:: On the Decomposition of a Simple Closed Curve into Two Arcs :: by Andrzej Trybulec and Yatsuka Nakamura :: :: Received September 16, 2002 :: Copyright (c) 2002-2012 Association of Mizar Users begin theorem :: JORDAN16:1 for C being Simple_closed_curve holds Lower_Arc C <> Upper_Arc C proofend; theorem Th2: :: JORDAN16:2 for A being Subset of (TOP-REAL 2) for p1, p2, q1, q2 being Point of (TOP-REAL 2) holds Segment (A,p1,p2,q1,q2) c= A proofend; theorem Th3: :: JORDAN16:3 for A being Subset of (TOP-REAL 2) for q, p1, p2 being Point of (TOP-REAL 2) st q in A holds q in L_Segment (A,p1,p2,q) proofend; theorem Th4: :: JORDAN16:4 for A being Subset of (TOP-REAL 2) for q, p1, p2 being Point of (TOP-REAL 2) st q in A holds q in R_Segment (A,p1,p2,q) proofend; theorem Th5: :: JORDAN16:5 for A being Subset of (TOP-REAL 2) for q1, q2, p1, p2 being Point of (TOP-REAL 2) st LE q1,q2,A,p1,p2 holds ( q1 in Segment (A,p1,p2,q1,q2) & q2 in Segment (A,p1,p2,q1,q2) ) proofend; theorem :: JORDAN16:6 for C being Simple_closed_curve for p, q being Point of (TOP-REAL 2) holds Segment (p,q,C) c= C proofend; theorem :: JORDAN16:7 for C being Simple_closed_curve for p, q being Point of (TOP-REAL 2) st p in C & q in C & not LE p,q,C holds LE q,p,C proofend; theorem Th8: :: JORDAN16:8 for X, Y being non empty TopSpace for Y0 being non empty SubSpace of Y for f being Function of X,Y for g being Function of X,Y0 st f = g & f is continuous holds g is continuous proofend; theorem Th9: :: JORDAN16:9 for S, T being non empty TopSpace for S0 being non empty SubSpace of S for T0 being non empty SubSpace of T for f being Function of S,T st f is being_homeomorphism holds for g being Function of S0,T0 st g = f | S0 & g is onto holds g is being_homeomorphism proofend; theorem Th10: :: JORDAN16:10 for P1, P2, P3 being Subset of (TOP-REAL 2) for p1, p2 being Point of (TOP-REAL 2) st P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P3 is_an_arc_of p1,p2 & P2 /\ P3 = {p1,p2} & P1 c= P2 \/ P3 & not P1 = P2 holds P1 = P3 proofend; theorem Th11: :: JORDAN16:11 for C being Simple_closed_curve for A1, A2 being Subset of (TOP-REAL 2) for p1, p2 being Point of (TOP-REAL 2) st A1 is_an_arc_of p1,p2 & A2 is_an_arc_of p1,p2 & A1 c= C & A2 c= C & A1 <> A2 holds ( A1 \/ A2 = C & A1 /\ A2 = {p1,p2} ) proofend; theorem Th12: :: JORDAN16:12 for A1, A2 being Subset of (TOP-REAL 2) for p1, p2, q1, q2 being Point of (TOP-REAL 2) st A1 is_an_arc_of p1,p2 & A1 /\ A2 = {q1,q2} holds A1 <> A2 proofend; theorem :: JORDAN16:13 for C being Simple_closed_curve for A1, A2 being Subset of (TOP-REAL 2) for p1, p2 being Point of (TOP-REAL 2) st A1 is_an_arc_of p1,p2 & A2 is_an_arc_of p1,p2 & A1 c= C & A2 c= C & A1 /\ A2 = {p1,p2} holds A1 \/ A2 = C proofend; theorem :: JORDAN16:14 for C being Simple_closed_curve for A1, A2 being Subset of (TOP-REAL 2) for p1, p2 being Point of (TOP-REAL 2) st A1 c= C & A2 c= C & A1 <> A2 & A1 is_an_arc_of p1,p2 & A2 is_an_arc_of p1,p2 holds for A being Subset of (TOP-REAL 2) st A is_an_arc_of p1,p2 & A c= C & not A = A1 holds A = A2 proofend; theorem Th15: :: JORDAN16:15 for C being Simple_closed_curve for A being non empty Subset of (TOP-REAL 2) st A is_an_arc_of W-min C, E-max C & A c= C & not A = Lower_Arc C holds A = Upper_Arc C proofend; theorem Th16: :: JORDAN16:16 for A being Subset of (TOP-REAL 2) for p1, p2, q1, q2 being Point of (TOP-REAL 2) st A is_an_arc_of p1,p2 & LE q1,q2,A,p1,p2 holds ex g being Function of I[01],((TOP-REAL 2) | A) ex s1, s2 being Real st ( g is being_homeomorphism & g . 0 = p1 & g . 1 = p2 & g . s1 = q1 & g . s2 = q2 & 0 <= s1 & s1 <= s2 & s2 <= 1 ) proofend; theorem Th17: :: JORDAN16:17 for A being Subset of (TOP-REAL 2) for p1, p2, q1, q2 being Point of (TOP-REAL 2) st A is_an_arc_of p1,p2 & LE q1,q2,A,p1,p2 & q1 <> q2 holds ex g being Function of I[01],((TOP-REAL 2) | A) ex s1, s2 being Real st ( g is being_homeomorphism & g . 0 = p1 & g . 1 = p2 & g . s1 = q1 & g . s2 = q2 & 0 <= s1 & s1 < s2 & s2 <= 1 ) proofend; theorem :: JORDAN16:18 for A being Subset of (TOP-REAL 2) for q1, q2, p1, p2 being Point of (TOP-REAL 2) st LE q1,q2,A,p1,p2 holds not Segment (A,p1,p2,q1,q2) is empty proofend; theorem :: JORDAN16:19 for C being Simple_closed_curve for p being Point of (TOP-REAL 2) st p in C holds ( p in Segment (p,(W-min C),C) & W-min C in Segment (p,(W-min C),C) ) proofend; theorem Th20: :: JORDAN16:20 for f being Function of R^1,R^1 for a, b being Real st a <> 0 & f = AffineMap (a,b) holds f is being_homeomorphism proofend; theorem Th21: :: JORDAN16:21 for A being Subset of (TOP-REAL 2) for p1, p2, q1, q2 being Point of (TOP-REAL 2) st A is_an_arc_of p1,p2 & LE q1,q2,A,p1,p2 & q1 <> q2 holds Segment (A,p1,p2,q1,q2) is_an_arc_of q1,q2 proofend; theorem :: JORDAN16:22 for C being Simple_closed_curve for p1, p2 being Point of (TOP-REAL 2) for P being Subset of (TOP-REAL 2) st P c= C & P is_an_arc_of p1,p2 & W-min C in P & E-max C in P & not Upper_Arc C c= P holds Lower_Arc C c= P proofend; :: Moved from JORDAN18, AK, 23.02.2006 theorem :: JORDAN16:23 for P being 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 & q1 <> p1 & q1 <> p2 & q2 <> p1 & q2 <> p2 & q1 <> q2 holds ex Q being non empty Subset of (TOP-REAL 2) st ( Q is_an_arc_of q1,q2 & Q c= P & Q misses {p1,p2} ) proofend; :: moved from JORDAN20, AG 1.04.2006 theorem :: JORDAN16:24 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 & p1 <> q1 holds Segment (P,p1,p2,p1,q1) is_an_arc_of p1,q1 proofend;