:: 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 Association of Mizar Users

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;
canceled;
pred P is_an_arc_of p1,p2 means :Def2: :: TOPREAL1:def 2
ex f being Function of I[01] ,(T | P) st
( f is being_homeomorphism & f . 0 = p1 & f . 1 = p2 );

end;

:: deftheorem TOPREAL1:def 1 :

canceled;

:: deftheorem Def2 defines is_an_arc_of TOPREAL1:def 2 :

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 :: TOPREAL1:1

canceled;

theorem :: TOPREAL1:2

canceled;

theorem :: TOPREAL1:3

canceled;

theorem Th4: :: TOPREAL1:4

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 Th5: :: TOPREAL1:5

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

proof end;

definition

canceled;
func R^2-unit_square -> Subset of (TOP-REAL 2) equals :: TOPREAL1:def 4
((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 TOPREAL1:def 3 :

canceled;

:: deftheorem defines R^2-unit_square TOPREAL1:def 4 :

R^2-unit_square = ((LSeg |[0 ,0 ]|,|[0 ,1]|) \/ (LSeg |[0 ,1]|,|[1,1]|)) \/ ((LSeg |[1,1]|,|[1,0 ]|) \/ (LSeg |[1,0 ]|,|[0 ,0 ]|));

theorem :: TOPREAL1:6

canceled;

theorem :: TOPREAL1:7

canceled;

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 :: TOPREAL1:8

canceled;

theorem :: TOPREAL1:9

for p1, p2, p 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 :: TOPREAL1:10

for p1, p2, p 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 Th11: :: TOPREAL1:11

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 Th12: :: TOPREAL1:12

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 :: TOPREAL1:13

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 :: TOPREAL1:14

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 -> Relation-like Function-like Element of the carrier of (TOP-REAL n);
coherence
for b₁ being Point of (TOP-REAL n) holds
( b₁ is Function-like & b₁ is Relation-like ) by EUCLID:26;

end;

theorem Th15: :: TOPREAL1:15

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 Th16: :: TOPREAL1:16

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 Th17: :: TOPREAL1:17

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 :: TOPREAL1:18

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 Th19: :: TOPREAL1:19

( 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 :: TOPREAL1:20

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 :: TOPREAL1:21

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

proof end;

theorem :: TOPREAL1:22

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

proof end;

theorem Th23: :: TOPREAL1:23

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

proof end;

theorem Th24: :: TOPREAL1:24

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

proof end;

theorem :: TOPREAL1:25

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

proof end;

theorem Th26: :: TOPREAL1:26

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 :Def5: :: TOPREAL1:def 5
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 Def5 defines LSeg TOPREAL1:def 5 :

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 Th27: :: TOPREAL1:27

for n, i 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 6
union { (LSeg f,i) where i is Element of NAT : ( 1 <= i & i + 1 <= len f ) } ;
coherence
union { (LSeg f,i) where i is Element of NAT : ( 1 <= i & i + 1 <= len f ) } is Subset of (TOP-REAL n)

proof end;

end;

:: deftheorem defines L~ TOPREAL1:def 6 :

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

theorem Th28: :: TOPREAL1:28

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 Th29: :: TOPREAL1:29

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 7
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 :Def8: :: TOPREAL1:def 8
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 :Def9: :: TOPREAL1:def 9
for i, j being Nat st i + 1 < j holds
LSeg IT,i misses LSeg IT,j;

end;

:: deftheorem defines special TOPREAL1:def 7 :

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 Def8 defines unfolded TOPREAL1:def 8 :

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 Def9 defines s.n.c. TOPREAL1:def 9 :

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 :Def10: :: TOPREAL1:def 10
( f is one-to-one & len f >= 2 & f is unfolded & f is s.n.c. & f is special );

end;

:: deftheorem Def10 defines being_S-Seq TOPREAL1:def 10 :

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 Th30: :: TOPREAL1:30

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 Th31: :: TOPREAL1:31

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 :Def11: :: TOPREAL1:def 11
ex f being FinSequence of (TOP-REAL 2) st
( f is being_S-Seq & P = L~ f );

end;

:: deftheorem Def11 defines being_S-P_arc TOPREAL1:def 11 :

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 Th32: :: TOPREAL1:32

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 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 Th32;

end;

theorem :: TOPREAL1:33

canceled;

theorem :: TOPREAL1:34

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 Th35: :: TOPREAL1:35

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 :: TOPREAL1:36

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;

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

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[ set ] } :

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 :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 east_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;

func south_halfline p -> Subset of (TOP-REAL 2) means :Def14: :: TOPREAL1:def 14
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 :Def15: :: TOPREAL1:def 15
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 Def12 defines north_halfline TOPREAL1:def 12 :

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 Def13 defines east_halfline TOPREAL1:def 13 :

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 Def14 defines south_halfline TOPREAL1:def 14 :

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 Def15 defines west_halfline TOPREAL1:def 15 :

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 :: TOPREAL1:37

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 Th38: :: TOPREAL1:38

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

proof end;

theorem :: TOPREAL1:39

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 Th40: :: TOPREAL1:40

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

proof end;

theorem :: TOPREAL1:41

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 Th42: :: TOPREAL1:42

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

proof end;

theorem :: TOPREAL1:43

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 Th44: :: TOPREAL1:44

for p being Point of (TOP-REAL 2) holds west_halfline p = { |[r,(p `2 )]| where r is Element of 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;

theorem :: TOPREAL1:45

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;