:: The Topological Space ${\calE}^2_{\rm T}$. Arcs, Line Segments and Special Polygonal Arcs
:: by Agata Darmochwa{\l} and Yatsuka Nakamura
::
:: Received November 21, 1991
:: Copyright (c) 1991-2021 Association of Mizar Users

:: PRELIMINARIES

scheme :: TOPREAL1:sch 1
FraenkelAlt{ F₁() -> non empty set , P₁[ set ], P₂[ set ] } :

{ v where v is Element of F₁() : ( P₁[v] or P₂[v] ) } = { v1 where v1 is Element of F₁() : P₁[v1] } \/ { v2 where v2 is Element of F₁() : P₂[v2] }

proof end;

definition

let T be TopSpace;
let p1, p2 be Point of T;
let P be Subset of T;

pred P is_an_arc_of p1,p2 means :: TOPREAL1:def 1
ex f being Function of I[01],(T | P) st
( f is being_homeomorphism & f . 0 = p1 & f . 1 = p2 );

end;

:: deftheorem defines is_an_arc_of TOPREAL1:def 1 :
for T being TopSpace
for p1, p2 being Point of T
for P being Subset of T holds
( P is_an_arc_of p1,p2 iff ex f being Function of I[01],(T | P) st
( f is being_homeomorphism & f . 0 = p1 & f . 1 = p2 ) );

theorem Th1: :: TOPREAL1:1

for T being TopSpace
for P being Subset of T
for p1, p2 being Point of T st P is_an_arc_of p1,p2 holds
( p1 in P & p2 in P )

proof end;

theorem Th2: :: TOPREAL1:2

for T being T_2 TopSpace
for P, Q being Subset of T
for p1, p2, q1 being Point of T st P is_an_arc_of p1,p2 & Q is_an_arc_of p2,q1 & P /\ Q = {p2} holds
P \/ Q is_an_arc_of p1,q1 by TOPMETR2:3;

definition

func R^2-unit_square -> Subset of (TOP-REAL 2) equals :: TOPREAL1:def 2
((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ ((LSeg (|[1,1]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[0,0]|)));

coherence
((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ ((LSeg (|[1,1]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[0,0]|))) is Subset of (TOP-REAL 2) ;

end;

:: deftheorem defines R^2-unit_square TOPREAL1:def 2 :
R^2-unit_square = ((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ ((LSeg (|[1,1]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[0,0]|)));

Lm1: for n being Nat
for p, p1, p2 being Point of (TOP-REAL n) st p in LSeg (p1,p2) holds
LSeg (p1,p) c= LSeg (p1,p2)

proof end;

theorem :: TOPREAL1:3

for p, p1, p2 being Point of (TOP-REAL 2) st p1 `1 <= p2 `1 & p in LSeg (p1,p2) holds
( p1 `1 <= p `1 & p `1 <= p2 `1 )

proof end;

theorem :: TOPREAL1:4

for p, p1, p2 being Point of (TOP-REAL 2) st p1 `2 <= p2 `2 & p in LSeg (p1,p2) holds
( p1 `2 <= p `2 & p `2 <= p2 `2 )

proof end;

theorem Th5: :: TOPREAL1:5

for n being Nat
for p, p1, p2 being Point of (TOP-REAL n) st p in LSeg (p1,p2) holds
LSeg (p1,p2) = (LSeg (p1,p)) \/ (LSeg (p,p2))

proof end;

theorem Th6: :: TOPREAL1:6

for n being Nat
for p1, p2, q1, q2 being Point of (TOP-REAL n) st q1 in LSeg (p1,p2) & q2 in LSeg (p1,p2) holds
LSeg (q1,q2) c= LSeg (p1,p2)

proof end;

theorem :: TOPREAL1:7

for n being Nat
for p, q, p1, p2 being Point of (TOP-REAL n) st p in LSeg (p1,p2) & q in LSeg (p1,p2) holds
LSeg (p1,p2) = ((LSeg (p1,p)) \/ (LSeg (p,q))) \/ (LSeg (q,p2))

proof end;

theorem :: TOPREAL1:8

for n being Nat
for p, p1, p2 being Point of (TOP-REAL n) st p in LSeg (p1,p2) holds
(LSeg (p1,p)) /\ (LSeg (p,p2)) = {p}

proof end;

Lm2: for T being TopSpace holds
( T is T_2 iff TopStruct(# the carrier of T, the topology of T #) is T_2 )

proof end;

registration

let n be Nat;

cluster the carrier of (TOP-REAL n) -> functional ;

coherence
the carrier of (TOP-REAL n) is functional by EUCLID:23;

end;

theorem Th9: :: TOPREAL1:9

for n being Nat
for p1, p2 being Point of (TOP-REAL n) st p1 <> p2 holds
LSeg (p1,p2) is_an_arc_of p1,p2

proof end;

registration

let n be Nat;

cluster TOP-REAL n -> T_2 ;

coherence
TOP-REAL n is T_2

proof end;

end;

theorem Th10: :: TOPREAL1:10

for n being Nat
for P being Subset of (TOP-REAL n)
for p1, p2, q1 being Point of (TOP-REAL n) st P is_an_arc_of p1,p2 & P /\ (LSeg (p2,q1)) = {p2} holds
P \/ (LSeg (p2,q1)) is_an_arc_of p1,q1

proof end;

theorem Th11: :: TOPREAL1:11

for n being Nat
for P being Subset of (TOP-REAL n)
for p1, p2, q1 being Point of (TOP-REAL n) st P is_an_arc_of p2,p1 & (LSeg (q1,p2)) /\ P = {p2} holds
(LSeg (q1,p2)) \/ P is_an_arc_of q1,p1

proof end;

theorem :: TOPREAL1:12

for n being Nat
for p1, p2, q1 being Point of (TOP-REAL n) st ( p1 <> p2 or p2 <> q1 ) & (LSeg (p1,p2)) /\ (LSeg (p2,q1)) = {p2} holds
(LSeg (p1,p2)) \/ (LSeg (p2,q1)) is_an_arc_of p1,q1

proof end;

theorem Th13: :: TOPREAL1:13

( LSeg (|[0,0]|,|[0,1]|) = { p1 where p1 is Point of (TOP-REAL 2) : ( p1 `1 = 0 & p1 `2 <= 1 & p1 `2 >= 0 ) } & LSeg (|[0,1]|,|[1,1]|) = { p2 where p2 is Point of (TOP-REAL 2) : ( p2 `1 <= 1 & p2 `1 >= 0 & p2 `2 = 1 ) } & LSeg (|[0,0]|,|[1,0]|) = { q1 where q1 is Point of (TOP-REAL 2) : ( q1 `1 <= 1 & q1 `1 >= 0 & q1 `2 = 0 ) } & LSeg (|[1,0]|,|[1,1]|) = { q2 where q2 is Point of (TOP-REAL 2) : ( q2 `1 = 1 & q2 `2 <= 1 & q2 `2 >= 0 ) } )

proof end;

theorem :: TOPREAL1:14

R^2-unit_square = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = 0 & p `2 <= 1 & p `2 >= 0 ) or ( p `1 <= 1 & p `1 >= 0 & p `2 = 1 ) or ( p `1 <= 1 & p `1 >= 0 & p `2 = 0 ) or ( p `1 = 1 & p `2 <= 1 & p `2 >= 0 ) ) }

proof end;

registration

cluster R^2-unit_square -> non empty ;

coherence
not R^2-unit_square is empty ;

end;

theorem :: TOPREAL1:15

(LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {|[0,1]|}

proof end;

theorem :: TOPREAL1:16

(LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {|[1,0]|}

proof end;

theorem Th17: :: TOPREAL1:17

(LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) = {|[0,0]|}

proof end;

theorem Th18: :: TOPREAL1:18

(LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {|[1,1]|}

proof end;

theorem :: TOPREAL1:19

LSeg (|[0,0]|,|[1,0]|) misses LSeg (|[0,1]|,|[1,1]|)

proof end;

theorem Th20: :: TOPREAL1:20

LSeg (|[0,0]|,|[0,1]|) misses LSeg (|[1,0]|,|[1,1]|)

proof end;

definition

let n be Nat;
let f be FinSequence of (TOP-REAL n);
let i be Nat;

func LSeg (f,i) -> Subset of (TOP-REAL n) equals :Def3: :: TOPREAL1:def 3
LSeg ((f /. i),(f /. (i + 1))) if ( 1 <= i & i + 1 <= len f )
otherwise {} ;

coherence
( ( 1 <= i & i + 1 <= len f implies LSeg ((f /. i),(f /. (i + 1))) is Subset of (TOP-REAL n) ) & ( ( not 1 <= i or not i + 1 <= len f ) implies {} is Subset of (TOP-REAL n) ) )

proof end;

correctness
consistency
for b₁ being Subset of (TOP-REAL n) holds verum;
;

end;

:: deftheorem Def3 defines LSeg TOPREAL1:def 3 :
for n being Nat
for f being FinSequence of (TOP-REAL n)
for i being Nat holds
( ( 1 <= i & i + 1 <= len f implies LSeg (f,i) = LSeg ((f /. i),(f /. (i + 1))) ) & ( ( not 1 <= i or not i + 1 <= len f ) implies LSeg (f,i) = {} ) );

theorem Th21: :: TOPREAL1:21

for i, n being Nat
for f being FinSequence of (TOP-REAL n) st 1 <= i & i + 1 <= len f holds
( f /. i in LSeg (f,i) & f /. (i + 1) in LSeg (f,i) )

proof end;

definition

let n be Nat;
let f be FinSequence of (TOP-REAL n);

func L~ f -> Subset of (TOP-REAL n) equals :: TOPREAL1:def 4
union { (LSeg (f,i)) where i is Nat : ( 1 <= i & i + 1 <= len f ) } ;

coherence
union { (LSeg (f,i)) where i is Nat : ( 1 <= i & i + 1 <= len f ) } is Subset of (TOP-REAL n)

proof end;

end;

:: deftheorem defines L~ TOPREAL1:def 4 :
for n being Nat
for f being FinSequence of (TOP-REAL n) holds L~ f = union { (LSeg (f,i)) where i is Nat : ( 1 <= i & i + 1 <= len f ) } ;

theorem Th22: :: TOPREAL1:22

for n being Nat
for f being FinSequence of (TOP-REAL n) holds
( ( len f = 0 or len f = 1 ) iff L~ f = {} )

proof end;

theorem Th23: :: TOPREAL1:23

for n being Nat
for f being FinSequence of (TOP-REAL n) st len f >= 2 holds
L~ f <> {}

proof end;

definition

let IT be FinSequence of (TOP-REAL 2);

attr IT is special means :: TOPREAL1:def 5
for i being Nat st 1 <= i & i + 1 <= len IT & not (IT /. i) `1 = (IT /. (i + 1)) `1 holds
(IT /. i) `2 = (IT /. (i + 1)) `2 ;

attr IT is unfolded means :: TOPREAL1:def 6
for i being Nat st 1 <= i & i + 2 <= len IT holds
(LSeg (IT,i)) /\ (LSeg (IT,(i + 1))) = {(IT /. (i + 1))};

attr IT is s.n.c. means :: TOPREAL1:def 7
for i, j being Nat st i + 1 < j holds
LSeg (IT,i) misses LSeg (IT,j);

end;

:: deftheorem defines special TOPREAL1:def 5 :
for IT being FinSequence of (TOP-REAL 2) holds
( IT is special iff for i being Nat st 1 <= i & i + 1 <= len IT & not (IT /. i) `1 = (IT /. (i + 1)) `1 holds
(IT /. i) `2 = (IT /. (i + 1)) `2 );

:: deftheorem defines unfolded TOPREAL1:def 6 :
for IT being FinSequence of (TOP-REAL 2) holds
( IT is unfolded iff for i being Nat st 1 <= i & i + 2 <= len IT holds
(LSeg (IT,i)) /\ (LSeg (IT,(i + 1))) = {(IT /. (i + 1))} );

:: deftheorem defines s.n.c. TOPREAL1:def 7 :
for IT being FinSequence of (TOP-REAL 2) holds
( IT is s.n.c. iff for i, j being Nat st i + 1 < j holds
LSeg (IT,i) misses LSeg (IT,j) );

definition

let f be FinSequence of (TOP-REAL 2);

attr f is being_S-Seq means :: TOPREAL1:def 8
( f is one-to-one & len f >= 2 & f is unfolded & f is s.n.c. & f is special );

end;

:: deftheorem defines being_S-Seq TOPREAL1:def 8 :
for f being FinSequence of (TOP-REAL 2) holds
( f is being_S-Seq iff ( f is one-to-one & len f >= 2 & f is unfolded & f is s.n.c. & f is special ) );

theorem Th24: :: TOPREAL1:24

ex f1, f2 being FinSequence of (TOP-REAL 2) st
( f1 is being_S-Seq & f2 is being_S-Seq & R^2-unit_square = (L~ f1) \/ (L~ f2) & (L~ f1) /\ (L~ f2) = {|[0,0]|,|[1,1]|} & f1 /. 1 = |[0,0]| & f1 /. (len f1) = |[1,1]| & f2 /. 1 = |[0,0]| & f2 /. (len f2) = |[1,1]| )

proof end;

theorem Th25: :: TOPREAL1:25

for h being FinSequence of (TOP-REAL 2) st h is being_S-Seq holds
L~ h is_an_arc_of h /. 1,h /. (len h)

proof end;

definition

let P be Subset of (TOP-REAL 2);

attr P is being_S-P_arc means :: TOPREAL1:def 9
ex f being FinSequence of (TOP-REAL 2) st
( f is being_S-Seq & P = L~ f );

end;

:: deftheorem defines being_S-P_arc TOPREAL1:def 9 :
for P being Subset of (TOP-REAL 2) holds
( P is being_S-P_arc iff ex f being FinSequence of (TOP-REAL 2) st
( f is being_S-Seq & P = L~ f ) );

theorem Th26: :: TOPREAL1:26

for P1 being Subset of (TOP-REAL 2) st P1 is being_S-P_arc holds
P1 <> {}

proof end;

registration

cluster being_S-P_arc -> non empty for Element of K10( the carrier of (TOP-REAL 2));

coherence
for b₁ being Subset of (TOP-REAL 2) st b₁ is being_S-P_arc holds
not b₁ is empty by Th26;

end;

theorem :: TOPREAL1:27

ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is being_S-P_arc & P2 is being_S-P_arc & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {|[0,0]|,|[1,1]|} )

proof end;

theorem Th28: :: TOPREAL1:28

for P being Subset of (TOP-REAL 2) st P is being_S-P_arc holds
ex p1, p2 being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2

proof end;

theorem :: TOPREAL1:29

for P being Subset of (TOP-REAL 2) st P is being_S-P_arc holds
ex f being Function of I[01],((TOP-REAL 2) | P) st f is being_homeomorphism

proof end;

:: Moved from JORDAN1A, AK, 23.02.2006

scheme :: TOPREAL1:sch 2
TRSubsetEx{ F₁() -> Nat, P₁[ object ] } :

ex A being Subset of (TOP-REAL F₁()) st
for p being Point of (TOP-REAL F₁()) holds
( p in A iff P₁[p] )

proof end;

scheme :: TOPREAL1:sch 3
TRSubsetUniq{ F₁() -> Nat, P₁[ object ] } :

for A, B being Subset of (TOP-REAL F₁()) st ( for p being Point of (TOP-REAL F₁()) holds
( p in A iff P₁[p] ) ) & ( for p being Point of (TOP-REAL F₁()) holds
( p in B iff P₁[p] ) ) holds
A = B

proof end;

definition

let p be Point of (TOP-REAL 2);

func north_halfline p -> Subset of (TOP-REAL 2) means :Def10: :: TOPREAL1:def 10
for x being Point of (TOP-REAL 2) holds
( x in it iff ( x `1 = p `1 & x `2 >= p `2 ) );

existence
ex b₁ being Subset of (TOP-REAL 2) st
for x being Point of (TOP-REAL 2) holds
( x in b₁ iff ( x `1 = p `1 & x `2 >= p `2 ) )

proof end;

uniqueness
for b₁, b₂ being Subset of (TOP-REAL 2) st ( for x being Point of (TOP-REAL 2) holds
( x in b₁ iff ( x `1 = p `1 & x `2 >= p `2 ) ) ) & ( for x being Point of (TOP-REAL 2) holds
( x in b₂ iff ( x `1 = p `1 & x `2 >= p `2 ) ) ) holds
b₁ = b₂

proof end;

func east_halfline p -> Subset of (TOP-REAL 2) means :Def11: :: TOPREAL1:def 11
for x being Point of (TOP-REAL 2) holds
( x in it iff ( x `1 >= p `1 & x `2 = p `2 ) );

existence
ex b₁ being Subset of (TOP-REAL 2) st
for x being Point of (TOP-REAL 2) holds
( x in b₁ iff ( x `1 >= p `1 & x `2 = p `2 ) )

proof end;

uniqueness
for b₁, b₂ being Subset of (TOP-REAL 2) st ( for x being Point of (TOP-REAL 2) holds
( x in b₁ iff ( x `1 >= p `1 & x `2 = p `2 ) ) ) & ( for x being Point of (TOP-REAL 2) holds
( x in b₂ iff ( x `1 >= p `1 & x `2 = p `2 ) ) ) holds
b₁ = b₂

proof end;

func south_halfline p -> Subset of (TOP-REAL 2) means :Def12: :: TOPREAL1:def 12
for x being Point of (TOP-REAL 2) holds
( x in it iff ( x `1 = p `1 & x `2 <= p `2 ) );

existence
ex b₁ being Subset of (TOP-REAL 2) st
for x being Point of (TOP-REAL 2) holds
( x in b₁ iff ( x `1 = p `1 & x `2 <= p `2 ) )

proof end;

uniqueness
for b₁, b₂ being Subset of (TOP-REAL 2) st ( for x being Point of (TOP-REAL 2) holds
( x in b₁ iff ( x `1 = p `1 & x `2 <= p `2 ) ) ) & ( for x being Point of (TOP-REAL 2) holds
( x in b₂ iff ( x `1 = p `1 & x `2 <= p `2 ) ) ) holds
b₁ = b₂

proof end;

func west_halfline p -> Subset of (TOP-REAL 2) means :Def13: :: TOPREAL1:def 13
for x being Point of (TOP-REAL 2) holds
( x in it iff ( x `1 <= p `1 & x `2 = p `2 ) );

existence
ex b₁ being Subset of (TOP-REAL 2) st
for x being Point of (TOP-REAL 2) holds
( x in b₁ iff ( x `1 <= p `1 & x `2 = p `2 ) )

proof end;

uniqueness
for b₁, b₂ being Subset of (TOP-REAL 2) st ( for x being Point of (TOP-REAL 2) holds
( x in b₁ iff ( x `1 <= p `1 & x `2 = p `2 ) ) ) & ( for x being Point of (TOP-REAL 2) holds
( x in b₂ iff ( x `1 <= p `1 & x `2 = p `2 ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def10 defines north_halfline TOPREAL1:def 10 :
for p being Point of (TOP-REAL 2)
for b₂ being Subset of (TOP-REAL 2) holds
( b₂ = north_halfline p iff for x being Point of (TOP-REAL 2) holds
( x in b₂ iff ( x `1 = p `1 & x `2 >= p `2 ) ) );

:: deftheorem Def11 defines east_halfline TOPREAL1:def 11 :
for p being Point of (TOP-REAL 2)
for b₂ being Subset of (TOP-REAL 2) holds
( b₂ = east_halfline p iff for x being Point of (TOP-REAL 2) holds
( x in b₂ iff ( x `1 >= p `1 & x `2 = p `2 ) ) );

:: deftheorem Def12 defines south_halfline TOPREAL1:def 12 :
for p being Point of (TOP-REAL 2)
for b₂ being Subset of (TOP-REAL 2) holds
( b₂ = south_halfline p iff for x being Point of (TOP-REAL 2) holds
( x in b₂ iff ( x `1 = p `1 & x `2 <= p `2 ) ) );

:: deftheorem Def13 defines west_halfline TOPREAL1:def 13 :
for p being Point of (TOP-REAL 2)
for b₂ being Subset of (TOP-REAL 2) holds
( b₂ = west_halfline p iff for x being Point of (TOP-REAL 2) holds
( x in b₂ iff ( x `1 <= p `1 & x `2 = p `2 ) ) );

theorem :: TOPREAL1:30

for p being Point of (TOP-REAL 2) holds north_halfline p = { q where q is Point of (TOP-REAL 2) : ( q `1 = p `1 & q `2 >= p `2 ) }

proof end;

theorem Th31: :: TOPREAL1:31

for p being Point of (TOP-REAL 2) holds north_halfline p = { |[(p `1),r]| where r is Real : r >= p `2 }

proof end;

theorem :: TOPREAL1:32

for p being Point of (TOP-REAL 2) holds east_halfline p = { q where q is Point of (TOP-REAL 2) : ( q `1 >= p `1 & q `2 = p `2 ) }

proof end;

theorem Th33: :: TOPREAL1:33

for p being Point of (TOP-REAL 2) holds east_halfline p = { |[r,(p `2)]| where r is Real : r >= p `1 }

proof end;

theorem :: TOPREAL1:34

for p being Point of (TOP-REAL 2) holds south_halfline p = { q where q is Point of (TOP-REAL 2) : ( q `1 = p `1 & q `2 <= p `2 ) }

proof end;

theorem Th35: :: TOPREAL1:35

for p being Point of (TOP-REAL 2) holds south_halfline p = { |[(p `1),r]| where r is Real : r <= p `2 }

proof end;

theorem :: TOPREAL1:36

for p being Point of (TOP-REAL 2) holds west_halfline p = { q where q is Point of (TOP-REAL 2) : ( q `1 <= p `1 & q `2 = p `2 ) }

proof end;

theorem Th37: :: TOPREAL1:37

for p being Point of (TOP-REAL 2) holds west_halfline p = { |[r,(p `2)]| where r is Real : r <= p `1 }

proof end;

registration

let p be Point of (TOP-REAL 2);

cluster north_halfline p -> non empty ;

coherence
not north_halfline p is empty

proof end;

cluster east_halfline p -> non empty ;

coherence
not east_halfline p is empty

proof end;

cluster south_halfline p -> non empty ;

coherence
not south_halfline p is empty

proof end;

cluster west_halfline p -> non empty ;

coherence
not west_halfline p is empty

proof end;

end;

:: Moved from JORDAN1C, AK, 23.02.2006

theorem :: TOPREAL1:38

for p being Point of (TOP-REAL 2) holds
( p in west_halfline p & p in east_halfline p & p in north_halfline p & p in south_halfline p )

proof end;