:: The Topological Space ${\calE}^2_{\rm T}$. Simple Closed Curves
:: by Agata Darmochwa{\l} and Yatsuka Nakamura
::
:: Received December 30, 1991
:: Copyright (c) 1991 Association of Mizar Users

Lm1: now

let x, X be set ; :: thesis: ( not x in X implies {x} /\ X = {} )
assume not x in X ; :: thesis: {x} /\ X = {}
then {x} misses X by ZFMISC_1:56;
hence {x} /\ X = {} by XBOOLE_0:def 7; :: thesis: verum

end;

Lm2: (LSeg |[0 ,0 ]|,|[1,0 ]|) /\ (LSeg |[0 ,1]|,|[1,1]|) = {}
by TOPREAL1:25, XBOOLE_0:def 7;

Lm3: (LSeg |[0 ,0 ]|,|[0 ,1]|) /\ (LSeg |[1,0 ]|,|[1,1]|) = {}
by TOPREAL1:26, XBOOLE_0:def 7;

set p00 = |[0 ,0 ]|;

set p01 = |[0 ,1]|;

set p10 = |[1,0 ]|;

set p11 = |[1,1]|;

set L1 = LSeg |[0 ,0 ]|,|[0 ,1]|;

set L2 = LSeg |[0 ,1]|,|[1,1]|;

set L3 = LSeg |[0 ,0 ]|,|[1,0 ]|;

set L4 = LSeg |[1,0 ]|,|[1,1]|;

Lm4: ( |[0 ,0 ]| `1 = 0 & |[0 ,0 ]| `2 = 0 & |[0 ,1]| `1 = 0 & |[0 ,1]| `2 = 1 & |[1,0 ]| `1 = 1 & |[1,0 ]| `2 = 0 & |[1,1]| `1 = 1 & |[1,1]| `2 = 1 )
by EUCLID:56;

then Lm5: not |[0 ,0 ]| in LSeg |[1,0 ]|,|[1,1]|
by TOPREAL1:9;

Lm6: not |[0 ,0 ]| in LSeg |[0 ,1]|,|[1,1]|
by Lm4, TOPREAL1:10;

Lm7: not |[0 ,1]| in LSeg |[0 ,0 ]|,|[1,0 ]|
by Lm4, TOPREAL1:10;

Lm8: not |[0 ,1]| in LSeg |[1,0 ]|,|[1,1]|
by Lm4, TOPREAL1:9;

Lm9: not |[1,0 ]| in LSeg |[0 ,0 ]|,|[0 ,1]|
by Lm4, TOPREAL1:9;

Lm10: not |[1,0 ]| in LSeg |[0 ,1]|,|[1,1]|
by Lm4, TOPREAL1:10;

Lm11: not |[1,1]| in LSeg |[0 ,0 ]|,|[0 ,1]|
by Lm4, TOPREAL1:9;

Lm12: not |[1,1]| in LSeg |[0 ,0 ]|,|[1,0 ]|
by Lm4, TOPREAL1:10;

Lm13: |[0 ,0 ]| in LSeg |[0 ,0 ]|,|[0 ,1]|
by RLTOPSP1:69;

Lm14: |[0 ,0 ]| in LSeg |[0 ,0 ]|,|[1,0 ]|
by RLTOPSP1:69;

Lm15: |[0 ,1]| in LSeg |[0 ,0 ]|,|[0 ,1]|
by RLTOPSP1:69;

Lm16: |[0 ,1]| in LSeg |[0 ,1]|,|[1,1]|
by RLTOPSP1:69;

Lm17: |[1,0 ]| in LSeg |[0 ,0 ]|,|[1,0 ]|
by RLTOPSP1:69;

Lm18: |[1,0 ]| in LSeg |[1,0 ]|,|[1,1]|
by RLTOPSP1:69;

Lm19: |[1,1]| in LSeg |[0 ,1]|,|[1,1]|
by RLTOPSP1:69;

Lm20: |[1,1]| in LSeg |[1,0 ]|,|[1,1]|
by RLTOPSP1:69;

set L = { 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 ) ) } ;

Lm21: |[0 ,0 ]| in { 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 ) ) }
by Lm4;

Lm22: |[1,1]| in { 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 ) ) }
by Lm4;

Lm23: for p1, p2 being Point of (TOP-REAL 2) st p1 <> p2 & p2 in R^2-unit_square & p1 in LSeg |[0 ,0 ]|,|[0 ,1]| holds
ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )

proof end;

Lm24: for p1, p2 being Point of (TOP-REAL 2) st p1 <> p2 & p2 in R^2-unit_square & p1 in LSeg |[0 ,1]|,|[1,1]| holds
ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )

proof end;

Lm25: for p1, p2 being Point of (TOP-REAL 2) st p1 <> p2 & p2 in R^2-unit_square & p1 in LSeg |[0 ,0 ]|,|[1,0 ]| holds
ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )

proof end;

Lm26: for p1, p2 being Point of (TOP-REAL 2) st p1 <> p2 & p2 in R^2-unit_square & p1 in LSeg |[1,0 ]|,|[1,1]| holds
ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )

proof end;

theorem Th1: :: TOPREAL2:1

for p1, p2 being Point of (TOP-REAL 2) st p1 <> p2 & p1 in R^2-unit_square & p2 in R^2-unit_square holds
ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )

proof end;

theorem Th2: :: TOPREAL2:2

R^2-unit_square is compact

proof end;

theorem Th3: :: TOPREAL2:3

for q1, q2 being Point of (TOP-REAL 2)
for Q, P being non empty Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | Q),((TOP-REAL 2) | P) st f is being_homeomorphism & Q is_an_arc_of q1,q2 holds
for p1, p2 being Point of (TOP-REAL 2) st p1 = f . q1 & p2 = f . q2 holds
P is_an_arc_of p1,p2

proof end;

definition

let P be Subset of (TOP-REAL 2);
attr P is being_simple_closed_curve means :Def1: :: TOPREAL2:def 1
ex f being Function of ((TOP-REAL 2) | R^2-unit_square ),((TOP-REAL 2) | P) st f is being_homeomorphism ;

end;

:: deftheorem Def1 defines being_simple_closed_curve TOPREAL2:def 1 :

for P being Subset of (TOP-REAL 2) holds
( P is being_simple_closed_curve iff ex f being Function of ((TOP-REAL 2) | R^2-unit_square ),((TOP-REAL 2) | P) st f is being_homeomorphism );

registration

cluster R^2-unit_square -> being_simple_closed_curve ;
coherence
R^2-unit_square is being_simple_closed_curve

proof end;

end;

registration

cluster non empty being_simple_closed_curve Element of K21(the carrier of (TOP-REAL 2));
existence
ex b₁ being Subset of (TOP-REAL 2) st
( b₁ is being_simple_closed_curve & not b₁ is empty )

proof end;

end;

definition

mode Simple_closed_curve is being_simple_closed_curve Subset of (TOP-REAL 2);

end;

theorem Th4: :: TOPREAL2:4

for P being non empty Subset of (TOP-REAL 2) st P is being_simple_closed_curve holds
ex p1, p2 being Point of (TOP-REAL 2) st
( p1 <> p2 & p1 in P & p2 in P )

proof end;

Lm27: for p1, p2 being Point of (TOP-REAL 2)
for P, P1, P2 being non empty Subset of (TOP-REAL 2) st p1 <> p2 & p1 in P & p2 in P & P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P = P1 \/ P2 & P1 /\ P2 = {p1,p2} holds
P is being_simple_closed_curve

proof end;

theorem Th5: :: TOPREAL2:5

for P being non empty Subset of (TOP-REAL 2) holds
( P is being_simple_closed_curve iff ( ex p1, p2 being Point of (TOP-REAL 2) st
( p1 <> p2 & p1 in P & p2 in P ) & ( for p1, p2 being Point of (TOP-REAL 2) st p1 <> p2 & p1 in P & p2 in P holds
ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P = P1 \/ P2 & P1 /\ P2 = {p1,p2} ) ) ) )

proof end;

theorem :: TOPREAL2:6

for P being non empty Subset of (TOP-REAL 2) holds
( P is being_simple_closed_curve iff ex p1, p2 being Point of (TOP-REAL 2) ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( p1 <> p2 & p1 in P & p2 in P & P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P = P1 \/ P2 & P1 /\ P2 = {p1,p2} ) )

proof end;

Lm28: for S, T being 1-sorted
for f being Function of S,T st S is empty & rng f = [#] T holds
T is empty

proof end;

Lm29: for S, T being 1-sorted
for f being Function of S,T st T is empty & dom f = [#] S holds
S is empty

proof end;

Lm30: for S, T being TopStruct st ex f being Function of S,T st f is being_homeomorphism holds
( S is empty iff T is empty )

proof end;

registration

cluster being_simple_closed_curve -> non empty compact Element of K21(the carrier of (TOP-REAL 2));
coherence
for b₁ being Subset of (TOP-REAL 2) st b₁ is being_simple_closed_curve holds
( not b₁ is empty & b₁ is compact )

proof end;

end;