:: On a Dividing Function of the Simple Closed Curve into Segments
:: by Yatsuka Nakamura
::
:: Copyright (c) 1998-2021 Association of Mizar Users

Lm1: 2 -' 1 = 2 - 1 by XREAL_1:233
.= 1 ;

Lm2: for i, j, k being Nat st i -' k <= j holds
i <= j + k

proof end;

Lm3: for i, j, k being Nat st j + k <= i holds
k <= i -' j

proof end;

theorem Th1: :: JORDAN7:1
for P being non empty compact Subset of () st P is being_simple_closed_curve holds
( W-min P in Lower_Arc P & E-max P in Lower_Arc P & W-min P in Upper_Arc P & E-max P in Upper_Arc P )
proof end;

theorem Th2: :: JORDAN7:2
for P being non empty compact Subset of ()
for q being Point of () st P is being_simple_closed_curve & LE q, W-min P,P holds
q = W-min P
proof end;

theorem Th3: :: JORDAN7:3
for P being non empty compact Subset of ()
for q being Point of () st P is being_simple_closed_curve & q in P holds
LE W-min P,q,P
proof end;

definition
let P be non empty compact Subset of ();
let q1, q2 be Point of ();
func Segment (q1,q2,P) -> Subset of () equals :Def1: :: JORDAN7:def 1
{ p where p is Point of () : ( LE q1,p,P & LE p,q2,P ) } if q2 <> W-min P
otherwise { p1 where p1 is Point of () : ( LE q1,p1,P or ( q1 in P & p1 = W-min P ) ) } ;
correctness
coherence
( ( q2 <> W-min P implies { p where p is Point of () : ( LE q1,p,P & LE p,q2,P ) } is Subset of () ) & ( not q2 <> W-min P implies { p1 where p1 is Point of () : ( LE q1,p1,P or ( q1 in P & p1 = W-min P ) ) } is Subset of () ) )
;
consistency
for b1 being Subset of () holds verum
;
proof end;
end;

:: deftheorem Def1 defines Segment JORDAN7:def 1 :
for P being non empty compact Subset of ()
for q1, q2 being Point of () holds
( ( q2 <> W-min P implies Segment (q1,q2,P) = { p where p is Point of () : ( LE q1,p,P & LE p,q2,P ) } ) & ( not q2 <> W-min P implies Segment (q1,q2,P) = { p1 where p1 is Point of () : ( LE q1,p1,P or ( q1 in P & p1 = W-min P ) ) } ) );

theorem :: JORDAN7:4
for P being non empty compact Subset of () st P is being_simple_closed_curve holds
( Segment ((),(),P) = Upper_Arc P & Segment ((),(),P) = Lower_Arc P )
proof end;

theorem Th5: :: JORDAN7:5
for P being non empty compact Subset of ()
for q1, q2 being Point of () st P is being_simple_closed_curve & LE q1,q2,P holds
( q1 in P & q2 in P )
proof end;

theorem Th6: :: JORDAN7:6
for P being non empty compact Subset of ()
for q1, q2 being Point of () st P is being_simple_closed_curve & LE q1,q2,P holds
( q1 in Segment (q1,q2,P) & q2 in Segment (q1,q2,P) )
proof end;

theorem Th7: :: JORDAN7:7
for P being non empty compact Subset of ()
for q1 being Point of () st q1 in P & P is being_simple_closed_curve holds
q1 in Segment (q1,(),P)
proof end;

theorem :: JORDAN7:8
for P being non empty compact Subset of ()
for q being Point of () st P is being_simple_closed_curve & q in P & q <> W-min P holds
Segment (q,q,P) = {q}
proof end;

theorem :: JORDAN7:9
for P being non empty compact Subset of ()
for q1, q2 being Point of () st P is being_simple_closed_curve & q1 <> W-min P & q2 <> W-min P holds
not W-min P in Segment (q1,q2,P)
proof end;

theorem Th10: :: JORDAN7:10
for P being non empty compact Subset of ()
for q1, q2, q3 being Point of () st P is being_simple_closed_curve & LE q1,q2,P & LE q2,q3,P & ( not q1 = q2 or not q1 = W-min P ) & ( not q2 = q3 or not q2 = W-min P ) holds
(Segment (q1,q2,P)) /\ (Segment (q2,q3,P)) = {q2}
proof end;

theorem Th11: :: JORDAN7:11
for P being non empty compact Subset of ()
for q1, q2 being Point of () st P is being_simple_closed_curve & LE q1,q2,P & q1 <> W-min P & q2 <> W-min P holds
(Segment (q1,q2,P)) /\ (Segment (q2,(),P)) = {q2}
proof end;

theorem Th12: :: JORDAN7:12
for P being non empty compact Subset of ()
for q1, q2 being Point of () st P is being_simple_closed_curve & LE q1,q2,P & q1 <> q2 & q1 <> W-min P holds
(Segment (q2,(),P)) /\ (Segment ((),q1,P)) = {()}
proof end;

theorem Th13: :: JORDAN7:13
for P being non empty compact Subset of ()
for q1, q2, q3, q4 being Point of () st P is being_simple_closed_curve & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P & q1 <> q2 & q2 <> q3 holds
Segment (q1,q2,P) misses Segment (q3,q4,P)
proof end;

theorem Th14: :: JORDAN7:14
for P being non empty compact Subset of ()
for q1, q2, q3 being Point of () st P is being_simple_closed_curve & LE q1,q2,P & LE q2,q3,P & q1 <> W-min P & q2 <> q3 holds
Segment (q1,q2,P) misses Segment (q3,(),P)
proof end;

theorem Th15: :: JORDAN7:15
for n being Nat
for P being non empty Subset of ()
for f being Function of I[01],(() | P) st f is being_homeomorphism holds
ex g being Function of I[01],() st
( f = g & g is continuous & g is one-to-one )
proof end;

theorem Th16: :: JORDAN7:16
for n being Nat
for P being non empty Subset of ()
for g being Function of I[01],() st g is continuous & g is one-to-one & rng g = P holds
ex f being Function of I[01],(() | P) st
( f = g & f is being_homeomorphism )
proof end;

Lm4: now :: thesis: for h2 being Nat holds (h2 - 1) - 1 < h2
let h2 be Nat; :: thesis: (h2 - 1) - 1 < h2
h2 < h2 + 1 by NAT_1:13;
then A1: h2 - 1 < (h2 + 1) - 1 by XREAL_1:9;
then (h2 - 1) - 1 < h2 - 1 by XREAL_1:9;
hence (h2 - 1) - 1 < h2 by ; :: thesis: verum
end;

Lm5: 0 in [.0,1.]
by XXREAL_1:1;

Lm6: 1 in [.0,1.]
by XXREAL_1:1;

theorem Th17: :: JORDAN7:17
for A being Subset of ()
for p1, p2 being Point of () st A is_an_arc_of p1,p2 holds
ex g being Function of I[01],() st
( g is continuous & g is one-to-one & rng g = A & g . 0 = p1 & g . 1 = p2 )
proof end;

theorem Th18: :: JORDAN7:18
for P being non empty Subset of ()
for p1, p2, q1, q2 being Point of ()
for g being Function of I[01],()
for s1, s2 being Real st P is_an_arc_of p1,p2 & g is continuous & g is one-to-one & rng g = P & g . 0 = p1 & g . 1 = p2 & g . s1 = q1 & 0 <= s1 & s1 <= 1 & g . s2 = q2 & 0 <= s2 & s2 <= 1 & s1 <= s2 holds
LE q1,q2,P,p1,p2
proof end;

theorem Th19: :: JORDAN7:19
for P being non empty Subset of ()
for p1, p2, q1, q2 being Point of ()
for g being Function of I[01],()
for s1, s2 being Real st g is continuous & g is one-to-one & rng g = P & g . 0 = p1 & g . 1 = p2 & g . s1 = q1 & 0 <= s1 & s1 <= 1 & g . s2 = q2 & 0 <= s2 & s2 <= 1 & LE q1,q2,P,p1,p2 holds
s1 <= s2
proof end;

theorem :: JORDAN7:20
for P being non empty compact Subset of ()
for e being Real st P is being_simple_closed_curve & e > 0 holds
ex h being FinSequence of the carrier of () st
( h . 1 = W-min P & h is one-to-one & 8 <= len h & rng h c= P & ( for i being Nat st 1 <= i & i < len h holds
LE h /. i,h /. (i + 1),P ) & ( for i being Nat
for W being Subset of () st 1 <= i & i < len h & W = Segment ((h /. i),(h /. (i + 1)),P) holds
diameter W < e ) & ( for W being Subset of () st W = Segment ((h /. (len h)),(h /. 1),P) holds
diameter W < e ) & ( for i being Nat st 1 <= i & i + 1 < len h holds
(Segment ((h /. i),(h /. (i + 1)),P)) /\ (Segment ((h /. (i + 1)),(h /. (i + 2)),P)) = {(h /. (i + 1))} ) & (Segment ((h /. (len h)),(h /. 1),P)) /\ (Segment ((h /. 1),(h /. 2),P)) = {(h /. 1)} & (Segment ((h /. ((len h) -' 1)),(h /. (len h)),P)) /\ (Segment ((h /. (len h)),(h /. 1),P)) = {(h /. (len h))} & Segment ((h /. ((len h) -' 1)),(h /. (len h)),P) misses Segment ((h /. 1),(h /. 2),P) & ( for i, j being Nat st 1 <= i & i < j & j < len h & i,j aren't_adjacent holds
Segment ((h /. i),(h /. (i + 1)),P) misses Segment ((h /. j),(h /. (j + 1)),P) ) & ( for i being Nat st 1 < i & i + 1 < len h holds
Segment ((h /. (len h)),(h /. 1),P) misses Segment ((h /. i),(h /. (i + 1)),P) ) )
proof end;