:: The Ordering of Points on a Curve, Part { I }
:: by Adam Grabowski and Yatsuka Nakamura
::
:: Received September 10, 1997
:: Copyright (c) 1997-2011 Association of Mizar Users

begin

theorem Th1: :: JORDAN5B:1

for i1 being Nat st 1 <= i1 holds
i1 -' 1 < i1

proof end;

theorem :: JORDAN5B:2

for i, k being Nat st i + 1 <= k holds
1 <= k -' i

proof end;

theorem :: JORDAN5B:3

for i, k being Nat st 1 <= i & 1 <= k holds
(k -' i) + 1 <= k

proof end;

Lm1: for r being real number st 0 <= r & r <= 1 holds
( 0 <= 1 - r & 1 - r <= 1 )

proof end;

theorem :: JORDAN5B:4

for r being real number st r in the carrier of I[01] holds
1 - r in the carrier of I[01]

proof end;

theorem :: JORDAN5B:5

for p, q, p1 being Point of (TOP-REAL 2) st p `2 <> q `2 & p1 in LSeg (p,q) & p1 `2 = p `2 holds
p1 = p

proof end;

theorem :: JORDAN5B:6

for p, q, p1 being Point of (TOP-REAL 2) st p `1 <> q `1 & p1 in LSeg (p,q) & p1 `1 = p `1 holds
p1 = p

proof end;

Lm2: for P being Point of I[01] holds P is Real

proof end;

theorem Th7: :: JORDAN5B:7

for f being FinSequence of (TOP-REAL 2)
for P being non empty Subset of (TOP-REAL 2)
for F being Function of I[01],((TOP-REAL 2) | P)
for i being Element of NAT st 1 <= i & i + 1 <= len f & f is being_S-Seq & P = L~ f & F is being_homeomorphism & F . 0 = f /. 1 & F . 1 = f /. (len f) holds
ex p1, p2 being Real st
( p1 < p2 & 0 <= p1 & p1 <= 1 & 0 <= p2 & p2 <= 1 & LSeg (f,i) = F .: [.p1,p2.] & F . p1 = f /. i & F . p2 = f /. (i + 1) )

proof end;

theorem :: JORDAN5B:8

for f being FinSequence of (TOP-REAL 2)
for Q, R being non empty Subset of (TOP-REAL 2)
for F being Function of I[01],((TOP-REAL 2) | Q)
for i being Element of NAT
for P being non empty Subset of I[01] st f is being_S-Seq & F is being_homeomorphism & F . 0 = f /. 1 & F . 1 = f /. (len f) & 1 <= i & i + 1 <= len f & F .: P = LSeg (f,i) & Q = L~ f & R = LSeg (f,i) holds
ex G being Function of (I[01] | P),((TOP-REAL 2) | R) st
( G = F | P & G is being_homeomorphism )

proof end;

begin

theorem Th9: :: JORDAN5B:9

for p1, p2, p being Point of (TOP-REAL 2) st p1 <> p2 & p in LSeg (p1,p2) holds
LE p,p,p1,p2

proof end;

theorem Th10: :: JORDAN5B:10

for p, p1, p2 being Point of (TOP-REAL 2) st p1 <> p2 & p in LSeg (p1,p2) holds
LE p1,p,p1,p2

proof end;

theorem Th11: :: JORDAN5B:11

for p, p1, p2 being Point of (TOP-REAL 2) st p in LSeg (p1,p2) & p1 <> p2 holds
LE p,p2,p1,p2

proof end;

theorem :: JORDAN5B:12

for p1, p2, q1, q2, q3 being Point of (TOP-REAL 2) st p1 <> p2 & LE q1,q2,p1,p2 & LE q2,q3,p1,p2 holds
LE q1,q3,p1,p2

proof end;

theorem :: JORDAN5B:13

for p, q being Point of (TOP-REAL 2) st p <> q holds
LSeg (p,q) = { p1 where p1 is Point of (TOP-REAL 2) : ( LE p,p1,p,q & LE p1,q,p,q ) }

proof end;

theorem :: JORDAN5B:14

for n being Element of NAT
for P being Subset of (TOP-REAL n)
for p1, p2 being Point of (TOP-REAL n) st P is_an_arc_of p1,p2 holds
P is_an_arc_of p2,p1

proof end;

theorem :: JORDAN5B:15

for i being Element of NAT
for f being FinSequence of (TOP-REAL 2)
for P being Subset of (TOP-REAL 2) st f is being_S-Seq & 1 <= i & i + 1 <= len f & P = LSeg (f,i) holds
P is_an_arc_of f /. i,f /. (i + 1)

proof end;

begin

theorem :: JORDAN5B:16

for g1 being FinSequence of (TOP-REAL 2)
for i being Element of NAT st 1 <= i & i <= len g1 & g1 is being_S-Seq & g1 /. 1 in L~ (mid (g1,i,(len g1))) holds
i = 1

proof end;

theorem :: JORDAN5B:17

for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st f is being_S-Seq & p = f . (len f) holds
L_Cut (f,p) = <*p*>

proof end;

theorem :: JORDAN5B:18

canceled;

theorem :: JORDAN5B:19

canceled;

theorem :: JORDAN5B:20

canceled;

theorem :: JORDAN5B:21

for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in L~ f & p <> f . (len f) & f is being_S-Seq holds
Index (p,(L_Cut (f,p))) = 1

proof end;

theorem Th22: :: JORDAN5B:22

for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in L~ f & f is being_S-Seq & p <> f . (len f) holds
p in L~ (L_Cut (f,p))

proof end;

theorem :: JORDAN5B:23

for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in L~ f & f is being_S-Seq & p <> f . 1 holds
p in L~ (R_Cut (f,p))

proof end;

theorem Th24: :: JORDAN5B:24

for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in L~ f & f is one-to-one holds
B_Cut (f,p,p) = <*p*>

proof end;

Lm3: for f being FinSequence of (TOP-REAL 2)
for p, q being Point of (TOP-REAL 2) st p in L~ f & q in L~ f & p <> f . (len f) & f is being_S-Seq & not p in L~ (L_Cut (f,q)) holds
q in L~ (L_Cut (f,p))

proof end;

theorem Th25: :: JORDAN5B:25

for f being FinSequence of (TOP-REAL 2)
for p, q being Point of (TOP-REAL 2) st p in L~ f & q in L~ f & q <> f . (len f) & p = f . (len f) & f is being_S-Seq holds
p in L~ (L_Cut (f,q))

proof end;

theorem :: JORDAN5B:26

for f being FinSequence of (TOP-REAL 2)
for p, q being Point of (TOP-REAL 2) st p <> f . (len f) & p in L~ f & q in L~ f & f is being_S-Seq & not p in L~ (L_Cut (f,q)) holds
q in L~ (L_Cut (f,p))

proof end;

Lm4: for f being FinSequence of (TOP-REAL 2)
for p, q being Point of (TOP-REAL 2) st p in L~ f & q in L~ f & ( Index (p,f) < Index (q,f) or ( Index (p,f) = Index (q,f) & LE p,q,f /. (Index (p,f)),f /. ((Index (p,f)) + 1) ) ) & p <> q holds
L~ (B_Cut (f,p,q)) c= L~ f

proof end;

theorem :: JORDAN5B:27

for f being FinSequence of (TOP-REAL 2)
for p, q being Point of (TOP-REAL 2) st p in L~ f & q in L~ f & f is being_S-Seq holds
L~ (B_Cut (f,p,q)) c= L~ f

proof end;

theorem :: JORDAN5B:28

for f being non constant standard special_circular_sequence
for i, j being Element of NAT st 1 <= i & j <= len (GoB f) & i < j holds
(LSeg (((GoB f) * (1,(width (GoB f)))),((GoB f) * (i,(width (GoB f)))))) /\ (LSeg (((GoB f) * (j,(width (GoB f)))),((GoB f) * ((len (GoB f)),(width (GoB f)))))) = {}

proof end;

theorem :: JORDAN5B:29

for f being non constant standard special_circular_sequence
for i, j being Element of NAT st 1 <= i & j <= width (GoB f) & i < j holds
(LSeg (((GoB f) * ((len (GoB f)),1)),((GoB f) * ((len (GoB f)),i)))) /\ (LSeg (((GoB f) * ((len (GoB f)),j)),((GoB f) * ((len (GoB f)),(width (GoB f)))))) = {}

proof end;

theorem Th30: :: JORDAN5B:30

for f being FinSequence of (TOP-REAL 2) st f is being_S-Seq holds
L_Cut (f,(f /. 1)) = f

proof end;

theorem :: JORDAN5B:31

for f being FinSequence of (TOP-REAL 2) st f is being_S-Seq holds
R_Cut (f,(f /. (len f))) = f

proof end;

theorem :: JORDAN5B:32

for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in L~ f holds
p in LSeg ((f /. (Index (p,f))),(f /. ((Index (p,f)) + 1)))

proof end;

theorem :: JORDAN5B:33

for f being FinSequence of (TOP-REAL 2)
for i being Element of NAT st f is unfolded & f is s.n.c. & f is one-to-one & len f >= 2 & f /. 1 in LSeg (f,i) holds
i = 1

proof end;

theorem :: JORDAN5B:34

for f being non constant standard special_circular_sequence
for j being Element of NAT
for P being Subset of (TOP-REAL 2) st 1 <= j & j <= width (GoB f) & P = LSeg (((GoB f) * (1,j)),((GoB f) * ((len (GoB f)),j))) holds
P is_S-P_arc_joining (GoB f) * (1,j),(GoB f) * ((len (GoB f)),j)

proof end;

theorem :: JORDAN5B:35

for f being non constant standard special_circular_sequence
for j being Element of NAT
for P being Subset of (TOP-REAL 2) st 1 <= j & j <= len (GoB f) & P = LSeg (((GoB f) * (j,1)),((GoB f) * (j,(width (GoB f))))) holds
P is_S-P_arc_joining (GoB f) * (j,1),(GoB f) * (j,(width (GoB f)))

proof end;

theorem :: JORDAN5B:36

for f being FinSequence of (TOP-REAL 2)
for p, q being Point of (TOP-REAL 2) st p in L~ f & q in L~ f & ( Index (p,f) < Index (q,f) or ( Index (p,f) = Index (q,f) & LE p,q,f /. (Index (p,f)),f /. ((Index (p,f)) + 1) ) ) & p <> q holds
L~ (B_Cut (f,p,q)) c= L~ f by Lm4;