:: The Ordering of Points on a Curve, Part { I } :: by Adam Grabowski and Yatsuka Nakamura :: :: Received September 10, 1997 :: Copyright (c) 1997-2012 Association of Mizar Users begin theorem Th1: :: JORDAN5B:1 for i1 being Nat st 1 <= i1 holds i1 -' 1 < i1 proofend; theorem :: JORDAN5B:2 for i, k being Nat st i + 1 <= k holds 1 <= k -' i proofend; theorem :: JORDAN5B:3 for i, k being Nat st 1 <= i & 1 <= k holds (k -' i) + 1 <= k proofend; Lm1: for r being real number st 0 <= r & r <= 1 holds ( 0 <= 1 - r & 1 - r <= 1 ) proofend; 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] proofend; 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 proofend; 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 proofend; Lm2: for P being Point of I[01] holds P is Real proofend; 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) ) proofend; 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 ) proofend; 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 proofend; 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 proofend; 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 proofend; 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 proofend; 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 ) } proofend; 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 proofend; 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) proofend; 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 proofend; 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*> proofend; theorem :: JORDAN5B:18 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 proofend; theorem Th19: :: JORDAN5B:19 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)) proofend; theorem :: JORDAN5B:20 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)) proofend; theorem Th21: :: JORDAN5B:21 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*> proofend; 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)) proofend; theorem Th22: :: JORDAN5B:22 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)) proofend; theorem :: JORDAN5B:23 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)) proofend; 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 proofend; theorem :: JORDAN5B:24 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 proofend; theorem :: JORDAN5B:25 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)))))) = {} proofend; theorem :: JORDAN5B:26 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)))))) = {} proofend; theorem Th27: :: JORDAN5B:27 for f being FinSequence of (TOP-REAL 2) st f is being_S-Seq holds L_Cut (f,(f /. 1)) = f proofend; theorem :: JORDAN5B:28 for f being FinSequence of (TOP-REAL 2) st f is being_S-Seq holds R_Cut (f,(f /. (len f))) = f proofend; theorem :: JORDAN5B:29 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))) proofend; theorem :: JORDAN5B:30 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 proofend; theorem :: JORDAN5B:31 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) proofend; theorem :: JORDAN5B:32 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))) proofend; theorem :: JORDAN5B:33 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;