:: Behaviour of an Arc Crossing a Line :: by Yatsuka Nakamura :: :: Received January 26, 2004 :: Copyright (c) 2004-2012 Association of Mizar Users begin theorem Th1: :: JORDAN20:1 for P being Subset of (TOP-REAL 2) for p1, p2, p being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & p in P holds Segment (P,p1,p2,p,p) = {p} proofend; theorem Th2: :: JORDAN20:2 for p1, p2, p being Point of (TOP-REAL 2) for a being Real st p in LSeg (p1,p2) & p1 `1 <= a & p2 `1 <= a holds p `1 <= a proofend; theorem Th3: :: JORDAN20:3 for p1, p2, p being Point of (TOP-REAL 2) for a being Real st p in LSeg (p1,p2) & p1 `1 >= a & p2 `1 >= a holds p `1 >= a proofend; theorem :: JORDAN20:4 for p1, p2, p being Point of (TOP-REAL 2) for a being Real st p in LSeg (p1,p2) & p1 `1 < a & p2 `1 < a holds p `1 < a proofend; theorem :: JORDAN20:5 for p1, p2, p being Point of (TOP-REAL 2) for a being Real st p in LSeg (p1,p2) & p1 `1 > a & p2 `1 > a holds p `1 > a proofend; theorem Th6: :: JORDAN20:6 for j being Element of NAT for f being S-Sequence_in_R2 for p, q being Point of (TOP-REAL 2) st 1 <= j & j < len f & p in LSeg (f,j) & q in LSeg (f,j) & (f /. j) `2 = (f /. (j + 1)) `2 & (f /. j) `1 > (f /. (j + 1)) `1 & LE p,q, L~ f,f /. 1,f /. (len f) holds p `1 >= q `1 proofend; theorem Th7: :: JORDAN20:7 for j being Element of NAT for f being S-Sequence_in_R2 for p, q being Point of (TOP-REAL 2) st 1 <= j & j < len f & p in LSeg (f,j) & q in LSeg (f,j) & (f /. j) `2 = (f /. (j + 1)) `2 & (f /. j) `1 < (f /. (j + 1)) `1 & LE p,q, L~ f,f /. 1,f /. (len f) holds p `1 <= q `1 proofend; definition let P be Subset of (TOP-REAL 2); let p1, p2, p be Point of (TOP-REAL 2); let e be Real; predp is_Lin P,p1,p2,e means :Def1: :: JORDAN20:def 1 ( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p4 being Point of (TOP-REAL 2) st ( p4 `1 < e & LE p4,p,P,p1,p2 & ( for p5 being Point of (TOP-REAL 2) st LE p4,p5,P,p1,p2 & LE p5,p,P,p1,p2 holds p5 `1 <= e ) ) ); predp is_Rin P,p1,p2,e means :Def2: :: JORDAN20:def 2 ( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p4 being Point of (TOP-REAL 2) st ( p4 `1 > e & LE p4,p,P,p1,p2 & ( for p5 being Point of (TOP-REAL 2) st LE p4,p5,P,p1,p2 & LE p5,p,P,p1,p2 holds p5 `1 >= e ) ) ); predp is_Lout P,p1,p2,e means :Def3: :: JORDAN20:def 3 ( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p4 being Point of (TOP-REAL 2) st ( p4 `1 < e & LE p,p4,P,p1,p2 & ( for p5 being Point of (TOP-REAL 2) st LE p5,p4,P,p1,p2 & LE p,p5,P,p1,p2 holds p5 `1 <= e ) ) ); predp is_Rout P,p1,p2,e means :Def4: :: JORDAN20:def 4 ( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p4 being Point of (TOP-REAL 2) st ( p4 `1 > e & LE p,p4,P,p1,p2 & ( for p5 being Point of (TOP-REAL 2) st LE p5,p4,P,p1,p2 & LE p,p5,P,p1,p2 holds p5 `1 >= e ) ) ); predp is_OSin P,p1,p2,e means :Def5: :: JORDAN20:def 5 ( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p7 being Point of (TOP-REAL 2) st ( LE p7,p,P,p1,p2 & ( for p8 being Point of (TOP-REAL 2) st LE p7,p8,P,p1,p2 & LE p8,p,P,p1,p2 holds p8 `1 = e ) & ( for p4 being Point of (TOP-REAL 2) st LE p4,p7,P,p1,p2 & p4 <> p7 holds ( ex p5 being Point of (TOP-REAL 2) st ( LE p4,p5,P,p1,p2 & LE p5,p7,P,p1,p2 & p5 `1 > e ) & ex p6 being Point of (TOP-REAL 2) st ( LE p4,p6,P,p1,p2 & LE p6,p7,P,p1,p2 & p6 `1 < e ) ) ) ) ); predp is_OSout P,p1,p2,e means :Def6: :: JORDAN20:def 6 ( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p7 being Point of (TOP-REAL 2) st ( LE p,p7,P,p1,p2 & ( for p8 being Point of (TOP-REAL 2) st LE p8,p7,P,p1,p2 & LE p,p8,P,p1,p2 holds p8 `1 = e ) & ( for p4 being Point of (TOP-REAL 2) st LE p7,p4,P,p1,p2 & p4 <> p7 holds ( ex p5 being Point of (TOP-REAL 2) st ( LE p5,p4,P,p1,p2 & LE p7,p5,P,p1,p2 & p5 `1 > e ) & ex p6 being Point of (TOP-REAL 2) st ( LE p6,p4,P,p1,p2 & LE p7,p6,P,p1,p2 & p6 `1 < e ) ) ) ) ); correctness ; end; :: deftheorem Def1 defines is_Lin JORDAN20:def_1_:_ for P being Subset of (TOP-REAL 2) for p1, p2, p being Point of (TOP-REAL 2) for e being Real holds ( p is_Lin P,p1,p2,e iff ( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p4 being Point of (TOP-REAL 2) st ( p4 `1 < e & LE p4,p,P,p1,p2 & ( for p5 being Point of (TOP-REAL 2) st LE p4,p5,P,p1,p2 & LE p5,p,P,p1,p2 holds p5 `1 <= e ) ) ) ); :: deftheorem Def2 defines is_Rin JORDAN20:def_2_:_ for P being Subset of (TOP-REAL 2) for p1, p2, p being Point of (TOP-REAL 2) for e being Real holds ( p is_Rin P,p1,p2,e iff ( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p4 being Point of (TOP-REAL 2) st ( p4 `1 > e & LE p4,p,P,p1,p2 & ( for p5 being Point of (TOP-REAL 2) st LE p4,p5,P,p1,p2 & LE p5,p,P,p1,p2 holds p5 `1 >= e ) ) ) ); :: deftheorem Def3 defines is_Lout JORDAN20:def_3_:_ for P being Subset of (TOP-REAL 2) for p1, p2, p being Point of (TOP-REAL 2) for e being Real holds ( p is_Lout P,p1,p2,e iff ( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p4 being Point of (TOP-REAL 2) st ( p4 `1 < e & LE p,p4,P,p1,p2 & ( for p5 being Point of (TOP-REAL 2) st LE p5,p4,P,p1,p2 & LE p,p5,P,p1,p2 holds p5 `1 <= e ) ) ) ); :: deftheorem Def4 defines is_Rout JORDAN20:def_4_:_ for P being Subset of (TOP-REAL 2) for p1, p2, p being Point of (TOP-REAL 2) for e being Real holds ( p is_Rout P,p1,p2,e iff ( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p4 being Point of (TOP-REAL 2) st ( p4 `1 > e & LE p,p4,P,p1,p2 & ( for p5 being Point of (TOP-REAL 2) st LE p5,p4,P,p1,p2 & LE p,p5,P,p1,p2 holds p5 `1 >= e ) ) ) ); :: deftheorem Def5 defines is_OSin JORDAN20:def_5_:_ for P being Subset of (TOP-REAL 2) for p1, p2, p being Point of (TOP-REAL 2) for e being Real holds ( p is_OSin P,p1,p2,e iff ( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p7 being Point of (TOP-REAL 2) st ( LE p7,p,P,p1,p2 & ( for p8 being Point of (TOP-REAL 2) st LE p7,p8,P,p1,p2 & LE p8,p,P,p1,p2 holds p8 `1 = e ) & ( for p4 being Point of (TOP-REAL 2) st LE p4,p7,P,p1,p2 & p4 <> p7 holds ( ex p5 being Point of (TOP-REAL 2) st ( LE p4,p5,P,p1,p2 & LE p5,p7,P,p1,p2 & p5 `1 > e ) & ex p6 being Point of (TOP-REAL 2) st ( LE p4,p6,P,p1,p2 & LE p6,p7,P,p1,p2 & p6 `1 < e ) ) ) ) ) ); :: deftheorem Def6 defines is_OSout JORDAN20:def_6_:_ for P being Subset of (TOP-REAL 2) for p1, p2, p being Point of (TOP-REAL 2) for e being Real holds ( p is_OSout P,p1,p2,e iff ( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p7 being Point of (TOP-REAL 2) st ( LE p,p7,P,p1,p2 & ( for p8 being Point of (TOP-REAL 2) st LE p8,p7,P,p1,p2 & LE p,p8,P,p1,p2 holds p8 `1 = e ) & ( for p4 being Point of (TOP-REAL 2) st LE p7,p4,P,p1,p2 & p4 <> p7 holds ( ex p5 being Point of (TOP-REAL 2) st ( LE p5,p4,P,p1,p2 & LE p7,p5,P,p1,p2 & p5 `1 > e ) & ex p6 being Point of (TOP-REAL 2) st ( LE p6,p4,P,p1,p2 & LE p7,p6,P,p1,p2 & p6 `1 < e ) ) ) ) ) ); theorem :: JORDAN20:8 for P being Subset of (TOP-REAL 2) for p1, p2, p being Point of (TOP-REAL 2) for e being Real st P is_an_arc_of p1,p2 & p1 `1 <= e & p2 `1 >= e holds ex p3 being Point of (TOP-REAL 2) st ( p3 in P & p3 `1 = e ) proofend; theorem :: JORDAN20:9 for P being non empty Subset of (TOP-REAL 2) for p1, p2, p being Point of (TOP-REAL 2) for e being Real st P is_an_arc_of p1,p2 & p1 `1 < e & p in P & p `1 = e & not p is_Lin P,p1,p2,e & not p is_Rin P,p1,p2,e holds p is_OSin P,p1,p2,e proofend; theorem :: JORDAN20:10 for P being non empty Subset of (TOP-REAL 2) for p1, p2, p being Point of (TOP-REAL 2) for e being Real st P is_an_arc_of p1,p2 & p2 `1 > e & p in P & p `1 = e & not p is_Lout P,p1,p2,e & not p is_Rout P,p1,p2,e holds p is_OSout P,p1,p2,e proofend; theorem Th11: :: JORDAN20:11 for P being Subset of I[01] for s being Real st P = [.0,s.[ holds P is open proofend; theorem Th12: :: JORDAN20:12 for P being Subset of I[01] for s being Real st P = ].s,1.] holds P is open proofend; theorem Th13: :: JORDAN20:13 for P being non empty Subset of (TOP-REAL 2) for P1 being Subset of ((TOP-REAL 2) | P) for Q being Subset of I[01] for f being Function of I[01],((TOP-REAL 2) | P) for s being Real st s <= 1 & P1 = { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st ( 0 <= ss & ss < s & q0 = f . ss ) } & Q = [.0,s.[ holds f .: Q = P1 proofend; theorem Th14: :: JORDAN20:14 for P being non empty Subset of (TOP-REAL 2) for P1 being Subset of ((TOP-REAL 2) | P) for Q being Subset of I[01] for f being Function of I[01],((TOP-REAL 2) | P) for s being Real st s >= 0 & P1 = { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st ( s < ss & ss <= 1 & q0 = f . ss ) } & Q = ].s,1.] holds f .: Q = P1 proofend; Lm1: [#] I[01] <> {} ; theorem Th15: :: JORDAN20:15 for P being non empty Subset of (TOP-REAL 2) for P1 being Subset of ((TOP-REAL 2) | P) for f being Function of I[01],((TOP-REAL 2) | P) for s being Real st s <= 1 & f is being_homeomorphism & P1 = { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st ( 0 <= ss & ss < s & q0 = f . ss ) } holds P1 is open proofend; theorem Th16: :: JORDAN20:16 for P being non empty Subset of (TOP-REAL 2) for P1 being Subset of ((TOP-REAL 2) | P) for f being Function of I[01],((TOP-REAL 2) | P) for s being Real st s >= 0 & f is being_homeomorphism & P1 = { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st ( s < ss & ss <= 1 & q0 = f . ss ) } holds P1 is open proofend; theorem Th17: :: JORDAN20:17 for T being non empty TopStruct for Q1, Q2 being Subset of T for p1, p2 being Point of T st Q1 /\ Q2 = {} & Q1 \/ Q2 = the carrier of T & p1 in Q1 & p2 in Q2 & Q1 is open & Q2 is open holds for P being Function of I[01],T holds ( not P is continuous or not P . 0 = p1 or not P . 1 = p2 ) proofend; theorem Th18: :: JORDAN20:18 for P being non empty Subset of (TOP-REAL 2) for Q being Subset of ((TOP-REAL 2) | P) for p1, p2, q being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & q in P & q <> p1 & q <> p2 & Q = P \ {q} holds ( not Q is connected & ( for R being Function of I[01],(((TOP-REAL 2) | P) | Q) holds ( not R is continuous or not R . 0 = p1 or not R . 1 = p2 ) ) ) proofend; theorem Th19: :: JORDAN20:19 for P being non empty Subset of (TOP-REAL 2) for p1, p2, q1, q2 being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & q1 in P & q2 in P & not LE q1,q2,P,p1,p2 holds LE q2,q1,P,p1,p2 proofend; theorem Th20: :: JORDAN20:20 for n being Element of NAT for p1, p2 being Point of (TOP-REAL n) for P, P1 being non empty Subset of (TOP-REAL n) st P is_an_arc_of p1,p2 & P1 is_an_arc_of p1,p2 & P1 c= P holds P1 = P proofend; theorem Th21: :: JORDAN20:21 for P being non empty Subset of (TOP-REAL 2) for p1, p2, q1 being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & q1 in P & p2 <> q1 holds Segment (P,p1,p2,q1,p2) is_an_arc_of q1,p2 proofend; theorem Th22: :: JORDAN20:22 for P being non empty Subset of (TOP-REAL 2) for p1, p2, q1, q2, q3 being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & LE q1,q2,P,p1,p2 & LE q2,q3,P,p1,p2 holds (Segment (P,p1,p2,q1,q2)) \/ (Segment (P,p1,p2,q2,q3)) = Segment (P,p1,p2,q1,q3) proofend; theorem :: JORDAN20:23 for P being non empty Subset of (TOP-REAL 2) for p1, p2, q1, q2, q3 being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & LE q1,q2,P,p1,p2 & LE q2,q3,P,p1,p2 holds (Segment (P,p1,p2,q1,q2)) /\ (Segment (P,p1,p2,q2,q3)) = {q2} proofend; theorem Th24: :: JORDAN20:24 for P being non empty Subset of (TOP-REAL 2) for p1, p2 being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 holds Segment (P,p1,p2,p1,p2) = P proofend; theorem Th25: :: JORDAN20:25 for P, Q1 being non empty Subset of (TOP-REAL 2) for p1, p2, q1, q2 being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & Q1 is_an_arc_of q1,q2 & LE q1,q2,P,p1,p2 & Q1 c= P holds Q1 = Segment (P,p1,p2,q1,q2) proofend; theorem :: JORDAN20:26 for P being non empty Subset of (TOP-REAL 2) for p1, p2, q1, q2, p being Point of (TOP-REAL 2) for e being Real st q1 is_Lin P,p1,p2,e & q2 `1 = e & LSeg (q1,q2) c= P & p in LSeg (q1,q2) holds p is_Lin P,p1,p2,e proofend; theorem :: JORDAN20:27 for P being non empty Subset of (TOP-REAL 2) for p1, p2, q1, q2, p being Point of (TOP-REAL 2) for e being Real st q1 is_Rin P,p1,p2,e & q2 `1 = e & LSeg (q1,q2) c= P & p in LSeg (q1,q2) holds p is_Rin P,p1,p2,e proofend; theorem :: JORDAN20:28 for P being non empty Subset of (TOP-REAL 2) for p1, p2, q1, q2, p being Point of (TOP-REAL 2) for e being Real st q1 is_Lout P,p1,p2,e & q2 `1 = e & LSeg (q1,q2) c= P & p in LSeg (q1,q2) holds p is_Lout P,p1,p2,e proofend; theorem :: JORDAN20:29 for P being non empty Subset of (TOP-REAL 2) for p1, p2, q1, q2, p being Point of (TOP-REAL 2) for e being Real st q1 is_Rout P,p1,p2,e & q2 `1 = e & LSeg (q1,q2) c= P & p in LSeg (q1,q2) holds p is_Rout P,p1,p2,e proofend; theorem :: JORDAN20:30 for P being non empty Subset of (TOP-REAL 2) for p1, p2, p being Point of (TOP-REAL 2) for e being Real st P is_S-P_arc_joining p1,p2 & p1 `1 < e & p in P & p `1 = e & not p is_Lin P,p1,p2,e holds p is_Rin P,p1,p2,e proofend; theorem :: JORDAN20:31 for P being non empty Subset of (TOP-REAL 2) for p1, p2, p being Point of (TOP-REAL 2) for e being Real st P is_S-P_arc_joining p1,p2 & p2 `1 > e & p in P & p `1 = e & not p is_Lout P,p1,p2,e holds p is_Rout P,p1,p2,e proofend;