:: The {F}ashoda Meet Theorem for Continuous Mappings
:: by Yatsuka Nakamura , Andrzej Trybulec and Artur Korni{\l}owicz
::
:: Received September 14, 2005
:: Copyright (c) 2005-2021 Association of Mizar Users

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) )

proof end;

Lm3: for f being FinSequence of REAL st ( for k being Nat st 1 <= k & k < len f holds
f /. k < f /. (k + 1) ) holds
f is increasing

proof end;

registration

let a, b, c, d be Real;

cluster closed_inside_of_rectangle (a,b,c,d) -> convex ;

coherence
closed_inside_of_rectangle (a,b,c,d) is convex

proof end;

end;

registration

let a, b, c, d be Real;

cluster Trectangle (a,b,c,d) -> convex ;

coherence
Trectangle (a,b,c,d) is convex by PRE_TOPC:8;

end;

theorem Th1: :: JGRAPH_8:1

for n being Nat
for e being positive Real
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 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 ) )

proof end;

theorem Th2: :: JGRAPH_8:2

for n being 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)

proof end;

theorem Th3: :: JGRAPH_8:3

for n being 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)

proof end;

:: 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

proof end;

:: Fashoda Meet Theorem

theorem Th5: :: JGRAPH_8:5

for a, b, c, d being Real
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

proof end;

theorem :: JGRAPH_8:6

for a, b, c, d being Real
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

proof end;