:: The {F}ashoda Meet Theorem for Continuous Mappings :: by Yatsuka Nakamura , Andrzej Trybulec and Artur Korni{\l}owicz :: :: Received September 14, 2005 :: Copyright (c) 2005-2012 Association of Mizar Users begin Lm1: I[01] = TopSpaceMetr (Closed-Interval-MSpace (0,1)) by TOPMETR:20, TOPMETR:def_7; Lm2: for x being set for f being FinSequence st 1 <= len f holds ( (f ^ <*x*>) . 1 = f . 1 & (<*x*> ^ f) . ((len f) + 1) = f . (len f) ) proofend; Lm3: for f being FinSequence of REAL st ( for k being Element of NAT st 1 <= k & k < len f holds f /. k < f /. (k + 1) ) holds f is increasing proofend; registration let a, b, c, d be real number ; cluster closed_inside_of_rectangle (a,b,c,d) -> convex ; coherence closed_inside_of_rectangle (a,b,c,d) is convex proofend; end; registration let a, b, c, d be real number ; cluster Trectangle (a,b,c,d) -> convex ; coherence Trectangle (a,b,c,d) is convex proofend; end; theorem Th1: :: JGRAPH_8:1 for n being Element of NAT for e being real positive number for g being continuous Function of I[01],(TOP-REAL n) ex h being FinSequence of REAL st ( h . 1 = 0 & h . (len h) = 1 & 5 <= len h & rng h c= the carrier of I[01] & h is increasing & ( for i being Element of NAT for Q being Subset of I[01] for W being Subset of (Euclid n) st 1 <= i & i < len h & Q = [.(h /. i),(h /. (i + 1)).] & W = g .: Q holds diameter W < e ) ) proofend; theorem Th2: :: JGRAPH_8:2 for n being Element of NAT for p1, p2 being Point of (TOP-REAL n) for P being Subset of (TOP-REAL n) st P c= LSeg (p1,p2) & p1 in P & p2 in P & P is connected holds P = LSeg (p1,p2) proofend; theorem Th3: :: JGRAPH_8:3 for n being Element of NAT for p1, p2 being Point of (TOP-REAL n) for g being Path of p1,p2 st rng g c= LSeg (p1,p2) holds rng g = LSeg (p1,p2) proofend; :: Goboard Theorem in continuous case theorem Th4: :: JGRAPH_8:4 for P, Q being non empty Subset of (TOP-REAL 2) for p1, p2, q1, q2 being Point of (TOP-REAL 2) for f being Path of p1,p2 for g being Path of q1,q2 st rng f = P & rng g = Q & ( for p being Point of (TOP-REAL 2) st p in P holds ( p1 `1 <= p `1 & p `1 <= p2 `1 ) ) & ( for p being Point of (TOP-REAL 2) st p in Q holds ( p1 `1 <= p `1 & p `1 <= p2 `1 ) ) & ( for p being Point of (TOP-REAL 2) st p in P holds ( q1 `2 <= p `2 & p `2 <= q2 `2 ) ) & ( for p being Point of (TOP-REAL 2) st p in Q holds ( q1 `2 <= p `2 & p `2 <= q2 `2 ) ) holds P meets Q proofend; :: Fashoda Meet Theorem theorem Th5: :: JGRAPH_8:5 for a, b, c, d being real number for f, g being continuous Function of I[01],(TOP-REAL 2) for O, I being Point of I[01] st O = 0 & I = 1 & (f . O) `1 = a & (f . I) `1 = b & (g . O) `2 = c & (g . I) `2 = d & ( for r being Point of I[01] holds ( a <= (f . r) `1 & (f . r) `1 <= b & a <= (g . r) `1 & (g . r) `1 <= b & c <= (f . r) `2 & (f . r) `2 <= d & c <= (g . r) `2 & (g . r) `2 <= d ) ) holds rng f meets rng g proofend; theorem :: JGRAPH_8:6 for a, b, c, d being real number for ar, br, cr, dr being Point of (Trectangle (a,b,c,d)) for h being Path of ar,br for v being Path of dr,cr for Ar, Br, Cr, Dr being Point of (TOP-REAL 2) st Ar `1 = a & Br `1 = b & Cr `2 = c & Dr `2 = d & ar = Ar & br = Br & cr = Cr & dr = Dr holds ex s, t being Point of I[01] st h . s = v . t proofend;