FinSequence of (TOP-REAL 2) by A80;
set godo = (go ^' pion1) ^' do;
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) ^' do) >= len (go ^' pion1) by TOPREAL8:7;
then A90: 1 + 1 <= len ((go ^' pion1) ^' do) 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)) * (j,i)),((Gauge (C,n)) * (k,i))*> /. (len <*((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))*>) by A83, GRAPH_2:54
.= <*((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))*> /. 2 by FINSEQ_1:44
.= do /. 1 by A42, FINSEQ_4:17 ;
then A95: (go ^' pion1) ^' do is_sequence_on Gauge (C,n) by A38, A93, TOPREAL8:12;
LSeg (pion1,1) c= L~ <*((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))*> by A81, TOPREAL3:19;
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 A44, A51, XBOOLE_1:27;
A97: len pion1 >= 1 + 1 by A84, FINSEQ_1:44;
{((Gauge (C,n)) * (j,i))} c= (LSeg (go,m)) /\ (LSeg (pion1,1))
proof
let x be set ; :: 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 A48, RLTOPSP1:68;
(Gauge (C,n)) * (j,i) 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; ::_thesis: verum
end;
then (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) = {(go /. (len go))} by A41, A44, A96, XBOOLE_0:def_10;
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 GRAPH_2:13;
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)) * (j,i)),((Gauge (C,n)) * (k,i))*> by A81, TOPREAL3:19;
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 (do,1)) c= {((Gauge (C,n)) * (k,i))} by A58, XBOOLE_1:27;
{((Gauge (C,n)) * (k,i))} c= (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (k,i))} or x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) )
assume x in {((Gauge (C,n)) * (k,i))} ; ::_thesis: x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,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 A83, A104, FINSEQ_1:44
.= (Gauge (C,n)) * (k,i) by FINSEQ_4:17 ;
then A109: (Gauge (C,n)) * (k,i) in LSeg (pion1,((len pion1) -' 1)) by A103, A104, TOPREAL1:21;
(Gauge (C,n)) * (k,i) in LSeg (do,1) by A55, RLTOPSP1:68;
hence x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) by A108, A109, XBOOLE_0:def_4; ::_thesis: verum
end;
then (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) = {((Gauge (C,n)) * (k,i))} by A107, XBOOLE_0:def_10;
then A110: (LSeg ((go ^' pion1),((len go) + ((len pion1) -' 2)))) /\ (LSeg (do,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)
proof
let x be set ; :: 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 A92, A114, REVROT_1:3;
hence x in (L~ go) /\ (L~ pion1) by A61, A112, A115, XBOOLE_0:def_4; ::_thesis: verum
end;
(L~ go) /\ (L~ pion1) c= {(pion1 /. 1)}
proof
let x be set ; :: 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 A116, XBOOLE_0:def_4;
then x in (L~ pion1) /\ (L~ (Upper_Seq (C,n))) by A46, A117, XBOOLE_0:def_4;
hence x in {(pion1 /. 1)} by A6, A41, A81, A92, SPPOL_2:21; ::_thesis: verum
end;
then A118: (L~ go) /\ (L~ pion1) = {(pion1 /. 1)} by A113, XBOOLE_0:def_10;
then A119: go ^' pion1 is s.n.c. by A92, JORDAN1J:54;
(rng go) /\ (rng pion1) c= {(pion1 /. 1)} by A61, A112, A118, XBOOLE_1:27;
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
.= do /. 1 by A42, FINSEQ_4:17 ;
A122: {(pion1 /. (len pion1))} c= (L~ do) /\ (L~ pion1)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(pion1 /. (len pion1))} or x in (L~ do) /\ (L~ pion1) )
assume x in {(pion1 /. (len pion1))} ; ::_thesis: x in (L~ do) /\ (L~ pion1)
then A123: x = pion1 /. (len pion1) by TARSKI:def_1;
then A124: x in rng pion1 by REVROT_1:3;
x in rng do by A83, A121, A123, FINSEQ_6:42;
hence x in (L~ do) /\ (L~ pion1) by A62, A112, A124, XBOOLE_0:def_4; ::_thesis: verum
end;
(L~ do) /\ (L~ pion1) c= {(pion1 /. (len pion1))}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ do) /\ (L~ pion1) or x in {(pion1 /. (len pion1))} )
assume A125: x in (L~ do) /\ (L~ pion1) ; ::_thesis: x in {(pion1 /. (len pion1))}
then A126: x in L~ pion1 by XBOOLE_0:def_4;
x in L~ do by A125, XBOOLE_0:def_4;
then x in (L~ pion1) /\ (L~ (Lower_Seq (C,n))) by A53, A126, XBOOLE_0:def_4;
hence x in {(pion1 /. (len pion1))} by A7, A42, A81, A83, A121, SPPOL_2:21; ::_thesis: verum
end;
then A127: (L~ do) /\ (L~ pion1) = {(pion1 /. (len pion1))} by A122, XBOOLE_0:def_10;
A128: (L~ (go ^' pion1)) /\ (L~ do) = ((L~ go) \/ (L~ pion1)) /\ (L~ do) by A92, TOPREAL8:35
.= {(go /. 1)} \/ {(do /. 1)} by A72, A83, A121, A127, XBOOLE_1:23
.= {((go ^' pion1) /. 1)} \/ {(do /. 1)} by GRAPH_2:53
.= {((go ^' pion1) /. 1),(do /. 1)} by ENUMSET1:1 ;
do /. (len do) = (go ^' pion1) /. 1 by A60, GRAPH_2:53;
then reconsider godo = (go ^' pion1) ^' do as non constant 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__::_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 A34, FINSEQ_4:21;
then ((Gauge (C,n)) * (j,i)) .. (Upper_Seq (C,n)) = 1 by A137, XXREAL_0:1;
then (Gauge (C,n)) * (j,i) = (Upper_Seq (C,n)) /. 1 by A34, FINSEQ_5:38;
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 A90, A95, JORDAN9:27;
A141: L~ godo = (L~ (go ^' pion1)) \/ (L~ do) by A94, TOPREAL8:35
.= ((L~ go) \/ (L~ pion1)) \/ (L~ do) 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, XBOOLE_1:1;
A146: L~ do c= L~ (Cage (C,n)) by A53, A144, XBOOLE_1:1;
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~ do ) by A141, XBOOLE_0:def_3;
percases ( W-min C in L~ go or W-min C in L~ pion1 or W-min C in L~ do ) by A150, XBOOLE_0:def_3;
suppose W-min C in L~ go ; ::_thesis: contradiction
then C meets L~ (Cage (C,n)) by A145, A147, XBOOLE_0:3;
hence contradiction by JORDAN10:5; ::_thesis: verum
end;
suppose W-min C in L~ pion1 ; ::_thesis: contradiction
hence contradiction by A8, A81, A131, A148, XBOOLE_0:3; ::_thesis: verum
end;
suppose W-min C in L~ do ; ::_thesis: contradiction
then C meets L~ (Cage (C,n)) by A146, A147, XBOOLE_0:3;
hence contradiction by JORDAN10:5; ::_thesis: verum
end;
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 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, 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)) * (j,i)))),1,(Gauge (C,n))) by A34, A91, A136, JORDAN1J:52
.= right_cell ((go ^' pion1),1,(Gauge (C,n))) by A39, A93, JORDAN1J:51
.= right_cell (godo,1,(Gauge (C,n))) by A88, A95, JORDAN1J:51 ;
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 GRAPH_2:53
.= W-min (L~ (Cage (C,n))) by A59, GRAPH_2:53 ;
A153: len (Upper_Seq (C,n)) >= 2 by A18, XXREAL_0:2;
A154: godo /. 2 = (go ^' pion1) /. 2 by A87, GRAPH_2:57
.= (Upper_Seq (C,n)) /. 2 by A33, A75, GRAPH_2:57
.= ((Upper_Seq (C,n)) ^' (Lower_Seq (C,n))) /. 2 by A153, GRAPH_2:57
.= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 2 by JORDAN1E:11 ;
A155: (L~ go) \/ (L~ do) is compact by COMPTS_1:10;
W-min (L~ (Cage (C,n))) in (L~ go) \/ (L~ do) by A61, A76, XBOOLE_0:def_3;
then A156: W-min ((L~ go) \/ (L~ do)) = W-min (L~ (Cage (C,n))) by A145, A146, A155, JORDAN1J:21, XBOOLE_1:8;
A157: (W-min ((L~ go) \/ (L~ do))) `1 = W-bound ((L~ go) \/ (L~ do)) 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 A1, A2, A3, A4, A5, SPRECT_3:13;
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 A81, SPPOL_2:21;
((Gauge (C,n)) * (j,i)) `1 >= W-bound (L~ (Cage (C,n))) by A10, A143, PSCOMP_1:24;
then ((Gauge (C,n)) * (j,i)) `1 > W-bound (L~ (Cage (C,n))) by A74, XXREAL_0:1;
then W-min (((L~ go) \/ (L~ do)) \/ (L~ pion1)) = W-min ((L~ go) \/ (L~ do)) by A155, A156, A157, A158, A160, JORDAN1J:33;
then A161: W-min (L~ godo) = W-min (L~ (Cage (C,n))) by A141, A156, XBOOLE_1:4;
A162: rng godo c= L~ godo by A87, A89, SPPOL_2:18, XXREAL_0:2;
2 in dom godo by A90, FINSEQ_3:25;
then A163: 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 A161, PSCOMP_1:31
.= W-bound (L~ godo) by EUCLID:52 ;
then godo /. 2 in W-most (L~ godo) by A162, A163, SPRECT_2:12;
then (Rotate (godo,(W-min (L~ godo)))) /. 2 in W-most (L~ godo) by A152, A161, FINSEQ_6:89;
then reconsider godo = godo as non 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 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 (E-max C) misses L~ go
proof
assume east_halfline (E-max C) meets L~ go ; ::_thesis: contradiction
then consider p being set such that
A166: p in east_halfline (E-max C) and
A167: p in L~ go by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A166;
p in L~ (Upper_Seq (C,n)) by A46, A167;
then p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A143, A166, XBOOLE_0:def_4;
then A168: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;
then A169: p = E-max (L~ (Cage (C,n))) by A46, A167, JORDAN1J:46;
then E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (j,i) by A10, A164, A167, JORDAN1J:43;
then ((Gauge (C,n)) * (j,i)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by A3, A14, A168, A169, JORDAN1A:71;
hence contradiction by A2, A3, A16, A30, JORDAN1G:7; ::_thesis: verum
end;
now__::_thesis:_not_east_halfline_(E-max_C)_meets_L~_godo
assume east_halfline (E-max C) meets L~ godo ; ::_thesis: contradiction
then A170: ( east_halfline (E-max C) meets (L~ go) \/ (L~ pion1) or east_halfline (E-max C) meets L~ do ) by A141, XBOOLE_1:70;
percases ( 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~ do ) by A170, XBOOLE_1:70;
suppose east_halfline (E-max C) meets L~ go ; ::_thesis: contradiction
hence contradiction by A165; ::_thesis: verum
end;
suppose east_halfline (E-max C) meets L~ pion1 ; ::_thesis: contradiction
then consider p being set such that
A171: p in east_halfline (E-max C) and
A172: p in L~ pion1 by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A171;
A173: p `2 = (E-max C) `2 by A171, TOPREAL1:def_11;
k + 1 <= len (Gauge (C,n)) by A3, NAT_1:13;
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 A4, A5, A11, A14, A21, A174, JORDAN1A:18;
p `1 <= ((Gauge (C,n)) * (k,i)) `1 by A81, A148, A159, A172, TOPREAL1:3;
then p `1 <= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),1)) `1 by A175, XXREAL_0:2;
then p `1 <= E-bound C by A21, JORDAN8:12;
then A176: p `1 <= (E-max C) `1 by EUCLID:52;
p `1 >= (E-max C) `1 by A171, TOPREAL1:def_11;
then p `1 = (E-max C) `1 by A176, XXREAL_0:1;
then p = E-max C by A173, TOPREAL3:6;
hence contradiction by A8, A81, A132, A148, A172, XBOOLE_0:3; ::_thesis: verum
end;
suppose east_halfline (E-max C) meets L~ do ; ::_thesis: contradiction
then consider p being set such that
A177: p in east_halfline (E-max C) and
A178: p in L~ do by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A177;
A179: p in LSeg (do,(Index (p,do))) by A178, JORDAN3:9;
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 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)) * (k,i)),(Lower_Seq (C,n)))) + 1 = t by A181, A182, JORDAN3:12;
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 A9, A181, JORDAN3:26;
Index (p,do) < len do by A178, JORDAN3:8;
then Index (p,do) < (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (k,i)),(Lower_Seq (C,n)))) by A183, XREAL_0:def_2;
then (Index (p,do)) + 1 <= (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (k,i)),(Lower_Seq (C,n)))) by NAT_1:13;
then A184: Index (p,do) <= ((len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (k,i)),(Lower_Seq (C,n))))) - 1 by XREAL_1:19;
A185: do = mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n)))) by A37, JORDAN1J:37;
p in L~ (Lower_Seq (C,n)) by A53, A178;
then p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A144, A177, XBOOLE_0:def_4;
then A186: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;
A187: (Index (((Gauge (C,n)) * (k,i)),(Lower_Seq (C,n)))) + 1 = ((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)) by A32, A37, JORDAN1J:56;
0 + (Index (((Gauge (C,n)) * (k,i)),(Lower_Seq (C,n)))) < len (Lower_Seq (C,n)) by A9, JORDAN3:8;
then (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (k,i)),(Lower_Seq (C,n)))) > 0 by XREAL_1:20;
then Index (p,do) <= ((len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (k,i)),(Lower_Seq (C,n))))) - 1 by A184, XREAL_0:def_2;
then Index (p,do) <= (len (Lower_Seq (C,n))) - (((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n))) by A187;
then Index (p,do) <= (len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n))) by XREAL_0:def_2;
then A188: Index (p,do) < ((len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)))) + 1 by NAT_1:13;
A189: 1 <= Index (p,do) by A178, JORDAN3:8;
A190: ((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)) <= len (Lower_Seq (C,n)) by A37, FINSEQ_4:21;
((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)) <> len (Lower_Seq (C,n)) by A29, A37, FINSEQ_4:19;
then A191: ((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)) < len (Lower_Seq (C,n)) by A190, XXREAL_0:1;
A192: 1 + 1 <= len (Lower_Seq (C,n)) by A25, XXREAL_0:2;
then A193: 2 in dom (Lower_Seq (C,n)) by FINSEQ_3:25;
set tt = ((Index (p,do)) + (((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 Element of 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 A197, A198, MATRIX_1:36;
A202: len (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = len (Cage (C,n)) by REVROT_1:14;
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 A192, SPPOL_2:9;
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 A138, REVROT_1:34;
then consider ii, g being Element of 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 A85, A196, A202, A204, FINSEQ_6:92, JORDAN1I:23;
A209: (g + 1) + 1 <> g ;
A210: 1 <= g by A206, MATRIX_1:38;
(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 A196, A204, FINSEQ_6:92;
then A211: ii = len (Gauge (C,n)) by A196, A205, A207, A199, A201, GOBOARD1:5;
then ii - 1 >= 4 - 1 by A200, XREAL_1:9;
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 A206, MATRIX_1:38;
then A215: ((Gauge (C,n)) * ((len (Gauge (C,n))),g)) `1 = E-bound (L~ (Cage (C,n))) by A11, A210, JORDAN1A:71;
A216: g + 1 <= width (Gauge (C,n)) by A205, MATRIX_1:38;
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 A85, A202, A195, A205, A206, A207, A208, A209, GOBOARD5:def_6;
A218: ii <= len (Gauge (C,n)) by A206, MATRIX_1:38;
A219: 1 <= ii by A206, MATRIX_1:38;
A220: ii <= len (Gauge (C,n)) by A205, MATRIX_1:38;
A221: 1 <= g + 1 by A205, MATRIX_1:38;
then A222: E-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * ((len (Gauge (C,n))),(g + 1))) `1 by A11, A216, JORDAN1A:71;
A223: 1 <= ii by A205, MATRIX_1:38;
then A224: (ii -' 1) + 1 = ii by XREAL_1:235;
then A225: ii -' 1 < len (Gauge (C,n)) by A220, NAT_1:13;
then A226: ((Gauge (C,n)) * ((ii -' 1),(g + 1))) `2 = ((Gauge (C,n)) * (1,(g + 1))) `2 by A221, A216, A213, GOBOARD5:1
.= ((Gauge (C,n)) * (ii,(g + 1))) `2 by A223, A220, A221, A216, GOBOARD5:1 ;
A227: (E-max C) `2 = p `2 by A177, TOPREAL1:def_11;
then A228: p `2 <= ((Gauge (C,n)) * ((ii -' 1),(g + 1))) `2 by A194, A220, A216, A210, A217, A224, A212, JORDAN9:17;
A229: ((Gauge (C,n)) * ((ii -' 1),g)) `2 = ((Gauge (C,n)) * (1,g)) `2 by A210, A214, A213, A225, GOBOARD5:1
.= ((Gauge (C,n)) * (ii,g)) `2 by A219, A218, A210, A214, GOBOARD5:1 ;
((Gauge (C,n)) * ((ii -' 1),g)) `2 <= p `2 by A227, A194, A220, A216, A210, A217, A224, A212, JORDAN9:17;
then p in LSeg (((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1),((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1))) by A186, A207, A208, A211, A228, A229, A226, A215, A222, GOBOARD7:7;
then A230: p in LSeg ((Lower_Seq (C,n)),1) by A85, A203, A202, TOPREAL1:def_3;
1 <= ((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)) by A37, FINSEQ_4:21;
then A231: LSeg ((mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n))))),(Index (p,do))) = LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)))) -' 1)) by A191, A189, A188, JORDAN4:19;
1 <= Index (((Gauge (C,n)) * (k,i)),(Lower_Seq (C,n))) by A9, JORDAN3:8;
then A232: 1 + 1 <= ((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)) by A187, XREAL_1:7;
then (Index (p,do)) + (((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n))) >= (1 + 1) + 1 by A189, XREAL_1:7;
then ((Index (p,do)) + (((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)))) - 1 >= ((1 + 1) + 1) - 1 by XREAL_1:9;
then A233: ((Index (p,do)) + (((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)))) -' 1 >= 1 + 1 by XREAL_0:def_2;
now__::_thesis:_contradiction
percases ( ((Index (p,do)) + (((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)))) -' 1 > 1 + 1 or ((Index (p,do)) + (((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ) by A233, XXREAL_0:1;
suppose ((Index (p,do)) + (((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,do)) + (((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;
supposeA234: ((Index (p,do)) + (((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ; ::_thesis: contradiction
then 1 + 1 = ((Index (p,do)) + (((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)))) - 1 by XREAL_0:def_2;
then (1 + 1) + 1 = (Index (p,do)) + (((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n))) ;
then A235: ((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)) = 2 by A189, A232, JORDAN1E:6;
(LSeg ((Lower_Seq (C,n)),1)) /\ (LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (k,i)) .. (Lower_Seq (C,n)))) -' 1))) = {((Lower_Seq (C,n)) /. 2)} by A25, A234, TOPREAL1:def_6;
then p in {((Lower_Seq (C,n)) /. 2)} by A230, A179, A185, A231, XBOOLE_0:def_4;
then A236: p = (Lower_Seq (C,n)) /. 2 by TARSKI:def_1;
then A237: p in rng (Lower_Seq (C,n)) by A193, PARTFUN2:2;
p .. (Lower_Seq (C,n)) = 2 by A193, A236, FINSEQ_5:41;
then p = (Gauge (C,n)) * (k,i) by A37, A235, A237, FINSEQ_5:9;
then ((Gauge (C,n)) * (k,i)) `1 = E-bound (L~ (Cage (C,n))) by A236, JORDAN1G:32;
then ((Gauge (C,n)) * (k,i)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 by A1, A11, A13, JORDAN1A:71;
hence contradiction by A3, A17, A67, JORDAN1G:7; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
end;
end;
then east_halfline (E-max C) c= (L~ godo) ` by SUBSET_1:23;
then consider W being Subset of (TOP-REAL 2) such that
A238: W is_a_component_of (L~ godo) ` and
A239: east_halfline (E-max C) c= W by GOBOARD9:3;
not W is bounded by A239, JORDAN2C:121, RLTOPSP1:42;
then W is_outside_component_of L~ godo by A238, JORDAN2C:def_3;
then W c= UBD (L~ godo) by JORDAN2C:23;
then A240: east_halfline (E-max C) c= UBD (L~ godo) by A239, XBOOLE_1:1;
E-max C in east_halfline (E-max C) by TOPREAL1:38;
then E-max C in UBD (L~ godo) by A240;
then E-max C in LeftComp godo by GOBRD14:36;
then Upper_Arc C meets L~ godo by A130, A131, A132, A140, A151, JORDAN1J:36;
then A241: ( Upper_Arc C meets (L~ go) \/ (L~ pion1) or Upper_Arc C meets L~ do ) by A141, XBOOLE_1:70;
A242: Upper_Arc C c= C by JORDAN6:61;
percases ( Upper_Arc C meets L~ go or Upper_Arc C meets L~ pion1 or Upper_Arc C meets L~ do ) by A241, XBOOLE_1:70;
suppose Upper_Arc C meets L~ go ; ::_thesis: contradiction
then Upper_Arc C meets L~ (Cage (C,n)) by A46, A143, XBOOLE_1:1, XBOOLE_1:63;
hence contradiction by A242, JORDAN10:5, XBOOLE_1:63; ::_thesis: verum
end;
suppose Upper_Arc C meets L~ pion1 ; ::_thesis: contradiction
hence contradiction by A8, A81, A148; ::_thesis: verum
end;
suppose Upper_Arc C meets L~ do ; ::_thesis: contradiction
then Upper_Arc C meets L~ (Cage (C,n)) by A53, A144, XBOOLE_1:1, XBOOLE_1:63;
hence contradiction by A242, JORDAN10:5, XBOOLE_1:63; ::_thesis: verum
end;
end;
end;
theorem Th38: :: JORDAN15:38
for n being Element of NAT
for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (k,i) in L~ (Lower_Seq (C,n)) holds
LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Lower_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (k,i) in L~ (Lower_Seq (C,n)) 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 Element of NAT st 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (k,i) in L~ (Lower_Seq (C,n)) holds
LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Lower_Arc C
let i, j, k be Element of NAT ; ::_thesis: ( 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (k,i) in L~ (Lower_Seq (C,n)) implies LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Lower_Arc C )
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: (Gauge (C,n)) * (j,i) in L~ (Upper_Seq (C,n)) and
A7: (Gauge (C,n)) * (k,i) in L~ (Lower_Seq (C,n)) ; ::_thesis: LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Lower_Arc C
consider j1, k1 being Element of NAT such that
A8: j <= j1 and
A9: j1 <= k1 and
A10: k1 <= k and
A11: (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))} and
A12: (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (k1,i))} by A1, A2, A3, A4, A5, A6, A7, Th20;
A13: k1 < len (Gauge (C,n)) by A3, A10, XXREAL_0:2;
1 < j1 by A1, A8, XXREAL_0:2;
then LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i))) meets Lower_Arc C by A4, A5, A9, A11, A12, A13, Th36;
hence LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Lower_Arc C by A1, A3, A4, A5, A8, A9, A10, Th6, XBOOLE_1:63; ::_thesis: verum
end;
theorem Th39: :: JORDAN15:39
for n being Element of NAT
for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (k,i) in L~ (Lower_Seq (C,n)) holds
LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Upper_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (k,i) in L~ (Lower_Seq (C,n)) holds
LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Upper_Arc C
let C be Simple_closed_curve; ::_thesis: for i, j, k being Element of NAT st 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (k,i) in L~ (Lower_Seq (C,n)) holds
LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Upper_Arc C
let i, j, k be Element of NAT ; ::_thesis: ( 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (k,i) in L~ (Lower_Seq (C,n)) implies LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Upper_Arc C )
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: (Gauge (C,n)) * (j,i) in L~ (Upper_Seq (C,n)) and
A7: (Gauge (C,n)) * (k,i) in L~ (Lower_Seq (C,n)) ; ::_thesis: LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Upper_Arc C
consider j1, k1 being Element of NAT such that
A8: j <= j1 and
A9: j1 <= k1 and
A10: k1 <= k and
A11: (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))} and
A12: (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (k1,i))} by A1, A2, A3, A4, A5, A6, A7, Th20;
A13: k1 < len (Gauge (C,n)) by A3, A10, XXREAL_0:2;
1 < j1 by A1, A8, XXREAL_0:2;
then LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i))) meets Upper_Arc C by A4, A5, A9, A11, A12, A13, Th37;
hence LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Upper_Arc C by A1, A3, A4, A5, A8, A9, A10, Th6, XBOOLE_1:63; ::_thesis: verum
end;
theorem Th40: :: JORDAN15:40
for n being Element of NAT
for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & n > 0 & (Gauge (C,n)) * (j,i) in Upper_Arc (L~ (Cage (C,n))) & (Gauge (C,n)) * (k,i) in Lower_Arc (L~ (Cage (C,n))) holds
LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Lower_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & n > 0 & (Gauge (C,n)) * (j,i) in Upper_Arc (L~ (Cage (C,n))) & (Gauge (C,n)) * (k,i) in Lower_Arc (L~ (Cage (C,n))) 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 Element of NAT st 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & n > 0 & (Gauge (C,n)) * (j,i) in Upper_Arc (L~ (Cage (C,n))) & (Gauge (C,n)) * (k,i) in Lower_Arc (L~ (Cage (C,n))) holds
LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Lower_Arc C
let i, j, k be Element of NAT ; ::_thesis: ( 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & n > 0 & (Gauge (C,n)) * (j,i) in Upper_Arc (L~ (Cage (C,n))) & (Gauge (C,n)) * (k,i) in Lower_Arc (L~ (Cage (C,n))) implies LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Lower_Arc C )
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: n > 0 and
A7: (Gauge (C,n)) * (j,i) in Upper_Arc (L~ (Cage (C,n))) and
A8: (Gauge (C,n)) * (k,i) in Lower_Arc (L~ (Cage (C,n))) ; ::_thesis: LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Lower_Arc C
A9: L~ (Lower_Seq (C,n)) = Lower_Arc (L~ (Cage (C,n))) by A6, JORDAN1G:56;
L~ (Upper_Seq (C,n)) = Upper_Arc (L~ (Cage (C,n))) by A6, JORDAN1G:55;
hence LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Lower_Arc C by A1, A2, A3, A4, A5, A7, A8, A9, Th38; ::_thesis: verum
end;
theorem Th41: :: JORDAN15:41
for n being Element of NAT
for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & n > 0 & (Gauge (C,n)) * (j,i) in Upper_Arc (L~ (Cage (C,n))) & (Gauge (C,n)) * (k,i) in Lower_Arc (L~ (Cage (C,n))) holds
LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Upper_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & n > 0 & (Gauge (C,n)) * (j,i) in Upper_Arc (L~ (Cage (C,n))) & (Gauge (C,n)) * (k,i) in Lower_Arc (L~ (Cage (C,n))) holds
LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Upper_Arc C
let C be Simple_closed_curve; ::_thesis: for i, j, k being Element of NAT st 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & n > 0 & (Gauge (C,n)) * (j,i) in Upper_Arc (L~ (Cage (C,n))) & (Gauge (C,n)) * (k,i) in Lower_Arc (L~ (Cage (C,n))) holds
LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Upper_Arc C
let i, j, k be Element of NAT ; ::_thesis: ( 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & n > 0 & (Gauge (C,n)) * (j,i) in Upper_Arc (L~ (Cage (C,n))) & (Gauge (C,n)) * (k,i) in Lower_Arc (L~ (Cage (C,n))) implies LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Upper_Arc C )
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: n > 0 and
A7: (Gauge (C,n)) * (j,i) in Upper_Arc (L~ (Cage (C,n))) and
A8: (Gauge (C,n)) * (k,i) in Lower_Arc (L~ (Cage (C,n))) ; ::_thesis: LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Upper_Arc C
A9: L~ (Lower_Seq (C,n)) = Lower_Arc (L~ (Cage (C,n))) by A6, JORDAN1G:56;
L~ (Upper_Seq (C,n)) = Upper_Arc (L~ (Cage (C,n))) by A6, JORDAN1G:55;
hence LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Upper_Arc C by A1, A2, A3, A4, A5, A7, A8, A9, Th39; ::_thesis: verum
end;
theorem :: JORDAN15:42
for n being Element of NAT
for C being Simple_closed_curve
for j, k being Element of NAT st 1 < j & j <= k & k < len (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1))))) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))) in Lower_Arc (L~ (Cage (C,(n + 1)))) holds
LSeg (((Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1)))))),((Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))))) meets Lower_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for j, k being Element of NAT st 1 < j & j <= k & k < len (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1))))) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))) in Lower_Arc (L~ (Cage (C,(n + 1)))) holds
LSeg (((Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1)))))),((Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))))) meets Lower_Arc C
let C be Simple_closed_curve; ::_thesis: for j, k being Element of NAT st 1 < j & j <= k & k < len (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1))))) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))) in Lower_Arc (L~ (Cage (C,(n + 1)))) holds
LSeg (((Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1)))))),((Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))))) meets Lower_Arc C
let j, k be Element of NAT ; ::_thesis: ( 1 < j & j <= k & k < len (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1))))) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))) in Lower_Arc (L~ (Cage (C,(n + 1)))) implies LSeg (((Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1)))))),((Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))))) meets Lower_Arc C )
assume that
A1: 1 < j and
A2: j <= k and
A3: k < len (Gauge (C,(n + 1))) and
A4: (Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1))))) in Upper_Arc (L~ (Cage (C,(n + 1)))) and
A5: (Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))) in Lower_Arc (L~ (Cage (C,(n + 1)))) ; ::_thesis: LSeg (((Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1)))))),((Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))))) meets Lower_Arc C
A6: len (Gauge (C,(n + 1))) >= 4 by JORDAN8:10;
then len (Gauge (C,(n + 1))) >= 3 by XXREAL_0:2;
then Center (Gauge (C,(n + 1))) < len (Gauge (C,(n + 1))) by JORDAN1B:15;
then A7: Center (Gauge (C,(n + 1))) < width (Gauge (C,(n + 1))) by JORDAN8:def_1;
len (Gauge (C,(n + 1))) >= 2 by A6, XXREAL_0:2;
then 1 < Center (Gauge (C,(n + 1))) by JORDAN1B:14;
hence LSeg (((Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1)))))),((Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))))) meets Lower_Arc C by A1, A2, A3, A4, A5, A7, Th40; ::_thesis: verum
end;
theorem :: JORDAN15:43
for n being Element of NAT
for C being Simple_closed_curve
for j, k being Element of NAT st 1 < j & j <= k & k < len (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1))))) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))) in Lower_Arc (L~ (Cage (C,(n + 1)))) holds
LSeg (((Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1)))))),((Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))))) meets Upper_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for j, k being Element of NAT st 1 < j & j <= k & k < len (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1))))) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))) in Lower_Arc (L~ (Cage (C,(n + 1)))) holds
LSeg (((Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1)))))),((Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))))) meets Upper_Arc C
let C be Simple_closed_curve; ::_thesis: for j, k being Element of NAT st 1 < j & j <= k & k < len (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1))))) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))) in Lower_Arc (L~ (Cage (C,(n + 1)))) holds
LSeg (((Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1)))))),((Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))))) meets Upper_Arc C
let j, k be Element of NAT ; ::_thesis: ( 1 < j & j <= k & k < len (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1))))) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))) in Lower_Arc (L~ (Cage (C,(n + 1)))) implies LSeg (((Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1)))))),((Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))))) meets Upper_Arc C )
assume that
A1: 1 < j and
A2: j <= k and
A3: k < len (Gauge (C,(n + 1))) and
A4: (Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1))))) in Upper_Arc (L~ (Cage (C,(n + 1)))) and
A5: (Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))) in Lower_Arc (L~ (Cage (C,(n + 1)))) ; ::_thesis: LSeg (((Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1)))))),((Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))))) meets Upper_Arc C
A6: len (Gauge (C,(n + 1))) >= 4 by JORDAN8:10;
then len (Gauge (C,(n + 1))) >= 3 by XXREAL_0:2;
then Center (Gauge (C,(n + 1))) < len (Gauge (C,(n + 1))) by JORDAN1B:15;
then A7: Center (Gauge (C,(n + 1))) < width (Gauge (C,(n + 1))) by JORDAN8:def_1;
len (Gauge (C,(n + 1))) >= 2 by A6, XXREAL_0:2;
then 1 < Center (Gauge (C,(n + 1))) by JORDAN1B:14;
hence LSeg (((Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1)))))),((Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))))) meets Upper_Arc C by A1, A2, A3, A4, A5, A7, Th41; ::_thesis: verum
end;
theorem Th44: :: JORDAN15:44
for n being Element of NAT
for C being Simple_closed_curve
for i1, i2, j, k being Element of NAT st 1 < i1 & i1 <= i2 & i2 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} holds
(LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Upper_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i1, i2, j, k being Element of NAT st 1 < i1 & i1 <= i2 & i2 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} holds
(LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Upper_Arc C
let C be Simple_closed_curve; ::_thesis: for i1, i2, j, k being Element of NAT st 1 < i1 & i1 <= i2 & i2 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} holds
(LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Upper_Arc C
let i1, i2, j, k be Element of NAT ; ::_thesis: ( 1 < i1 & i1 <= i2 & i2 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} implies (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Upper_Arc C )
set G = Gauge (C,n);
set pio = LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)));
set poz = LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)));
set US = Upper_Seq (C,n);
set LS = Lower_Seq (C,n);
assume that
A1: 1 < i1 and
A2: i1 <= i2 and
A3: i2 < len (Gauge (C,n)) and
A4: 1 <= j and
A5: j <= k and
A6: k <= width (Gauge (C,n)) and
A7: ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} and
A8: ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} and
A9: (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) misses Upper_Arc C ; ::_thesis: contradiction
set Gij = (Gauge (C,n)) * (i1,j);
A10: j <= width (Gauge (C,n)) by A5, A6, XXREAL_0:2;
A11: i1 < len (Gauge (C,n)) by A2, A3, XXREAL_0:2;
then A12: [i1,j] in Indices (Gauge (C,n)) by A1, A4, A10, MATRIX_1:36;
set Gi1k = (Gauge (C,n)) * (i1,k);
set Gik = (Gauge (C,n)) * (i2,k);
A13: L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> = (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) by TOPREAL3:16;
len (Gauge (C,n)) >= 4 by JORDAN8:10;
then A14: len (Gauge (C,n)) >= 1 by XXREAL_0:2;
then A15: [(len (Gauge (C,n))),j] in Indices (Gauge (C,n)) by A4, A10, MATRIX_1:36;
A16: 1 <= k by A4, A5, XXREAL_0:2;
then A17: [1,k] in Indices (Gauge (C,n)) by A6, A14, MATRIX_1:36;
A18: 1 < i2 by A1, A2, XXREAL_0:2;
then A19: [i2,k] in Indices (Gauge (C,n)) by A3, A6, A16, MATRIX_1:36;
A20: ((Gauge (C,n)) * (i1,k)) `2 = ((Gauge (C,n)) * (1,k)) `2 by A1, A6, A11, A16, GOBOARD5:1
.= ((Gauge (C,n)) * (i2,k)) `2 by A3, A6, A18, A16, GOBOARD5:1 ;
((Gauge (C,n)) * (i1,k)) `1 = ((Gauge (C,n)) * (i1,1)) `1 by A1, A6, A11, A16, GOBOARD5:2
.= ((Gauge (C,n)) * (i1,j)) `1 by A1, A4, A11, A10, GOBOARD5:2 ;
then A21: (Gauge (C,n)) * (i1,k) = |[(((Gauge (C,n)) * (i1,j)) `1),(((Gauge (C,n)) * (i2,k)) `2)]| by A20, EUCLID:53;
A22: [(len (Gauge (C,n))),k] in Indices (Gauge (C,n)) by A6, A16, A14, MATRIX_1:36;
A23: [i1,j] in Indices (Gauge (C,n)) by A1, A4, A11, A10, MATRIX_1:36;
set Wbo = W-bound (L~ (Cage (C,n)));
set Wmin = W-min (L~ (Cage (C,n)));
A24: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
set Ebo = E-bound (L~ (Cage (C,n)));
set Emax = E-max (L~ (Cage (C,n)));
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;
then A28: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by PARTFUN1:def_6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;
len (Lower_Seq (C,n)) in dom (Lower_Seq (C,n)) by A26, FINSEQ_3:25;
then A29: (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by PARTFUN1:def_6
.= W-min (L~ (Cage (C,n))) by JORDAN1F:8 ;
set do = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i1,j)));
(Gauge (C,n)) * (i1,j) in {((Gauge (C,n)) * (i1,j))} by TARSKI:def_1;
then A30: (Gauge (C,n)) * (i1,j) in L~ (Lower_Seq (C,n)) by A8, XBOOLE_0:def_4;
(W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * (1,k)) `1 by A6, A16, A24, JORDAN1A:73 ;
then A31: (Gauge (C,n)) * (i1,j) <> (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) by A1, A17, A29, A12, JORDAN1G:7;
then reconsider do = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i1,j))) as being_S-Seq FinSequence of (TOP-REAL 2) by A30, JORDAN3:34;
A32: (Gauge (C,n)) * (i1,j) in rng (Lower_Seq (C,n)) by A1, A4, A11, A30, A10, JORDAN1G:5, JORDAN1J:40;
then A33: do is_sequence_on Gauge (C,n) by JORDAN1G:5, JORDAN1J:39;
(E-max (L~ (Cage (C,n)))) `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by A6, A16, A24, JORDAN1A:71 ;
then A34: (Gauge (C,n)) * (i1,j) <> (Lower_Seq (C,n)) . 1 by A2, A3, A12, A22, A28, JORDAN1G:7;
A35: len do >= 1 + 1 by TOPREAL1:def_8;
then reconsider do = do as non constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A33, JGRAPH_1:12, JORDAN8:5;
A36: L~ do c= L~ (Lower_Seq (C,n)) by A30, JORDAN3:42;
A37: [1,j] in Indices (Gauge (C,n)) by A4, A10, A14, MATRIX_1:36;
A38: now__::_thesis:_not_((Gauge_(C,n))_*_(i1,j))_`1_=_W-bound_(L~_(Cage_(C,n)))
assume ((Gauge (C,n)) * (i1,j)) `1 = W-bound (L~ (Cage (C,n))) ; ::_thesis: contradiction
then ((Gauge (C,n)) * (1,j)) `1 = ((Gauge (C,n)) * (i1,j)) `1 by A4, A10, A24, JORDAN1A:73;
hence contradiction by A1, A23, A37, JORDAN1G:7; ::_thesis: verum
end;
set pion = <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>;
A39: (Gauge (C,n)) * (i1,k) in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) by RLTOPSP1:68;
set UA = Upper_Arc C;
A40: (Gauge (C,n)) * (i1,k) in LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) by RLTOPSP1:68;
set go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i2,k)));
A41: 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 A42: (Upper_Seq (C,n)) . 1 = (Upper_Seq (C,n)) /. 1 by PARTFUN1:def_6
.= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ;
A43: [i1,k] in Indices (Gauge (C,n)) by A1, A6, A11, A16, MATRIX_1:36;
A44: now__::_thesis:_for_n_being_Element_of_NAT_st_n_in_dom_<*((Gauge_(C,n))_*_(i2,k)),((Gauge_(C,n))_*_(i1,k)),((Gauge_(C,n))_*_(i1,j))*>_holds_
ex_i,_j_being_Element_of_NAT_st_
(_[i,j]_in_Indices_(Gauge_(C,n))_&_<*((Gauge_(C,n))_*_(i2,k)),((Gauge_(C,n))_*_(i1,k)),((Gauge_(C,n))_*_(i1,j))*>_/._n_=_(Gauge_(C,n))_*_(i,j)_)
let n be Element of NAT ; ::_thesis: ( n in dom <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> implies ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. n = (Gauge (C,n)) * (i,j) ) )
assume n in dom <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. n = (Gauge (C,n)) * (i,j) )
then n in {1,2,3} by FINSEQ_1:89, FINSEQ_3:1;
then ( n = 1 or n = 2 or n = 3 ) by ENUMSET1:def_1;
hence ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. n = (Gauge (C,n)) * (i,j) ) by A23, A19, A43, FINSEQ_4:18; ::_thesis: verum
end;
(Gauge (C,n)) * (i2,k) in {((Gauge (C,n)) * (i2,k))} by TARSKI:def_1;
then A45: (Gauge (C,n)) * (i2,k) in L~ (Upper_Seq (C,n)) by A7, XBOOLE_0:def_4;
(W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * (1,k)) `1 by A6, A16, A24, JORDAN1A:73 ;
then A46: (Gauge (C,n)) * (i2,k) <> (Upper_Seq (C,n)) . 1 by A1, A2, A19, A42, A17, JORDAN1G:7;
then reconsider go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i2,k))) as being_S-Seq FinSequence of (TOP-REAL 2) by A45, JORDAN3:35;
A47: (Gauge (C,n)) * (i2,k) in rng (Upper_Seq (C,n)) by A3, A6, A18, A45, A16, JORDAN1G:4, JORDAN1J:40;
then A48: go is_sequence_on Gauge (C,n) by JORDAN1G:4, JORDAN1J:38;
len do >= 1 by A35, XXREAL_0:2;
then 1 in dom do by FINSEQ_3:25;
then A49: do /. 1 = do . 1 by PARTFUN1:def_6
.= (Gauge (C,n)) * (i1,j) by A30, JORDAN3:23 ;
then A50: LSeg (do,1) = LSeg (((Gauge (C,n)) * (i1,j)),(do /. (1 + 1))) by A35, TOPREAL1:def_3;
A51: {((Gauge (C,n)) * (i1,j))} c= (LSeg (do,1)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i1,j))} or x in (LSeg (do,1)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) )
assume x in {((Gauge (C,n)) * (i1,j))} ; ::_thesis: x in (LSeg (do,1)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>)
then A52: x = (Gauge (C,n)) * (i1,j) by TARSKI:def_1;
(Gauge (C,n)) * (i1,j) in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))) by RLTOPSP1:68;
then (Gauge (C,n)) * (i1,j) in (LSeg (((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j)))) by XBOOLE_0:def_3;
then A53: (Gauge (C,n)) * (i1,j) in L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> by SPRECT_1:8;
(Gauge (C,n)) * (i1,j) in LSeg (do,1) by A50, RLTOPSP1:68;
hence x in (LSeg (do,1)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) by A52, A53, XBOOLE_0:def_4; ::_thesis: verum
end;
LSeg (do,1) c= L~ do by TOPREAL3:19;
then LSeg (do,1) c= L~ (Lower_Seq (C,n)) by A36, XBOOLE_1:1;
then (LSeg (do,1)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) c= {((Gauge (C,n)) * (i1,j))} by A8, A13, XBOOLE_1:26;
then A54: (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) /\ (LSeg (do,1)) = {((Gauge (C,n)) * (i1,j))} by A51, XBOOLE_0:def_10;
A55: rng do c= L~ do by A35, SPPOL_2:18;
A56: len go >= 1 + 1 by TOPREAL1:def_8;
then reconsider go = go as non constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A48, JGRAPH_1:12, JORDAN8:5;
A57: L~ go c= L~ (Upper_Seq (C,n)) by A45, JORDAN3:41;
A58: len go > 1 by A56, NAT_1:13;
then A59: len go in dom go by FINSEQ_3:25;
then A60: go /. (len go) = go . (len go) by PARTFUN1:def_6
.= (Gauge (C,n)) * (i2,k) by A45, JORDAN3:24 ;
reconsider m = (len go) - 1 as Element of NAT by A59, FINSEQ_3:26;
A61: m + 1 = len go ;
then A62: (len go) -' 1 = m by NAT_D:34;
m >= 1 by A56, XREAL_1:19;
then A63: LSeg (go,m) = LSeg ((go /. m),((Gauge (C,n)) * (i2,k))) by A60, A61, TOPREAL1:def_3;
A64: {((Gauge (C,n)) * (i2,k))} c= (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i2,k))} or x in (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) )
assume x in {((Gauge (C,n)) * (i2,k))} ; ::_thesis: x in (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>)
then A65: x = (Gauge (C,n)) * (i2,k) by TARSKI:def_1;
(Gauge (C,n)) * (i2,k) in LSeg (((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k))) by RLTOPSP1:68;
then (Gauge (C,n)) * (i2,k) in (LSeg (((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j)))) by XBOOLE_0:def_3;
then A66: (Gauge (C,n)) * (i2,k) in L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> by SPRECT_1:8;
(Gauge (C,n)) * (i2,k) in LSeg (go,m) by A63, RLTOPSP1:68;
hence x in (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) by A65, A66, XBOOLE_0:def_4; ::_thesis: verum
end;
LSeg (go,m) c= L~ go by TOPREAL3:19;
then LSeg (go,m) c= L~ (Upper_Seq (C,n)) by A57, XBOOLE_1:1;
then (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) c= {((Gauge (C,n)) * (i2,k))} by A7, A13, XBOOLE_1:26;
then A67: (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) = {((Gauge (C,n)) * (i2,k))} by A64, XBOOLE_0:def_10;
A68: go /. 1 = (Upper_Seq (C,n)) /. 1 by A45, SPRECT_3:22
.= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ;
A69: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by A27, PARTFUN1:def_6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;
A70: (L~ go) /\ (L~ do) c= {(go /. 1)}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ go) /\ (L~ do) or x in {(go /. 1)} )
assume A71: x in (L~ go) /\ (L~ do) ; ::_thesis: x in {(go /. 1)}
then A72: x in L~ do by XBOOLE_0:def_4;
A73: now__::_thesis:_not_x_=_E-max_(L~_(Cage_(C,n)))
assume x = E-max (L~ (Cage (C,n))) ; ::_thesis: contradiction
then A74: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i1,j) by A30, A69, A72, JORDAN1E:7;
((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 = E-bound (L~ (Cage (C,n))) by A4, A10, A24, JORDAN1A:71;
then (E-max (L~ (Cage (C,n)))) `1 <> E-bound (L~ (Cage (C,n))) by A2, A3, A23, A15, A74, JORDAN1G:7;
hence contradiction by EUCLID:52; ::_thesis: verum
end;
x in L~ go by A71, XBOOLE_0:def_4;
then x in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A57, A36, A72, XBOOLE_0:def_4;
then x in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16;
then ( x = W-min (L~ (Cage (C,n))) or x = E-max (L~ (Cage (C,n))) ) by TARSKI:def_2;
hence x in {(go /. 1)} by A68, A73, TARSKI:def_1; ::_thesis: verum
end;
set W2 = go /. 2;
A75: 2 in dom go by A56, FINSEQ_3:25;
go = mid ((Upper_Seq (C,n)),1,(((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n)))) by A47, JORDAN1G:49
.= (Upper_Seq (C,n)) | (((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n))) by A47, FINSEQ_4:21, FINSEQ_6:116 ;
then A76: go /. 2 = (Upper_Seq (C,n)) /. 2 by A75, FINSEQ_4:70;
A77: rng go c= L~ go by A56, SPPOL_2:18;
A78: go /. 1 = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by A68, JORDAN1F:8
.= do /. (len do) by A30, JORDAN1J:35 ;
{(go /. 1)} c= (L~ go) /\ (L~ do)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(go /. 1)} or x in (L~ go) /\ (L~ do) )
assume x in {(go /. 1)} ; ::_thesis: x in (L~ go) /\ (L~ do)
then A79: x = go /. 1 by TARSKI:def_1;
then A80: x in rng go by FINSEQ_6:42;
x in rng do by A78, A79, REVROT_1:3;
hence x in (L~ go) /\ (L~ do) by A77, A55, A80, XBOOLE_0:def_4; ::_thesis: verum
end;
then A81: (L~ go) /\ (L~ do) = {(go /. 1)} by A70, XBOOLE_0:def_10;
now__::_thesis:_contradiction
percases ( ( ((Gauge (C,n)) * (i1,j)) `1 <> ((Gauge (C,n)) * (i2,k)) `1 & ((Gauge (C,n)) * (i1,j)) `2 <> ((Gauge (C,n)) * (i2,k)) `2 ) or ((Gauge (C,n)) * (i1,j)) `1 = ((Gauge (C,n)) * (i2,k)) `1 or ((Gauge (C,n)) * (i1,j)) `2 = ((Gauge (C,n)) * (i2,k)) `2 ) ;
suppose ( ((Gauge (C,n)) * (i1,j)) `1 <> ((Gauge (C,n)) * (i2,k)) `1 & ((Gauge (C,n)) * (i1,j)) `2 <> ((Gauge (C,n)) * (i2,k)) `2 ) ; ::_thesis: contradiction
then <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> is being_S-Seq by A21, TOPREAL3:35;
then consider pion1 being FinSequence of (TOP-REAL 2) such that
A82: pion1 is_sequence_on Gauge (C,n) and
A83: pion1 is being_S-Seq and
A84: L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> = L~ pion1 and
A85: <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. 1 = pion1 /. 1 and
A86: <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. (len <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) = pion1 /. (len pion1) and
A87: len <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> <= len pion1 by A44, GOBOARD3:2;
reconsider pion1 = pion1 as being_S-Seq FinSequence of (TOP-REAL 2) by A83;
A88: (go ^' pion1) /. (len (go ^' pion1)) = <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. (len <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) by A86, GRAPH_2:54
.= <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. 3 by FINSEQ_1:45
.= do /. 1 by A49, FINSEQ_4:18 ;
A89: go /. (len go) = pion1 /. 1 by A60, A85, FINSEQ_4:18;
A90: (L~ go) /\ (L~ pion1) c= {(pion1 /. 1)}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ go) /\ (L~ pion1) or x in {(pion1 /. 1)} )
assume A91: x in (L~ go) /\ (L~ pion1) ; ::_thesis: x in {(pion1 /. 1)}
then A92: x in L~ pion1 by XBOOLE_0:def_4;
x in L~ go by A91, XBOOLE_0:def_4;
hence x in {(pion1 /. 1)} by A7, A13, A60, A57, A84, A89, A92, XBOOLE_0:def_4; ::_thesis: verum
end;
len pion1 >= 2 + 1 by A87, FINSEQ_1:45;
then A93: len pion1 > 1 + 1 by NAT_1:13;
then A94: rng pion1 c= L~ pion1 by SPPOL_2:18;
{(pion1 /. 1)} c= (L~ go) /\ (L~ pion1)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(pion1 /. 1)} or x in (L~ go) /\ (L~ pion1) )
assume x in {(pion1 /. 1)} ; ::_thesis: x in (L~ go) /\ (L~ pion1)
then A95: x = pion1 /. 1 by TARSKI:def_1;
then A96: x in rng pion1 by FINSEQ_6:42;
x in rng go by A89, A95, REVROT_1:3;
hence x in (L~ go) /\ (L~ pion1) by A77, A94, A96, XBOOLE_0:def_4; ::_thesis: verum
end;
then A97: (L~ go) /\ (L~ pion1) = {(pion1 /. 1)} by A90, XBOOLE_0:def_10;
then A98: go ^' pion1 is s.n.c. by A89, JORDAN1J:54;
A99: <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. (len <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) = <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. 3 by FINSEQ_1:45
.= do /. 1 by A49, FINSEQ_4:18 ;
A100: {(pion1 /. (len pion1))} c= (L~ do) /\ (L~ pion1)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(pion1 /. (len pion1))} or x in (L~ do) /\ (L~ pion1) )
assume x in {(pion1 /. (len pion1))} ; ::_thesis: x in (L~ do) /\ (L~ pion1)
then A101: x = pion1 /. (len pion1) by TARSKI:def_1;
then A102: x in rng pion1 by REVROT_1:3;
x in rng do by A86, A99, A101, FINSEQ_6:42;
hence x in (L~ do) /\ (L~ pion1) by A55, A94, A102, XBOOLE_0:def_4; ::_thesis: verum
end;
(L~ do) /\ (L~ pion1) c= {(pion1 /. (len pion1))}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ do) /\ (L~ pion1) or x in {(pion1 /. (len pion1))} )
assume A103: x in (L~ do) /\ (L~ pion1) ; ::_thesis: x in {(pion1 /. (len pion1))}
then A104: x in L~ pion1 by XBOOLE_0:def_4;
x in L~ do by A103, XBOOLE_0:def_4;
hence x in {(pion1 /. (len pion1))} by A8, A13, A49, A36, A84, A86, A99, A104, XBOOLE_0:def_4; ::_thesis: verum
end;
then A105: (L~ do) /\ (L~ pion1) = {(pion1 /. (len pion1))} by A100, XBOOLE_0:def_10;
A106: (L~ (go ^' pion1)) /\ (L~ do) = ((L~ go) \/ (L~ pion1)) /\ (L~ do) by A89, TOPREAL8:35
.= {(go /. 1)} \/ {(do /. 1)} by A81, A86, A99, A105, XBOOLE_1:23
.= {((go ^' pion1) /. 1)} \/ {(do /. 1)} by GRAPH_2:53
.= {((go ^' pion1) /. 1),(do /. 1)} by ENUMSET1:1 ;
A107: Upper_Arc C is_an_arc_of W-min C, E-max C by JORDAN6:def_8;
then A108: Upper_Arc C is connected by JORDAN6:10;
set godo = (go ^' pion1) ^' do;
A109: do /. (len do) = (go ^' pion1) /. 1 by A78, GRAPH_2:53;
A110: go ^' pion1 is_sequence_on Gauge (C,n) by A48, A82, A89, TOPREAL8:12;
then A111: (go ^' pion1) ^' do is_sequence_on Gauge (C,n) by A33, A88, TOPREAL8:12;
A112: (len pion1) - 1 >= 1 by A93, XREAL_1:19;
then A113: (len pion1) -' 1 = (len pion1) - 1 by XREAL_0:def_2;
A114: {((Gauge (C,n)) * (i1,j))} c= (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i1,j))} or x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) )
assume x in {((Gauge (C,n)) * (i1,j))} ; ::_thesis: x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1))
then A115: x = (Gauge (C,n)) * (i1,j) by TARSKI:def_1;
pion1 /. (((len pion1) -' 1) + 1) = <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. 3 by A86, A113, FINSEQ_1:45
.= (Gauge (C,n)) * (i1,j) by FINSEQ_4:18 ;
then A116: (Gauge (C,n)) * (i1,j) in LSeg (pion1,((len pion1) -' 1)) by A112, A113, TOPREAL1:21;
(Gauge (C,n)) * (i1,j) in LSeg (do,1) by A50, RLTOPSP1:68;
hence x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) by A115, A116, XBOOLE_0:def_4; ::_thesis: verum
end;
LSeg (pion1,((len pion1) -' 1)) c= L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> by A84, TOPREAL3:19;
then (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) c= {((Gauge (C,n)) * (i1,j))} by A54, XBOOLE_1:27;
then A117: (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) = {((Gauge (C,n)) * (i1,j))} by A114, XBOOLE_0:def_10;
((len pion1) - 1) + 1 <= len pion1 ;
then A118: (len pion1) -' 1 < len pion1 by A113, NAT_1:13;
len pion1 >= 2 + 1 by A87, FINSEQ_1:45;
then A119: (len pion1) - 2 >= 0 by XREAL_1:19;
then ((len pion1) -' 2) + 1 = ((len pion1) - 2) + 1 by XREAL_0:def_2
.= (len pion1) -' 1 by A112, XREAL_0:def_2 ;
then A120: (LSeg ((go ^' pion1),((len go) + ((len pion1) -' 2)))) /\ (LSeg (do,1)) = {((go ^' pion1) /. (len (go ^' pion1)))} by A49, A89, A88, A118, A117, TOPREAL8:31;
(rng go) /\ (rng pion1) c= {(pion1 /. 1)} by A77, A94, A97, XBOOLE_1:27;
then A121: go ^' pion1 is one-to-one by JORDAN1J:55;
((len (go ^' pion1)) + 1) - 1 = ((len go) + (len pion1)) - 1 by GRAPH_2:13;
then (len (go ^' pion1)) - 1 = (len go) + ((len pion1) - 2)
.= (len go) + ((len pion1) -' 2) by A119, XREAL_0:def_2 ;
then A122: (len (go ^' pion1)) -' 1 = (len go) + ((len pion1) -' 2) by XREAL_0:def_2;
A123: L~ (Cage (C,n)) = (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n))) by JORDAN1E:13;
then A124: L~ (Upper_Seq (C,n)) c= L~ (Cage (C,n)) by XBOOLE_1:7;
then A125: L~ go c= L~ (Cage (C,n)) by A57, XBOOLE_1:1;
A126: {((Gauge (C,n)) * (i2,k))} c= (LSeg (go,m)) /\ (LSeg (pion1,1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i2,k))} or x in (LSeg (go,m)) /\ (LSeg (pion1,1)) )
assume x in {((Gauge (C,n)) * (i2,k))} ; ::_thesis: x in (LSeg (go,m)) /\ (LSeg (pion1,1))
then A127: x = (Gauge (C,n)) * (i2,k) by TARSKI:def_1;
A128: (Gauge (C,n)) * (i2,k) in LSeg (go,m) by A63, RLTOPSP1:68;
(Gauge (C,n)) * (i2,k) in LSeg (pion1,1) by A60, A89, A93, TOPREAL1:21;
hence x in (LSeg (go,m)) /\ (LSeg (pion1,1)) by A127, A128, XBOOLE_0:def_4; ::_thesis: verum
end;
LSeg (pion1,1) c= L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> by A84, TOPREAL3:19;
then (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) c= {((Gauge (C,n)) * (i2,k))} by A62, A67, XBOOLE_1:27;
then (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) = {(go /. (len go))} by A60, A62, A126, XBOOLE_0:def_10;
then A129: go ^' pion1 is unfolded by A89, TOPREAL8:34;
len (go ^' pion1) >= len go by TOPREAL8:7;
then A130: len (go ^' pion1) >= 1 + 1 by A56, XXREAL_0:2;
then A131: len (go ^' pion1) > 1 + 0 by NAT_1:13;
A132: now__::_thesis:_not_((Gauge_(C,n))_*_(i2,k))_.._(Upper_Seq_(C,n))_<=_1
assume A133: ((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n)) <= 1 ; ::_thesis: contradiction
((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n)) >= 1 by A47, FINSEQ_4:21;
then ((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n)) = 1 by A133, XXREAL_0:1;
then (Gauge (C,n)) * (i2,k) = (Upper_Seq (C,n)) /. 1 by A47, FINSEQ_5:38;
hence contradiction by A42, A46, JORDAN1F:5; ::_thesis: verum
end;
A134: Upper_Seq (C,n) is_sequence_on Gauge (C,n) by JORDAN1G:4;
A135: (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52;
set ff = Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))));
A136: 1 + 1 <= len (Cage (C,n)) by GOBOARD7:34, XXREAL_0:2;
A137: len ((go ^' pion1) ^' do) >= len (go ^' pion1) by TOPREAL8:7;
then A138: 1 + 1 <= len ((go ^' pion1) ^' do) by A130, XXREAL_0:2;
not go ^' pion1 is trivial by A130, NAT_D:60;
then reconsider godo = (go ^' pion1) ^' do as non constant standard special_circular_sequence by A138, A88, A111, A129, A122, A120, A98, A121, A106, A109, JORDAN8:4, JORDAN8:5, TOPREAL8:11, TOPREAL8:33, TOPREAL8:34;
A139: L~ godo = (L~ (go ^' pion1)) \/ (L~ do) by A88, TOPREAL8:35
.= ((L~ go) \/ (L~ pion1)) \/ (L~ do) by A89, TOPREAL8:35 ;
A140: (right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo) c= RightComp godo by A138, A111, JORDAN9:27;
2 in dom godo by A138, FINSEQ_3:25;
then A141: godo /. 2 in rng godo by PARTFUN2:2;
A142: W-min C in Upper_Arc C by A107, TOPREAL1:1;
W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
then A143: (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 = W-min (L~ (Cage (C,n))) by FINSEQ_6:92;
A144: 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 A143, 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 A143, A144, 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 A143, A144, SPRECT_5:24, XXREAL_0:2;
then A145: (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A143, A144, SPRECT_5:25, XXREAL_0:2;
A146: Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def_1;
then A147: Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))) is_sequence_on Gauge (C,n) by REVROT_1:34;
A148: ((Gauge (C,n)) * (i1,k)) `1 = ((Gauge (C,n)) * (i1,1)) `1 by A1, A6, A11, A16, GOBOARD5:2
.= ((Gauge (C,n)) * (i1,j)) `1 by A1, A4, A11, A10, GOBOARD5:2 ;
then A149: W-bound (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) = ((Gauge (C,n)) * (i1,j)) `1 by SPRECT_1:54;
A150: L~ (Lower_Seq (C,n)) c= L~ (Cage (C,n)) by A123, XBOOLE_1:7;
then A151: L~ do c= L~ (Cage (C,n)) by A36, XBOOLE_1:1;
A152: W-min C in C by SPRECT_1:13;
A153: now__::_thesis:_not_W-min_C_in_L~_godo
assume W-min C in L~ godo ; ::_thesis: contradiction
then A154: ( W-min C in (L~ go) \/ (L~ pion1) or W-min C in L~ do ) by A139, XBOOLE_0:def_3;
percases ( W-min C in L~ go or W-min C in L~ pion1 or W-min C in L~ do ) by A154, XBOOLE_0:def_3;
suppose W-min C in L~ go ; ::_thesis: contradiction
then C meets L~ (Cage (C,n)) by A125, A152, XBOOLE_0:3;
hence contradiction by JORDAN10:5; ::_thesis: verum
end;
suppose W-min C in L~ pion1 ; ::_thesis: contradiction
hence contradiction by A9, A13, A84, A142, XBOOLE_0:3; ::_thesis: verum
end;
suppose W-min C in L~ do ; ::_thesis: contradiction
then C meets L~ (Cage (C,n)) by A151, A152, XBOOLE_0:3;
hence contradiction by JORDAN10:5; ::_thesis: verum
end;
end;
end;
A155: len (Upper_Seq (C,n)) >= 2 by A41, XXREAL_0:2;
A156: (L~ go) \/ (L~ do) is compact by COMPTS_1:10;
1 + 1 <= len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by GOBOARD7:34, XXREAL_0:2;
then right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1) = right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(GoB (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) by JORDAN1H:23
.= right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(GoB (Cage (C,n)))) by REVROT_1:28
.= right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(Gauge (C,n))) by JORDAN1H:44
.= right_cell (((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n))))),1,(Gauge (C,n))) by A145, A147, JORDAN1J:53
.= right_cell ((Upper_Seq (C,n)),1,(Gauge (C,n))) by JORDAN1E:def_1
.= right_cell ((R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i2,k)))),1,(Gauge (C,n))) by A47, A134, A132, JORDAN1J:52
.= right_cell ((go ^' pion1),1,(Gauge (C,n))) by A58, A110, JORDAN1J:51
.= right_cell (godo,1,(Gauge (C,n))) by A131, A111, JORDAN1J:51 ;
then W-min C in right_cell (godo,1,(Gauge (C,n))) by JORDAN1I:6;
then A157: W-min C in (right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo) by A153, XBOOLE_0:def_5;
A158: rng godo c= L~ godo by A130, A137, SPPOL_2:18, XXREAL_0:2;
A159: godo /. 1 = (go ^' pion1) /. 1 by GRAPH_2:53
.= W-min (L~ (Cage (C,n))) by A68, GRAPH_2:53 ;
A160: ((Gauge (C,n)) * (i1,k)) `1 <= ((Gauge (C,n)) * (i2,k)) `1 by A1, A2, A3, A6, A16, JORDAN1A:18;
then A161: W-bound (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) = ((Gauge (C,n)) * (i1,k)) `1 by SPRECT_1:54;
W-bound ((LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))))) = min ((W-bound (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))),(W-bound (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))))) by SPRECT_1:47
.= ((Gauge (C,n)) * (i1,j)) `1 by A148, A161, A149 ;
then A162: W-bound (L~ pion1) = ((Gauge (C,n)) * (i1,j)) `1 by A84, TOPREAL3:16;
A163: Upper_Arc C c= C by JORDAN6:61;
((Gauge (C,n)) * (i1,j)) `1 >= W-bound (L~ (Cage (C,n))) by A30, A150, PSCOMP_1:24;
then A164: ((Gauge (C,n)) * (i1,j)) `1 > W-bound (L~ (Cage (C,n))) by A38, XXREAL_0:1;
A165: E-max C in Upper_Arc C by A107, TOPREAL1:1;
W-min (L~ (Cage (C,n))) in rng go by A68, FINSEQ_6:42;
then W-min (L~ (Cage (C,n))) in (L~ go) \/ (L~ do) by A77, XBOOLE_0:def_3;
then A166: W-min ((L~ go) \/ (L~ do)) = W-min (L~ (Cage (C,n))) by A125, A151, A156, JORDAN1J:21, XBOOLE_1:8;
(W-min ((L~ go) \/ (L~ do))) `1 = W-bound ((L~ go) \/ (L~ do)) by EUCLID:52;
then W-min (((L~ go) \/ (L~ do)) \/ (L~ pion1)) = W-min ((L~ go) \/ (L~ do)) by A162, A156, A166, A135, A164, JORDAN1J:33;
then A167: W-min (L~ godo) = W-min (L~ (Cage (C,n))) by A139, A166, XBOOLE_1:4;
godo /. 2 = (go ^' pion1) /. 2 by A130, GRAPH_2:57
.= (Upper_Seq (C,n)) /. 2 by A56, A76, GRAPH_2:57
.= ((Upper_Seq (C,n)) ^' (Lower_Seq (C,n))) /. 2 by A155, GRAPH_2:57
.= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 2 by JORDAN1E:11 ;
then godo /. 2 in W-most (L~ (Cage (C,n))) by JORDAN1I:25;
then (godo /. 2) `1 = (W-min (L~ godo)) `1 by A167, PSCOMP_1:31
.= W-bound (L~ godo) by EUCLID:52 ;
then godo /. 2 in W-most (L~ godo) by A158, A141, SPRECT_2:12;
then (Rotate (godo,(W-min (L~ godo)))) /. 2 in W-most (L~ godo) by A159, A167, FINSEQ_6:89;
then reconsider godo = godo as non constant standard clockwise_oriented special_circular_sequence by JORDAN1I:25;
len (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by FINSEQ_5:6;
then A168: (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) = (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) by PARTFUN1:def_6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:7 ;
A169: east_halfline (E-max C) misses L~ go
proof
assume east_halfline (E-max C) meets L~ go ; ::_thesis: contradiction
then consider p being set such that
A170: p in east_halfline (E-max C) and
A171: p in L~ go by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A170;
p in L~ (Upper_Seq (C,n)) by A57, A171;
then p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A124, A170, XBOOLE_0:def_4;
then A172: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;
then A173: p = E-max (L~ (Cage (C,n))) by A57, A171, JORDAN1J:46;
then E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i2,k) by A45, A168, A171, JORDAN1J:43;
then ((Gauge (C,n)) * (i2,k)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by A6, A16, A24, A172, A173, JORDAN1A:71;
hence contradiction by A3, A19, A22, JORDAN1G:7; ::_thesis: verum
end;
now__::_thesis:_not_east_halfline_(E-max_C)_meets_L~_godo
assume east_halfline (E-max C) meets L~ godo ; ::_thesis: contradiction
then A174: ( east_halfline (E-max C) meets (L~ go) \/ (L~ pion1) or east_halfline (E-max C) meets L~ do ) by A139, XBOOLE_1:70;
percases ( 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~ do ) by A174, XBOOLE_1:70;
suppose east_halfline (E-max C) meets L~ go ; ::_thesis: contradiction
hence contradiction by A169; ::_thesis: verum
end;
suppose east_halfline (E-max C) meets L~ pion1 ; ::_thesis: contradiction
then consider p being set such that
A175: p in east_halfline (E-max C) and
A176: p in L~ pion1 by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A175;
A177: p `2 = (E-max C) `2 by A175, TOPREAL1:def_11;
A178: now__::_thesis:_p_`1_<=_((Gauge_(C,n))_*_(i2,k))_`1
percases ( p in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) or p in LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) ) by A13, A84, A176, XBOOLE_0:def_3;
suppose p in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) ; ::_thesis: p `1 <= ((Gauge (C,n)) * (i2,k)) `1
hence p `1 <= ((Gauge (C,n)) * (i2,k)) `1 by A160, TOPREAL1:3; ::_thesis: verum
end;
suppose p in LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) ; ::_thesis: p `1 <= ((Gauge (C,n)) * (i2,k)) `1
hence p `1 <= ((Gauge (C,n)) * (i2,k)) `1 by A148, A160, GOBOARD7:5; ::_thesis: verum
end;
end;
end;
i2 + 1 <= len (Gauge (C,n)) by A3, NAT_1:13;
then (i2 + 1) - 1 <= (len (Gauge (C,n))) - 1 by XREAL_1:9;
then A179: i2 <= (len (Gauge (C,n))) -' 1 by XREAL_0:def_2;
(len (Gauge (C,n))) -' 1 <= len (Gauge (C,n)) by NAT_D:35;
then ((Gauge (C,n)) * (i2,k)) `1 <= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),1)) `1 by A6, A18, A16, A24, A14, A179, JORDAN1A:18;
then p `1 <= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),1)) `1 by A178, XXREAL_0:2;
then p `1 <= E-bound C by A14, JORDAN8:12;
then A180: p `1 <= (E-max C) `1 by EUCLID:52;
p `1 >= (E-max C) `1 by A175, TOPREAL1:def_11;
then p `1 = (E-max C) `1 by A180, XXREAL_0:1;
then p = E-max C by A177, TOPREAL3:6;
hence contradiction by A9, A13, A84, A165, A176, XBOOLE_0:3; ::_thesis: verum
end;
suppose east_halfline (E-max C) meets L~ do ; ::_thesis: contradiction
then consider p being set such that
A181: p in east_halfline (E-max C) and
A182: p in L~ do by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A181;
A183: p in LSeg (do,(Index (p,do))) by A182, JORDAN3:9;
consider t being Nat such that
A184: t in dom (Lower_Seq (C,n)) and
A185: (Lower_Seq (C,n)) . t = (Gauge (C,n)) * (i1,j) by A32, FINSEQ_2:10;
1 <= t by A184, FINSEQ_3:25;
then A186: 1 < t by A34, A185, XXREAL_0:1;
t <= len (Lower_Seq (C,n)) by A184, FINSEQ_3:25;
then (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) + 1 = t by A185, A186, JORDAN3:12;
then A187: len (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i1,j)))) = (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) by A30, A185, JORDAN3:26;
Index (p,do) < len do by A182, JORDAN3:8;
then Index (p,do) < (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) by A187, XREAL_0:def_2;
then (Index (p,do)) + 1 <= (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) by NAT_1:13;
then A188: Index (p,do) <= ((len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n))))) - 1 by XREAL_1:19;
A189: do = mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n)))) by A32, JORDAN1J:37;
p in L~ (Lower_Seq (C,n)) by A36, A182;
then p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A150, A181, XBOOLE_0:def_4;
then A190: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;
A191: (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) + 1 = ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) by A34, A32, JORDAN1J:56;
0 + (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) < len (Lower_Seq (C,n)) by A30, JORDAN3:8;
then (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) > 0 by XREAL_1:20;
then Index (p,do) <= ((len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n))))) - 1 by A188, XREAL_0:def_2;
then Index (p,do) <= (len (Lower_Seq (C,n))) - (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))) by A191;
then Index (p,do) <= (len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))) by XREAL_0:def_2;
then A192: Index (p,do) < ((len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) + 1 by NAT_1:13;
A193: 1 <= Index (p,do) by A182, JORDAN3:8;
A194: ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) <= len (Lower_Seq (C,n)) by A32, FINSEQ_4:21;
((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) <> len (Lower_Seq (C,n)) by A31, A32, FINSEQ_4:19;
then A195: ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) < len (Lower_Seq (C,n)) by A194, XXREAL_0:1;
A196: 1 + 1 <= len (Lower_Seq (C,n)) by A25, XXREAL_0:2;
then A197: 2 in dom (Lower_Seq (C,n)) by FINSEQ_3:25;
set tt = ((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1;
set RC = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))));
A198: E-max C in right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by JORDAN1I:7;
A199: GoB (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = GoB (Cage (C,n)) by REVROT_1:28
.= Gauge (C,n) by JORDAN1H:44 ;
A200: L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33;
consider jj2 being Element of NAT such that
A201: 1 <= jj2 and
A202: jj2 <= width (Gauge (C,n)) and
A203: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * ((len (Gauge (C,n))),jj2) by JORDAN1D:25;
A204: len (Gauge (C,n)) >= 4 by JORDAN8:10;
then len (Gauge (C,n)) >= 1 by XXREAL_0:2;
then A205: [(len (Gauge (C,n))),jj2] in Indices (Gauge (C,n)) by A201, A202, MATRIX_1:36;
A206: len (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = len (Cage (C,n)) by REVROT_1:14;
Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) by JORDAN1G:18;
then A207: LSeg ((Lower_Seq (C,n)),1) = LSeg ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by A196, SPPOL_2:9;
A208: E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))) is_sequence_on Gauge (C,n) by A146, REVROT_1:34;
then consider ii, jj being Element of NAT such that
A209: [ii,(jj + 1)] in Indices (Gauge (C,n)) and
A210: [ii,jj] in Indices (Gauge (C,n)) and
A211: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = (Gauge (C,n)) * (ii,(jj + 1)) and
A212: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1) = (Gauge (C,n)) * (ii,jj) by A136, A200, A206, A208, FINSEQ_6:92, JORDAN1I:23;
A213: (jj + 1) + 1 <> jj ;
A214: 1 <= jj by A210, MATRIX_1:38;
(Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = E-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))))) by A200, A208, FINSEQ_6:92;
then A215: ii = len (Gauge (C,n)) by A200, A209, A211, A203, A205, GOBOARD1:5;
then ii - 1 >= 4 - 1 by A204, XREAL_1:9;
then A216: ii - 1 >= 1 by XXREAL_0:2;
then A217: 1 <= ii -' 1 by XREAL_0:def_2;
A218: jj <= width (Gauge (C,n)) by A210, MATRIX_1:38;
then A219: ((Gauge (C,n)) * ((len (Gauge (C,n))),jj)) `1 = E-bound (L~ (Cage (C,n))) by A24, A214, JORDAN1A:71;
A220: jj + 1 <= width (Gauge (C,n)) by A209, MATRIX_1:38;
ii + 1 <> ii ;
then A221: right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) = cell ((Gauge (C,n)),(ii -' 1),jj) by A136, A206, A199, A209, A210, A211, A212, A213, GOBOARD5:def_6;
A222: ii <= len (Gauge (C,n)) by A210, MATRIX_1:38;
A223: 1 <= ii by A210, MATRIX_1:38;
A224: ii <= len (Gauge (C,n)) by A209, MATRIX_1:38;
A225: 1 <= jj + 1 by A209, MATRIX_1:38;
then A226: E-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * ((len (Gauge (C,n))),(jj + 1))) `1 by A24, A220, JORDAN1A:71;
A227: 1 <= ii by A209, MATRIX_1:38;
then A228: (ii -' 1) + 1 = ii by XREAL_1:235;
then A229: ii -' 1 < len (Gauge (C,n)) by A224, NAT_1:13;
then A230: ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 = ((Gauge (C,n)) * (1,(jj + 1))) `2 by A225, A220, A217, GOBOARD5:1
.= ((Gauge (C,n)) * (ii,(jj + 1))) `2 by A227, A224, A225, A220, GOBOARD5:1 ;
A231: (E-max C) `2 = p `2 by A181, TOPREAL1:def_11;
then A232: p `2 <= ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 by A198, A224, A220, A214, A221, A228, A216, JORDAN9:17;
A233: ((Gauge (C,n)) * ((ii -' 1),jj)) `2 = ((Gauge (C,n)) * (1,jj)) `2 by A214, A218, A217, A229, GOBOARD5:1
.= ((Gauge (C,n)) * (ii,jj)) `2 by A223, A222, A214, A218, GOBOARD5:1 ;
((Gauge (C,n)) * ((ii -' 1),jj)) `2 <= p `2 by A231, A198, A224, A220, A214, A221, A228, A216, JORDAN9:17;
then p in LSeg (((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1),((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1))) by A190, A211, A212, A215, A232, A233, A230, A219, A226, GOBOARD7:7;
then A234: p in LSeg ((Lower_Seq (C,n)),1) by A136, A207, A206, TOPREAL1:def_3;
1 <= ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) by A32, FINSEQ_4:21;
then A235: LSeg ((mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n))))),(Index (p,do))) = LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1)) by A195, A193, A192, JORDAN4:19;
1 <= Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n))) by A30, JORDAN3:8;
then A236: 1 + 1 <= ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) by A191, XREAL_1:7;
then (Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))) >= (1 + 1) + 1 by A193, XREAL_1:7;
then ((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) - 1 >= ((1 + 1) + 1) - 1 by XREAL_1:9;
then A237: ((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1 >= 1 + 1 by XREAL_0:def_2;
now__::_thesis:_contradiction
percases ( ((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1 > 1 + 1 or ((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ) by A237, XXREAL_0:1;
suppose ((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1 > 1 + 1 ; ::_thesis: contradiction
then LSeg ((Lower_Seq (C,n)),1) misses LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1)) by TOPREAL1:def_7;
hence contradiction by A234, A183, A189, A235, XBOOLE_0:3; ::_thesis: verum
end;
supposeA238: ((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ; ::_thesis: contradiction
then 1 + 1 = ((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) - 1 by XREAL_0:def_2;
then (1 + 1) + 1 = (Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))) ;
then A239: ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) = 2 by A193, A236, JORDAN1E:6;
(LSeg ((Lower_Seq (C,n)),1)) /\ (LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1))) = {((Lower_Seq (C,n)) /. 2)} by A25, A238, TOPREAL1:def_6;
then p in {((Lower_Seq (C,n)) /. 2)} by A234, A183, A189, A235, XBOOLE_0:def_4;
then A240: p = (Lower_Seq (C,n)) /. 2 by TARSKI:def_1;
then A241: p in rng (Lower_Seq (C,n)) by A197, PARTFUN2:2;
p .. (Lower_Seq (C,n)) = 2 by A197, A240, FINSEQ_5:41;
then p = (Gauge (C,n)) * (i1,j) by A32, A239, A241, FINSEQ_5:9;
then ((Gauge (C,n)) * (i1,j)) `1 = E-bound (L~ (Cage (C,n))) by A240, JORDAN1G:32;
then ((Gauge (C,n)) * (i1,j)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 by A4, A10, A24, JORDAN1A:71;
hence contradiction by A2, A3, A23, A15, JORDAN1G:7; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
end;
end;
then east_halfline (E-max C) c= (L~ godo) ` by SUBSET_1:23;
then consider W being Subset of (TOP-REAL 2) such that
A242: W is_a_component_of (L~ godo) ` and
A243: east_halfline (E-max C) c= W by GOBOARD9:3;
not W is bounded by A243, JORDAN2C:121, RLTOPSP1:42;
then W is_outside_component_of L~ godo by A242, JORDAN2C:def_3;
then W c= UBD (L~ godo) by JORDAN2C:23;
then A244: east_halfline (E-max C) c= UBD (L~ godo) by A243, XBOOLE_1:1;
E-max C in east_halfline (E-max C) by TOPREAL1:38;
then E-max C in UBD (L~ godo) by A244;
then E-max C in LeftComp godo by GOBRD14:36;
then Upper_Arc C meets L~ godo by A108, A142, A165, A140, A157, JORDAN1J:36;
then A245: ( Upper_Arc C meets (L~ go) \/ (L~ pion1) or Upper_Arc C meets L~ do ) by A139, XBOOLE_1:70;
now__::_thesis:_contradiction
percases ( Upper_Arc C meets L~ go or Upper_Arc C meets L~ pion1 or Upper_Arc C meets L~ do ) by A245, XBOOLE_1:70;
suppose Upper_Arc C meets L~ go ; ::_thesis: contradiction
then Upper_Arc C meets L~ (Cage (C,n)) by A57, A124, XBOOLE_1:1, XBOOLE_1:63;
hence contradiction by A163, JORDAN10:5, XBOOLE_1:63; ::_thesis: verum
end;
suppose Upper_Arc C meets L~ pion1 ; ::_thesis: contradiction
hence contradiction by A9, A13, A84; ::_thesis: verum
end;
suppose Upper_Arc C meets L~ do ; ::_thesis: contradiction
then Upper_Arc C meets L~ (Cage (C,n)) by A36, A150, XBOOLE_1:1, XBOOLE_1:63;
hence contradiction by A163, JORDAN10:5, XBOOLE_1:63; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
suppose ((Gauge (C,n)) * (i1,j)) `1 = ((Gauge (C,n)) * (i2,k)) `1 ; ::_thesis: contradiction
then A246: i1 = i2 by A23, A19, JORDAN1G:7;
then LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) = {((Gauge (C,n)) * (i1,k))} by RLTOPSP1:70;
then LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) c= LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) by A40, ZFMISC_1:31;
then (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) = LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) by XBOOLE_1:12;
hence contradiction by A1, A3, A4, A5, A6, A7, A8, A9, A246, JORDAN1J:59; ::_thesis: verum
end;
suppose ((Gauge (C,n)) * (i1,j)) `2 = ((Gauge (C,n)) * (i2,k)) `2 ; ::_thesis: contradiction
then A247: j = k by A23, A19, JORDAN1G:6;
then LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) = {((Gauge (C,n)) * (i1,k))} by RLTOPSP1:70;
then LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) c= LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) by A39, ZFMISC_1:31;
then (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) = LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) by XBOOLE_1:12;
hence contradiction by A1, A2, A3, A4, A6, A7, A8, A9, A247, Th29; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
theorem Th45: :: JORDAN15:45
for n being Element of NAT
for C being Simple_closed_curve
for i1, i2, j, k being Element of NAT st 1 < i1 & i1 <= i2 & i2 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} holds
(LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Lower_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i1, i2, j, k being Element of NAT st 1 < i1 & i1 <= i2 & i2 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} holds
(LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Lower_Arc C
let C be Simple_closed_curve; ::_thesis: for i1, i2, j, k being Element of NAT st 1 < i1 & i1 <= i2 & i2 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} holds
(LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Lower_Arc C
let i1, i2, j, k be Element of NAT ; ::_thesis: ( 1 < i1 & i1 <= i2 & i2 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} implies (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Lower_Arc C )
set G = Gauge (C,n);
set pio = LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)));
set poz = LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)));
set US = Upper_Seq (C,n);
set LS = Lower_Seq (C,n);
assume that
A1: 1 < i1 and
A2: i1 <= i2 and
A3: i2 < len (Gauge (C,n)) and
A4: 1 <= j and
A5: j <= k and
A6: k <= width (Gauge (C,n)) and
A7: ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} and
A8: ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} and
A9: (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) misses Lower_Arc C ; ::_thesis: contradiction
set Gij = (Gauge (C,n)) * (i1,j);
A10: j <= width (Gauge (C,n)) by A5, A6, XXREAL_0:2;
A11: i1 < len (Gauge (C,n)) by A2, A3, XXREAL_0:2;
then A12: [i1,j] in Indices (Gauge (C,n)) by A1, A4, A10, MATRIX_1:36;
set Gi1k = (Gauge (C,n)) * (i1,k);
set Gik = (Gauge (C,n)) * (i2,k);
A13: L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> = (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) by TOPREAL3:16;
len (Gauge (C,n)) >= 4 by JORDAN8:10;
then A14: len (Gauge (C,n)) >= 1 by XXREAL_0:2;
then A15: [(len (Gauge (C,n))),j] in Indices (Gauge (C,n)) by A4, A10, MATRIX_1:36;
A16: 1 <= k by A4, A5, XXREAL_0:2;
then A17: [1,k] in Indices (Gauge (C,n)) by A6, A14, MATRIX_1:36;
A18: 1 < i2 by A1, A2, XXREAL_0:2;
then A19: [i2,k] in Indices (Gauge (C,n)) by A3, A6, A16, MATRIX_1:36;
A20: ((Gauge (C,n)) * (i1,k)) `2 = ((Gauge (C,n)) * (1,k)) `2 by A1, A6, A11, A16, GOBOARD5:1
.= ((Gauge (C,n)) * (i2,k)) `2 by A3, A6, A18, A16, GOBOARD5:1 ;
((Gauge (C,n)) * (i1,k)) `1 = ((Gauge (C,n)) * (i1,1)) `1 by A1, A6, A11, A16, GOBOARD5:2
.= ((Gauge (C,n)) * (i1,j)) `1 by A1, A4, A11, A10, GOBOARD5:2 ;
then A21: (Gauge (C,n)) * (i1,k) = |[(((Gauge (C,n)) * (i1,j)) `1),(((Gauge (C,n)) * (i2,k)) `2)]| by A20, EUCLID:53;
A22: [(len (Gauge (C,n))),k] in Indices (Gauge (C,n)) by A6, A16, A14, MATRIX_1:36;
A23: [i1,j] in Indices (Gauge (C,n)) by A1, A4, A11, A10, MATRIX_1:36;
set Wbo = W-bound (L~ (Cage (C,n)));
set Wmin = W-min (L~ (Cage (C,n)));
A24: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
set Ebo = E-bound (L~ (Cage (C,n)));
set Emax = E-max (L~ (Cage (C,n)));
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;
then A28: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by PARTFUN1:def_6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;
len (Lower_Seq (C,n)) in dom (Lower_Seq (C,n)) by A26, FINSEQ_3:25;
then A29: (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by PARTFUN1:def_6
.= W-min (L~ (Cage (C,n))) by JORDAN1F:8 ;
set do = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i1,j)));
(Gauge (C,n)) * (i1,j) in {((Gauge (C,n)) * (i1,j))} by TARSKI:def_1;
then A30: (Gauge (C,n)) * (i1,j) in L~ (Lower_Seq (C,n)) by A8, XBOOLE_0:def_4;
(W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * (1,k)) `1 by A6, A16, A24, JORDAN1A:73 ;
then A31: (Gauge (C,n)) * (i1,j) <> (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) by A1, A17, A29, A12, JORDAN1G:7;
then reconsider do = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i1,j))) as being_S-Seq FinSequence of (TOP-REAL 2) by A30, JORDAN3:34;
A32: (Gauge (C,n)) * (i1,j) in rng (Lower_Seq (C,n)) by A1, A4, A11, A30, A10, JORDAN1G:5, JORDAN1J:40;
then A33: do is_sequence_on Gauge (C,n) by JORDAN1G:5, JORDAN1J:39;
(E-max (L~ (Cage (C,n)))) `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by A6, A16, A24, JORDAN1A:71 ;
then A34: (Gauge (C,n)) * (i1,j) <> (Lower_Seq (C,n)) . 1 by A2, A3, A12, A22, A28, JORDAN1G:7;
A35: len do >= 1 + 1 by TOPREAL1:def_8;
then reconsider do = do as non constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A33, JGRAPH_1:12, JORDAN8:5;
A36: L~ do c= L~ (Lower_Seq (C,n)) by A30, JORDAN3:42;
A37: [1,j] in Indices (Gauge (C,n)) by A4, A10, A14, MATRIX_1:36;
A38: now__::_thesis:_not_((Gauge_(C,n))_*_(i1,j))_`1_=_W-bound_(L~_(Cage_(C,n)))
assume ((Gauge (C,n)) * (i1,j)) `1 = W-bound (L~ (Cage (C,n))) ; ::_thesis: contradiction
then ((Gauge (C,n)) * (1,j)) `1 = ((Gauge (C,n)) * (i1,j)) `1 by A4, A10, A24, JORDAN1A:73;
hence contradiction by A1, A23, A37, JORDAN1G:7; ::_thesis: verum
end;
set pion = <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>;
A39: (Gauge (C,n)) * (i1,k) in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) by RLTOPSP1:68;
set LA = Lower_Arc C;
A40: (Gauge (C,n)) * (i1,k) in LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) by RLTOPSP1:68;
set go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i2,k)));
A41: 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 A42: (Upper_Seq (C,n)) . 1 = (Upper_Seq (C,n)) /. 1 by PARTFUN1:def_6
.= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ;
A43: [i1,k] in Indices (Gauge (C,n)) by A1, A6, A11, A16, MATRIX_1:36;
A44: now__::_thesis:_for_n_being_Element_of_NAT_st_n_in_dom_<*((Gauge_(C,n))_*_(i2,k)),((Gauge_(C,n))_*_(i1,k)),((Gauge_(C,n))_*_(i1,j))*>_holds_
ex_i,_j_being_Element_of_NAT_st_
(_[i,j]_in_Indices_(Gauge_(C,n))_&_<*((Gauge_(C,n))_*_(i2,k)),((Gauge_(C,n))_*_(i1,k)),((Gauge_(C,n))_*_(i1,j))*>_/._n_=_(Gauge_(C,n))_*_(i,j)_)
let n be Element of NAT ; ::_thesis: ( n in dom <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> implies ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. n = (Gauge (C,n)) * (i,j) ) )
assume n in dom <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. n = (Gauge (C,n)) * (i,j) )
then n in {1,2,3} by FINSEQ_1:89, FINSEQ_3:1;
then ( n = 1 or n = 2 or n = 3 ) by ENUMSET1:def_1;
hence ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. n = (Gauge (C,n)) * (i,j) ) by A23, A19, A43, FINSEQ_4:18; ::_thesis: verum
end;
(Gauge (C,n)) * (i2,k) in {((Gauge (C,n)) * (i2,k))} by TARSKI:def_1;
then A45: (Gauge (C,n)) * (i2,k) in L~ (Upper_Seq (C,n)) by A7, XBOOLE_0:def_4;
(W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * (1,k)) `1 by A6, A16, A24, JORDAN1A:73 ;
then A46: (Gauge (C,n)) * (i2,k) <> (Upper_Seq (C,n)) . 1 by A1, A2, A19, A42, A17, JORDAN1G:7;
then reconsider go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i2,k))) as being_S-Seq FinSequence of (TOP-REAL 2) by A45, JORDAN3:35;
A47: (Gauge (C,n)) * (i2,k) in rng (Upper_Seq (C,n)) by A3, A6, A18, A45, A16, JORDAN1G:4, JORDAN1J:40;
then A48: go is_sequence_on Gauge (C,n) by JORDAN1G:4, JORDAN1J:38;
len do >= 1 by A35, XXREAL_0:2;
then 1 in dom do by FINSEQ_3:25;
then A49: do /. 1 = do . 1 by PARTFUN1:def_6
.= (Gauge (C,n)) * (i1,j) by A30, JORDAN3:23 ;
then A50: LSeg (do,1) = LSeg (((Gauge (C,n)) * (i1,j)),(do /. (1 + 1))) by A35, TOPREAL1:def_3;
A51: {((Gauge (C,n)) * (i1,j))} c= (LSeg (do,1)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i1,j))} or x in (LSeg (do,1)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) )
assume x in {((Gauge (C,n)) * (i1,j))} ; ::_thesis: x in (LSeg (do,1)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>)
then A52: x = (Gauge (C,n)) * (i1,j) by TARSKI:def_1;
(Gauge (C,n)) * (i1,j) in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))) by RLTOPSP1:68;
then (Gauge (C,n)) * (i1,j) in (LSeg (((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j)))) by XBOOLE_0:def_3;
then A53: (Gauge (C,n)) * (i1,j) in L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> by SPRECT_1:8;
(Gauge (C,n)) * (i1,j) in LSeg (do,1) by A50, RLTOPSP1:68;
hence x in (LSeg (do,1)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) by A52, A53, XBOOLE_0:def_4; ::_thesis: verum
end;
LSeg (do,1) c= L~ do by TOPREAL3:19;
then LSeg (do,1) c= L~ (Lower_Seq (C,n)) by A36, XBOOLE_1:1;
then (LSeg (do,1)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) c= {((Gauge (C,n)) * (i1,j))} by A8, A13, XBOOLE_1:26;
then A54: (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) /\ (LSeg (do,1)) = {((Gauge (C,n)) * (i1,j))} by A51, XBOOLE_0:def_10;
A55: rng do c= L~ do by A35, SPPOL_2:18;
A56: len go >= 1 + 1 by TOPREAL1:def_8;
then reconsider go = go as non constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A48, JGRAPH_1:12, JORDAN8:5;
A57: L~ go c= L~ (Upper_Seq (C,n)) by A45, JORDAN3:41;
A58: len go > 1 by A56, NAT_1:13;
then A59: len go in dom go by FINSEQ_3:25;
then A60: go /. (len go) = go . (len go) by PARTFUN1:def_6
.= (Gauge (C,n)) * (i2,k) by A45, JORDAN3:24 ;
reconsider m = (len go) - 1 as Element of NAT by A59, FINSEQ_3:26;
A61: m + 1 = len go ;
then A62: (len go) -' 1 = m by NAT_D:34;
m >= 1 by A56, XREAL_1:19;
then A63: LSeg (go,m) = LSeg ((go /. m),((Gauge (C,n)) * (i2,k))) by A60, A61, TOPREAL1:def_3;
A64: {((Gauge (C,n)) * (i2,k))} c= (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i2,k))} or x in (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) )
assume x in {((Gauge (C,n)) * (i2,k))} ; ::_thesis: x in (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>)
then A65: x = (Gauge (C,n)) * (i2,k) by TARSKI:def_1;
(Gauge (C,n)) * (i2,k) in LSeg (((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k))) by RLTOPSP1:68;
then (Gauge (C,n)) * (i2,k) in (LSeg (((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j)))) by XBOOLE_0:def_3;
then A66: (Gauge (C,n)) * (i2,k) in L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> by SPRECT_1:8;
(Gauge (C,n)) * (i2,k) in LSeg (go,m) by A63, RLTOPSP1:68;
hence x in (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) by A65, A66, XBOOLE_0:def_4; ::_thesis: verum
end;
LSeg (go,m) c= L~ go by TOPREAL3:19;
then LSeg (go,m) c= L~ (Upper_Seq (C,n)) by A57, XBOOLE_1:1;
then (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) c= {((Gauge (C,n)) * (i2,k))} by A7, A13, XBOOLE_1:26;
then A67: (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) = {((Gauge (C,n)) * (i2,k))} by A64, XBOOLE_0:def_10;
A68: go /. 1 = (Upper_Seq (C,n)) /. 1 by A45, SPRECT_3:22
.= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ;
then A69: W-min (L~ (Cage (C,n))) in rng go by FINSEQ_6:42;
A70: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by A27, PARTFUN1:def_6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;
A71: (L~ go) /\ (L~ do) c= {(go /. 1)}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ go) /\ (L~ do) or x in {(go /. 1)} )
assume A72: x in (L~ go) /\ (L~ do) ; ::_thesis: x in {(go /. 1)}
then A73: x in L~ do by XBOOLE_0:def_4;
A74: now__::_thesis:_not_x_=_E-max_(L~_(Cage_(C,n)))
assume x = E-max (L~ (Cage (C,n))) ; ::_thesis: contradiction
then A75: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i1,j) by A30, A70, A73, JORDAN1E:7;
((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 = E-bound (L~ (Cage (C,n))) by A4, A10, A24, JORDAN1A:71;
then (E-max (L~ (Cage (C,n)))) `1 <> E-bound (L~ (Cage (C,n))) by A2, A3, A23, A15, A75, JORDAN1G:7;
hence contradiction by EUCLID:52; ::_thesis: verum
end;
x in L~ go by A72, XBOOLE_0:def_4;
then x in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A57, A36, A73, 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 A68, A74, TARSKI:def_1; ::_thesis: verum
end;
set W2 = go /. 2;
A76: 2 in dom go by A56, FINSEQ_3:25;
go = mid ((Upper_Seq (C,n)),1,(((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n)))) by A47, JORDAN1G:49
.= (Upper_Seq (C,n)) | (((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n))) by A47, FINSEQ_4:21, FINSEQ_6:116 ;
then A77: go /. 2 = (Upper_Seq (C,n)) /. 2 by A76, FINSEQ_4:70;
A78: rng go c= L~ go by A56, SPPOL_2:18;
A79: go /. 1 = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by A68, JORDAN1F:8
.= do /. (len do) by A30, JORDAN1J:35 ;
{(go /. 1)} c= (L~ go) /\ (L~ do)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(go /. 1)} or x in (L~ go) /\ (L~ do) )
assume x in {(go /. 1)} ; ::_thesis: x in (L~ go) /\ (L~ do)
then A80: x = go /. 1 by TARSKI:def_1;
then A81: x in rng go by FINSEQ_6:42;
x in rng do by A79, A80, REVROT_1:3;
hence x in (L~ go) /\ (L~ do) by A78, A55, A81, XBOOLE_0:def_4; ::_thesis: verum
end;
then A82: (L~ go) /\ (L~ do) = {(go /. 1)} by A71, XBOOLE_0:def_10;
now__::_thesis:_contradiction
percases ( ( ((Gauge (C,n)) * (i1,j)) `1 <> ((Gauge (C,n)) * (i2,k)) `1 & ((Gauge (C,n)) * (i1,j)) `2 <> ((Gauge (C,n)) * (i2,k)) `2 ) or ((Gauge (C,n)) * (i1,j)) `1 = ((Gauge (C,n)) * (i2,k)) `1 or ((Gauge (C,n)) * (i1,j)) `2 = ((Gauge (C,n)) * (i2,k)) `2 ) ;
suppose ( ((Gauge (C,n)) * (i1,j)) `1 <> ((Gauge (C,n)) * (i2,k)) `1 & ((Gauge (C,n)) * (i1,j)) `2 <> ((Gauge (C,n)) * (i2,k)) `2 ) ; ::_thesis: contradiction
then <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> is being_S-Seq by A21, TOPREAL3:35;
then consider pion1 being FinSequence of (TOP-REAL 2) such that
A83: pion1 is_sequence_on Gauge (C,n) and
A84: pion1 is being_S-Seq and
A85: L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> = L~ pion1 and
A86: <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. 1 = pion1 /. 1 and
A87: <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. (len <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) = pion1 /. (len pion1) and
A88: len <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> <= len pion1 by A44, GOBOARD3:2;
reconsider pion1 = pion1 as being_S-Seq FinSequence of (TOP-REAL 2) by A84;
A89: (go ^' pion1) /. (len (go ^' pion1)) = <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. (len <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) by A87, GRAPH_2:54
.= <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. 3 by FINSEQ_1:45
.= do /. 1 by A49, FINSEQ_4:18 ;
A90: go /. (len go) = pion1 /. 1 by A60, A86, FINSEQ_4:18;
A91: (L~ go) /\ (L~ pion1) c= {(pion1 /. 1)}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ go) /\ (L~ pion1) or x in {(pion1 /. 1)} )
assume A92: x in (L~ go) /\ (L~ pion1) ; ::_thesis: x in {(pion1 /. 1)}
then A93: x in L~ pion1 by XBOOLE_0:def_4;
x in L~ go by A92, XBOOLE_0:def_4;
hence x in {(pion1 /. 1)} by A7, A13, A60, A57, A85, A90, A93, XBOOLE_0:def_4; ::_thesis: verum
end;
len pion1 >= 2 + 1 by A88, FINSEQ_1:45;
then A94: len pion1 > 1 + 1 by NAT_1:13;
then A95: rng pion1 c= L~ pion1 by SPPOL_2:18;
{(pion1 /. 1)} c= (L~ go) /\ (L~ pion1)
proof
let x be set ; :: 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 A96: x = pion1 /. 1 by TARSKI:def_1;
then A97: x in rng pion1 by FINSEQ_6:42;
x in rng go by A90, A96, REVROT_1:3;
hence x in (L~ go) /\ (L~ pion1) by A78, A95, A97, XBOOLE_0:def_4; ::_thesis: verum
end;
then A98: (L~ go) /\ (L~ pion1) = {(pion1 /. 1)} by A91, XBOOLE_0:def_10;
then A99: go ^' pion1 is s.n.c. by A90, JORDAN1J:54;
A100: <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. (len <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) = <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. 3 by FINSEQ_1:45
.= do /. 1 by A49, FINSEQ_4:18 ;
A101: {(pion1 /. (len pion1))} c= (L~ do) /\ (L~ pion1)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(pion1 /. (len pion1))} or x in (L~ do) /\ (L~ pion1) )
assume x in {(pion1 /. (len pion1))} ; ::_thesis: x in (L~ do) /\ (L~ pion1)
then A102: x = pion1 /. (len pion1) by TARSKI:def_1;
then A103: x in rng pion1 by REVROT_1:3;
x in rng do by A87, A100, A102, FINSEQ_6:42;
hence x in (L~ do) /\ (L~ pion1) by A55, A95, A103, XBOOLE_0:def_4; ::_thesis: verum
end;
(L~ do) /\ (L~ pion1) c= {(pion1 /. (len pion1))}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ do) /\ (L~ pion1) or x in {(pion1 /. (len pion1))} )
assume A104: x in (L~ do) /\ (L~ pion1) ; ::_thesis: x in {(pion1 /. (len pion1))}
then A105: x in L~ pion1 by XBOOLE_0:def_4;
x in L~ do by A104, XBOOLE_0:def_4;
hence x in {(pion1 /. (len pion1))} by A8, A13, A49, A36, A85, A87, A100, A105, XBOOLE_0:def_4; ::_thesis: verum
end;
then A106: (L~ do) /\ (L~ pion1) = {(pion1 /. (len pion1))} by A101, XBOOLE_0:def_10;
A107: (L~ (go ^' pion1)) /\ (L~ do) = ((L~ go) \/ (L~ pion1)) /\ (L~ do) by A90, TOPREAL8:35
.= {(go /. 1)} \/ {(do /. 1)} by A82, A87, A100, A106, XBOOLE_1:23
.= {((go ^' pion1) /. 1)} \/ {(do /. 1)} by GRAPH_2:53
.= {((go ^' pion1) /. 1),(do /. 1)} by ENUMSET1:1 ;
A108: Lower_Arc C is_an_arc_of E-max C, W-min C by JORDAN6:def_9;
then A109: Lower_Arc C is connected by JORDAN6:10;
set godo = (go ^' pion1) ^' do;
A110: do /. (len do) = (go ^' pion1) /. 1 by A79, GRAPH_2:53;
A111: go ^' pion1 is_sequence_on Gauge (C,n) by A48, A83, A90, TOPREAL8:12;
then A112: (go ^' pion1) ^' do is_sequence_on Gauge (C,n) by A33, A89, TOPREAL8:12;
A113: (len pion1) - 1 >= 1 by A94, XREAL_1:19;
then A114: (len pion1) -' 1 = (len pion1) - 1 by XREAL_0:def_2;
A115: {((Gauge (C,n)) * (i1,j))} c= (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i1,j))} or x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) )
assume x in {((Gauge (C,n)) * (i1,j))} ; ::_thesis: x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1))
then A116: x = (Gauge (C,n)) * (i1,j) by TARSKI:def_1;
pion1 /. (((len pion1) -' 1) + 1) = <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. 3 by A87, A114, FINSEQ_1:45
.= (Gauge (C,n)) * (i1,j) by FINSEQ_4:18 ;
then A117: (Gauge (C,n)) * (i1,j) in LSeg (pion1,((len pion1) -' 1)) by A113, A114, TOPREAL1:21;
(Gauge (C,n)) * (i1,j) in LSeg (do,1) by A50, RLTOPSP1:68;
hence x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) by A116, A117, XBOOLE_0:def_4; ::_thesis: verum
end;
LSeg (pion1,((len pion1) -' 1)) c= L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> by A85, TOPREAL3:19;
then (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) c= {((Gauge (C,n)) * (i1,j))} by A54, XBOOLE_1:27;
then A118: (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) = {((Gauge (C,n)) * (i1,j))} by A115, XBOOLE_0:def_10;
((len pion1) - 1) + 1 <= len pion1 ;
then A119: (len pion1) -' 1 < len pion1 by A114, NAT_1:13;
len pion1 >= 2 + 1 by A88, FINSEQ_1:45;
then A120: (len pion1) - 2 >= 0 by XREAL_1:19;
then ((len pion1) -' 2) + 1 = ((len pion1) - 2) + 1 by XREAL_0:def_2
.= (len pion1) -' 1 by A113, XREAL_0:def_2 ;
then A121: (LSeg ((go ^' pion1),((len go) + ((len pion1) -' 2)))) /\ (LSeg (do,1)) = {((go ^' pion1) /. (len (go ^' pion1)))} by A49, A90, A89, A119, A118, TOPREAL8:31;
(rng go) /\ (rng pion1) c= {(pion1 /. 1)} by A78, A95, A98, XBOOLE_1:27;
then A122: go ^' pion1 is one-to-one by JORDAN1J:55;
((len (go ^' pion1)) + 1) - 1 = ((len go) + (len pion1)) - 1 by GRAPH_2:13;
then (len (go ^' pion1)) - 1 = (len go) + ((len pion1) - 2)
.= (len go) + ((len pion1) -' 2) by A120, XREAL_0:def_2 ;
then A123: (len (go ^' pion1)) -' 1 = (len go) + ((len pion1) -' 2) by XREAL_0:def_2;
A124: L~ (Cage (C,n)) = (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n))) by JORDAN1E:13;
then A125: L~ (Upper_Seq (C,n)) c= L~ (Cage (C,n)) by XBOOLE_1:7;
then A126: L~ go c= L~ (Cage (C,n)) by A57, XBOOLE_1:1;
A127: {((Gauge (C,n)) * (i2,k))} c= (LSeg (go,m)) /\ (LSeg (pion1,1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i2,k))} or x in (LSeg (go,m)) /\ (LSeg (pion1,1)) )
assume x in {((Gauge (C,n)) * (i2,k))} ; ::_thesis: x in (LSeg (go,m)) /\ (LSeg (pion1,1))
then A128: x = (Gauge (C,n)) * (i2,k) by TARSKI:def_1;
A129: (Gauge (C,n)) * (i2,k) in LSeg (go,m) by A63, RLTOPSP1:68;
(Gauge (C,n)) * (i2,k) in LSeg (pion1,1) by A60, A90, A94, TOPREAL1:21;
hence x in (LSeg (go,m)) /\ (LSeg (pion1,1)) by A128, A129, XBOOLE_0:def_4; ::_thesis: verum
end;
LSeg (pion1,1) c= L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> by A85, TOPREAL3:19;
then (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) c= {((Gauge (C,n)) * (i2,k))} by A62, A67, XBOOLE_1:27;
then (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) = {(go /. (len go))} by A60, A62, A127, XBOOLE_0:def_10;
then A130: go ^' pion1 is unfolded by A90, TOPREAL8:34;
len (go ^' pion1) >= len go by TOPREAL8:7;
then A131: len (go ^' pion1) >= 1 + 1 by A56, XXREAL_0:2;
then A132: len (go ^' pion1) > 1 + 0 by NAT_1:13;
A133: now__::_thesis:_not_((Gauge_(C,n))_*_(i2,k))_.._(Upper_Seq_(C,n))_<=_1
assume A134: ((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n)) <= 1 ; ::_thesis: contradiction
((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n)) >= 1 by A47, FINSEQ_4:21;
then ((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n)) = 1 by A134, XXREAL_0:1;
then (Gauge (C,n)) * (i2,k) = (Upper_Seq (C,n)) /. 1 by A47, FINSEQ_5:38;
hence contradiction by A42, A46, JORDAN1F:5; ::_thesis: verum
end;
A135: Upper_Seq (C,n) is_sequence_on Gauge (C,n) by JORDAN1G:4;
A136: (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52;
set ff = Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))));
A137: 1 + 1 <= len (Cage (C,n)) by GOBOARD7:34, XXREAL_0:2;
A138: len ((go ^' pion1) ^' do) >= len (go ^' pion1) by TOPREAL8:7;
then A139: 1 + 1 <= len ((go ^' pion1) ^' do) by A131, XXREAL_0:2;
not go ^' pion1 is trivial by A131, NAT_D:60;
then reconsider godo = (go ^' pion1) ^' do as non constant standard special_circular_sequence by A139, A89, A112, A130, A123, A121, A99, A122, A107, A110, JORDAN8:4, JORDAN8:5, TOPREAL8:11, TOPREAL8:33, TOPREAL8:34;
A140: L~ godo = (L~ (go ^' pion1)) \/ (L~ do) by A89, TOPREAL8:35
.= ((L~ go) \/ (L~ pion1)) \/ (L~ do) by A90, TOPREAL8:35 ;
A141: (right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo) c= RightComp godo by A139, A112, JORDAN9:27;
2 in dom godo by A139, FINSEQ_3:25;
then A142: godo /. 2 in rng godo by PARTFUN2:2;
A143: W-min C in Lower_Arc C by A108, TOPREAL1:1;
W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
then A144: (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 = W-min (L~ (Cage (C,n))) by FINSEQ_6:92;
A145: 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 A144, 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 A144, A145, 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 A144, A145, SPRECT_5:24, XXREAL_0:2;
then A146: (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A144, A145, SPRECT_5:25, XXREAL_0:2;
A147: Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def_1;
then A148: Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))) is_sequence_on Gauge (C,n) by REVROT_1:34;
A149: ((Gauge (C,n)) * (i1,k)) `1 = ((Gauge (C,n)) * (i1,1)) `1 by A1, A6, A11, A16, GOBOARD5:2
.= ((Gauge (C,n)) * (i1,j)) `1 by A1, A4, A11, A10, GOBOARD5:2 ;
then A150: W-bound (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) = ((Gauge (C,n)) * (i1,j)) `1 by SPRECT_1:54;
A151: L~ (Lower_Seq (C,n)) c= L~ (Cage (C,n)) by A124, XBOOLE_1:7;
then A152: L~ do c= L~ (Cage (C,n)) by A36, XBOOLE_1:1;
A153: W-min C in C by SPRECT_1:13;
A154: now__::_thesis:_not_W-min_C_in_L~_godo
assume W-min C in L~ godo ; ::_thesis: contradiction
then A155: ( W-min C in (L~ go) \/ (L~ pion1) or W-min C in L~ do ) by A140, XBOOLE_0:def_3;
percases ( W-min C in L~ go or W-min C in L~ pion1 or W-min C in L~ do ) by A155, XBOOLE_0:def_3;
suppose W-min C in L~ go ; ::_thesis: contradiction
then C meets L~ (Cage (C,n)) by A126, A153, XBOOLE_0:3;
hence contradiction by JORDAN10:5; ::_thesis: verum
end;
suppose W-min C in L~ pion1 ; ::_thesis: contradiction
hence contradiction by A9, A13, A85, A143, XBOOLE_0:3; ::_thesis: verum
end;
suppose W-min C in L~ do ; ::_thesis: contradiction
then C meets L~ (Cage (C,n)) by A152, A153, XBOOLE_0:3;
hence contradiction by JORDAN10:5; ::_thesis: verum
end;
end;
end;
A156: len (Upper_Seq (C,n)) >= 2 by A41, XXREAL_0:2;
A157: (L~ go) \/ (L~ do) is compact by COMPTS_1:10;
1 + 1 <= len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by GOBOARD7:34, XXREAL_0:2;
then right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1) = right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(GoB (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) by JORDAN1H:23
.= right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(GoB (Cage (C,n)))) by REVROT_1:28
.= right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(Gauge (C,n))) by JORDAN1H:44
.= right_cell (((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n))))),1,(Gauge (C,n))) by A146, A148, JORDAN1J:53
.= right_cell ((Upper_Seq (C,n)),1,(Gauge (C,n))) by JORDAN1E:def_1
.= right_cell ((R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i2,k)))),1,(Gauge (C,n))) by A47, A135, A133, JORDAN1J:52
.= right_cell ((go ^' pion1),1,(Gauge (C,n))) by A58, A111, JORDAN1J:51
.= right_cell (godo,1,(Gauge (C,n))) by A132, A112, JORDAN1J:51 ;
then W-min C in right_cell (godo,1,(Gauge (C,n))) by JORDAN1I:6;
then A158: W-min C in (right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo) by A154, XBOOLE_0:def_5;
A159: rng godo c= L~ godo by A131, A138, SPPOL_2:18, XXREAL_0:2;
A160: godo /. 1 = (go ^' pion1) /. 1 by GRAPH_2:53
.= W-min (L~ (Cage (C,n))) by A68, GRAPH_2:53 ;
A161: ((Gauge (C,n)) * (i1,k)) `1 <= ((Gauge (C,n)) * (i2,k)) `1 by A1, A2, A3, A6, A16, JORDAN1A:18;
then A162: W-bound (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) = ((Gauge (C,n)) * (i1,k)) `1 by SPRECT_1:54;
W-bound ((LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))))) = min ((W-bound (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))),(W-bound (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))))) by SPRECT_1:47
.= ((Gauge (C,n)) * (i1,j)) `1 by A149, A162, A150 ;
then A163: W-bound (L~ pion1) = ((Gauge (C,n)) * (i1,j)) `1 by A85, TOPREAL3:16;
A164: Lower_Arc C c= C by JORDAN6:61;
((Gauge (C,n)) * (i1,j)) `1 >= W-bound (L~ (Cage (C,n))) by A30, A151, PSCOMP_1:24;
then A165: ((Gauge (C,n)) * (i1,j)) `1 > W-bound (L~ (Cage (C,n))) by A38, XXREAL_0:1;
A166: E-max C in Lower_Arc C by A108, TOPREAL1:1;
W-min (L~ (Cage (C,n))) in (L~ go) \/ (L~ do) by A78, A69, XBOOLE_0:def_3;
then A167: W-min ((L~ go) \/ (L~ do)) = W-min (L~ (Cage (C,n))) by A126, A152, A157, JORDAN1J:21, XBOOLE_1:8;
(W-min ((L~ go) \/ (L~ do))) `1 = W-bound ((L~ go) \/ (L~ do)) by EUCLID:52;
then W-min (((L~ go) \/ (L~ do)) \/ (L~ pion1)) = W-min ((L~ go) \/ (L~ do)) by A163, A157, A167, A136, A165, JORDAN1J:33;
then A168: W-min (L~ godo) = W-min (L~ (Cage (C,n))) by A140, A167, XBOOLE_1:4;
godo /. 2 = (go ^' pion1) /. 2 by A131, GRAPH_2:57
.= (Upper_Seq (C,n)) /. 2 by A56, A77, GRAPH_2:57
.= ((Upper_Seq (C,n)) ^' (Lower_Seq (C,n))) /. 2 by A156, GRAPH_2:57
.= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 2 by JORDAN1E:11 ;
then godo /. 2 in W-most (L~ (Cage (C,n))) by JORDAN1I:25;
then (godo /. 2) `1 = (W-min (L~ godo)) `1 by A168, PSCOMP_1:31
.= W-bound (L~ godo) by EUCLID:52 ;
then godo /. 2 in W-most (L~ godo) by A159, A142, SPRECT_2:12;
then (Rotate (godo,(W-min (L~ godo)))) /. 2 in W-most (L~ godo) by A160, A168, FINSEQ_6:89;
then reconsider godo = godo as non 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 A169: (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 ;
A170: east_halfline (E-max C) misses L~ go
proof
assume east_halfline (E-max C) meets L~ go ; ::_thesis: contradiction
then consider p being set such that
A171: p in east_halfline (E-max C) and
A172: p in L~ go by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A171;
p in L~ (Upper_Seq (C,n)) by A57, A172;
then p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A125, A171, XBOOLE_0:def_4;
then A173: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;
then A174: p = E-max (L~ (Cage (C,n))) by A57, A172, JORDAN1J:46;
then E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i2,k) by A45, A169, A172, JORDAN1J:43;
then ((Gauge (C,n)) * (i2,k)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by A6, A16, A24, A173, A174, JORDAN1A:71;
hence contradiction by A3, A19, A22, JORDAN1G:7; ::_thesis: verum
end;
now__::_thesis:_not_east_halfline_(E-max_C)_meets_L~_godo
assume east_halfline (E-max C) meets L~ godo ; ::_thesis: contradiction
then A175: ( east_halfline (E-max C) meets (L~ go) \/ (L~ pion1) or east_halfline (E-max C) meets L~ do ) by A140, XBOOLE_1:70;
percases ( 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~ do ) by A175, XBOOLE_1:70;
suppose east_halfline (E-max C) meets L~ go ; ::_thesis: contradiction
hence contradiction by A170; ::_thesis: verum
end;
suppose east_halfline (E-max C) meets L~ pion1 ; ::_thesis: contradiction
then consider p being set such that
A176: p in east_halfline (E-max C) and
A177: p in L~ pion1 by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A176;
A178: p `2 = (E-max C) `2 by A176, TOPREAL1:def_11;
A179: now__::_thesis:_p_`1_<=_((Gauge_(C,n))_*_(i2,k))_`1
percases ( p in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) or p in LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) ) by A13, A85, A177, XBOOLE_0:def_3;
suppose p in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) ; ::_thesis: p `1 <= ((Gauge (C,n)) * (i2,k)) `1
hence p `1 <= ((Gauge (C,n)) * (i2,k)) `1 by A161, TOPREAL1:3; ::_thesis: verum
end;
suppose p in LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) ; ::_thesis: p `1 <= ((Gauge (C,n)) * (i2,k)) `1
hence p `1 <= ((Gauge (C,n)) * (i2,k)) `1 by A149, A161, GOBOARD7:5; ::_thesis: verum
end;
end;
end;
i2 + 1 <= len (Gauge (C,n)) by A3, NAT_1:13;
then (i2 + 1) - 1 <= (len (Gauge (C,n))) - 1 by XREAL_1:9;
then A180: i2 <= (len (Gauge (C,n))) -' 1 by XREAL_0:def_2;
(len (Gauge (C,n))) -' 1 <= len (Gauge (C,n)) by NAT_D:35;
then ((Gauge (C,n)) * (i2,k)) `1 <= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),1)) `1 by A6, A18, A16, A24, A14, A180, JORDAN1A:18;
then p `1 <= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),1)) `1 by A179, XXREAL_0:2;
then p `1 <= E-bound C by A14, JORDAN8:12;
then A181: p `1 <= (E-max C) `1 by EUCLID:52;
p `1 >= (E-max C) `1 by A176, TOPREAL1:def_11;
then p `1 = (E-max C) `1 by A181, XXREAL_0:1;
then p = E-max C by A178, TOPREAL3:6;
hence contradiction by A9, A13, A85, A166, A177, XBOOLE_0:3; ::_thesis: verum
end;
suppose east_halfline (E-max C) meets L~ do ; ::_thesis: contradiction
then consider p being set such that
A182: p in east_halfline (E-max C) and
A183: p in L~ do by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A182;
A184: p in LSeg (do,(Index (p,do))) by A183, JORDAN3:9;
consider t being Nat such that
A185: t in dom (Lower_Seq (C,n)) and
A186: (Lower_Seq (C,n)) . t = (Gauge (C,n)) * (i1,j) by A32, FINSEQ_2:10;
1 <= t by A185, FINSEQ_3:25;
then A187: 1 < t by A34, A186, XXREAL_0:1;
t <= len (Lower_Seq (C,n)) by A185, FINSEQ_3:25;
then (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) + 1 = t by A186, A187, JORDAN3:12;
then A188: len (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i1,j)))) = (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) by A30, A186, JORDAN3:26;
Index (p,do) < len do by A183, JORDAN3:8;
then Index (p,do) < (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) by A188, XREAL_0:def_2;
then (Index (p,do)) + 1 <= (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) by NAT_1:13;
then A189: Index (p,do) <= ((len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n))))) - 1 by XREAL_1:19;
A190: do = mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n)))) by A32, JORDAN1J:37;
p in L~ (Lower_Seq (C,n)) by A36, A183;
then p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A151, A182, XBOOLE_0:def_4;
then A191: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;
A192: (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) + 1 = ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) by A34, A32, JORDAN1J:56;
0 + (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) < len (Lower_Seq (C,n)) by A30, JORDAN3:8;
then (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) > 0 by XREAL_1:20;
then Index (p,do) <= ((len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n))))) - 1 by A189, XREAL_0:def_2;
then Index (p,do) <= (len (Lower_Seq (C,n))) - (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))) by A192;
then Index (p,do) <= (len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))) by XREAL_0:def_2;
then A193: Index (p,do) < ((len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) + 1 by NAT_1:13;
A194: 1 <= Index (p,do) by A183, JORDAN3:8;
A195: ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) <= len (Lower_Seq (C,n)) by A32, FINSEQ_4:21;
((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) <> len (Lower_Seq (C,n)) by A31, A32, FINSEQ_4:19;
then A196: ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) < len (Lower_Seq (C,n)) by A195, XXREAL_0:1;
A197: 1 + 1 <= len (Lower_Seq (C,n)) by A25, XXREAL_0:2;
then A198: 2 in dom (Lower_Seq (C,n)) by FINSEQ_3:25;
set tt = ((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1;
set RC = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))));
A199: E-max C in right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by JORDAN1I:7;
A200: 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 ;
A201: L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33;
consider jj2 being Element of NAT such that
A202: 1 <= jj2 and
A203: jj2 <= width (Gauge (C,n)) and
A204: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * ((len (Gauge (C,n))),jj2) by JORDAN1D:25;
A205: len (Gauge (C,n)) >= 4 by JORDAN8:10;
then len (Gauge (C,n)) >= 1 by XXREAL_0:2;
then A206: [(len (Gauge (C,n))),jj2] in Indices (Gauge (C,n)) by A202, A203, MATRIX_1:36;
A207: len (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = len (Cage (C,n)) by REVROT_1:14;
Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) by JORDAN1G:18;
then A208: LSeg ((Lower_Seq (C,n)),1) = LSeg ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by A197, SPPOL_2:9;
A209: 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 A147, REVROT_1:34;
then consider ii, jj being Element of NAT such that
A210: [ii,(jj + 1)] in Indices (Gauge (C,n)) and
A211: [ii,jj] in Indices (Gauge (C,n)) and
A212: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = (Gauge (C,n)) * (ii,(jj + 1)) and
A213: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1) = (Gauge (C,n)) * (ii,jj) by A137, A201, A207, A209, FINSEQ_6:92, JORDAN1I:23;
A214: (jj + 1) + 1 <> jj ;
A215: 1 <= jj by A211, MATRIX_1:38;
(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 A201, A209, FINSEQ_6:92;
then A216: ii = len (Gauge (C,n)) by A201, A210, A212, A204, A206, GOBOARD1:5;
then ii - 1 >= 4 - 1 by A205, XREAL_1:9;
then A217: ii - 1 >= 1 by XXREAL_0:2;
then A218: 1 <= ii -' 1 by XREAL_0:def_2;
A219: jj <= width (Gauge (C,n)) by A211, MATRIX_1:38;
then A220: ((Gauge (C,n)) * ((len (Gauge (C,n))),jj)) `1 = E-bound (L~ (Cage (C,n))) by A24, A215, JORDAN1A:71;
A221: jj + 1 <= width (Gauge (C,n)) by A210, MATRIX_1:38;
ii + 1 <> ii ;
then A222: right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) = cell ((Gauge (C,n)),(ii -' 1),jj) by A137, A207, A200, A210, A211, A212, A213, A214, GOBOARD5:def_6;
A223: ii <= len (Gauge (C,n)) by A211, MATRIX_1:38;
A224: 1 <= ii by A211, MATRIX_1:38;
A225: ii <= len (Gauge (C,n)) by A210, MATRIX_1:38;
A226: 1 <= jj + 1 by A210, MATRIX_1:38;
then A227: E-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * ((len (Gauge (C,n))),(jj + 1))) `1 by A24, A221, JORDAN1A:71;
A228: 1 <= ii by A210, MATRIX_1:38;
then A229: (ii -' 1) + 1 = ii by XREAL_1:235;
then A230: ii -' 1 < len (Gauge (C,n)) by A225, NAT_1:13;
then A231: ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 = ((Gauge (C,n)) * (1,(jj + 1))) `2 by A226, A221, A218, GOBOARD5:1
.= ((Gauge (C,n)) * (ii,(jj + 1))) `2 by A228, A225, A226, A221, GOBOARD5:1 ;
A232: (E-max C) `2 = p `2 by A182, TOPREAL1:def_11;
then A233: p `2 <= ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 by A199, A225, A221, A215, A222, A229, A217, JORDAN9:17;
A234: ((Gauge (C,n)) * ((ii -' 1),jj)) `2 = ((Gauge (C,n)) * (1,jj)) `2 by A215, A219, A218, A230, GOBOARD5:1
.= ((Gauge (C,n)) * (ii,jj)) `2 by A224, A223, A215, A219, GOBOARD5:1 ;
((Gauge (C,n)) * ((ii -' 1),jj)) `2 <= p `2 by A232, A199, A225, A221, A215, A222, A229, A217, JORDAN9:17;
then p in LSeg (((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1),((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1))) by A191, A212, A213, A216, A233, A234, A231, A220, A227, GOBOARD7:7;
then A235: p in LSeg ((Lower_Seq (C,n)),1) by A137, A208, A207, TOPREAL1:def_3;
1 <= ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) by A32, FINSEQ_4:21;
then A236: LSeg ((mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n))))),(Index (p,do))) = LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1)) by A196, A194, A193, JORDAN4:19;
1 <= Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n))) by A30, JORDAN3:8;
then A237: 1 + 1 <= ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) by A192, XREAL_1:7;
then (Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))) >= (1 + 1) + 1 by A194, XREAL_1:7;
then ((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) - 1 >= ((1 + 1) + 1) - 1 by XREAL_1:9;
then A238: ((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1 >= 1 + 1 by XREAL_0:def_2;
now__::_thesis:_contradiction
percases ( ((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1 > 1 + 1 or ((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ) by A238, XXREAL_0:1;
suppose ((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1 > 1 + 1 ; ::_thesis: contradiction
then LSeg ((Lower_Seq (C,n)),1) misses LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1)) by TOPREAL1:def_7;
hence contradiction by A235, A184, A190, A236, XBOOLE_0:3; ::_thesis: verum
end;
supposeA239: ((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ; ::_thesis: contradiction
then 1 + 1 = ((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) - 1 by XREAL_0:def_2;
then (1 + 1) + 1 = (Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))) ;
then A240: ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) = 2 by A194, A237, JORDAN1E:6;
(LSeg ((Lower_Seq (C,n)),1)) /\ (LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1))) = {((Lower_Seq (C,n)) /. 2)} by A25, A239, TOPREAL1:def_6;
then p in {((Lower_Seq (C,n)) /. 2)} by A235, A184, A190, A236, XBOOLE_0:def_4;
then A241: p = (Lower_Seq (C,n)) /. 2 by TARSKI:def_1;
then A242: p in rng (Lower_Seq (C,n)) by A198, PARTFUN2:2;
p .. (Lower_Seq (C,n)) = 2 by A198, A241, FINSEQ_5:41;
then p = (Gauge (C,n)) * (i1,j) by A32, A240, A242, FINSEQ_5:9;
then ((Gauge (C,n)) * (i1,j)) `1 = E-bound (L~ (Cage (C,n))) by A241, JORDAN1G:32;
then ((Gauge (C,n)) * (i1,j)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 by A4, A10, A24, JORDAN1A:71;
hence contradiction by A2, A3, A23, A15, JORDAN1G:7; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
end;
end;
then east_halfline (E-max C) c= (L~ godo) ` by SUBSET_1:23;
then consider W being Subset of (TOP-REAL 2) such that
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, XBOOLE_1:1;
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 Lower_Arc C meets L~ godo by A109, A143, A166, A141, A158, JORDAN1J:36;
then A246: ( Lower_Arc C meets (L~ go) \/ (L~ pion1) or Lower_Arc C meets L~ do ) by A140, XBOOLE_1:70;
now__::_thesis:_contradiction
percases ( Lower_Arc C meets L~ go or Lower_Arc C meets L~ pion1 or Lower_Arc C meets L~ do ) by A246, XBOOLE_1:70;
suppose Lower_Arc C meets L~ go ; ::_thesis: contradiction
then Lower_Arc C meets L~ (Cage (C,n)) by A57, A125, XBOOLE_1:1, XBOOLE_1:63;
hence contradiction by A164, JORDAN10:5, XBOOLE_1:63; ::_thesis: verum
end;
suppose Lower_Arc C meets L~ pion1 ; ::_thesis: contradiction
hence contradiction by A9, A13, A85; ::_thesis: verum
end;
suppose Lower_Arc C meets L~ do ; ::_thesis: contradiction
then Lower_Arc C meets L~ (Cage (C,n)) by A36, A151, XBOOLE_1:1, XBOOLE_1:63;
hence contradiction by A164, JORDAN10:5, XBOOLE_1:63; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
suppose ((Gauge (C,n)) * (i1,j)) `1 = ((Gauge (C,n)) * (i2,k)) `1 ; ::_thesis: contradiction
then A247: i1 = i2 by A23, A19, JORDAN1G:7;
then LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) = {((Gauge (C,n)) * (i1,k))} by RLTOPSP1:70;
then LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) c= LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) by A40, ZFMISC_1:31;
then (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) = LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) by XBOOLE_1:12;
hence contradiction by A1, A3, A4, A5, A6, A7, A8, A9, A247, JORDAN1J:58; ::_thesis: verum
end;
suppose ((Gauge (C,n)) * (i1,j)) `2 = ((Gauge (C,n)) * (i2,k)) `2 ; ::_thesis: contradiction
then A248: j = k by A23, A19, JORDAN1G:6;
then LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) = {((Gauge (C,n)) * (i1,k))} by RLTOPSP1:70;
then LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) c= LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) by A39, ZFMISC_1:31;
then (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) = LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) by XBOOLE_1:12;
hence contradiction by A1, A2, A3, A4, A6, A7, A8, A9, A248, Th28; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
theorem Th46: :: JORDAN15:46
for n being Element of NAT
for C being Simple_closed_curve
for i1, i2, j, k being Element of NAT st 1 < i2 & i2 <= i1 & i1 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} holds
(LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Upper_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i1, i2, j, k being Element of NAT st 1 < i2 & i2 <= i1 & i1 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} holds
(LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Upper_Arc C
let C be Simple_closed_curve; ::_thesis: for i1, i2, j, k being Element of NAT st 1 < i2 & i2 <= i1 & i1 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} holds
(LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Upper_Arc C
let i1, i2, j, k be Element of NAT ; ::_thesis: ( 1 < i2 & i2 <= i1 & i1 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} implies (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Upper_Arc C )
set G = Gauge (C,n);
set pio = LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)));
set poz = LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)));
set US = Upper_Seq (C,n);
set LS = Lower_Seq (C,n);
assume that
A1: 1 < i2 and
A2: i2 <= i1 and
A3: i1 < len (Gauge (C,n)) and
A4: 1 <= j and
A5: j <= k and
A6: k <= width (Gauge (C,n)) and
A7: ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} and
A8: ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} and
A9: (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) misses Upper_Arc C ; ::_thesis: contradiction
set Gi1k = (Gauge (C,n)) * (i1,k);
set Gik = (Gauge (C,n)) * (i2,k);
A10: 1 <= k by A4, A5, XXREAL_0:2;
A11: i2 < len (Gauge (C,n)) by A2, A3, XXREAL_0:2;
then A12: [i2,k] in Indices (Gauge (C,n)) by A1, A6, A10, MATRIX_1:36;
set Wmin = W-min (L~ (Cage (C,n)));
set Wbo = W-bound (L~ (Cage (C,n)));
A13: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
set go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i2,k)));
A14: len (Upper_Seq (C,n)) >= 3 by JORDAN1E:15;
then len (Upper_Seq (C,n)) >= 1 by XXREAL_0:2;
then 1 in dom (Upper_Seq (C,n)) by FINSEQ_3:25;
then A15: (Upper_Seq (C,n)) . 1 = (Upper_Seq (C,n)) /. 1 by PARTFUN1:def_6
.= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ;
set Gij = (Gauge (C,n)) * (i1,j);
set do = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i1,j)));
(Gauge (C,n)) * (i1,j) in {((Gauge (C,n)) * (i1,j))} by TARSKI:def_1;
then A16: (Gauge (C,n)) * (i1,j) in L~ (Lower_Seq (C,n)) by A8, XBOOLE_0:def_4;
A17: 1 < i1 by A1, A2, XXREAL_0:2;
then A18: ((Gauge (C,n)) * (i1,k)) `2 = ((Gauge (C,n)) * (1,k)) `2 by A3, A6, A10, GOBOARD5:1
.= ((Gauge (C,n)) * (i2,k)) `2 by A1, A6, A11, A10, GOBOARD5:1 ;
A19: j <= width (Gauge (C,n)) by A5, A6, XXREAL_0:2;
then A20: [i1,j] in Indices (Gauge (C,n)) by A3, A4, A17, MATRIX_1:36;
len (Gauge (C,n)) >= 4 by JORDAN8:10;
then A21: len (Gauge (C,n)) >= 1 by XXREAL_0:2;
then A22: [(len (Gauge (C,n))),j] in Indices (Gauge (C,n)) by A4, A19, MATRIX_1:36;
A23: [1,k] in Indices (Gauge (C,n)) by A6, A10, A21, MATRIX_1:36;
A24: now__::_thesis:_not_((Gauge_(C,n))_*_(i2,k))_`1_=_W-bound_(L~_(Cage_(C,n)))
assume ((Gauge (C,n)) * (i2,k)) `1 = W-bound (L~ (Cage (C,n))) ; ::_thesis: contradiction
then ((Gauge (C,n)) * (1,k)) `1 = ((Gauge (C,n)) * (i2,k)) `1 by A6, A10, A13, JORDAN1A:73;
hence contradiction by A1, A12, A23, JORDAN1G:7; ::_thesis: verum
end;
A25: [i1,j] in Indices (Gauge (C,n)) by A3, A4, A17, A19, MATRIX_1:36;
set pion = <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>;
A26: (Gauge (C,n)) * (i1,k) in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) by RLTOPSP1:68;
set UA = Upper_Arc C;
A27: (Gauge (C,n)) * (i1,k) in LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) by RLTOPSP1:68;
A28: [i1,k] in Indices (Gauge (C,n)) by A3, A6, A17, A10, MATRIX_1:36;
A29: now__::_thesis:_for_n_being_Element_of_NAT_st_n_in_dom_<*((Gauge_(C,n))_*_(i2,k)),((Gauge_(C,n))_*_(i1,k)),((Gauge_(C,n))_*_(i1,j))*>_holds_
ex_i,_j_being_Element_of_NAT_st_
(_[i,j]_in_Indices_(Gauge_(C,n))_&_<*((Gauge_(C,n))_*_(i2,k)),((Gauge_(C,n))_*_(i1,k)),((Gauge_(C,n))_*_(i1,j))*>_/._n_=_(Gauge_(C,n))_*_(i,j)_)
let n be Element of NAT ; ::_thesis: ( n in dom <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> implies ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. n = (Gauge (C,n)) * (i,j) ) )
assume n in dom <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. n = (Gauge (C,n)) * (i,j) )
then n in {1,2,3} by FINSEQ_1:89, FINSEQ_3:1;
then ( n = 1 or n = 2 or n = 3 ) by ENUMSET1:def_1;
hence ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. n = (Gauge (C,n)) * (i,j) ) by A25, A12, A28, FINSEQ_4:18; ::_thesis: verum
end;
(Gauge (C,n)) * (i2,k) in {((Gauge (C,n)) * (i2,k))} by TARSKI:def_1;
then A30: (Gauge (C,n)) * (i2,k) in L~ (Upper_Seq (C,n)) by A7, XBOOLE_0:def_4;
set Emax = E-max (L~ (Cage (C,n)));
A31: len (Lower_Seq (C,n)) >= 1 + 2 by JORDAN1E:15;
then A32: len (Lower_Seq (C,n)) >= 1 by XXREAL_0:2;
then A33: 1 in dom (Lower_Seq (C,n)) by FINSEQ_3:25;
then A34: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by PARTFUN1:def_6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;
len (Lower_Seq (C,n)) in dom (Lower_Seq (C,n)) by A32, FINSEQ_3:25;
then A35: (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by PARTFUN1:def_6
.= W-min (L~ (Cage (C,n))) by JORDAN1F:8 ;
set Ebo = E-bound (L~ (Cage (C,n)));
A36: L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> = (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) by TOPREAL3:16;
(W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * (1,k)) `1 by A6, A10, A13, JORDAN1A:73 ;
then A37: (Gauge (C,n)) * (i2,k) <> (Upper_Seq (C,n)) . 1 by A1, A12, A15, A23, JORDAN1G:7;
then reconsider go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i2,k))) as being_S-Seq FinSequence of (TOP-REAL 2) by A30, JORDAN3:35;
A38: (Gauge (C,n)) * (i2,k) in rng (Upper_Seq (C,n)) by A1, A6, A11, A30, A10, JORDAN1G:4, JORDAN1J:40;
then A39: go is_sequence_on Gauge (C,n) by JORDAN1G:4, JORDAN1J:38;
((Gauge (C,n)) * (i1,k)) `1 = ((Gauge (C,n)) * (i1,1)) `1 by A3, A6, A17, A10, GOBOARD5:2
.= ((Gauge (C,n)) * (i1,j)) `1 by A3, A4, A17, A19, GOBOARD5:2 ;
then A40: (Gauge (C,n)) * (i1,k) = |[(((Gauge (C,n)) * (i1,j)) `1),(((Gauge (C,n)) * (i2,k)) `2)]| by A18, EUCLID:53;
A41: [(len (Gauge (C,n))),k] in Indices (Gauge (C,n)) by A6, A10, A21, MATRIX_1:36;
A42: len go >= 1 + 1 by TOPREAL1:def_8;
(W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * (1,k)) `1 by A6, A10, A13, JORDAN1A:73 ;
then A43: (Gauge (C,n)) * (i1,j) <> (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) by A1, A2, A23, A35, A20, JORDAN1G:7;
then reconsider do = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i1,j))) as being_S-Seq FinSequence of (TOP-REAL 2) by A16, JORDAN3:34;
A44: (Gauge (C,n)) * (i1,j) in rng (Lower_Seq (C,n)) by A3, A4, A17, A16, A19, JORDAN1G:5, JORDAN1J:40;
then A45: do is_sequence_on Gauge (C,n) by JORDAN1G:5, JORDAN1J:39;
(E-max (L~ (Cage (C,n)))) `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by A6, A10, A13, JORDAN1A:71 ;
then A46: (Gauge (C,n)) * (i1,j) <> (Lower_Seq (C,n)) . 1 by A3, A20, A41, A34, JORDAN1G:7;
A47: len do >= 1 + 1 by TOPREAL1:def_8;
then reconsider do = do as non constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A45, JGRAPH_1:12, JORDAN8:5;
A48: L~ do c= L~ (Lower_Seq (C,n)) by A16, JORDAN3:42;
len do >= 1 by A47, XXREAL_0:2;
then 1 in dom do by FINSEQ_3:25;
then A49: do /. 1 = do . 1 by PARTFUN1:def_6
.= (Gauge (C,n)) * (i1,j) by A16, JORDAN3:23 ;
then A50: LSeg (do,1) = LSeg (((Gauge (C,n)) * (i1,j)),(do /. (1 + 1))) by A47, TOPREAL1:def_3;
A51: {((Gauge (C,n)) * (i1,j))} c= (LSeg (do,1)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i1,j))} or x in (LSeg (do,1)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) )
assume x in {((Gauge (C,n)) * (i1,j))} ; ::_thesis: x in (LSeg (do,1)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>)
then A52: x = (Gauge (C,n)) * (i1,j) by TARSKI:def_1;
(Gauge (C,n)) * (i1,j) in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))) by RLTOPSP1:68;
then (Gauge (C,n)) * (i1,j) in (LSeg (((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j)))) by XBOOLE_0:def_3;
then A53: (Gauge (C,n)) * (i1,j) in L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> by SPRECT_1:8;
(Gauge (C,n)) * (i1,j) in LSeg (do,1) by A50, RLTOPSP1:68;
hence x in (LSeg (do,1)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) by A52, A53, XBOOLE_0:def_4; ::_thesis: verum
end;
LSeg (do,1) c= L~ do by TOPREAL3:19;
then LSeg (do,1) c= L~ (Lower_Seq (C,n)) by A48, XBOOLE_1:1;
then (LSeg (do,1)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) c= {((Gauge (C,n)) * (i1,j))} by A8, A36, XBOOLE_1:26;
then A54: (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) /\ (LSeg (do,1)) = {((Gauge (C,n)) * (i1,j))} by A51, XBOOLE_0:def_10;
A55: rng do c= L~ do by A47, SPPOL_2:18;
reconsider go = go as non constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A42, A39, JGRAPH_1:12, JORDAN8:5;
A56: L~ go c= L~ (Upper_Seq (C,n)) by A30, JORDAN3:41;
A57: len go > 1 by A42, NAT_1:13;
then A58: len go in dom go by FINSEQ_3:25;
then A59: go /. (len go) = go . (len go) by PARTFUN1:def_6
.= (Gauge (C,n)) * (i2,k) by A30, JORDAN3:24 ;
reconsider m = (len go) - 1 as Element of NAT by A58, FINSEQ_3:26;
A60: m + 1 = len go ;
then A61: (len go) -' 1 = m by NAT_D:34;
m >= 1 by A42, XREAL_1:19;
then A62: LSeg (go,m) = LSeg ((go /. m),((Gauge (C,n)) * (i2,k))) by A59, A60, TOPREAL1:def_3;
A63: {((Gauge (C,n)) * (i2,k))} c= (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i2,k))} or x in (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) )
assume x in {((Gauge (C,n)) * (i2,k))} ; ::_thesis: x in (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>)
then A64: x = (Gauge (C,n)) * (i2,k) by TARSKI:def_1;
(Gauge (C,n)) * (i2,k) in LSeg (((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k))) by RLTOPSP1:68;
then (Gauge (C,n)) * (i2,k) in (LSeg (((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j)))) by XBOOLE_0:def_3;
then A65: (Gauge (C,n)) * (i2,k) in L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> by SPRECT_1:8;
(Gauge (C,n)) * (i2,k) in LSeg (go,m) by A62, RLTOPSP1:68;
hence x in (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) by A64, A65, XBOOLE_0:def_4; ::_thesis: verum
end;
LSeg (go,m) c= L~ go by TOPREAL3:19;
then LSeg (go,m) c= L~ (Upper_Seq (C,n)) by A56, XBOOLE_1:1;
then (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) c= {((Gauge (C,n)) * (i2,k))} by A7, A36, XBOOLE_1:26;
then A66: (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) = {((Gauge (C,n)) * (i2,k))} by A63, XBOOLE_0:def_10;
A67: go /. 1 = (Upper_Seq (C,n)) /. 1 by A30, SPRECT_3:22
.= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ;
then A68: W-min (L~ (Cage (C,n))) in rng go by FINSEQ_6:42;
A69: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by A33, PARTFUN1:def_6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;
A70: (L~ go) /\ (L~ do) c= {(go /. 1)}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ go) /\ (L~ do) or x in {(go /. 1)} )
assume A71: x in (L~ go) /\ (L~ do) ; ::_thesis: x in {(go /. 1)}
then A72: x in L~ do by XBOOLE_0:def_4;
A73: now__::_thesis:_not_x_=_E-max_(L~_(Cage_(C,n)))
assume x = E-max (L~ (Cage (C,n))) ; ::_thesis: contradiction
then A74: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i1,j) by A16, A69, A72, JORDAN1E:7;
((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 = E-bound (L~ (Cage (C,n))) by A4, A19, A13, JORDAN1A:71;
then (E-max (L~ (Cage (C,n)))) `1 <> E-bound (L~ (Cage (C,n))) by A3, A25, A22, A74, JORDAN1G:7;
hence contradiction by EUCLID:52; ::_thesis: verum
end;
x in L~ go by A71, XBOOLE_0:def_4;
then x in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A56, A48, A72, XBOOLE_0:def_4;
then x in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16;
then ( x = W-min (L~ (Cage (C,n))) or x = E-max (L~ (Cage (C,n))) ) by TARSKI:def_2;
hence x in {(go /. 1)} by A67, A73, TARSKI:def_1; ::_thesis: verum
end;
set W2 = go /. 2;
A75: 2 in dom go by A42, FINSEQ_3:25;
go = mid ((Upper_Seq (C,n)),1,(((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n)))) by A38, JORDAN1G:49
.= (Upper_Seq (C,n)) | (((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n))) by A38, FINSEQ_4:21, FINSEQ_6:116 ;
then A76: go /. 2 = (Upper_Seq (C,n)) /. 2 by A75, FINSEQ_4:70;
A77: rng go c= L~ go by A42, SPPOL_2:18;
A78: go /. 1 = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by A67, JORDAN1F:8
.= do /. (len do) by A16, JORDAN1J:35 ;
{(go /. 1)} c= (L~ go) /\ (L~ do)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(go /. 1)} or x in (L~ go) /\ (L~ do) )
assume x in {(go /. 1)} ; ::_thesis: x in (L~ go) /\ (L~ do)
then A79: x = go /. 1 by TARSKI:def_1;
then A80: x in rng go by FINSEQ_6:42;
x in rng do by A78, A79, REVROT_1:3;
hence x in (L~ go) /\ (L~ do) by A77, A55, A80, XBOOLE_0:def_4; ::_thesis: verum
end;
then A81: (L~ go) /\ (L~ do) = {(go /. 1)} by A70, XBOOLE_0:def_10;
now__::_thesis:_contradiction
percases ( ( ((Gauge (C,n)) * (i1,j)) `1 <> ((Gauge (C,n)) * (i2,k)) `1 & ((Gauge (C,n)) * (i1,j)) `2 <> ((Gauge (C,n)) * (i2,k)) `2 ) or ((Gauge (C,n)) * (i1,j)) `1 = ((Gauge (C,n)) * (i2,k)) `1 or ((Gauge (C,n)) * (i1,j)) `2 = ((Gauge (C,n)) * (i2,k)) `2 ) ;
suppose ( ((Gauge (C,n)) * (i1,j)) `1 <> ((Gauge (C,n)) * (i2,k)) `1 & ((Gauge (C,n)) * (i1,j)) `2 <> ((Gauge (C,n)) * (i2,k)) `2 ) ; ::_thesis: contradiction
then <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> is being_S-Seq by A40, TOPREAL3:35;
then consider pion1 being FinSequence of (TOP-REAL 2) such that
A82: pion1 is_sequence_on Gauge (C,n) and
A83: pion1 is being_S-Seq and
A84: L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> = L~ pion1 and
A85: <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. 1 = pion1 /. 1 and
A86: <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. (len <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) = pion1 /. (len pion1) and
A87: len <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> <= len pion1 by A29, GOBOARD3:2;
reconsider pion1 = pion1 as being_S-Seq FinSequence of (TOP-REAL 2) by A83;
A88: (go ^' pion1) /. (len (go ^' pion1)) = <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. (len <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) by A86, GRAPH_2:54
.= <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. 3 by FINSEQ_1:45
.= do /. 1 by A49, FINSEQ_4:18 ;
A89: go /. (len go) = pion1 /. 1 by A59, A85, FINSEQ_4:18;
A90: (L~ go) /\ (L~ pion1) c= {(pion1 /. 1)}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ go) /\ (L~ pion1) or x in {(pion1 /. 1)} )
assume A91: x in (L~ go) /\ (L~ pion1) ; ::_thesis: x in {(pion1 /. 1)}
then A92: x in L~ pion1 by XBOOLE_0:def_4;
x in L~ go by A91, XBOOLE_0:def_4;
hence x in {(pion1 /. 1)} by A7, A36, A59, A56, A84, A89, A92, XBOOLE_0:def_4; ::_thesis: verum
end;
len pion1 >= 2 + 1 by A87, FINSEQ_1:45;
then A93: len pion1 > 1 + 1 by NAT_1:13;
then A94: rng pion1 c= L~ pion1 by SPPOL_2:18;
{(pion1 /. 1)} c= (L~ go) /\ (L~ pion1)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(pion1 /. 1)} or x in (L~ go) /\ (L~ pion1) )
assume x in {(pion1 /. 1)} ; ::_thesis: x in (L~ go) /\ (L~ pion1)
then A95: x = pion1 /. 1 by TARSKI:def_1;
then A96: x in rng pion1 by FINSEQ_6:42;
x in rng go by A89, A95, REVROT_1:3;
hence x in (L~ go) /\ (L~ pion1) by A77, A94, A96, XBOOLE_0:def_4; ::_thesis: verum
end;
then A97: (L~ go) /\ (L~ pion1) = {(pion1 /. 1)} by A90, XBOOLE_0:def_10;
then A98: go ^' pion1 is s.n.c. by A89, JORDAN1J:54;
A99: {((Gauge (C,n)) * (i2,k))} c= (LSeg (go,m)) /\ (LSeg (pion1,1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i2,k))} or x in (LSeg (go,m)) /\ (LSeg (pion1,1)) )
assume x in {((Gauge (C,n)) * (i2,k))} ; ::_thesis: x in (LSeg (go,m)) /\ (LSeg (pion1,1))
then A100: x = (Gauge (C,n)) * (i2,k) by TARSKI:def_1;
A101: (Gauge (C,n)) * (i2,k) in LSeg (go,m) by A62, RLTOPSP1:68;
(Gauge (C,n)) * (i2,k) in LSeg (pion1,1) by A59, A89, A93, TOPREAL1:21;
hence x in (LSeg (go,m)) /\ (LSeg (pion1,1)) by A100, A101, XBOOLE_0:def_4; ::_thesis: verum
end;
LSeg (pion1,1) c= L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> by A84, TOPREAL3:19;
then (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) c= {((Gauge (C,n)) * (i2,k))} by A61, A66, XBOOLE_1:27;
then (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) = {(go /. (len go))} by A59, A61, A99, XBOOLE_0:def_10;
then A102: go ^' pion1 is unfolded by A89, TOPREAL8:34;
len (go ^' pion1) >= len go by TOPREAL8:7;
then A103: len (go ^' pion1) >= 1 + 1 by A42, XXREAL_0:2;
then A104: len (go ^' pion1) > 1 + 0 by NAT_1:13;
A105: <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. (len <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) = <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. 3 by FINSEQ_1:45
.= do /. 1 by A49, FINSEQ_4:18 ;
A106: {(pion1 /. (len pion1))} c= (L~ do) /\ (L~ pion1)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(pion1 /. (len pion1))} or x in (L~ do) /\ (L~ pion1) )
assume x in {(pion1 /. (len pion1))} ; ::_thesis: x in (L~ do) /\ (L~ pion1)
then A107: x = pion1 /. (len pion1) by TARSKI:def_1;
then A108: x in rng pion1 by REVROT_1:3;
x in rng do by A86, A105, A107, FINSEQ_6:42;
hence x in (L~ do) /\ (L~ pion1) by A55, A94, A108, XBOOLE_0:def_4; ::_thesis: verum
end;
(L~ do) /\ (L~ pion1) c= {(pion1 /. (len pion1))}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ do) /\ (L~ pion1) or x in {(pion1 /. (len pion1))} )
assume A109: x in (L~ do) /\ (L~ pion1) ; ::_thesis: x in {(pion1 /. (len pion1))}
then A110: x in L~ pion1 by XBOOLE_0:def_4;
x in L~ do by A109, XBOOLE_0:def_4;
hence x in {(pion1 /. (len pion1))} by A8, A36, A49, A48, A84, A86, A105, A110, XBOOLE_0:def_4; ::_thesis: verum
end;
then A111: (L~ do) /\ (L~ pion1) = {(pion1 /. (len pion1))} by A106, XBOOLE_0:def_10;
A112: (L~ (go ^' pion1)) /\ (L~ do) = ((L~ go) \/ (L~ pion1)) /\ (L~ do) by A89, TOPREAL8:35
.= {(go /. 1)} \/ {(do /. 1)} by A81, A86, A105, A111, XBOOLE_1:23
.= {((go ^' pion1) /. 1)} \/ {(do /. 1)} by GRAPH_2:53
.= {((go ^' pion1) /. 1),(do /. 1)} by ENUMSET1:1 ;
A113: Upper_Arc C is_an_arc_of W-min C, E-max C by JORDAN6:def_8;
then A114: Upper_Arc C is connected by JORDAN6:10;
set godo = (go ^' pion1) ^' do;
A115: do /. (len do) = (go ^' pion1) /. 1 by A78, GRAPH_2:53;
A116: go ^' pion1 is_sequence_on Gauge (C,n) by A39, A82, A89, TOPREAL8:12;
then A117: (go ^' pion1) ^' do is_sequence_on Gauge (C,n) by A45, A88, TOPREAL8:12;
A118: (len pion1) - 1 >= 1 by A93, XREAL_1:19;
then A119: (len pion1) -' 1 = (len pion1) - 1 by XREAL_0:def_2;
A120: {((Gauge (C,n)) * (i1,j))} c= (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i1,j))} or x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) )
assume x in {((Gauge (C,n)) * (i1,j))} ; ::_thesis: x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1))
then A121: x = (Gauge (C,n)) * (i1,j) by TARSKI:def_1;
pion1 /. (((len pion1) -' 1) + 1) = <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. 3 by A86, A119, FINSEQ_1:45
.= (Gauge (C,n)) * (i1,j) by FINSEQ_4:18 ;
then A122: (Gauge (C,n)) * (i1,j) in LSeg (pion1,((len pion1) -' 1)) by A118, A119, TOPREAL1:21;
(Gauge (C,n)) * (i1,j) in LSeg (do,1) by A50, RLTOPSP1:68;
hence x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) by A121, A122, XBOOLE_0:def_4; ::_thesis: verum
end;
LSeg (pion1,((len pion1) -' 1)) c= L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> by A84, TOPREAL3:19;
then (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) c= {((Gauge (C,n)) * (i1,j))} by A54, XBOOLE_1:27;
then A123: (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) = {((Gauge (C,n)) * (i1,j))} by A120, XBOOLE_0:def_10;
((len pion1) - 1) + 1 <= len pion1 ;
then A124: (len pion1) -' 1 < len pion1 by A119, NAT_1:13;
len pion1 >= 2 + 1 by A87, FINSEQ_1:45;
then A125: (len pion1) - 2 >= 0 by XREAL_1:19;
then ((len pion1) -' 2) + 1 = ((len pion1) - 2) + 1 by XREAL_0:def_2
.= (len pion1) -' 1 by A118, XREAL_0:def_2 ;
then A126: (LSeg ((go ^' pion1),((len go) + ((len pion1) -' 2)))) /\ (LSeg (do,1)) = {((go ^' pion1) /. (len (go ^' pion1)))} by A49, A89, A88, A124, A123, TOPREAL8:31;
(rng go) /\ (rng pion1) c= {(pion1 /. 1)} by A77, A94, A97, XBOOLE_1:27;
then A127: go ^' pion1 is one-to-one by JORDAN1J:55;
((len (go ^' pion1)) + 1) - 1 = ((len go) + (len pion1)) - 1 by GRAPH_2:13;
then (len (go ^' pion1)) - 1 = (len go) + ((len pion1) - 2)
.= (len go) + ((len pion1) -' 2) by A125, XREAL_0:def_2 ;
then A128: (len (go ^' pion1)) -' 1 = (len go) + ((len pion1) -' 2) by XREAL_0:def_2;
A129: L~ (Cage (C,n)) = (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n))) by JORDAN1E:13;
then A130: L~ (Upper_Seq (C,n)) c= L~ (Cage (C,n)) by XBOOLE_1:7;
then A131: L~ go c= L~ (Cage (C,n)) by A56, XBOOLE_1:1;
A132: len ((go ^' pion1) ^' do) >= len (go ^' pion1) by TOPREAL8:7;
then A133: 1 + 1 <= len ((go ^' pion1) ^' do) by A103, XXREAL_0:2;
not go ^' pion1 is trivial by A103, NAT_D:60;
then reconsider godo = (go ^' pion1) ^' do as non constant standard special_circular_sequence by A133, A88, A117, A102, A128, A126, A98, A127, A112, A115, JORDAN8:4, JORDAN8:5, TOPREAL8:11, TOPREAL8:33, TOPREAL8:34;
A134: L~ godo = (L~ (go ^' pion1)) \/ (L~ do) by A88, TOPREAL8:35
.= ((L~ go) \/ (L~ pion1)) \/ (L~ do) by A89, TOPREAL8:35 ;
A135: now__::_thesis:_not_((Gauge_(C,n))_*_(i2,k))_.._(Upper_Seq_(C,n))_<=_1
assume A136: ((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n)) <= 1 ; ::_thesis: contradiction
((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n)) >= 1 by A38, FINSEQ_4:21;
then ((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n)) = 1 by A136, XXREAL_0:1;
then (Gauge (C,n)) * (i2,k) = (Upper_Seq (C,n)) /. 1 by A38, FINSEQ_5:38;
hence contradiction by A15, A37, JORDAN1F:5; ::_thesis: verum
end;
A137: Upper_Seq (C,n) is_sequence_on Gauge (C,n) by JORDAN1G:4;
A138: ((Gauge (C,n)) * (i2,k)) `1 <= ((Gauge (C,n)) * (i1,k)) `1 by A1, A2, A3, A6, A10, JORDAN1A:18;
then A139: W-bound (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) = ((Gauge (C,n)) * (i2,k)) `1 by SPRECT_1:54;
A140: ((Gauge (C,n)) * (i1,k)) `1 = ((Gauge (C,n)) * (i1,1)) `1 by A3, A6, A17, A10, GOBOARD5:2
.= ((Gauge (C,n)) * (i1,j)) `1 by A3, A4, A17, A19, GOBOARD5:2 ;
then A141: W-bound (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) = ((Gauge (C,n)) * (i1,j)) `1 by SPRECT_1:54;
W-bound ((LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))))) = min ((W-bound (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))),(W-bound (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))))) by SPRECT_1:47
.= ((Gauge (C,n)) * (i2,k)) `1 by A140, A138, A139, A141, XXREAL_0:def_9 ;
then A142: W-bound (L~ pion1) = ((Gauge (C,n)) * (i2,k)) `1 by A84, TOPREAL3:16;
A143: Upper_Arc C c= C by JORDAN6:61;
((Gauge (C,n)) * (i2,k)) `1 >= W-bound (L~ (Cage (C,n))) by A30, A130, PSCOMP_1:24;
then A144: ((Gauge (C,n)) * (i2,k)) `1 > W-bound (L~ (Cage (C,n))) by A24, XXREAL_0:1;
A145: len (Upper_Seq (C,n)) >= 2 by A14, XXREAL_0:2;
A146: (L~ go) \/ (L~ do) is compact by COMPTS_1:10;
A147: L~ (Lower_Seq (C,n)) c= L~ (Cage (C,n)) by A129, XBOOLE_1:7;
then A148: L~ do c= L~ (Cage (C,n)) by A48, XBOOLE_1:1;
A149: (right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo) c= RightComp godo by A133, A117, JORDAN9:27;
2 in dom godo by A133, FINSEQ_3:25;
then A150: godo /. 2 in rng godo by PARTFUN2:2;
A151: rng godo c= L~ godo by A103, A132, SPPOL_2:18, XXREAL_0:2;
A152: godo /. 1 = (go ^' pion1) /. 1 by GRAPH_2:53
.= W-min (L~ (Cage (C,n))) by A67, GRAPH_2:53 ;
A153: W-min C in Upper_Arc C by A113, TOPREAL1:1;
A154: W-min C in C by SPRECT_1:13;
A155: now__::_thesis:_not_W-min_C_in_L~_godo
assume W-min C in L~ godo ; ::_thesis: contradiction
then A156: ( W-min C in (L~ go) \/ (L~ pion1) or W-min C in L~ do ) by A134, XBOOLE_0:def_3;
percases ( W-min C in L~ go or W-min C in L~ pion1 or W-min C in L~ do ) by A156, XBOOLE_0:def_3;
suppose W-min C in L~ go ; ::_thesis: contradiction
then C meets L~ (Cage (C,n)) by A131, A154, XBOOLE_0:3;
hence contradiction by JORDAN10:5; ::_thesis: verum
end;
suppose W-min C in L~ pion1 ; ::_thesis: contradiction
hence contradiction by A9, A36, A84, A153, XBOOLE_0:3; ::_thesis: verum
end;
suppose W-min C in L~ do ; ::_thesis: contradiction
then C meets L~ (Cage (C,n)) by A148, A154, XBOOLE_0:3;
hence contradiction by JORDAN10:5; ::_thesis: verum
end;
end;
end;
A157: (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52;
set ff = Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))));
A158: 1 + 1 <= len (Cage (C,n)) by GOBOARD7:34, XXREAL_0:2;
W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
then A159: (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 = W-min (L~ (Cage (C,n))) by FINSEQ_6:92;
A160: L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33;
then (W-max (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A159, SPRECT_5:22;
then (N-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A159, A160, SPRECT_5:23, XXREAL_0:2;
then (N-max (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A159, A160, SPRECT_5:24, XXREAL_0:2;
then A161: (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A159, A160, SPRECT_5:25, XXREAL_0:2;
A162: Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def_1;
then A163: Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))) is_sequence_on Gauge (C,n) by REVROT_1:34;
1 + 1 <= len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by GOBOARD7:34, XXREAL_0:2;
then right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1) = right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(GoB (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) by JORDAN1H:23
.= right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(GoB (Cage (C,n)))) by REVROT_1:28
.= right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(Gauge (C,n))) by JORDAN1H:44
.= right_cell (((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n))))),1,(Gauge (C,n))) by A161, A163, JORDAN1J:53
.= right_cell ((Upper_Seq (C,n)),1,(Gauge (C,n))) by JORDAN1E:def_1
.= right_cell ((R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i2,k)))),1,(Gauge (C,n))) by A38, A137, A135, JORDAN1J:52
.= right_cell ((go ^' pion1),1,(Gauge (C,n))) by A57, A116, JORDAN1J:51
.= right_cell (godo,1,(Gauge (C,n))) by A104, A117, JORDAN1J:51 ;
then W-min C in right_cell (godo,1,(Gauge (C,n))) by JORDAN1I:6;
then A164: W-min C in (right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo) by A155, XBOOLE_0:def_5;
A165: E-max C in Upper_Arc C by A113, TOPREAL1:1;
W-min (L~ (Cage (C,n))) in (L~ go) \/ (L~ do) by A77, A68, XBOOLE_0:def_3;
then A166: W-min ((L~ go) \/ (L~ do)) = W-min (L~ (Cage (C,n))) by A131, A148, A146, JORDAN1J:21, XBOOLE_1:8;
(W-min ((L~ go) \/ (L~ do))) `1 = W-bound ((L~ go) \/ (L~ do)) by EUCLID:52;
then W-min (((L~ go) \/ (L~ do)) \/ (L~ pion1)) = W-min ((L~ go) \/ (L~ do)) by A142, A146, A166, A157, A144, JORDAN1J:33;
then A167: W-min (L~ godo) = W-min (L~ (Cage (C,n))) by A134, A166, XBOOLE_1:4;
godo /. 2 = (go ^' pion1) /. 2 by A103, GRAPH_2:57
.= (Upper_Seq (C,n)) /. 2 by A42, A76, GRAPH_2:57
.= ((Upper_Seq (C,n)) ^' (Lower_Seq (C,n))) /. 2 by A145, GRAPH_2:57
.= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 2 by JORDAN1E:11 ;
then godo /. 2 in W-most (L~ (Cage (C,n))) by JORDAN1I:25;
then (godo /. 2) `1 = (W-min (L~ godo)) `1 by A167, PSCOMP_1:31
.= W-bound (L~ godo) by EUCLID:52 ;
then godo /. 2 in W-most (L~ godo) by A151, A150, SPRECT_2:12;
then (Rotate (godo,(W-min (L~ godo)))) /. 2 in W-most (L~ godo) by A152, A167, FINSEQ_6:89;
then reconsider godo = godo as non constant standard clockwise_oriented special_circular_sequence by JORDAN1I:25;
len (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by FINSEQ_5:6;
then A168: (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) = (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) by PARTFUN1:def_6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:7 ;
A169: east_halfline (E-max C) misses L~ go
proof
assume east_halfline (E-max C) meets L~ go ; ::_thesis: contradiction
then consider p being set such that
A170: p in east_halfline (E-max C) and
A171: p in L~ go by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A170;
p in L~ (Upper_Seq (C,n)) by A56, A171;
then p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A130, A170, XBOOLE_0:def_4;
then A172: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;
then A173: p = E-max (L~ (Cage (C,n))) by A56, A171, JORDAN1J:46;
then E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i2,k) by A30, A168, A171, JORDAN1J:43;
then ((Gauge (C,n)) * (i2,k)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by A6, A10, A13, A172, A173, JORDAN1A:71;
hence contradiction by A2, A3, A12, A41, JORDAN1G:7; ::_thesis: verum
end;
now__::_thesis:_not_east_halfline_(E-max_C)_meets_L~_godo
assume east_halfline (E-max C) meets L~ godo ; ::_thesis: contradiction
then A174: ( east_halfline (E-max C) meets (L~ go) \/ (L~ pion1) or east_halfline (E-max C) meets L~ do ) by A134, XBOOLE_1:70;
percases ( 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~ do ) by A174, XBOOLE_1:70;
suppose east_halfline (E-max C) meets L~ go ; ::_thesis: contradiction
hence contradiction by A169; ::_thesis: verum
end;
suppose east_halfline (E-max C) meets L~ pion1 ; ::_thesis: contradiction
then consider p being set such that
A175: p in east_halfline (E-max C) and
A176: p in L~ pion1 by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A175;
A177: p `2 = (E-max C) `2 by A175, TOPREAL1:def_11;
A178: now__::_thesis:_p_`1_<=_((Gauge_(C,n))_*_(i1,k))_`1
percases ( p in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) or p in LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) ) by A36, A84, A176, XBOOLE_0:def_3;
suppose p in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) ; ::_thesis: p `1 <= ((Gauge (C,n)) * (i1,k)) `1
hence p `1 <= ((Gauge (C,n)) * (i1,k)) `1 by A138, TOPREAL1:3; ::_thesis: verum
end;
suppose p in LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) ; ::_thesis: p `1 <= ((Gauge (C,n)) * (i1,k)) `1
hence p `1 <= ((Gauge (C,n)) * (i1,k)) `1 by A140, GOBOARD7:5; ::_thesis: verum
end;
end;
end;
i1 + 1 <= len (Gauge (C,n)) by A3, NAT_1:13;
then (i1 + 1) - 1 <= (len (Gauge (C,n))) - 1 by XREAL_1:9;
then A179: i1 <= (len (Gauge (C,n))) -' 1 by XREAL_0:def_2;
(len (Gauge (C,n))) -' 1 <= len (Gauge (C,n)) by NAT_D:35;
then ((Gauge (C,n)) * (i1,k)) `1 <= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),1)) `1 by A6, A17, A10, A13, A21, A179, JORDAN1A:18;
then p `1 <= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),1)) `1 by A178, XXREAL_0:2;
then p `1 <= E-bound C by A21, JORDAN8:12;
then A180: p `1 <= (E-max C) `1 by EUCLID:52;
p `1 >= (E-max C) `1 by A175, TOPREAL1:def_11;
then p `1 = (E-max C) `1 by A180, XXREAL_0:1;
then p = E-max C by A177, TOPREAL3:6;
hence contradiction by A9, A36, A84, A165, A176, XBOOLE_0:3; ::_thesis: verum
end;
suppose east_halfline (E-max C) meets L~ do ; ::_thesis: contradiction
then consider p being set such that
A181: p in east_halfline (E-max C) and
A182: p in L~ do by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A181;
A183: p in LSeg (do,(Index (p,do))) by A182, JORDAN3:9;
consider t being Nat such that
A184: t in dom (Lower_Seq (C,n)) and
A185: (Lower_Seq (C,n)) . t = (Gauge (C,n)) * (i1,j) by A44, FINSEQ_2:10;
1 <= t by A184, FINSEQ_3:25;
then A186: 1 < t by A46, A185, XXREAL_0:1;
t <= len (Lower_Seq (C,n)) by A184, FINSEQ_3:25;
then (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) + 1 = t by A185, A186, JORDAN3:12;
then A187: len (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i1,j)))) = (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) by A16, A185, JORDAN3:26;
Index (p,do) < len do by A182, JORDAN3:8;
then Index (p,do) < (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) by A187, XREAL_0:def_2;
then (Index (p,do)) + 1 <= (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) by NAT_1:13;
then A188: Index (p,do) <= ((len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n))))) - 1 by XREAL_1:19;
A189: do = mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n)))) by A44, JORDAN1J:37;
p in L~ (Lower_Seq (C,n)) by A48, A182;
then p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A147, A181, XBOOLE_0:def_4;
then A190: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;
A191: (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) + 1 = ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) by A46, A44, JORDAN1J:56;
0 + (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) < len (Lower_Seq (C,n)) by A16, JORDAN3:8;
then (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) > 0 by XREAL_1:20;
then Index (p,do) <= ((len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n))))) - 1 by A188, XREAL_0:def_2;
then Index (p,do) <= (len (Lower_Seq (C,n))) - (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))) by A191;
then Index (p,do) <= (len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))) by XREAL_0:def_2;
then A192: Index (p,do) < ((len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) + 1 by NAT_1:13;
A193: 1 <= Index (p,do) by A182, JORDAN3:8;
A194: ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) <= len (Lower_Seq (C,n)) by A44, FINSEQ_4:21;
((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) <> len (Lower_Seq (C,n)) by A43, A44, FINSEQ_4:19;
then A195: ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) < len (Lower_Seq (C,n)) by A194, XXREAL_0:1;
A196: 1 + 1 <= len (Lower_Seq (C,n)) by A31, XXREAL_0:2;
then A197: 2 in dom (Lower_Seq (C,n)) by FINSEQ_3:25;
set tt = ((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1;
set RC = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))));
A198: E-max C in right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by JORDAN1I:7;
A199: GoB (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = GoB (Cage (C,n)) by REVROT_1:28
.= Gauge (C,n) by JORDAN1H:44 ;
A200: L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33;
consider jj2 being Element of NAT such that
A201: 1 <= jj2 and
A202: jj2 <= width (Gauge (C,n)) and
A203: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * ((len (Gauge (C,n))),jj2) by JORDAN1D:25;
A204: len (Gauge (C,n)) >= 4 by JORDAN8:10;
then len (Gauge (C,n)) >= 1 by XXREAL_0:2;
then A205: [(len (Gauge (C,n))),jj2] in Indices (Gauge (C,n)) by A201, A202, MATRIX_1:36;
A206: len (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = len (Cage (C,n)) by REVROT_1:14;
Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) by JORDAN1G:18;
then A207: LSeg ((Lower_Seq (C,n)),1) = LSeg ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by A196, SPPOL_2:9;
A208: E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))) is_sequence_on Gauge (C,n) by A162, REVROT_1:34;
then consider ii, jj being Element of NAT such that
A209: [ii,(jj + 1)] in Indices (Gauge (C,n)) and
A210: [ii,jj] in Indices (Gauge (C,n)) and
A211: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = (Gauge (C,n)) * (ii,(jj + 1)) and
A212: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1) = (Gauge (C,n)) * (ii,jj) by A158, A200, A206, A208, FINSEQ_6:92, JORDAN1I:23;
A213: (jj + 1) + 1 <> jj ;
A214: 1 <= jj by A210, MATRIX_1:38;
(Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = E-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))))) by A200, A208, FINSEQ_6:92;
then A215: ii = len (Gauge (C,n)) by A200, A209, A211, A203, A205, GOBOARD1:5;
then ii - 1 >= 4 - 1 by A204, XREAL_1:9;
then A216: ii - 1 >= 1 by XXREAL_0:2;
then A217: 1 <= ii -' 1 by XREAL_0:def_2;
A218: jj <= width (Gauge (C,n)) by A210, MATRIX_1:38;
then A219: ((Gauge (C,n)) * ((len (Gauge (C,n))),jj)) `1 = E-bound (L~ (Cage (C,n))) by A13, A214, JORDAN1A:71;
A220: jj + 1 <= width (Gauge (C,n)) by A209, MATRIX_1:38;
ii + 1 <> ii ;
then A221: right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) = cell ((Gauge (C,n)),(ii -' 1),jj) by A158, A206, A199, A209, A210, A211, A212, A213, GOBOARD5:def_6;
A222: ii <= len (Gauge (C,n)) by A210, MATRIX_1:38;
A223: 1 <= ii by A210, MATRIX_1:38;
A224: ii <= len (Gauge (C,n)) by A209, MATRIX_1:38;
A225: 1 <= jj + 1 by A209, MATRIX_1:38;
then A226: E-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * ((len (Gauge (C,n))),(jj + 1))) `1 by A13, A220, JORDAN1A:71;
A227: 1 <= ii by A209, MATRIX_1:38;
then A228: (ii -' 1) + 1 = ii by XREAL_1:235;
then A229: ii -' 1 < len (Gauge (C,n)) by A224, NAT_1:13;
then A230: ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 = ((Gauge (C,n)) * (1,(jj + 1))) `2 by A225, A220, A217, GOBOARD5:1
.= ((Gauge (C,n)) * (ii,(jj + 1))) `2 by A227, A224, A225, A220, GOBOARD5:1 ;
A231: (E-max C) `2 = p `2 by A181, TOPREAL1:def_11;
then A232: p `2 <= ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 by A198, A224, A220, A214, A221, A228, A216, JORDAN9:17;
A233: ((Gauge (C,n)) * ((ii -' 1),jj)) `2 = ((Gauge (C,n)) * (1,jj)) `2 by A214, A218, A217, A229, GOBOARD5:1
.= ((Gauge (C,n)) * (ii,jj)) `2 by A223, A222, A214, A218, GOBOARD5:1 ;
((Gauge (C,n)) * ((ii -' 1),jj)) `2 <= p `2 by A231, A198, A224, A220, A214, A221, A228, A216, JORDAN9:17;
then p in LSeg (((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1),((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1))) by A190, A211, A212, A215, A232, A233, A230, A219, A226, GOBOARD7:7;
then A234: p in LSeg ((Lower_Seq (C,n)),1) by A158, A207, A206, TOPREAL1:def_3;
1 <= ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) by A44, FINSEQ_4:21;
then A235: LSeg ((mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n))))),(Index (p,do))) = LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1)) by A195, A193, A192, JORDAN4:19;
1 <= Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n))) by A16, JORDAN3:8;
then A236: 1 + 1 <= ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) by A191, XREAL_1:7;
then (Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))) >= (1 + 1) + 1 by A193, XREAL_1:7;
then ((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) - 1 >= ((1 + 1) + 1) - 1 by XREAL_1:9;
then A237: ((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1 >= 1 + 1 by XREAL_0:def_2;
now__::_thesis:_contradiction
percases ( ((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1 > 1 + 1 or ((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ) by A237, XXREAL_0:1;
suppose ((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1 > 1 + 1 ; ::_thesis: contradiction
then LSeg ((Lower_Seq (C,n)),1) misses LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1)) by TOPREAL1:def_7;
hence contradiction by A234, A183, A189, A235, XBOOLE_0:3; ::_thesis: verum
end;
supposeA238: ((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ; ::_thesis: contradiction
then 1 + 1 = ((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) - 1 by XREAL_0:def_2;
then (1 + 1) + 1 = (Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))) ;
then A239: ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) = 2 by A193, A236, JORDAN1E:6;
(LSeg ((Lower_Seq (C,n)),1)) /\ (LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1))) = {((Lower_Seq (C,n)) /. 2)} by A31, A238, TOPREAL1:def_6;
then p in {((Lower_Seq (C,n)) /. 2)} by A234, A183, A189, A235, XBOOLE_0:def_4;
then A240: p = (Lower_Seq (C,n)) /. 2 by TARSKI:def_1;
then A241: p in rng (Lower_Seq (C,n)) by A197, PARTFUN2:2;
p .. (Lower_Seq (C,n)) = 2 by A197, A240, FINSEQ_5:41;
then p = (Gauge (C,n)) * (i1,j) by A44, A239, A241, FINSEQ_5:9;
then ((Gauge (C,n)) * (i1,j)) `1 = E-bound (L~ (Cage (C,n))) by A240, JORDAN1G:32;
then ((Gauge (C,n)) * (i1,j)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 by A4, A19, A13, JORDAN1A:71;
hence contradiction by A3, A25, A22, JORDAN1G:7; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
end;
end;
then east_halfline (E-max C) c= (L~ godo) ` by SUBSET_1:23;
then consider W being Subset of (TOP-REAL 2) such that
A242: W is_a_component_of (L~ godo) ` and
A243: east_halfline (E-max C) c= W by GOBOARD9:3;
not W is bounded by A243, JORDAN2C:121, RLTOPSP1:42;
then W is_outside_component_of L~ godo by A242, JORDAN2C:def_3;
then W c= UBD (L~ godo) by JORDAN2C:23;
then A244: east_halfline (E-max C) c= UBD (L~ godo) by A243, XBOOLE_1:1;
E-max C in east_halfline (E-max C) by TOPREAL1:38;
then E-max C in UBD (L~ godo) by A244;
then E-max C in LeftComp godo by GOBRD14:36;
then Upper_Arc C meets L~ godo by A114, A153, A165, A149, A164, JORDAN1J:36;
then A245: ( Upper_Arc C meets (L~ go) \/ (L~ pion1) or Upper_Arc C meets L~ do ) by A134, XBOOLE_1:70;
now__::_thesis:_contradiction
percases ( Upper_Arc C meets L~ go or Upper_Arc C meets L~ pion1 or Upper_Arc C meets L~ do ) by A245, XBOOLE_1:70;
suppose Upper_Arc C meets L~ go ; ::_thesis: contradiction
then Upper_Arc C meets L~ (Cage (C,n)) by A56, A130, XBOOLE_1:1, XBOOLE_1:63;
hence contradiction by A143, JORDAN10:5, XBOOLE_1:63; ::_thesis: verum
end;
suppose Upper_Arc C meets L~ pion1 ; ::_thesis: contradiction
hence contradiction by A9, A36, A84; ::_thesis: verum
end;
suppose Upper_Arc C meets L~ do ; ::_thesis: contradiction
then Upper_Arc C meets L~ (Cage (C,n)) by A48, A147, XBOOLE_1:1, XBOOLE_1:63;
hence contradiction by A143, JORDAN10:5, XBOOLE_1:63; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
suppose ((Gauge (C,n)) * (i1,j)) `1 = ((Gauge (C,n)) * (i2,k)) `1 ; ::_thesis: contradiction
then A246: i1 = i2 by A25, A12, JORDAN1G:7;
then LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) = {((Gauge (C,n)) * (i1,k))} by RLTOPSP1:70;
then LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) c= LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) by A27, ZFMISC_1:31;
then (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) = LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) by XBOOLE_1:12;
hence contradiction by A1, A3, A4, A5, A6, A7, A8, A9, A246, JORDAN1J:59; ::_thesis: verum
end;
suppose ((Gauge (C,n)) * (i1,j)) `2 = ((Gauge (C,n)) * (i2,k)) `2 ; ::_thesis: contradiction
then A247: j = k by A25, A12, JORDAN1G:6;
then LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) = {((Gauge (C,n)) * (i1,k))} by RLTOPSP1:70;
then LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) c= LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) by A26, ZFMISC_1:31;
then (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) = LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) by XBOOLE_1:12;
hence contradiction by A1, A2, A3, A4, A6, A7, A8, A9, A247, Th37; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
theorem Th47: :: JORDAN15:47
for n being Element of NAT
for C being Simple_closed_curve
for i1, i2, j, k being Element of NAT st 1 < i2 & i2 <= i1 & i1 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} holds
(LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Lower_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i1, i2, j, k being Element of NAT st 1 < i2 & i2 <= i1 & i1 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} holds
(LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Lower_Arc C
let C be Simple_closed_curve; ::_thesis: for i1, i2, j, k being Element of NAT st 1 < i2 & i2 <= i1 & i1 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} holds
(LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Lower_Arc C
let i1, i2, j, k be Element of NAT ; ::_thesis: ( 1 < i2 & i2 <= i1 & i1 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} implies (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Lower_Arc C )
set G = Gauge (C,n);
set pio = LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)));
set poz = LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)));
set US = Upper_Seq (C,n);
set LS = Lower_Seq (C,n);
assume that
A1: 1 < i2 and
A2: i2 <= i1 and
A3: i1 < len (Gauge (C,n)) and
A4: 1 <= j and
A5: j <= k and
A6: k <= width (Gauge (C,n)) and
A7: ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} and
A8: ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} and
A9: (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) misses Lower_Arc C ; ::_thesis: contradiction
set Gi1k = (Gauge (C,n)) * (i1,k);
set Gik = (Gauge (C,n)) * (i2,k);
A10: 1 <= k by A4, A5, XXREAL_0:2;
A11: i2 < len (Gauge (C,n)) by A2, A3, XXREAL_0:2;
then A12: [i2,k] in Indices (Gauge (C,n)) by A1, A6, A10, MATRIX_1:36;
set Wmin = W-min (L~ (Cage (C,n)));
set Wbo = W-bound (L~ (Cage (C,n)));
A13: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
set go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i2,k)));
A14: len (Upper_Seq (C,n)) >= 3 by JORDAN1E:15;
then len (Upper_Seq (C,n)) >= 1 by XXREAL_0:2;
then 1 in dom (Upper_Seq (C,n)) by FINSEQ_3:25;
then A15: (Upper_Seq (C,n)) . 1 = (Upper_Seq (C,n)) /. 1 by PARTFUN1:def_6
.= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ;
set Gij = (Gauge (C,n)) * (i1,j);
set do = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i1,j)));
(Gauge (C,n)) * (i1,j) in {((Gauge (C,n)) * (i1,j))} by TARSKI:def_1;
then A16: (Gauge (C,n)) * (i1,j) in L~ (Lower_Seq (C,n)) by A8, XBOOLE_0:def_4;
A17: 1 < i1 by A1, A2, XXREAL_0:2;
then A18: ((Gauge (C,n)) * (i1,k)) `2 = ((Gauge (C,n)) * (1,k)) `2 by A3, A6, A10, GOBOARD5:1
.= ((Gauge (C,n)) * (i2,k)) `2 by A1, A6, A11, A10, GOBOARD5:1 ;
A19: j <= width (Gauge (C,n)) by A5, A6, XXREAL_0:2;
then A20: [i1,j] in Indices (Gauge (C,n)) by A3, A4, A17, MATRIX_1:36;
len (Gauge (C,n)) >= 4 by JORDAN8:10;
then A21: len (Gauge (C,n)) >= 1 by XXREAL_0:2;
then A22: [(len (Gauge (C,n))),j] in Indices (Gauge (C,n)) by A4, A19, MATRIX_1:36;
A23: [1,k] in Indices (Gauge (C,n)) by A6, A10, A21, MATRIX_1:36;
A24: now__::_thesis:_not_((Gauge_(C,n))_*_(i2,k))_`1_=_W-bound_(L~_(Cage_(C,n)))
assume ((Gauge (C,n)) * (i2,k)) `1 = W-bound (L~ (Cage (C,n))) ; ::_thesis: contradiction
then ((Gauge (C,n)) * (1,k)) `1 = ((Gauge (C,n)) * (i2,k)) `1 by A6, A10, A13, JORDAN1A:73;
hence contradiction by A1, A12, A23, JORDAN1G:7; ::_thesis: verum
end;
A25: [i1,j] in Indices (Gauge (C,n)) by A3, A4, A17, A19, MATRIX_1:36;
set pion = <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>;
A26: (Gauge (C,n)) * (i1,k) in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) by RLTOPSP1:68;
set LA = Lower_Arc C;
A27: (Gauge (C,n)) * (i1,k) in LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) by RLTOPSP1:68;
A28: [i1,k] in Indices (Gauge (C,n)) by A3, A6, A17, A10, MATRIX_1:36;
A29: now__::_thesis:_for_n_being_Element_of_NAT_st_n_in_dom_<*((Gauge_(C,n))_*_(i2,k)),((Gauge_(C,n))_*_(i1,k)),((Gauge_(C,n))_*_(i1,j))*>_holds_
ex_i,_j_being_Element_of_NAT_st_
(_[i,j]_in_Indices_(Gauge_(C,n))_&_<*((Gauge_(C,n))_*_(i2,k)),((Gauge_(C,n))_*_(i1,k)),((Gauge_(C,n))_*_(i1,j))*>_/._n_=_(Gauge_(C,n))_*_(i,j)_)
let n be Element of NAT ; ::_thesis: ( n in dom <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> implies ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. n = (Gauge (C,n)) * (i,j) ) )
assume n in dom <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. n = (Gauge (C,n)) * (i,j) )
then n in {1,2,3} by FINSEQ_1:89, FINSEQ_3:1;
then ( n = 1 or n = 2 or n = 3 ) by ENUMSET1:def_1;
hence ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. n = (Gauge (C,n)) * (i,j) ) by A25, A12, A28, FINSEQ_4:18; ::_thesis: verum
end;
(Gauge (C,n)) * (i2,k) in {((Gauge (C,n)) * (i2,k))} by TARSKI:def_1;
then A30: (Gauge (C,n)) * (i2,k) in L~ (Upper_Seq (C,n)) by A7, XBOOLE_0:def_4;
set Emax = E-max (L~ (Cage (C,n)));
A31: len (Lower_Seq (C,n)) >= 1 + 2 by JORDAN1E:15;
then A32: len (Lower_Seq (C,n)) >= 1 by XXREAL_0:2;
then A33: 1 in dom (Lower_Seq (C,n)) by FINSEQ_3:25;
then A34: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by PARTFUN1:def_6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;
len (Lower_Seq (C,n)) in dom (Lower_Seq (C,n)) by A32, FINSEQ_3:25;
then A35: (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by PARTFUN1:def_6
.= W-min (L~ (Cage (C,n))) by JORDAN1F:8 ;
set Ebo = E-bound (L~ (Cage (C,n)));
A36: L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> = (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) by TOPREAL3:16;
(W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * (1,k)) `1 by A6, A10, A13, JORDAN1A:73 ;
then A37: (Gauge (C,n)) * (i2,k) <> (Upper_Seq (C,n)) . 1 by A1, A12, A15, A23, JORDAN1G:7;
then reconsider go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i2,k))) as being_S-Seq FinSequence of (TOP-REAL 2) by A30, JORDAN3:35;
A38: (Gauge (C,n)) * (i2,k) in rng (Upper_Seq (C,n)) by A1, A6, A11, A30, A10, JORDAN1G:4, JORDAN1J:40;
then A39: go is_sequence_on Gauge (C,n) by JORDAN1G:4, JORDAN1J:38;
((Gauge (C,n)) * (i1,k)) `1 = ((Gauge (C,n)) * (i1,1)) `1 by A3, A6, A17, A10, GOBOARD5:2
.= ((Gauge (C,n)) * (i1,j)) `1 by A3, A4, A17, A19, GOBOARD5:2 ;
then A40: (Gauge (C,n)) * (i1,k) = |[(((Gauge (C,n)) * (i1,j)) `1),(((Gauge (C,n)) * (i2,k)) `2)]| by A18, EUCLID:53;
A41: [(len (Gauge (C,n))),k] in Indices (Gauge (C,n)) by A6, A10, A21, MATRIX_1:36;
A42: len go >= 1 + 1 by TOPREAL1:def_8;
(W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * (1,k)) `1 by A6, A10, A13, JORDAN1A:73 ;
then A43: (Gauge (C,n)) * (i1,j) <> (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) by A1, A2, A23, A35, A20, JORDAN1G:7;
then reconsider do = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i1,j))) as being_S-Seq FinSequence of (TOP-REAL 2) by A16, JORDAN3:34;
A44: (Gauge (C,n)) * (i1,j) in rng (Lower_Seq (C,n)) by A3, A4, A17, A16, A19, JORDAN1G:5, JORDAN1J:40;
then A45: do is_sequence_on Gauge (C,n) by JORDAN1G:5, JORDAN1J:39;
(E-max (L~ (Cage (C,n)))) `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by A6, A10, A13, JORDAN1A:71 ;
then A46: (Gauge (C,n)) * (i1,j) <> (Lower_Seq (C,n)) . 1 by A3, A20, A41, A34, JORDAN1G:7;
A47: len do >= 1 + 1 by TOPREAL1:def_8;
then reconsider do = do as non constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A45, JGRAPH_1:12, JORDAN8:5;
A48: L~ do c= L~ (Lower_Seq (C,n)) by A16, JORDAN3:42;
len do >= 1 by A47, XXREAL_0:2;
then 1 in dom do by FINSEQ_3:25;
then A49: do /. 1 = do . 1 by PARTFUN1:def_6
.= (Gauge (C,n)) * (i1,j) by A16, JORDAN3:23 ;
then A50: LSeg (do,1) = LSeg (((Gauge (C,n)) * (i1,j)),(do /. (1 + 1))) by A47, TOPREAL1:def_3;
A51: {((Gauge (C,n)) * (i1,j))} c= (LSeg (do,1)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i1,j))} or x in (LSeg (do,1)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) )
assume x in {((Gauge (C,n)) * (i1,j))} ; ::_thesis: x in (LSeg (do,1)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>)
then A52: x = (Gauge (C,n)) * (i1,j) by TARSKI:def_1;
(Gauge (C,n)) * (i1,j) in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))) by RLTOPSP1:68;
then (Gauge (C,n)) * (i1,j) in (LSeg (((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j)))) by XBOOLE_0:def_3;
then A53: (Gauge (C,n)) * (i1,j) in L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> by SPRECT_1:8;
(Gauge (C,n)) * (i1,j) in LSeg (do,1) by A50, RLTOPSP1:68;
hence x in (LSeg (do,1)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) by A52, A53, XBOOLE_0:def_4; ::_thesis: verum
end;
LSeg (do,1) c= L~ do by TOPREAL3:19;
then LSeg (do,1) c= L~ (Lower_Seq (C,n)) by A48, XBOOLE_1:1;
then (LSeg (do,1)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) c= {((Gauge (C,n)) * (i1,j))} by A8, A36, XBOOLE_1:26;
then A54: (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) /\ (LSeg (do,1)) = {((Gauge (C,n)) * (i1,j))} by A51, XBOOLE_0:def_10;
A55: rng do c= L~ do by A47, SPPOL_2:18;
reconsider go = go as non constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A42, A39, JGRAPH_1:12, JORDAN8:5;
A56: L~ go c= L~ (Upper_Seq (C,n)) by A30, JORDAN3:41;
A57: len go > 1 by A42, NAT_1:13;
then A58: len go in dom go by FINSEQ_3:25;
then A59: go /. (len go) = go . (len go) by PARTFUN1:def_6
.= (Gauge (C,n)) * (i2,k) by A30, JORDAN3:24 ;
reconsider m = (len go) - 1 as Element of NAT by A58, FINSEQ_3:26;
A60: m + 1 = len go ;
then A61: (len go) -' 1 = m by NAT_D:34;
m >= 1 by A42, XREAL_1:19;
then A62: LSeg (go,m) = LSeg ((go /. m),((Gauge (C,n)) * (i2,k))) by A59, A60, TOPREAL1:def_3;
A63: {((Gauge (C,n)) * (i2,k))} c= (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i2,k))} or x in (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) )
assume x in {((Gauge (C,n)) * (i2,k))} ; ::_thesis: x in (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>)
then A64: x = (Gauge (C,n)) * (i2,k) by TARSKI:def_1;
(Gauge (C,n)) * (i2,k) in LSeg (((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k))) by RLTOPSP1:68;
then (Gauge (C,n)) * (i2,k) in (LSeg (((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j)))) by XBOOLE_0:def_3;
then A65: (Gauge (C,n)) * (i2,k) in L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> by SPRECT_1:8;
(Gauge (C,n)) * (i2,k) in LSeg (go,m) by A62, RLTOPSP1:68;
hence x in (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) by A64, A65, XBOOLE_0:def_4; ::_thesis: verum
end;
LSeg (go,m) c= L~ go by TOPREAL3:19;
then LSeg (go,m) c= L~ (Upper_Seq (C,n)) by A56, XBOOLE_1:1;
then (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) c= {((Gauge (C,n)) * (i2,k))} by A7, A36, XBOOLE_1:26;
then A66: (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) = {((Gauge (C,n)) * (i2,k))} by A63, XBOOLE_0:def_10;
A67: go /. 1 = (Upper_Seq (C,n)) /. 1 by A30, SPRECT_3:22
.= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ;
then A68: W-min (L~ (Cage (C,n))) in rng go by FINSEQ_6:42;
A69: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by A33, PARTFUN1:def_6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;
A70: (L~ go) /\ (L~ do) c= {(go /. 1)}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ go) /\ (L~ do) or x in {(go /. 1)} )
assume A71: x in (L~ go) /\ (L~ do) ; ::_thesis: x in {(go /. 1)}
then A72: x in L~ do by XBOOLE_0:def_4;
A73: now__::_thesis:_not_x_=_E-max_(L~_(Cage_(C,n)))
assume x = E-max (L~ (Cage (C,n))) ; ::_thesis: contradiction
then A74: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i1,j) by A16, A69, A72, JORDAN1E:7;
((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 = E-bound (L~ (Cage (C,n))) by A4, A19, A13, JORDAN1A:71;
then (E-max (L~ (Cage (C,n)))) `1 <> E-bound (L~ (Cage (C,n))) by A3, A25, A22, A74, JORDAN1G:7;
hence contradiction by EUCLID:52; ::_thesis: verum
end;
x in L~ go by A71, XBOOLE_0:def_4;
then x in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A56, A48, A72, XBOOLE_0:def_4;
then x in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16;
then ( x = W-min (L~ (Cage (C,n))) or x = E-max (L~ (Cage (C,n))) ) by TARSKI:def_2;
hence x in {(go /. 1)} by A67, A73, TARSKI:def_1; ::_thesis: verum
end;
set W2 = go /. 2;
A75: 2 in dom go by A42, FINSEQ_3:25;
go = mid ((Upper_Seq (C,n)),1,(((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n)))) by A38, JORDAN1G:49
.= (Upper_Seq (C,n)) | (((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n))) by A38, FINSEQ_4:21, FINSEQ_6:116 ;
then A76: go /. 2 = (Upper_Seq (C,n)) /. 2 by A75, FINSEQ_4:70;
A77: rng go c= L~ go by A42, SPPOL_2:18;
A78: go /. 1 = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by A67, JORDAN1F:8
.= do /. (len do) by A16, JORDAN1J:35 ;
{(go /. 1)} c= (L~ go) /\ (L~ do)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(go /. 1)} or x in (L~ go) /\ (L~ do) )
assume x in {(go /. 1)} ; ::_thesis: x in (L~ go) /\ (L~ do)
then A79: x = go /. 1 by TARSKI:def_1;
then A80: x in rng go by FINSEQ_6:42;
x in rng do by A78, A79, REVROT_1:3;
hence x in (L~ go) /\ (L~ do) by A77, A55, A80, XBOOLE_0:def_4; ::_thesis: verum
end;
then A81: (L~ go) /\ (L~ do) = {(go /. 1)} by A70, XBOOLE_0:def_10;
now__::_thesis:_contradiction
percases ( ( ((Gauge (C,n)) * (i1,j)) `1 <> ((Gauge (C,n)) * (i2,k)) `1 & ((Gauge (C,n)) * (i1,j)) `2 <> ((Gauge (C,n)) * (i2,k)) `2 ) or ((Gauge (C,n)) * (i1,j)) `1 = ((Gauge (C,n)) * (i2,k)) `1 or ((Gauge (C,n)) * (i1,j)) `2 = ((Gauge (C,n)) * (i2,k)) `2 ) ;
suppose ( ((Gauge (C,n)) * (i1,j)) `1 <> ((Gauge (C,n)) * (i2,k)) `1 & ((Gauge (C,n)) * (i1,j)) `2 <> ((Gauge (C,n)) * (i2,k)) `2 ) ; ::_thesis: contradiction
then <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> is being_S-Seq by A40, TOPREAL3:35;
then consider pion1 being FinSequence of (TOP-REAL 2) such that
A82: pion1 is_sequence_on Gauge (C,n) and
A83: pion1 is being_S-Seq and
A84: L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> = L~ pion1 and
A85: <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. 1 = pion1 /. 1 and
A86: <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. (len <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) = pion1 /. (len pion1) and
A87: len <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> <= len pion1 by A29, GOBOARD3:2;
reconsider pion1 = pion1 as being_S-Seq FinSequence of (TOP-REAL 2) by A83;
A88: (go ^' pion1) /. (len (go ^' pion1)) = <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. (len <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) by A86, GRAPH_2:54
.= <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. 3 by FINSEQ_1:45
.= do /. 1 by A49, FINSEQ_4:18 ;
A89: go /. (len go) = pion1 /. 1 by A59, A85, FINSEQ_4:18;
A90: (L~ go) /\ (L~ pion1) c= {(pion1 /. 1)}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ go) /\ (L~ pion1) or x in {(pion1 /. 1)} )
assume A91: x in (L~ go) /\ (L~ pion1) ; ::_thesis: x in {(pion1 /. 1)}
then A92: x in L~ pion1 by XBOOLE_0:def_4;
x in L~ go by A91, XBOOLE_0:def_4;
hence x in {(pion1 /. 1)} by A7, A36, A59, A56, A84, A89, A92, XBOOLE_0:def_4; ::_thesis: verum
end;
len pion1 >= 2 + 1 by A87, FINSEQ_1:45;
then A93: len pion1 > 1 + 1 by NAT_1:13;
then A94: rng pion1 c= L~ pion1 by SPPOL_2:18;
{(pion1 /. 1)} c= (L~ go) /\ (L~ pion1)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(pion1 /. 1)} or x in (L~ go) /\ (L~ pion1) )
assume x in {(pion1 /. 1)} ; ::_thesis: x in (L~ go) /\ (L~ pion1)
then A95: x = pion1 /. 1 by TARSKI:def_1;
then A96: x in rng pion1 by FINSEQ_6:42;
x in rng go by A89, A95, REVROT_1:3;
hence x in (L~ go) /\ (L~ pion1) by A77, A94, A96, XBOOLE_0:def_4; ::_thesis: verum
end;
then A97: (L~ go) /\ (L~ pion1) = {(pion1 /. 1)} by A90, XBOOLE_0:def_10;
then A98: go ^' pion1 is s.n.c. by A89, JORDAN1J:54;
A99: {((Gauge (C,n)) * (i2,k))} c= (LSeg (go,m)) /\ (LSeg (pion1,1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i2,k))} or x in (LSeg (go,m)) /\ (LSeg (pion1,1)) )
assume x in {((Gauge (C,n)) * (i2,k))} ; ::_thesis: x in (LSeg (go,m)) /\ (LSeg (pion1,1))
then A100: x = (Gauge (C,n)) * (i2,k) by TARSKI:def_1;
A101: (Gauge (C,n)) * (i2,k) in LSeg (go,m) by A62, RLTOPSP1:68;
(Gauge (C,n)) * (i2,k) in LSeg (pion1,1) by A59, A89, A93, TOPREAL1:21;
hence x in (LSeg (go,m)) /\ (LSeg (pion1,1)) by A100, A101, XBOOLE_0:def_4; ::_thesis: verum
end;
LSeg (pion1,1) c= L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> by A84, TOPREAL3:19;
then (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) c= {((Gauge (C,n)) * (i2,k))} by A61, A66, XBOOLE_1:27;
then (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) = {(go /. (len go))} by A59, A61, A99, XBOOLE_0:def_10;
then A102: go ^' pion1 is unfolded by A89, TOPREAL8:34;
len (go ^' pion1) >= len go by TOPREAL8:7;
then A103: len (go ^' pion1) >= 1 + 1 by A42, XXREAL_0:2;
then A104: len (go ^' pion1) > 1 + 0 by NAT_1:13;
A105: <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. (len <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*>) = <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. 3 by FINSEQ_1:45
.= do /. 1 by A49, FINSEQ_4:18 ;
A106: {(pion1 /. (len pion1))} c= (L~ do) /\ (L~ pion1)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(pion1 /. (len pion1))} or x in (L~ do) /\ (L~ pion1) )
assume x in {(pion1 /. (len pion1))} ; ::_thesis: x in (L~ do) /\ (L~ pion1)
then A107: x = pion1 /. (len pion1) by TARSKI:def_1;
then A108: x in rng pion1 by REVROT_1:3;
x in rng do by A86, A105, A107, FINSEQ_6:42;
hence x in (L~ do) /\ (L~ pion1) by A55, A94, A108, XBOOLE_0:def_4; ::_thesis: verum
end;
(L~ do) /\ (L~ pion1) c= {(pion1 /. (len pion1))}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ do) /\ (L~ pion1) or x in {(pion1 /. (len pion1))} )
assume A109: x in (L~ do) /\ (L~ pion1) ; ::_thesis: x in {(pion1 /. (len pion1))}
then A110: x in L~ pion1 by XBOOLE_0:def_4;
x in L~ do by A109, XBOOLE_0:def_4;
hence x in {(pion1 /. (len pion1))} by A8, A36, A49, A48, A84, A86, A105, A110, XBOOLE_0:def_4; ::_thesis: verum
end;
then A111: (L~ do) /\ (L~ pion1) = {(pion1 /. (len pion1))} by A106, XBOOLE_0:def_10;
A112: (L~ (go ^' pion1)) /\ (L~ do) = ((L~ go) \/ (L~ pion1)) /\ (L~ do) by A89, TOPREAL8:35
.= {(go /. 1)} \/ {(do /. 1)} by A81, A86, A105, A111, XBOOLE_1:23
.= {((go ^' pion1) /. 1)} \/ {(do /. 1)} by GRAPH_2:53
.= {((go ^' pion1) /. 1),(do /. 1)} by ENUMSET1:1 ;
A113: Lower_Arc C is_an_arc_of E-max C, W-min C by JORDAN6:def_9;
then A114: Lower_Arc C is connected by JORDAN6:10;
set godo = (go ^' pion1) ^' do;
A115: do /. (len do) = (go ^' pion1) /. 1 by A78, GRAPH_2:53;
A116: go ^' pion1 is_sequence_on Gauge (C,n) by A39, A82, A89, TOPREAL8:12;
then A117: (go ^' pion1) ^' do is_sequence_on Gauge (C,n) by A45, A88, TOPREAL8:12;
A118: (len pion1) - 1 >= 1 by A93, XREAL_1:19;
then A119: (len pion1) -' 1 = (len pion1) - 1 by XREAL_0:def_2;
A120: {((Gauge (C,n)) * (i1,j))} c= (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i1,j))} or x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) )
assume x in {((Gauge (C,n)) * (i1,j))} ; ::_thesis: x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1))
then A121: x = (Gauge (C,n)) * (i1,j) by TARSKI:def_1;
pion1 /. (((len pion1) -' 1) + 1) = <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> /. 3 by A86, A119, FINSEQ_1:45
.= (Gauge (C,n)) * (i1,j) by FINSEQ_4:18 ;
then A122: (Gauge (C,n)) * (i1,j) in LSeg (pion1,((len pion1) -' 1)) by A118, A119, TOPREAL1:21;
(Gauge (C,n)) * (i1,j) in LSeg (do,1) by A50, RLTOPSP1:68;
hence x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) by A121, A122, XBOOLE_0:def_4; ::_thesis: verum
end;
LSeg (pion1,((len pion1) -' 1)) c= L~ <*((Gauge (C,n)) * (i2,k)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i1,j))*> by A84, TOPREAL3:19;
then (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) c= {((Gauge (C,n)) * (i1,j))} by A54, XBOOLE_1:27;
then A123: (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) = {((Gauge (C,n)) * (i1,j))} by A120, XBOOLE_0:def_10;
((len pion1) - 1) + 1 <= len pion1 ;
then A124: (len pion1) -' 1 < len pion1 by A119, NAT_1:13;
len pion1 >= 2 + 1 by A87, FINSEQ_1:45;
then A125: (len pion1) - 2 >= 0 by XREAL_1:19;
then ((len pion1) -' 2) + 1 = ((len pion1) - 2) + 1 by XREAL_0:def_2
.= (len pion1) -' 1 by A118, XREAL_0:def_2 ;
then A126: (LSeg ((go ^' pion1),((len go) + ((len pion1) -' 2)))) /\ (LSeg (do,1)) = {((go ^' pion1) /. (len (go ^' pion1)))} by A49, A89, A88, A124, A123, TOPREAL8:31;
(rng go) /\ (rng pion1) c= {(pion1 /. 1)} by A77, A94, A97, XBOOLE_1:27;
then A127: go ^' pion1 is one-to-one by JORDAN1J:55;
((len (go ^' pion1)) + 1) - 1 = ((len go) + (len pion1)) - 1 by GRAPH_2:13;
then (len (go ^' pion1)) - 1 = (len go) + ((len pion1) - 2)
.= (len go) + ((len pion1) -' 2) by A125, XREAL_0:def_2 ;
then A128: (len (go ^' pion1)) -' 1 = (len go) + ((len pion1) -' 2) by XREAL_0:def_2;
A129: L~ (Cage (C,n)) = (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n))) by JORDAN1E:13;
then A130: L~ (Upper_Seq (C,n)) c= L~ (Cage (C,n)) by XBOOLE_1:7;
then A131: L~ go c= L~ (Cage (C,n)) by A56, XBOOLE_1:1;
A132: len ((go ^' pion1) ^' do) >= len (go ^' pion1) by TOPREAL8:7;
then A133: 1 + 1 <= len ((go ^' pion1) ^' do) by A103, XXREAL_0:2;
not go ^' pion1 is trivial by A103, NAT_D:60;
then reconsider godo = (go ^' pion1) ^' do as non constant standard special_circular_sequence by A133, A88, A117, A102, A128, A126, A98, A127, A112, A115, JORDAN8:4, JORDAN8:5, TOPREAL8:11, TOPREAL8:33, TOPREAL8:34;
A134: L~ godo = (L~ (go ^' pion1)) \/ (L~ do) by A88, TOPREAL8:35
.= ((L~ go) \/ (L~ pion1)) \/ (L~ do) by A89, TOPREAL8:35 ;
A135: now__::_thesis:_not_((Gauge_(C,n))_*_(i2,k))_.._(Upper_Seq_(C,n))_<=_1
assume A136: ((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n)) <= 1 ; ::_thesis: contradiction
((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n)) >= 1 by A38, FINSEQ_4:21;
then ((Gauge (C,n)) * (i2,k)) .. (Upper_Seq (C,n)) = 1 by A136, XXREAL_0:1;
then (Gauge (C,n)) * (i2,k) = (Upper_Seq (C,n)) /. 1 by A38, FINSEQ_5:38;
hence contradiction by A15, A37, JORDAN1F:5; ::_thesis: verum
end;
A137: Upper_Seq (C,n) is_sequence_on Gauge (C,n) by JORDAN1G:4;
A138: ((Gauge (C,n)) * (i2,k)) `1 <= ((Gauge (C,n)) * (i1,k)) `1 by A1, A2, A3, A6, A10, JORDAN1A:18;
then A139: W-bound (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) = ((Gauge (C,n)) * (i2,k)) `1 by SPRECT_1:54;
A140: ((Gauge (C,n)) * (i1,k)) `1 = ((Gauge (C,n)) * (i1,1)) `1 by A3, A6, A17, A10, GOBOARD5:2
.= ((Gauge (C,n)) * (i1,j)) `1 by A3, A4, A17, A19, GOBOARD5:2 ;
then A141: W-bound (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) = ((Gauge (C,n)) * (i1,j)) `1 by SPRECT_1:54;
W-bound ((LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))))) = min ((W-bound (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))),(W-bound (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))))) by SPRECT_1:47
.= ((Gauge (C,n)) * (i2,k)) `1 by A140, A138, A139, A141, XXREAL_0:def_9 ;
then A142: W-bound (L~ pion1) = ((Gauge (C,n)) * (i2,k)) `1 by A84, TOPREAL3:16;
A143: Lower_Arc C c= C by JORDAN6:61;
((Gauge (C,n)) * (i2,k)) `1 >= W-bound (L~ (Cage (C,n))) by A30, A130, PSCOMP_1:24;
then A144: ((Gauge (C,n)) * (i2,k)) `1 > W-bound (L~ (Cage (C,n))) by A24, XXREAL_0:1;
A145: len (Upper_Seq (C,n)) >= 2 by A14, XXREAL_0:2;
A146: (L~ go) \/ (L~ do) is compact by COMPTS_1:10;
A147: L~ (Lower_Seq (C,n)) c= L~ (Cage (C,n)) by A129, XBOOLE_1:7;
then A148: L~ do c= L~ (Cage (C,n)) by A48, XBOOLE_1:1;
A149: (right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo) c= RightComp godo by A133, A117, JORDAN9:27;
2 in dom godo by A133, FINSEQ_3:25;
then A150: godo /. 2 in rng godo by PARTFUN2:2;
A151: rng godo c= L~ godo by A103, A132, SPPOL_2:18, XXREAL_0:2;
A152: godo /. 1 = (go ^' pion1) /. 1 by GRAPH_2:53
.= W-min (L~ (Cage (C,n))) by A67, GRAPH_2:53 ;
A153: W-min C in Lower_Arc C by A113, TOPREAL1:1;
A154: W-min C in C by SPRECT_1:13;
A155: now__::_thesis:_not_W-min_C_in_L~_godo
assume W-min C in L~ godo ; ::_thesis: contradiction
then A156: ( W-min C in (L~ go) \/ (L~ pion1) or W-min C in L~ do ) by A134, XBOOLE_0:def_3;
percases ( W-min C in L~ go or W-min C in L~ pion1 or W-min C in L~ do ) by A156, XBOOLE_0:def_3;
suppose W-min C in L~ go ; ::_thesis: contradiction
then C meets L~ (Cage (C,n)) by A131, A154, XBOOLE_0:3;
hence contradiction by JORDAN10:5; ::_thesis: verum
end;
suppose W-min C in L~ pion1 ; ::_thesis: contradiction
hence contradiction by A9, A36, A84, A153, XBOOLE_0:3; ::_thesis: verum
end;
suppose W-min C in L~ do ; ::_thesis: contradiction
then C meets L~ (Cage (C,n)) by A148, A154, XBOOLE_0:3;
hence contradiction by JORDAN10:5; ::_thesis: verum
end;
end;
end;
A157: (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52;
set ff = Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))));
A158: 1 + 1 <= len (Cage (C,n)) by GOBOARD7:34, XXREAL_0:2;
W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
then A159: (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 = W-min (L~ (Cage (C,n))) by FINSEQ_6:92;
A160: L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33;
then (W-max (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A159, SPRECT_5:22;
then (N-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A159, A160, SPRECT_5:23, XXREAL_0:2;
then (N-max (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A159, A160, SPRECT_5:24, XXREAL_0:2;
then A161: (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A159, A160, SPRECT_5:25, XXREAL_0:2;
A162: Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def_1;
then A163: Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))) is_sequence_on Gauge (C,n) by REVROT_1:34;
1 + 1 <= len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by GOBOARD7:34, XXREAL_0:2;
then right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1) = right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(GoB (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) by JORDAN1H:23
.= right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(GoB (Cage (C,n)))) by REVROT_1:28
.= right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(Gauge (C,n))) by JORDAN1H:44
.= right_cell (((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n))))),1,(Gauge (C,n))) by A161, A163, JORDAN1J:53
.= right_cell ((Upper_Seq (C,n)),1,(Gauge (C,n))) by JORDAN1E:def_1
.= right_cell ((R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i2,k)))),1,(Gauge (C,n))) by A38, A137, A135, JORDAN1J:52
.= right_cell ((go ^' pion1),1,(Gauge (C,n))) by A57, A116, JORDAN1J:51
.= right_cell (godo,1,(Gauge (C,n))) by A104, A117, JORDAN1J:51 ;
then W-min C in right_cell (godo,1,(Gauge (C,n))) by JORDAN1I:6;
then A164: W-min C in (right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo) by A155, XBOOLE_0:def_5;
A165: E-max C in Lower_Arc C by A113, TOPREAL1:1;
W-min (L~ (Cage (C,n))) in (L~ go) \/ (L~ do) by A77, A68, XBOOLE_0:def_3;
then A166: W-min ((L~ go) \/ (L~ do)) = W-min (L~ (Cage (C,n))) by A131, A148, A146, JORDAN1J:21, XBOOLE_1:8;
(W-min ((L~ go) \/ (L~ do))) `1 = W-bound ((L~ go) \/ (L~ do)) by EUCLID:52;
then W-min (((L~ go) \/ (L~ do)) \/ (L~ pion1)) = W-min ((L~ go) \/ (L~ do)) by A142, A146, A166, A157, A144, JORDAN1J:33;
then A167: W-min (L~ godo) = W-min (L~ (Cage (C,n))) by A134, A166, XBOOLE_1:4;
godo /. 2 = (go ^' pion1) /. 2 by A103, GRAPH_2:57
.= (Upper_Seq (C,n)) /. 2 by A42, A76, GRAPH_2:57
.= ((Upper_Seq (C,n)) ^' (Lower_Seq (C,n))) /. 2 by A145, GRAPH_2:57
.= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 2 by JORDAN1E:11 ;
then godo /. 2 in W-most (L~ (Cage (C,n))) by JORDAN1I:25;
then (godo /. 2) `1 = (W-min (L~ godo)) `1 by A167, PSCOMP_1:31
.= W-bound (L~ godo) by EUCLID:52 ;
then godo /. 2 in W-most (L~ godo) by A151, A150, SPRECT_2:12;
then (Rotate (godo,(W-min (L~ godo)))) /. 2 in W-most (L~ godo) by A152, A167, FINSEQ_6:89;
then reconsider godo = godo as non constant standard clockwise_oriented special_circular_sequence by JORDAN1I:25;
len (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by FINSEQ_5:6;
then A168: (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) = (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) by PARTFUN1:def_6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:7 ;
A169: east_halfline (E-max C) misses L~ go
proof
assume east_halfline (E-max C) meets L~ go ; ::_thesis: contradiction
then consider p being set such that
A170: p in east_halfline (E-max C) and
A171: p in L~ go by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A170;
p in L~ (Upper_Seq (C,n)) by A56, A171;
then p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A130, A170, XBOOLE_0:def_4;
then A172: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;
then A173: p = E-max (L~ (Cage (C,n))) by A56, A171, JORDAN1J:46;
then E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i2,k) by A30, A168, A171, JORDAN1J:43;
then ((Gauge (C,n)) * (i2,k)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by A6, A10, A13, A172, A173, JORDAN1A:71;
hence contradiction by A2, A3, A12, A41, JORDAN1G:7; ::_thesis: verum
end;
now__::_thesis:_not_east_halfline_(E-max_C)_meets_L~_godo
assume east_halfline (E-max C) meets L~ godo ; ::_thesis: contradiction
then A174: ( east_halfline (E-max C) meets (L~ go) \/ (L~ pion1) or east_halfline (E-max C) meets L~ do ) by A134, XBOOLE_1:70;
percases ( 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~ do ) by A174, XBOOLE_1:70;
suppose east_halfline (E-max C) meets L~ go ; ::_thesis: contradiction
hence contradiction by A169; ::_thesis: verum
end;
suppose east_halfline (E-max C) meets L~ pion1 ; ::_thesis: contradiction
then consider p being set such that
A175: p in east_halfline (E-max C) and
A176: p in L~ pion1 by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A175;
A177: p `2 = (E-max C) `2 by A175, TOPREAL1:def_11;
A178: now__::_thesis:_p_`1_<=_((Gauge_(C,n))_*_(i1,k))_`1
percases ( p in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) or p in LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) ) by A36, A84, A176, XBOOLE_0:def_3;
suppose p in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) ; ::_thesis: p `1 <= ((Gauge (C,n)) * (i1,k)) `1
hence p `1 <= ((Gauge (C,n)) * (i1,k)) `1 by A138, TOPREAL1:3; ::_thesis: verum
end;
suppose p in LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) ; ::_thesis: p `1 <= ((Gauge (C,n)) * (i1,k)) `1
hence p `1 <= ((Gauge (C,n)) * (i1,k)) `1 by A140, GOBOARD7:5; ::_thesis: verum
end;
end;
end;
i1 + 1 <= len (Gauge (C,n)) by A3, NAT_1:13;
then (i1 + 1) - 1 <= (len (Gauge (C,n))) - 1 by XREAL_1:9;
then A179: i1 <= (len (Gauge (C,n))) -' 1 by XREAL_0:def_2;
(len (Gauge (C,n))) -' 1 <= len (Gauge (C,n)) by NAT_D:35;
then ((Gauge (C,n)) * (i1,k)) `1 <= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),1)) `1 by A6, A17, A10, A13, A21, A179, JORDAN1A:18;
then p `1 <= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),1)) `1 by A178, XXREAL_0:2;
then p `1 <= E-bound C by A21, JORDAN8:12;
then A180: p `1 <= (E-max C) `1 by EUCLID:52;
p `1 >= (E-max C) `1 by A175, TOPREAL1:def_11;
then p `1 = (E-max C) `1 by A180, XXREAL_0:1;
then p = E-max C by A177, TOPREAL3:6;
hence contradiction by A9, A36, A84, A165, A176, XBOOLE_0:3; ::_thesis: verum
end;
suppose east_halfline (E-max C) meets L~ do ; ::_thesis: contradiction
then consider p being set such that
A181: p in east_halfline (E-max C) and
A182: p in L~ do by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A181;
A183: p in LSeg (do,(Index (p,do))) by A182, JORDAN3:9;
consider t being Nat such that
A184: t in dom (Lower_Seq (C,n)) and
A185: (Lower_Seq (C,n)) . t = (Gauge (C,n)) * (i1,j) by A44, FINSEQ_2:10;
1 <= t by A184, FINSEQ_3:25;
then A186: 1 < t by A46, A185, XXREAL_0:1;
t <= len (Lower_Seq (C,n)) by A184, FINSEQ_3:25;
then (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) + 1 = t by A185, A186, JORDAN3:12;
then A187: len (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i1,j)))) = (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) by A16, A185, JORDAN3:26;
Index (p,do) < len do by A182, JORDAN3:8;
then Index (p,do) < (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) by A187, XREAL_0:def_2;
then (Index (p,do)) + 1 <= (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) by NAT_1:13;
then A188: Index (p,do) <= ((len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n))))) - 1 by XREAL_1:19;
A189: do = mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n)))) by A44, JORDAN1J:37;
p in L~ (Lower_Seq (C,n)) by A48, A182;
then p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A147, A181, XBOOLE_0:def_4;
then A190: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;
A191: (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) + 1 = ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) by A46, A44, JORDAN1J:56;
0 + (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) < len (Lower_Seq (C,n)) by A16, JORDAN3:8;
then (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n)))) > 0 by XREAL_1:20;
then Index (p,do) <= ((len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n))))) - 1 by A188, XREAL_0:def_2;
then Index (p,do) <= (len (Lower_Seq (C,n))) - (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))) by A191;
then Index (p,do) <= (len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))) by XREAL_0:def_2;
then A192: Index (p,do) < ((len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) + 1 by NAT_1:13;
A193: 1 <= Index (p,do) by A182, JORDAN3:8;
A194: ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) <= len (Lower_Seq (C,n)) by A44, FINSEQ_4:21;
((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) <> len (Lower_Seq (C,n)) by A43, A44, FINSEQ_4:19;
then A195: ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) < len (Lower_Seq (C,n)) by A194, XXREAL_0:1;
A196: 1 + 1 <= len (Lower_Seq (C,n)) by A31, XXREAL_0:2;
then A197: 2 in dom (Lower_Seq (C,n)) by FINSEQ_3:25;
set tt = ((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1;
set RC = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))));
A198: E-max C in right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by JORDAN1I:7;
A199: GoB (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = GoB (Cage (C,n)) by REVROT_1:28
.= Gauge (C,n) by JORDAN1H:44 ;
A200: L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33;
consider jj2 being Element of NAT such that
A201: 1 <= jj2 and
A202: jj2 <= width (Gauge (C,n)) and
A203: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * ((len (Gauge (C,n))),jj2) by JORDAN1D:25;
A204: len (Gauge (C,n)) >= 4 by JORDAN8:10;
then len (Gauge (C,n)) >= 1 by XXREAL_0:2;
then A205: [(len (Gauge (C,n))),jj2] in Indices (Gauge (C,n)) by A201, A202, MATRIX_1:36;
A206: len (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = len (Cage (C,n)) by REVROT_1:14;
Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) by JORDAN1G:18;
then A207: LSeg ((Lower_Seq (C,n)),1) = LSeg ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by A196, SPPOL_2:9;
A208: E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))) is_sequence_on Gauge (C,n) by A162, REVROT_1:34;
then consider ii, jj being Element of NAT such that
A209: [ii,(jj + 1)] in Indices (Gauge (C,n)) and
A210: [ii,jj] in Indices (Gauge (C,n)) and
A211: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = (Gauge (C,n)) * (ii,(jj + 1)) and
A212: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1) = (Gauge (C,n)) * (ii,jj) by A158, A200, A206, A208, FINSEQ_6:92, JORDAN1I:23;
A213: (jj + 1) + 1 <> jj ;
A214: 1 <= jj by A210, MATRIX_1:38;
(Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = E-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))))) by A200, A208, FINSEQ_6:92;
then A215: ii = len (Gauge (C,n)) by A200, A209, A211, A203, A205, GOBOARD1:5;
then ii - 1 >= 4 - 1 by A204, XREAL_1:9;
then A216: ii - 1 >= 1 by XXREAL_0:2;
then A217: 1 <= ii -' 1 by XREAL_0:def_2;
A218: jj <= width (Gauge (C,n)) by A210, MATRIX_1:38;
then A219: ((Gauge (C,n)) * ((len (Gauge (C,n))),jj)) `1 = E-bound (L~ (Cage (C,n))) by A13, A214, JORDAN1A:71;
A220: jj + 1 <= width (Gauge (C,n)) by A209, MATRIX_1:38;
ii + 1 <> ii ;
then A221: right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) = cell ((Gauge (C,n)),(ii -' 1),jj) by A158, A206, A199, A209, A210, A211, A212, A213, GOBOARD5:def_6;
A222: ii <= len (Gauge (C,n)) by A210, MATRIX_1:38;
A223: 1 <= ii by A210, MATRIX_1:38;
A224: ii <= len (Gauge (C,n)) by A209, MATRIX_1:38;
A225: 1 <= jj + 1 by A209, MATRIX_1:38;
then A226: E-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * ((len (Gauge (C,n))),(jj + 1))) `1 by A13, A220, JORDAN1A:71;
A227: 1 <= ii by A209, MATRIX_1:38;
then A228: (ii -' 1) + 1 = ii by XREAL_1:235;
then A229: ii -' 1 < len (Gauge (C,n)) by A224, NAT_1:13;
then A230: ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 = ((Gauge (C,n)) * (1,(jj + 1))) `2 by A225, A220, A217, GOBOARD5:1
.= ((Gauge (C,n)) * (ii,(jj + 1))) `2 by A227, A224, A225, A220, GOBOARD5:1 ;
A231: (E-max C) `2 = p `2 by A181, TOPREAL1:def_11;
then A232: p `2 <= ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 by A198, A224, A220, A214, A221, A228, A216, JORDAN9:17;
A233: ((Gauge (C,n)) * ((ii -' 1),jj)) `2 = ((Gauge (C,n)) * (1,jj)) `2 by A214, A218, A217, A229, GOBOARD5:1
.= ((Gauge (C,n)) * (ii,jj)) `2 by A223, A222, A214, A218, GOBOARD5:1 ;
((Gauge (C,n)) * ((ii -' 1),jj)) `2 <= p `2 by A231, A198, A224, A220, A214, A221, A228, A216, JORDAN9:17;
then p in LSeg (((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1),((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1))) by A190, A211, A212, A215, A232, A233, A230, A219, A226, GOBOARD7:7;
then A234: p in LSeg ((Lower_Seq (C,n)),1) by A158, A207, A206, TOPREAL1:def_3;
1 <= ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) by A44, FINSEQ_4:21;
then A235: LSeg ((mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n))))),(Index (p,do))) = LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1)) by A195, A193, A192, JORDAN4:19;
1 <= Index (((Gauge (C,n)) * (i1,j)),(Lower_Seq (C,n))) by A16, JORDAN3:8;
then A236: 1 + 1 <= ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) by A191, XREAL_1:7;
then (Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))) >= (1 + 1) + 1 by A193, XREAL_1:7;
then ((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) - 1 >= ((1 + 1) + 1) - 1 by XREAL_1:9;
then A237: ((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1 >= 1 + 1 by XREAL_0:def_2;
now__::_thesis:_contradiction
percases ( ((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1 > 1 + 1 or ((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ) by A237, XXREAL_0:1;
suppose ((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1 > 1 + 1 ; ::_thesis: contradiction
then LSeg ((Lower_Seq (C,n)),1) misses LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1)) by TOPREAL1:def_7;
hence contradiction by A234, A183, A189, A235, XBOOLE_0:3; ::_thesis: verum
end;
supposeA238: ((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ; ::_thesis: contradiction
then 1 + 1 = ((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) - 1 by XREAL_0:def_2;
then (1 + 1) + 1 = (Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n))) ;
then A239: ((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)) = 2 by A193, A236, JORDAN1E:6;
(LSeg ((Lower_Seq (C,n)),1)) /\ (LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i1,j)) .. (Lower_Seq (C,n)))) -' 1))) = {((Lower_Seq (C,n)) /. 2)} by A31, A238, TOPREAL1:def_6;
then p in {((Lower_Seq (C,n)) /. 2)} by A234, A183, A189, A235, XBOOLE_0:def_4;
then A240: p = (Lower_Seq (C,n)) /. 2 by TARSKI:def_1;
then A241: p in rng (Lower_Seq (C,n)) by A197, PARTFUN2:2;
p .. (Lower_Seq (C,n)) = 2 by A197, A240, FINSEQ_5:41;
then p = (Gauge (C,n)) * (i1,j) by A44, A239, A241, FINSEQ_5:9;
then ((Gauge (C,n)) * (i1,j)) `1 = E-bound (L~ (Cage (C,n))) by A240, JORDAN1G:32;
then ((Gauge (C,n)) * (i1,j)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 by A4, A19, A13, JORDAN1A:71;
hence contradiction by A3, A25, A22, JORDAN1G:7; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
end;
end;
then east_halfline (E-max C) c= (L~ godo) ` by SUBSET_1:23;
then consider W being Subset of (TOP-REAL 2) such that
A242: W is_a_component_of (L~ godo) ` and
A243: east_halfline (E-max C) c= W by GOBOARD9:3;
not W is bounded by A243, JORDAN2C:121, RLTOPSP1:42;
then W is_outside_component_of L~ godo by A242, JORDAN2C:def_3;
then W c= UBD (L~ godo) by JORDAN2C:23;
then A244: east_halfline (E-max C) c= UBD (L~ godo) by A243, XBOOLE_1:1;
E-max C in east_halfline (E-max C) by TOPREAL1:38;
then E-max C in UBD (L~ godo) by A244;
then E-max C in LeftComp godo by GOBRD14:36;
then Lower_Arc C meets L~ godo by A114, A153, A165, A149, A164, JORDAN1J:36;
then A245: ( Lower_Arc C meets (L~ go) \/ (L~ pion1) or Lower_Arc C meets L~ do ) by A134, XBOOLE_1:70;
now__::_thesis:_contradiction
percases ( Lower_Arc C meets L~ go or Lower_Arc C meets L~ pion1 or Lower_Arc C meets L~ do ) by A245, XBOOLE_1:70;
suppose Lower_Arc C meets L~ go ; ::_thesis: contradiction
then Lower_Arc C meets L~ (Cage (C,n)) by A56, A130, XBOOLE_1:1, XBOOLE_1:63;
hence contradiction by A143, JORDAN10:5, XBOOLE_1:63; ::_thesis: verum
end;
suppose Lower_Arc C meets L~ pion1 ; ::_thesis: contradiction
hence contradiction by A9, A36, A84; ::_thesis: verum
end;
suppose Lower_Arc C meets L~ do ; ::_thesis: contradiction
then Lower_Arc C meets L~ (Cage (C,n)) by A48, A147, XBOOLE_1:1, XBOOLE_1:63;
hence contradiction by A143, JORDAN10:5, XBOOLE_1:63; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
suppose ((Gauge (C,n)) * (i1,j)) `1 = ((Gauge (C,n)) * (i2,k)) `1 ; ::_thesis: contradiction
then A246: i1 = i2 by A25, A12, JORDAN1G:7;
then LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) = {((Gauge (C,n)) * (i1,k))} by RLTOPSP1:70;
then LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) c= LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) by A27, ZFMISC_1:31;
then (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) = LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) by XBOOLE_1:12;
hence contradiction by A1, A3, A4, A5, A6, A7, A8, A9, A246, JORDAN1J:58; ::_thesis: verum
end;
suppose ((Gauge (C,n)) * (i1,j)) `2 = ((Gauge (C,n)) * (i2,k)) `2 ; ::_thesis: contradiction
then A247: j = k by A25, A12, JORDAN1G:6;
then LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) = {((Gauge (C,n)) * (i1,k))} by RLTOPSP1:70;
then LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) c= LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) by A26, ZFMISC_1:31;
then (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) = LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) by XBOOLE_1:12;
hence contradiction by A1, A2, A3, A4, A6, A7, A8, A9, A247, Th36; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
theorem Th48: :: JORDAN15:48
for n being Element of NAT
for C being Simple_closed_curve
for i1, i2, j, k being Element of NAT st 1 < i1 & i1 < len (Gauge (C,(n + 1))) & 1 < i2 & i2 < len (Gauge (C,(n + 1))) & 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (i1,k) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * (i2,j) in Lower_Arc (L~ (Cage (C,(n + 1)))) holds
(LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Upper_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i1, i2, j, k being Element of NAT st 1 < i1 & i1 < len (Gauge (C,(n + 1))) & 1 < i2 & i2 < len (Gauge (C,(n + 1))) & 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (i1,k) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * (i2,j) in Lower_Arc (L~ (Cage (C,(n + 1)))) holds
(LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Upper_Arc C
let C be Simple_closed_curve; ::_thesis: for i1, i2, j, k being Element of NAT st 1 < i1 & i1 < len (Gauge (C,(n + 1))) & 1 < i2 & i2 < len (Gauge (C,(n + 1))) & 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (i1,k) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * (i2,j) in Lower_Arc (L~ (Cage (C,(n + 1)))) holds
(LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Upper_Arc C
let i1, i2, j, k be Element of NAT ; ::_thesis: ( 1 < i1 & i1 < len (Gauge (C,(n + 1))) & 1 < i2 & i2 < len (Gauge (C,(n + 1))) & 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (i1,k) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * (i2,j) in Lower_Arc (L~ (Cage (C,(n + 1)))) implies (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Upper_Arc C )
set G = Gauge (C,(n + 1));
assume that
A1: 1 < i1 and
A2: i1 < len (Gauge (C,(n + 1))) and
A3: 1 < i2 and
A4: i2 < len (Gauge (C,(n + 1))) and
A5: 1 <= j and
A6: j <= k and
A7: k <= width (Gauge (C,(n + 1))) and
A8: (Gauge (C,(n + 1))) * (i1,k) in Upper_Arc (L~ (Cage (C,(n + 1)))) and
A9: (Gauge (C,(n + 1))) * (i2,j) in Lower_Arc (L~ (Cage (C,(n + 1)))) ; ::_thesis: (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Upper_Arc C
A10: 1 <= k by A5, A6, XXREAL_0:2;
then A11: [i2,k] in Indices (Gauge (C,(n + 1))) by A3, A4, A7, MATRIX_1:36;
A12: [i1,k] in Indices (Gauge (C,(n + 1))) by A1, A2, A7, A10, MATRIX_1:36;
((Gauge (C,(n + 1))) * (i2,k)) `2 = ((Gauge (C,(n + 1))) * (1,k)) `2 by A3, A4, A7, A10, GOBOARD5:1
.= ((Gauge (C,(n + 1))) * (i1,k)) `2 by A1, A2, A7, A10, GOBOARD5:1 ;
then A13: LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) is horizontal by SPPOL_1:15;
A14: Lower_Arc (L~ (Cage (C,(n + 1)))) = L~ (Lower_Seq (C,(n + 1))) by JORDAN1G:56;
A15: j <= width (Gauge (C,(n + 1))) by A6, A7, XXREAL_0:2;
then A16: [i2,j] in Indices (Gauge (C,(n + 1))) by A3, A4, A5, MATRIX_1:36;
((Gauge (C,(n + 1))) * (i2,j)) `1 = ((Gauge (C,(n + 1))) * (i2,1)) `1 by A3, A4, A5, A15, GOBOARD5:2
.= ((Gauge (C,(n + 1))) * (i2,k)) `1 by A3, A4, A7, A10, GOBOARD5:2 ;
then A17: LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k))) is vertical by SPPOL_1:16;
A18: Upper_Arc (L~ (Cage (C,(n + 1)))) = L~ (Upper_Seq (C,(n + 1))) by JORDAN1G:55;
A19: [i2,k] in Indices (Gauge (C,(n + 1))) by A3, A4, A7, A10, MATRIX_1:36;
now__::_thesis:_(LSeg_(((Gauge_(C,(n_+_1)))_*_(i2,j)),((Gauge_(C,(n_+_1)))_*_(i2,k))))_\/_(LSeg_(((Gauge_(C,(n_+_1)))_*_(i2,k)),((Gauge_(C,(n_+_1)))_*_(i1,k))))_meets_Upper_Arc_C
percases ( LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k))) meets Upper_Arc (L~ (Cage (C,(n + 1)))) or ( LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) meets Lower_Arc (L~ (Cage (C,(n + 1)))) & i2 <= i1 ) or ( LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) meets Lower_Arc (L~ (Cage (C,(n + 1)))) & i1 < i2 ) or ( LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k))) misses Upper_Arc (L~ (Cage (C,(n + 1)))) & LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) misses Lower_Arc (L~ (Cage (C,(n + 1)))) ) ) ;
supposeA20: LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k))) meets Upper_Arc (L~ (Cage (C,(n + 1)))) ; ::_thesis: (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Upper_Arc C
set X = (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1))));
ex x being set st
( x in LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k))) & x in L~ (Upper_Seq (C,(n + 1))) ) by A18, A20, XBOOLE_0:3;
then reconsider X1 = (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))) as non empty compact Subset of (TOP-REAL 2) by XBOOLE_0:def_4;
consider pp being set such that
A21: pp in S-most X1 by XBOOLE_0:def_1;
reconsider pp = pp as Point of (TOP-REAL 2) by A21;
A22: pp in (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))) by A21, XBOOLE_0:def_4;
then A23: pp in L~ (Upper_Seq (C,(n + 1))) by XBOOLE_0:def_4;
A24: pp in LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k))) by A22, XBOOLE_0:def_4;
consider m being Element of NAT such that
A25: j <= m and
A26: m <= k and
A27: ((Gauge (C,(n + 1))) * (i2,m)) `2 = lower_bound (proj2 .: ((LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))))) by A6, A18, A16, A19, A20, JORDAN1F:1, JORDAN1G:4;
A28: m <= width (Gauge (C,(n + 1))) by A7, A26, XXREAL_0:2;
1 <= m by A5, A25, XXREAL_0:2;
then A29: ((Gauge (C,(n + 1))) * (i2,m)) `1 = ((Gauge (C,(n + 1))) * (i2,1)) `1 by A3, A4, A28, GOBOARD5:2;
then A30: |[(((Gauge (C,(n + 1))) * (i2,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))))))]| = (Gauge (C,(n + 1))) * (i2,m) by A27, EUCLID:53;
then ((Gauge (C,(n + 1))) * (i2,j)) `1 = |[(((Gauge (C,(n + 1))) * (i2,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))))))]| `1 by A3, A4, A5, A15, A29, GOBOARD5:2;
then A31: pp `1 = |[(((Gauge (C,(n + 1))) * (i2,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))))))]| `1 by A17, A24, SPPOL_1:41;
|[(((Gauge (C,(n + 1))) * (i2,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))))))]| `2 = S-bound ((LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1))))) by A27, A30, SPRECT_1:44
.= (S-min ((LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))))) `2 by EUCLID:52
.= pp `2 by A21, PSCOMP_1:55 ;
then (Gauge (C,(n + 1))) * (i2,m) in Upper_Arc (L~ (Cage (C,(n + 1)))) by A18, A30, A23, A31, TOPREAL3:6;
then LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,m))) meets Upper_Arc C by A3, A4, A5, A9, A25, A28, Th24;
then LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k))) meets Upper_Arc C by A3, A4, A5, A7, A25, A26, Th5, XBOOLE_1:63;
hence (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Upper_Arc C by XBOOLE_1:70; ::_thesis: verum
end;
supposeA32: ( LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) meets Lower_Arc (L~ (Cage (C,(n + 1)))) & i2 <= i1 ) ; ::_thesis: (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Upper_Arc C
set X = (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) /\ (L~ (Lower_Seq (C,(n + 1))));
ex x being set st
( x in LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) & x in L~ (Lower_Seq (C,(n + 1))) ) by A14, A32, XBOOLE_0:3;
then reconsider X1 = (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))) as non empty compact Subset of (TOP-REAL 2) by XBOOLE_0:def_4;
consider pp being set such that
A33: pp in E-most X1 by XBOOLE_0:def_1;
reconsider pp = pp as Point of (TOP-REAL 2) by A33;
A34: pp in (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))) by A33, XBOOLE_0:def_4;
then A35: pp in L~ (Lower_Seq (C,(n + 1))) by XBOOLE_0:def_4;
A36: pp in LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) by A34, XBOOLE_0:def_4;
consider m being Element of NAT such that
A37: i2 <= m and
A38: m <= i1 and
A39: ((Gauge (C,(n + 1))) * (m,k)) `1 = upper_bound (proj1 .: ((LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))))) by A14, A11, A12, A32, JORDAN1F:4, JORDAN1G:5;
A40: 1 < m by A3, A37, XXREAL_0:2;
m < len (Gauge (C,(n + 1))) by A2, A38, XXREAL_0:2;
then A41: ((Gauge (C,(n + 1))) * (m,k)) `2 = ((Gauge (C,(n + 1))) * (1,k)) `2 by A7, A10, A40, GOBOARD5:1;
then A42: |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) /\ (L~ (Lower_Seq (C,(n + 1))))))),(((Gauge (C,(n + 1))) * (1,k)) `2)]| = (Gauge (C,(n + 1))) * (m,k) by A39, EUCLID:53;
then ((Gauge (C,(n + 1))) * (i2,k)) `2 = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) /\ (L~ (Lower_Seq (C,(n + 1))))))),(((Gauge (C,(n + 1))) * (1,k)) `2)]| `2 by A3, A4, A7, A10, A41, GOBOARD5:1;
then A43: pp `2 = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) /\ (L~ (Lower_Seq (C,(n + 1))))))),(((Gauge (C,(n + 1))) * (1,k)) `2)]| `2 by A13, A36, SPPOL_1:40;
|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) /\ (L~ (Lower_Seq (C,(n + 1))))))),(((Gauge (C,(n + 1))) * (1,k)) `2)]| `1 = E-bound ((LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) /\ (L~ (Lower_Seq (C,(n + 1))))) by A39, A42, SPRECT_1:46
.= (E-min ((LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))))) `1 by EUCLID:52
.= pp `1 by A33, PSCOMP_1:47 ;
then (Gauge (C,(n + 1))) * (m,k) in Lower_Arc (L~ (Cage (C,(n + 1)))) by A14, A42, A35, A43, TOPREAL3:6;
then LSeg (((Gauge (C,(n + 1))) * (m,k)),((Gauge (C,(n + 1))) * (i1,k))) meets Upper_Arc C by A2, A7, A8, A10, A38, A40, Th33;
then LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) meets Upper_Arc C by A2, A3, A7, A10, A37, A38, Th6, XBOOLE_1:63;
hence (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Upper_Arc C by XBOOLE_1:70; ::_thesis: verum
end;
supposeA44: ( LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) meets Lower_Arc (L~ (Cage (C,(n + 1)))) & i1 < i2 ) ; ::_thesis: (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Upper_Arc C
set X = (LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1))));
ex x being set st
( x in LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k))) & x in L~ (Lower_Seq (C,(n + 1))) ) by A14, A44, XBOOLE_0:3;
then reconsider X1 = (LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))) as non empty compact Subset of (TOP-REAL 2) by XBOOLE_0:def_4;
consider pp being set such that
A45: pp in W-most X1 by XBOOLE_0:def_1;
reconsider pp = pp as Point of (TOP-REAL 2) by A45;
A46: pp in (LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))) by A45, XBOOLE_0:def_4;
then A47: pp in L~ (Lower_Seq (C,(n + 1))) by XBOOLE_0:def_4;
A48: pp in LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k))) by A46, XBOOLE_0:def_4;
consider m being Element of NAT such that
A49: i1 <= m and
A50: m <= i2 and
A51: ((Gauge (C,(n + 1))) * (m,k)) `1 = lower_bound (proj1 .: ((LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))))) by A14, A11, A12, A44, JORDAN1F:3, JORDAN1G:5;
A52: m < len (Gauge (C,(n + 1))) by A4, A50, XXREAL_0:2;
1 < m by A1, A49, XXREAL_0:2;
then A53: ((Gauge (C,(n + 1))) * (m,k)) `2 = ((Gauge (C,(n + 1))) * (1,k)) `2 by A7, A10, A52, GOBOARD5:1;
then A54: |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1))))))),(((Gauge (C,(n + 1))) * (1,k)) `2)]| = (Gauge (C,(n + 1))) * (m,k) by A51, EUCLID:53;
then ((Gauge (C,(n + 1))) * (i1,k)) `2 = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1))))))),(((Gauge (C,(n + 1))) * (1,k)) `2)]| `2 by A1, A2, A7, A10, A53, GOBOARD5:1;
then A55: pp `2 = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1))))))),(((Gauge (C,(n + 1))) * (1,k)) `2)]| `2 by A13, A48, SPPOL_1:40;
|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1))))))),(((Gauge (C,(n + 1))) * (1,k)) `2)]| `1 = W-bound ((LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1))))) by A51, A54, SPRECT_1:43
.= (W-min ((LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))))) `1 by EUCLID:52
.= pp `1 by A45, PSCOMP_1:31 ;
then (Gauge (C,(n + 1))) * (m,k) in Lower_Arc (L~ (Cage (C,(n + 1)))) by A14, A54, A47, A55, TOPREAL3:6;
then LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (m,k))) meets Upper_Arc C by A1, A7, A8, A10, A49, A52, Th41;
then LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k))) meets Upper_Arc C by A1, A4, A7, A10, A49, A50, Th6, XBOOLE_1:63;
hence (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Upper_Arc C by XBOOLE_1:70; ::_thesis: verum
end;
supposeA56: ( LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k))) misses Upper_Arc (L~ (Cage (C,(n + 1)))) & LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) misses Lower_Arc (L~ (Cage (C,(n + 1)))) ) ; ::_thesis: (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Upper_Arc C
consider j1 being Element of NAT such that
A57: j <= j1 and
A58: j1 <= k and
A59: (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))) = {((Gauge (C,(n + 1))) * (i2,j1))} by A3, A4, A5, A6, A7, A9, A14, Th9;
(Gauge (C,(n + 1))) * (i2,j1) in (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))) by A59, TARSKI:def_1;
then A60: (Gauge (C,(n + 1))) * (i2,j1) in L~ (Lower_Seq (C,(n + 1))) by XBOOLE_0:def_4;
A61: 1 <= j1 by A5, A57, XXREAL_0:2;
now__::_thesis:_(LSeg_(((Gauge_(C,(n_+_1)))_*_(i2,j)),((Gauge_(C,(n_+_1)))_*_(i2,k))))_\/_(LSeg_(((Gauge_(C,(n_+_1)))_*_(i2,k)),((Gauge_(C,(n_+_1)))_*_(i1,k))))_meets_Upper_Arc_C
percases ( i2 <= i1 or i1 < i2 ) ;
supposeA62: i2 <= i1 ; ::_thesis: (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Upper_Arc C
A63: LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k))) c= LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k))) by A3, A4, A5, A7, A57, A58, Th5;
consider i3 being Element of NAT such that
A64: i2 <= i3 and
A65: i3 <= i1 and
A66: (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))) = {((Gauge (C,(n + 1))) * (i3,k))} by A2, A3, A7, A8, A18, A10, A62, Th13;
A67: LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))) c= LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) by A2, A3, A7, A10, A64, A65, Th6;
then A68: (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) c= (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) by A63, XBOOLE_1:13;
(Gauge (C,(n + 1))) * (i3,k) in (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))) by A66, TARSKI:def_1;
then A69: (Gauge (C,(n + 1))) * (i3,k) in L~ (Upper_Seq (C,(n + 1))) by XBOOLE_0:def_4;
A70: ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) = {((Gauge (C,(n + 1))) * (i3,k))}
proof
thus ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) c= {((Gauge (C,(n + 1))) * (i3,k))} :: according to XBOOLE_0:def_10 ::_thesis: {((Gauge (C,(n + 1))) * (i3,k))} c= ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) or x in {((Gauge (C,(n + 1))) * (i3,k))} )
assume A71: x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) ; ::_thesis: x in {((Gauge (C,(n + 1))) * (i3,k))}
then x in (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) by XBOOLE_0:def_4;
then A72: ( x in LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k))) or x in LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))) ) by XBOOLE_0:def_3;
x in L~ (Upper_Seq (C,(n + 1))) by A71, XBOOLE_0:def_4;
hence x in {((Gauge (C,(n + 1))) * (i3,k))} by A18, A56, A66, A63, A72, XBOOLE_0:3, XBOOLE_0:def_4; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,(n + 1))) * (i3,k))} or x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) )
(Gauge (C,(n + 1))) * (i3,k) in LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))) by RLTOPSP1:68;
then A73: (Gauge (C,(n + 1))) * (i3,k) in (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) by XBOOLE_0:def_3;
assume x in {((Gauge (C,(n + 1))) * (i3,k))} ; ::_thesis: x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1))))
then x = (Gauge (C,(n + 1))) * (i3,k) by TARSKI:def_1;
hence x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) by A69, A73, XBOOLE_0:def_4; ::_thesis: verum
end;
A74: ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) = {((Gauge (C,(n + 1))) * (i2,j1))}
proof
thus ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) c= {((Gauge (C,(n + 1))) * (i2,j1))} :: according to XBOOLE_0:def_10 ::_thesis: {((Gauge (C,(n + 1))) * (i2,j1))} c= ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) or x in {((Gauge (C,(n + 1))) * (i2,j1))} )
assume A75: x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) ; ::_thesis: x in {((Gauge (C,(n + 1))) * (i2,j1))}
then x in (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) by XBOOLE_0:def_4;
then A76: ( x in LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k))) or x in LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))) ) by XBOOLE_0:def_3;
x in L~ (Lower_Seq (C,(n + 1))) by A75, XBOOLE_0:def_4;
hence x in {((Gauge (C,(n + 1))) * (i2,j1))} by A14, A56, A59, A67, A76, XBOOLE_0:3, XBOOLE_0:def_4; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,(n + 1))) * (i2,j1))} or x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) )
(Gauge (C,(n + 1))) * (i2,j1) in LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k))) by RLTOPSP1:68;
then A77: (Gauge (C,(n + 1))) * (i2,j1) in (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) by XBOOLE_0:def_3;
assume x in {((Gauge (C,(n + 1))) * (i2,j1))} ; ::_thesis: x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1))))
then x = (Gauge (C,(n + 1))) * (i2,j1) by TARSKI:def_1;
hence x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) by A60, A77, XBOOLE_0:def_4; ::_thesis: verum
end;
i3 < len (Gauge (C,(n + 1))) by A2, A65, XXREAL_0:2;
hence (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Upper_Arc C by A3, A7, A58, A61, A64, A68, A70, A74, Th44, XBOOLE_1:63; ::_thesis: verum
end;
supposeA78: i1 < i2 ; ::_thesis: (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Upper_Arc C
A79: LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k))) c= LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k))) by A3, A4, A5, A7, A57, A58, Th5;
consider i3 being Element of NAT such that
A80: i1 <= i3 and
A81: i3 <= i2 and
A82: (LSeg (((Gauge (C,(n + 1))) * (i3,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))) = {((Gauge (C,(n + 1))) * (i3,k))} by A1, A4, A7, A8, A18, A10, A78, Th18;
A83: LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))) c= LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) by A1, A4, A7, A10, A80, A81, Th6;
then A84: (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) c= (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) by A79, XBOOLE_1:13;
(Gauge (C,(n + 1))) * (i3,k) in (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))) by A82, TARSKI:def_1;
then A85: (Gauge (C,(n + 1))) * (i3,k) in L~ (Upper_Seq (C,(n + 1))) by XBOOLE_0:def_4;
A86: ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) = {((Gauge (C,(n + 1))) * (i3,k))}
proof
thus ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) c= {((Gauge (C,(n + 1))) * (i3,k))} :: according to XBOOLE_0:def_10 ::_thesis: {((Gauge (C,(n + 1))) * (i3,k))} c= ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) or x in {((Gauge (C,(n + 1))) * (i3,k))} )
assume A87: x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) ; ::_thesis: x in {((Gauge (C,(n + 1))) * (i3,k))}
then x in (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) by XBOOLE_0:def_4;
then A88: ( x in LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k))) or x in LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))) ) by XBOOLE_0:def_3;
x in L~ (Upper_Seq (C,(n + 1))) by A87, XBOOLE_0:def_4;
hence x in {((Gauge (C,(n + 1))) * (i3,k))} by A18, A56, A82, A79, A88, XBOOLE_0:3, XBOOLE_0:def_4; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,(n + 1))) * (i3,k))} or x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) )
(Gauge (C,(n + 1))) * (i3,k) in LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))) by RLTOPSP1:68;
then A89: (Gauge (C,(n + 1))) * (i3,k) in (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) by XBOOLE_0:def_3;
assume x in {((Gauge (C,(n + 1))) * (i3,k))} ; ::_thesis: x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1))))
then x = (Gauge (C,(n + 1))) * (i3,k) by TARSKI:def_1;
hence x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) by A85, A89, XBOOLE_0:def_4; ::_thesis: verum
end;
A90: ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) = {((Gauge (C,(n + 1))) * (i2,j1))}
proof
thus ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) c= {((Gauge (C,(n + 1))) * (i2,j1))} :: according to XBOOLE_0:def_10 ::_thesis: {((Gauge (C,(n + 1))) * (i2,j1))} c= ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) or x in {((Gauge (C,(n + 1))) * (i2,j1))} )
assume A91: x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) ; ::_thesis: x in {((Gauge (C,(n + 1))) * (i2,j1))}
then x in (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) by XBOOLE_0:def_4;
then A92: ( x in LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k))) or x in LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))) ) by XBOOLE_0:def_3;
x in L~ (Lower_Seq (C,(n + 1))) by A91, XBOOLE_0:def_4;
hence x in {((Gauge (C,(n + 1))) * (i2,j1))} by A14, A56, A59, A83, A92, XBOOLE_0:3, XBOOLE_0:def_4; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,(n + 1))) * (i2,j1))} or x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) )
(Gauge (C,(n + 1))) * (i2,j1) in LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k))) by RLTOPSP1:68;
then A93: (Gauge (C,(n + 1))) * (i2,j1) in (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) by XBOOLE_0:def_3;
assume x in {((Gauge (C,(n + 1))) * (i2,j1))} ; ::_thesis: x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1))))
then x = (Gauge (C,(n + 1))) * (i2,j1) by TARSKI:def_1;
hence x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) by A60, A93, XBOOLE_0:def_4; ::_thesis: verum
end;
1 < i3 by A1, A80, XXREAL_0:2;
hence (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Upper_Arc C by A4, A7, A58, A61, A81, A84, A86, A90, Th46, XBOOLE_1:63; ::_thesis: verum
end;
end;
end;
hence (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Upper_Arc C ; ::_thesis: verum
end;
end;
end;
hence (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Upper_Arc C ; ::_thesis: verum
end;
theorem Th49: :: JORDAN15:49
for n being Element of NAT
for C being Simple_closed_curve
for i1, i2, j, k being Element of NAT st 1 < i1 & i1 < len (Gauge (C,(n + 1))) & 1 < i2 & i2 < len (Gauge (C,(n + 1))) & 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (i1,k) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * (i2,j) in Lower_Arc (L~ (Cage (C,(n + 1)))) holds
(LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Lower_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i1, i2, j, k being Element of NAT st 1 < i1 & i1 < len (Gauge (C,(n + 1))) & 1 < i2 & i2 < len (Gauge (C,(n + 1))) & 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (i1,k) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * (i2,j) in Lower_Arc (L~ (Cage (C,(n + 1)))) holds
(LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Lower_Arc C
let C be Simple_closed_curve; ::_thesis: for i1, i2, j, k being Element of NAT st 1 < i1 & i1 < len (Gauge (C,(n + 1))) & 1 < i2 & i2 < len (Gauge (C,(n + 1))) & 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (i1,k) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * (i2,j) in Lower_Arc (L~ (Cage (C,(n + 1)))) holds
(LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Lower_Arc C
let i1, i2, j, k be Element of NAT ; ::_thesis: ( 1 < i1 & i1 < len (Gauge (C,(n + 1))) & 1 < i2 & i2 < len (Gauge (C,(n + 1))) & 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (i1,k) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * (i2,j) in Lower_Arc (L~ (Cage (C,(n + 1)))) implies (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Lower_Arc C )
set G = Gauge (C,(n + 1));
assume that
A1: 1 < i1 and
A2: i1 < len (Gauge (C,(n + 1))) and
A3: 1 < i2 and
A4: i2 < len (Gauge (C,(n + 1))) and
A5: 1 <= j and
A6: j <= k and
A7: k <= width (Gauge (C,(n + 1))) and
A8: (Gauge (C,(n + 1))) * (i1,k) in Upper_Arc (L~ (Cage (C,(n + 1)))) and
A9: (Gauge (C,(n + 1))) * (i2,j) in Lower_Arc (L~ (Cage (C,(n + 1)))) ; ::_thesis: (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Lower_Arc C
A10: 1 <= k by A5, A6, XXREAL_0:2;
then A11: [i2,k] in Indices (Gauge (C,(n + 1))) by A3, A4, A7, MATRIX_1:36;
A12: [i1,k] in Indices (Gauge (C,(n + 1))) by A1, A2, A7, A10, MATRIX_1:36;
((Gauge (C,(n + 1))) * (i2,k)) `2 = ((Gauge (C,(n + 1))) * (1,k)) `2 by A3, A4, A7, A10, GOBOARD5:1
.= ((Gauge (C,(n + 1))) * (i1,k)) `2 by A1, A2, A7, A10, GOBOARD5:1 ;
then A13: LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) is horizontal by SPPOL_1:15;
A14: Lower_Arc (L~ (Cage (C,(n + 1)))) = L~ (Lower_Seq (C,(n + 1))) by JORDAN1G:56;
A15: j <= width (Gauge (C,(n + 1))) by A6, A7, XXREAL_0:2;
then A16: [i2,j] in Indices (Gauge (C,(n + 1))) by A3, A4, A5, MATRIX_1:36;
((Gauge (C,(n + 1))) * (i2,j)) `1 = ((Gauge (C,(n + 1))) * (i2,1)) `1 by A3, A4, A5, A15, GOBOARD5:2
.= ((Gauge (C,(n + 1))) * (i2,k)) `1 by A3, A4, A7, A10, GOBOARD5:2 ;
then A17: LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k))) is vertical by SPPOL_1:16;
A18: Upper_Arc (L~ (Cage (C,(n + 1)))) = L~ (Upper_Seq (C,(n + 1))) by JORDAN1G:55;
A19: [i2,k] in Indices (Gauge (C,(n + 1))) by A3, A4, A7, A10, MATRIX_1:36;
now__::_thesis:_(LSeg_(((Gauge_(C,(n_+_1)))_*_(i2,j)),((Gauge_(C,(n_+_1)))_*_(i2,k))))_\/_(LSeg_(((Gauge_(C,(n_+_1)))_*_(i2,k)),((Gauge_(C,(n_+_1)))_*_(i1,k))))_meets_Lower_Arc_C
percases ( LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k))) meets Upper_Arc (L~ (Cage (C,(n + 1)))) or ( LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) meets Lower_Arc (L~ (Cage (C,(n + 1)))) & i2 <= i1 ) or ( LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) meets Lower_Arc (L~ (Cage (C,(n + 1)))) & i1 < i2 ) or ( LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k))) misses Upper_Arc (L~ (Cage (C,(n + 1)))) & LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) misses Lower_Arc (L~ (Cage (C,(n + 1)))) ) ) ;
supposeA20: LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k))) meets Upper_Arc (L~ (Cage (C,(n + 1)))) ; ::_thesis: (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Lower_Arc C
set X = (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1))));
ex x being set st
( x in LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k))) & x in L~ (Upper_Seq (C,(n + 1))) ) by A18, A20, XBOOLE_0:3;
then reconsider X1 = (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))) as non empty compact Subset of (TOP-REAL 2) by XBOOLE_0:def_4;
consider pp being set such that
A21: pp in S-most X1 by XBOOLE_0:def_1;
reconsider pp = pp as Point of (TOP-REAL 2) by A21;
A22: pp in (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))) by A21, XBOOLE_0:def_4;
then A23: pp in L~ (Upper_Seq (C,(n + 1))) by XBOOLE_0:def_4;
A24: pp in LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k))) by A22, XBOOLE_0:def_4;
consider m being Element of NAT such that
A25: j <= m and
A26: m <= k and
A27: ((Gauge (C,(n + 1))) * (i2,m)) `2 = lower_bound (proj2 .: ((LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))))) by A6, A18, A16, A19, A20, JORDAN1F:1, JORDAN1G:4;
A28: m <= width (Gauge (C,(n + 1))) by A7, A26, XXREAL_0:2;
1 <= m by A5, A25, XXREAL_0:2;
then A29: ((Gauge (C,(n + 1))) * (i2,m)) `1 = ((Gauge (C,(n + 1))) * (i2,1)) `1 by A3, A4, A28, GOBOARD5:2;
then A30: |[(((Gauge (C,(n + 1))) * (i2,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))))))]| = (Gauge (C,(n + 1))) * (i2,m) by A27, EUCLID:53;
then ((Gauge (C,(n + 1))) * (i2,j)) `1 = |[(((Gauge (C,(n + 1))) * (i2,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))))))]| `1 by A3, A4, A5, A15, A29, GOBOARD5:2;
then A31: pp `1 = |[(((Gauge (C,(n + 1))) * (i2,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))))))]| `1 by A17, A24, SPPOL_1:41;
|[(((Gauge (C,(n + 1))) * (i2,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))))))]| `2 = S-bound ((LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1))))) by A27, A30, SPRECT_1:44
.= (S-min ((LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))))) `2 by EUCLID:52
.= pp `2 by A21, PSCOMP_1:55 ;
then (Gauge (C,(n + 1))) * (i2,m) in Upper_Arc (L~ (Cage (C,(n + 1)))) by A18, A30, A23, A31, TOPREAL3:6;
then LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,m))) meets Lower_Arc C by A3, A4, A5, A9, A25, A28, Th23;
then LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k))) meets Lower_Arc C by A3, A4, A5, A7, A25, A26, Th5, XBOOLE_1:63;
hence (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Lower_Arc C by XBOOLE_1:70; ::_thesis: verum
end;
supposeA32: ( LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) meets Lower_Arc (L~ (Cage (C,(n + 1)))) & i2 <= i1 ) ; ::_thesis: (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Lower_Arc C
set X = (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) /\ (L~ (Lower_Seq (C,(n + 1))));
ex x being set st
( x in LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) & x in L~ (Lower_Seq (C,(n + 1))) ) by A14, A32, XBOOLE_0:3;
then reconsider X1 = (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))) as non empty compact Subset of (TOP-REAL 2) by XBOOLE_0:def_4;
consider pp being set such that
A33: pp in E-most X1 by XBOOLE_0:def_1;
reconsider pp = pp as Point of (TOP-REAL 2) by A33;
A34: pp in (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))) by A33, XBOOLE_0:def_4;
then A35: pp in L~ (Lower_Seq (C,(n + 1))) by XBOOLE_0:def_4;
A36: pp in LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) by A34, XBOOLE_0:def_4;
consider m being Element of NAT such that
A37: i2 <= m and
A38: m <= i1 and
A39: ((Gauge (C,(n + 1))) * (m,k)) `1 = upper_bound (proj1 .: ((LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))))) by A14, A11, A12, A32, JORDAN1F:4, JORDAN1G:5;
A40: 1 < m by A3, A37, XXREAL_0:2;
m < len (Gauge (C,(n + 1))) by A2, A38, XXREAL_0:2;
then A41: ((Gauge (C,(n + 1))) * (m,k)) `2 = ((Gauge (C,(n + 1))) * (1,k)) `2 by A7, A10, A40, GOBOARD5:1;
then A42: |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) /\ (L~ (Lower_Seq (C,(n + 1))))))),(((Gauge (C,(n + 1))) * (1,k)) `2)]| = (Gauge (C,(n + 1))) * (m,k) by A39, EUCLID:53;
then ((Gauge (C,(n + 1))) * (i2,k)) `2 = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) /\ (L~ (Lower_Seq (C,(n + 1))))))),(((Gauge (C,(n + 1))) * (1,k)) `2)]| `2 by A3, A4, A7, A10, A41, GOBOARD5:1;
then A43: pp `2 = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) /\ (L~ (Lower_Seq (C,(n + 1))))))),(((Gauge (C,(n + 1))) * (1,k)) `2)]| `2 by A13, A36, SPPOL_1:40;
|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) /\ (L~ (Lower_Seq (C,(n + 1))))))),(((Gauge (C,(n + 1))) * (1,k)) `2)]| `1 = E-bound ((LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) /\ (L~ (Lower_Seq (C,(n + 1))))) by A39, A42, SPRECT_1:46
.= (E-min ((LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))))) `1 by EUCLID:52
.= pp `1 by A33, PSCOMP_1:47 ;
then (Gauge (C,(n + 1))) * (m,k) in Lower_Arc (L~ (Cage (C,(n + 1)))) by A14, A42, A35, A43, TOPREAL3:6;
then LSeg (((Gauge (C,(n + 1))) * (m,k)),((Gauge (C,(n + 1))) * (i1,k))) meets Lower_Arc C by A2, A7, A8, A10, A38, A40, Th32;
then LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) meets Lower_Arc C by A2, A3, A7, A10, A37, A38, Th6, XBOOLE_1:63;
hence (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Lower_Arc C by XBOOLE_1:70; ::_thesis: verum
end;
supposeA44: ( LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) meets Lower_Arc (L~ (Cage (C,(n + 1)))) & i1 < i2 ) ; ::_thesis: (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Lower_Arc C
set X = (LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1))));
ex x being set st
( x in LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k))) & x in L~ (Lower_Seq (C,(n + 1))) ) by A14, A44, XBOOLE_0:3;
then reconsider X1 = (LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))) as non empty compact Subset of (TOP-REAL 2) by XBOOLE_0:def_4;
consider pp being set such that
A45: pp in W-most X1 by XBOOLE_0:def_1;
reconsider pp = pp as Point of (TOP-REAL 2) by A45;
A46: pp in (LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))) by A45, XBOOLE_0:def_4;
then A47: pp in L~ (Lower_Seq (C,(n + 1))) by XBOOLE_0:def_4;
A48: pp in LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k))) by A46, XBOOLE_0:def_4;
consider m being Element of NAT such that
A49: i1 <= m and
A50: m <= i2 and
A51: ((Gauge (C,(n + 1))) * (m,k)) `1 = lower_bound (proj1 .: ((LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))))) by A14, A11, A12, A44, JORDAN1F:3, JORDAN1G:5;
A52: m < len (Gauge (C,(n + 1))) by A4, A50, XXREAL_0:2;
1 < m by A1, A49, XXREAL_0:2;
then A53: ((Gauge (C,(n + 1))) * (m,k)) `2 = ((Gauge (C,(n + 1))) * (1,k)) `2 by A7, A10, A52, GOBOARD5:1;
then A54: |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1))))))),(((Gauge (C,(n + 1))) * (1,k)) `2)]| = (Gauge (C,(n + 1))) * (m,k) by A51, EUCLID:53;
then ((Gauge (C,(n + 1))) * (i1,k)) `2 = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1))))))),(((Gauge (C,(n + 1))) * (1,k)) `2)]| `2 by A1, A2, A7, A10, A53, GOBOARD5:1;
then A55: pp `2 = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1))))))),(((Gauge (C,(n + 1))) * (1,k)) `2)]| `2 by A13, A48, SPPOL_1:40;
|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1))))))),(((Gauge (C,(n + 1))) * (1,k)) `2)]| `1 = W-bound ((LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1))))) by A51, A54, SPRECT_1:43
.= (W-min ((LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))))) `1 by EUCLID:52
.= pp `1 by A45, PSCOMP_1:31 ;
then (Gauge (C,(n + 1))) * (m,k) in Lower_Arc (L~ (Cage (C,(n + 1)))) by A14, A54, A47, A55, TOPREAL3:6;
then LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (m,k))) meets Lower_Arc C by A1, A7, A8, A10, A49, A52, Th40;
then LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k))) meets Lower_Arc C by A1, A4, A7, A10, A49, A50, Th6, XBOOLE_1:63;
hence (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Lower_Arc C by XBOOLE_1:70; ::_thesis: verum
end;
supposeA56: ( LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k))) misses Upper_Arc (L~ (Cage (C,(n + 1)))) & LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) misses Lower_Arc (L~ (Cage (C,(n + 1)))) ) ; ::_thesis: (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Lower_Arc C
consider j1 being Element of NAT such that
A57: j <= j1 and
A58: j1 <= k and
A59: (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))) = {((Gauge (C,(n + 1))) * (i2,j1))} by A3, A4, A5, A6, A7, A9, A14, Th9;
(Gauge (C,(n + 1))) * (i2,j1) in (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))) by A59, TARSKI:def_1;
then A60: (Gauge (C,(n + 1))) * (i2,j1) in L~ (Lower_Seq (C,(n + 1))) by XBOOLE_0:def_4;
A61: 1 <= j1 by A5, A57, XXREAL_0:2;
now__::_thesis:_(LSeg_(((Gauge_(C,(n_+_1)))_*_(i2,j)),((Gauge_(C,(n_+_1)))_*_(i2,k))))_\/_(LSeg_(((Gauge_(C,(n_+_1)))_*_(i2,k)),((Gauge_(C,(n_+_1)))_*_(i1,k))))_meets_Lower_Arc_C
percases ( i2 <= i1 or i1 < i2 ) ;
supposeA62: i2 <= i1 ; ::_thesis: (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Lower_Arc C
A63: LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k))) c= LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k))) by A3, A4, A5, A7, A57, A58, Th5;
consider i3 being Element of NAT such that
A64: i2 <= i3 and
A65: i3 <= i1 and
A66: (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))) = {((Gauge (C,(n + 1))) * (i3,k))} by A2, A3, A7, A8, A18, A10, A62, Th13;
A67: LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))) c= LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) by A2, A3, A7, A10, A64, A65, Th6;
then A68: (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) c= (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) by A63, XBOOLE_1:13;
(Gauge (C,(n + 1))) * (i3,k) in (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))) by A66, TARSKI:def_1;
then A69: (Gauge (C,(n + 1))) * (i3,k) in L~ (Upper_Seq (C,(n + 1))) by XBOOLE_0:def_4;
A70: ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) = {((Gauge (C,(n + 1))) * (i3,k))}
proof
thus ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) c= {((Gauge (C,(n + 1))) * (i3,k))} :: according to XBOOLE_0:def_10 ::_thesis: {((Gauge (C,(n + 1))) * (i3,k))} c= ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) or x in {((Gauge (C,(n + 1))) * (i3,k))} )
assume A71: x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) ; ::_thesis: x in {((Gauge (C,(n + 1))) * (i3,k))}
then x in (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) by XBOOLE_0:def_4;
then A72: ( x in LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k))) or x in LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))) ) by XBOOLE_0:def_3;
x in L~ (Upper_Seq (C,(n + 1))) by A71, XBOOLE_0:def_4;
hence x in {((Gauge (C,(n + 1))) * (i3,k))} by A18, A56, A66, A63, A72, XBOOLE_0:3, XBOOLE_0:def_4; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,(n + 1))) * (i3,k))} or x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) )
(Gauge (C,(n + 1))) * (i3,k) in LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))) by RLTOPSP1:68;
then A73: (Gauge (C,(n + 1))) * (i3,k) in (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) by XBOOLE_0:def_3;
assume x in {((Gauge (C,(n + 1))) * (i3,k))} ; ::_thesis: x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1))))
then x = (Gauge (C,(n + 1))) * (i3,k) by TARSKI:def_1;
hence x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) by A69, A73, XBOOLE_0:def_4; ::_thesis: verum
end;
A74: ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) = {((Gauge (C,(n + 1))) * (i2,j1))}
proof
thus ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) c= {((Gauge (C,(n + 1))) * (i2,j1))} :: according to XBOOLE_0:def_10 ::_thesis: {((Gauge (C,(n + 1))) * (i2,j1))} c= ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) or x in {((Gauge (C,(n + 1))) * (i2,j1))} )
assume A75: x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) ; ::_thesis: x in {((Gauge (C,(n + 1))) * (i2,j1))}
then x in (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) by XBOOLE_0:def_4;
then A76: ( x in LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k))) or x in LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))) ) by XBOOLE_0:def_3;
x in L~ (Lower_Seq (C,(n + 1))) by A75, XBOOLE_0:def_4;
hence x in {((Gauge (C,(n + 1))) * (i2,j1))} by A14, A56, A59, A67, A76, XBOOLE_0:3, XBOOLE_0:def_4; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,(n + 1))) * (i2,j1))} or x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) )
(Gauge (C,(n + 1))) * (i2,j1) in LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k))) by RLTOPSP1:68;
then A77: (Gauge (C,(n + 1))) * (i2,j1) in (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) by XBOOLE_0:def_3;
assume x in {((Gauge (C,(n + 1))) * (i2,j1))} ; ::_thesis: x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1))))
then x = (Gauge (C,(n + 1))) * (i2,j1) by TARSKI:def_1;
hence x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) by A60, A77, XBOOLE_0:def_4; ::_thesis: verum
end;
i3 < len (Gauge (C,(n + 1))) by A2, A65, XXREAL_0:2;
hence (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Lower_Arc C by A3, A7, A58, A61, A64, A68, A70, A74, Th45, XBOOLE_1:63; ::_thesis: verum
end;
supposeA78: i1 < i2 ; ::_thesis: (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Lower_Arc C
A79: LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k))) c= LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k))) by A3, A4, A5, A7, A57, A58, Th5;
consider i3 being Element of NAT such that
A80: i1 <= i3 and
A81: i3 <= i2 and
A82: (LSeg (((Gauge (C,(n + 1))) * (i3,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))) = {((Gauge (C,(n + 1))) * (i3,k))} by A1, A4, A7, A8, A18, A10, A78, Th18;
A83: LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))) c= LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) by A1, A4, A7, A10, A80, A81, Th6;
then A84: (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) c= (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) by A79, XBOOLE_1:13;
(Gauge (C,(n + 1))) * (i3,k) in (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))) by A82, TARSKI:def_1;
then A85: (Gauge (C,(n + 1))) * (i3,k) in L~ (Upper_Seq (C,(n + 1))) by XBOOLE_0:def_4;
A86: ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) = {((Gauge (C,(n + 1))) * (i3,k))}
proof
thus ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) c= {((Gauge (C,(n + 1))) * (i3,k))} :: according to XBOOLE_0:def_10 ::_thesis: {((Gauge (C,(n + 1))) * (i3,k))} c= ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) or x in {((Gauge (C,(n + 1))) * (i3,k))} )
assume A87: x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) ; ::_thesis: x in {((Gauge (C,(n + 1))) * (i3,k))}
then x in (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) by XBOOLE_0:def_4;
then A88: ( x in LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k))) or x in LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))) ) by XBOOLE_0:def_3;
x in L~ (Upper_Seq (C,(n + 1))) by A87, XBOOLE_0:def_4;
hence x in {((Gauge (C,(n + 1))) * (i3,k))} by A18, A56, A82, A79, A88, XBOOLE_0:3, XBOOLE_0:def_4; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,(n + 1))) * (i3,k))} or x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) )
(Gauge (C,(n + 1))) * (i3,k) in LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))) by RLTOPSP1:68;
then A89: (Gauge (C,(n + 1))) * (i3,k) in (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) by XBOOLE_0:def_3;
assume x in {((Gauge (C,(n + 1))) * (i3,k))} ; ::_thesis: x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1))))
then x = (Gauge (C,(n + 1))) * (i3,k) by TARSKI:def_1;
hence x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) by A85, A89, XBOOLE_0:def_4; ::_thesis: verum
end;
A90: ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) = {((Gauge (C,(n + 1))) * (i2,j1))}
proof
thus ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) c= {((Gauge (C,(n + 1))) * (i2,j1))} :: according to XBOOLE_0:def_10 ::_thesis: {((Gauge (C,(n + 1))) * (i2,j1))} c= ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) or x in {((Gauge (C,(n + 1))) * (i2,j1))} )
assume A91: x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) ; ::_thesis: x in {((Gauge (C,(n + 1))) * (i2,j1))}
then x in (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) by XBOOLE_0:def_4;
then A92: ( x in LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k))) or x in LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))) ) by XBOOLE_0:def_3;
x in L~ (Lower_Seq (C,(n + 1))) by A91, XBOOLE_0:def_4;
hence x in {((Gauge (C,(n + 1))) * (i2,j1))} by A14, A56, A59, A83, A92, XBOOLE_0:3, XBOOLE_0:def_4; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,(n + 1))) * (i2,j1))} or x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) )
(Gauge (C,(n + 1))) * (i2,j1) in LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k))) by RLTOPSP1:68;
then A93: (Gauge (C,(n + 1))) * (i2,j1) in (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) by XBOOLE_0:def_3;
assume x in {((Gauge (C,(n + 1))) * (i2,j1))} ; ::_thesis: x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1))))
then x = (Gauge (C,(n + 1))) * (i2,j1) by TARSKI:def_1;
hence x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) by A60, A93, XBOOLE_0:def_4; ::_thesis: verum
end;
1 < i3 by A1, A80, XXREAL_0:2;
hence (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Lower_Arc C by A4, A7, A58, A61, A81, A84, A86, A90, Th47, XBOOLE_1:63; ::_thesis: verum
end;
end;
end;
hence (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Lower_Arc C ; ::_thesis: verum
end;
end;
end;
hence (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Lower_Arc C ; ::_thesis: verum
end;
theorem :: JORDAN15:50
for n being Element of NAT
for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,(n + 1))) & 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (i,k) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j) in Lower_Arc (L~ (Cage (C,(n + 1)))) holds
(LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) \/ (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)),((Gauge (C,(n + 1))) * (i,k)))) meets Upper_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,(n + 1))) & 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (i,k) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j) in Lower_Arc (L~ (Cage (C,(n + 1)))) holds
(LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) \/ (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)),((Gauge (C,(n + 1))) * (i,k)))) meets Upper_Arc C
let C be Simple_closed_curve; ::_thesis: for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,(n + 1))) & 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (i,k) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j) in Lower_Arc (L~ (Cage (C,(n + 1)))) holds
(LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) \/ (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)),((Gauge (C,(n + 1))) * (i,k)))) meets Upper_Arc C
let i, j, k be Element of NAT ; ::_thesis: ( 1 < i & i < len (Gauge (C,(n + 1))) & 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (i,k) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j) in Lower_Arc (L~ (Cage (C,(n + 1)))) implies (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) \/ (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)),((Gauge (C,(n + 1))) * (i,k)))) meets Upper_Arc C )
assume that
A1: 1 < i and
A2: i < len (Gauge (C,(n + 1))) and
A3: 1 <= j and
A4: j <= k and
A5: k <= width (Gauge (C,(n + 1))) and
A6: (Gauge (C,(n + 1))) * (i,k) in Upper_Arc (L~ (Cage (C,(n + 1)))) and
A7: (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j) in Lower_Arc (L~ (Cage (C,(n + 1)))) ; ::_thesis: (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) \/ (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)),((Gauge (C,(n + 1))) * (i,k)))) meets Upper_Arc C
A8: len (Gauge (C,(n + 1))) >= 4 by JORDAN8:10;
then len (Gauge (C,(n + 1))) >= 3 by XXREAL_0:2;
then A9: Center (Gauge (C,(n + 1))) < len (Gauge (C,(n + 1))) by JORDAN1B:15;
len (Gauge (C,(n + 1))) >= 2 by A8, XXREAL_0:2;
then 1 < Center (Gauge (C,(n + 1))) by JORDAN1B:14;
hence (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) \/ (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)),((Gauge (C,(n + 1))) * (i,k)))) meets Upper_Arc C by A1, A2, A3, A4, A5, A6, A7, A9, Th48; ::_thesis: verum
end;
theorem :: JORDAN15:51
for n being Element of NAT
for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,(n + 1))) & 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (i,k) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j) in Lower_Arc (L~ (Cage (C,(n + 1)))) holds
(LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) \/ (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)),((Gauge (C,(n + 1))) * (i,k)))) meets Lower_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,(n + 1))) & 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (i,k) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j) in Lower_Arc (L~ (Cage (C,(n + 1)))) holds
(LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) \/ (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)),((Gauge (C,(n + 1))) * (i,k)))) meets Lower_Arc C
let C be Simple_closed_curve; ::_thesis: for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,(n + 1))) & 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (i,k) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j) in Lower_Arc (L~ (Cage (C,(n + 1)))) holds
(LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) \/ (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)),((Gauge (C,(n + 1))) * (i,k)))) meets Lower_Arc C
let i, j, k be Element of NAT ; ::_thesis: ( 1 < i & i < len (Gauge (C,(n + 1))) & 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (i,k) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j) in Lower_Arc (L~ (Cage (C,(n + 1)))) implies (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) \/ (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)),((Gauge (C,(n + 1))) * (i,k)))) meets Lower_Arc C )
assume that
A1: 1 < i and
A2: i < len (Gauge (C,(n + 1))) and
A3: 1 <= j and
A4: j <= k and
A5: k <= width (Gauge (C,(n + 1))) and
A6: (Gauge (C,(n + 1))) * (i,k) in Upper_Arc (L~ (Cage (C,(n + 1)))) and
A7: (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j) in Lower_Arc (L~ (Cage (C,(n + 1)))) ; ::_thesis: (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) \/ (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)),((Gauge (C,(n + 1))) * (i,k)))) meets Lower_Arc C
A8: len (Gauge (C,(n + 1))) >= 4 by JORDAN8:10;
then len (Gauge (C,(n + 1))) >= 3 by XXREAL_0:2;
then A9: Center (Gauge (C,(n + 1))) < len (Gauge (C,(n + 1))) by JORDAN1B:15;
len (Gauge (C,(n + 1))) >= 2 by A8, XXREAL_0:2;
then 1 < Center (Gauge (C,(n + 1))) by JORDAN1B:14;
hence (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) \/ (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)),((Gauge (C,(n + 1))) * (i,k)))) meets Lower_Arc C by A1, A2, A3, A4, A5, A6, A7, A9, Th49; ::_thesis: verum
end;