:: On Some Points of a Simple Closed Curve
:: by Artur Korni{\l}owicz
::
:: Received October 6, 2004
:: Copyright (c) 2004-2021 Association of Mizar Users

Lm1: dom proj2 = the carrier of (TOP-REAL 2)
by FUNCT_2:def 1;

Lm2: for r being Real
for X being Subset of (TOP-REAL 2) st r in proj2 .: X holds
ex x being Point of (TOP-REAL 2) st
( x in X & proj2 . x = r )

proof end;

Lm3: now :: thesis: for a, A, B, C being set holds
( not a in (A \/ B) \/ C or a in A or a in B or a in C )

let a, A, B, C be set ; :: thesis: ( not a in (A \/ B) \/ C or a in A or a in B or a in C )
assume a in (A \/ B) \/ C ; :: thesis: ( a in A or a in B or a in C )
then ( a in A \/ B or a in C ) by XBOOLE_0:def 3;
hence ( a in A or a in B or a in C ) by XBOOLE_0:def 3; :: thesis: verum

end;

Lm4: for A, B, C, D being set st A misses D & B misses D & C misses D holds
(A \/ B) \/ C misses D

proof end;

theorem :: JORDAN21:1

for n being Element of NAT
for p being Point of (TOP-REAL n) holds {p} is bounded

proof end;

theorem Th2: :: JORDAN21:2

for s1, t being Real
for P being Subset of (TOP-REAL 2) st P = { |[s,t]| where s is Real : s1 < s } holds
P is convex

proof end;

theorem Th3: :: JORDAN21:3

for s2, t being Real
for P being Subset of (TOP-REAL 2) st P = { |[s,t]| where s is Real : s < s2 } holds
P is convex

proof end;

theorem Th4: :: JORDAN21:4

for s, t1 being Real
for P being Subset of (TOP-REAL 2) st P = { |[s,t]| where t is Real : t1 < t } holds
P is convex

proof end;

theorem Th5: :: JORDAN21:5

for s, t2 being Real
for P being Subset of (TOP-REAL 2) st P = { |[s,t]| where t is Real : t < t2 } holds
P is convex

proof end;

theorem :: JORDAN21:6

for p being Point of (TOP-REAL 2) holds (north_halfline p) \ {p} is convex

proof end;

theorem :: JORDAN21:7

for p being Point of (TOP-REAL 2) holds (south_halfline p) \ {p} is convex

proof end;

theorem :: JORDAN21:8

for p being Point of (TOP-REAL 2) holds (west_halfline p) \ {p} is convex

proof end;

theorem :: JORDAN21:9

for p being Point of (TOP-REAL 2) holds (east_halfline p) \ {p} is convex

proof end;

theorem :: JORDAN21:10

for A being Subset of the carrier of (TOP-REAL 2) holds UBD A misses A

proof end;

theorem Th11: :: JORDAN21:11

for P being Subset of the carrier of (TOP-REAL 2)
for p1, p2, q1, q2 being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & p1 <> q1 & p2 <> q2 holds
( not p1 in Segment (P,p1,p2,q1,q2) & not p2 in Segment (P,p1,p2,q1,q2) )

proof end;

definition

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

attr D is with_the_max_arc means :Def1: :: JORDAN21:def 1
D meets Vertical_Line (((W-bound D) + (E-bound D)) / 2);

end;

:: deftheorem Def1 defines with_the_max_arc JORDAN21:def 1 :
for D being Subset of (TOP-REAL 2) holds
( D is with_the_max_arc iff D meets Vertical_Line (((W-bound D) + (E-bound D)) / 2) );

registration

cluster with_the_max_arc -> non empty for Element of K16( the carrier of (TOP-REAL 2));

coherence
for b₁ being Subset of (TOP-REAL 2) st b₁ is with_the_max_arc holds
not b₁ is empty

proof end;

end;

registration

cluster being_simple_closed_curve -> with_the_max_arc for Element of K16( the carrier of (TOP-REAL 2));

coherence
for b₁ being Simple_closed_curve holds b₁ is with_the_max_arc

proof end;

end;

registration

cluster non empty closed connected compact bounded non horizontal non vertical being_simple_closed_curve with_the_max_arc for Element of K16( the carrier of (TOP-REAL 2));

existence
not for b₁ being Simple_closed_curve holds b₁ is empty

proof end;

end;

theorem Th12: :: JORDAN21:12

for D being with_the_max_arc Subset of (TOP-REAL 2) holds not proj2 .: (D /\ (Vertical_Line (((W-bound D) + (E-bound D)) / 2))) is empty

proof end;

theorem Th13: :: JORDAN21:13

for C being compact Subset of (TOP-REAL 2) holds
( proj2 .: (C /\ (Vertical_Line (((W-bound C) + (E-bound C)) / 2))) is closed & proj2 .: (C /\ (Vertical_Line (((W-bound C) + (E-bound C)) / 2))) is bounded_below & proj2 .: (C /\ (Vertical_Line (((W-bound C) + (E-bound C)) / 2))) is bounded_above )

proof end;

theorem Th14: :: JORDAN21:14

for C being Simple_closed_curve holds Lower_Middle_Point C in C

proof end;

theorem Th15: :: JORDAN21:15

for C being Simple_closed_curve holds (Lower_Middle_Point C) `2 <> (Upper_Middle_Point C) `2

proof end;

theorem :: JORDAN21:16

for C being Simple_closed_curve holds Lower_Middle_Point C <> Upper_Middle_Point C

proof end;

theorem Th17: :: JORDAN21:17

for C being Simple_closed_curve holds W-bound C = W-bound (Upper_Arc C)

proof end;

theorem Th18: :: JORDAN21:18

for C being Simple_closed_curve holds E-bound C = E-bound (Upper_Arc C)

proof end;

theorem Th19: :: JORDAN21:19

for C being Simple_closed_curve holds W-bound C = W-bound (Lower_Arc C)

proof end;

theorem Th20: :: JORDAN21:20

for C being Simple_closed_curve holds E-bound C = E-bound (Lower_Arc C)

proof end;

theorem Th21: :: JORDAN21:21

for C being Simple_closed_curve holds
( not (Upper_Arc C) /\ (Vertical_Line (((W-bound C) + (E-bound C)) / 2)) is empty & not proj2 .: ((Upper_Arc C) /\ (Vertical_Line (((W-bound C) + (E-bound C)) / 2))) is empty )

proof end;

theorem Th22: :: JORDAN21:22

for C being Simple_closed_curve holds
( not (Lower_Arc C) /\ (Vertical_Line (((W-bound C) + (E-bound C)) / 2)) is empty & not proj2 .: ((Lower_Arc C) /\ (Vertical_Line (((W-bound C) + (E-bound C)) / 2))) is empty )

proof end;

theorem :: JORDAN21:23

for C being Simple_closed_curve
for P being connected compact Subset of (TOP-REAL 2) st P c= C & W-min C in P & E-max C in P & not Upper_Arc C c= P holds
Lower_Arc C c= P

proof end;

registration

let C be Simple_closed_curve;

cluster Lower_Arc C -> with_the_max_arc ;

coherence
Lower_Arc C is with_the_max_arc

proof end;

cluster Upper_Arc C -> with_the_max_arc ;

coherence
Upper_Arc C is with_the_max_arc

proof end;

end;

definition

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

func UMP P -> Point of (TOP-REAL 2) equals :: JORDAN21:def 2
|[(((E-bound P) + (W-bound P)) / 2),(upper_bound (proj2 .: (P /\ (Vertical_Line (((E-bound P) + (W-bound P)) / 2)))))]|;

correctness
coherence
|[(((E-bound P) + (W-bound P)) / 2),(upper_bound (proj2 .: (P /\ (Vertical_Line (((E-bound P) + (W-bound P)) / 2)))))]| is Point of (TOP-REAL 2);
;

func LMP P -> Point of (TOP-REAL 2) equals :: JORDAN21:def 3
|[(((E-bound P) + (W-bound P)) / 2),(lower_bound (proj2 .: (P /\ (Vertical_Line (((E-bound P) + (W-bound P)) / 2)))))]|;

correctness
coherence
|[(((E-bound P) + (W-bound P)) / 2),(lower_bound (proj2 .: (P /\ (Vertical_Line (((E-bound P) + (W-bound P)) / 2)))))]| is Point of (TOP-REAL 2);
;

end;

:: deftheorem defines UMP JORDAN21:def 2 :
for P being Subset of the carrier of (TOP-REAL 2) holds UMP P = |[(((E-bound P) + (W-bound P)) / 2),(upper_bound (proj2 .: (P /\ (Vertical_Line (((E-bound P) + (W-bound P)) / 2)))))]|;

:: deftheorem defines LMP JORDAN21:def 3 :
for P being Subset of the carrier of (TOP-REAL 2) holds LMP P = |[(((E-bound P) + (W-bound P)) / 2),(lower_bound (proj2 .: (P /\ (Vertical_Line (((E-bound P) + (W-bound P)) / 2)))))]|;

theorem Th24: :: JORDAN21:24

for C being compact non vertical Subset of (TOP-REAL 2) holds UMP C <> W-min C

proof end;

theorem Th25: :: JORDAN21:25

for C being compact non vertical Subset of (TOP-REAL 2) holds UMP C <> E-max C

proof end;

theorem Th26: :: JORDAN21:26

for C being compact non vertical Subset of (TOP-REAL 2) holds LMP C <> W-min C

proof end;

theorem Th27: :: JORDAN21:27

for C being compact non vertical Subset of (TOP-REAL 2) holds LMP C <> E-max C

proof end;

theorem :: JORDAN21:28

for p being Point of (TOP-REAL 2)
for C being compact Subset of (TOP-REAL 2) st p in C /\ (Vertical_Line (((W-bound C) + (E-bound C)) / 2)) holds
p `2 <= (UMP C) `2

proof end;

theorem :: JORDAN21:29

for p being Point of (TOP-REAL 2)
for C being compact Subset of (TOP-REAL 2) st p in C /\ (Vertical_Line (((W-bound C) + (E-bound C)) / 2)) holds
(LMP C) `2 <= p `2

proof end;

theorem Th30: :: JORDAN21:30

for D being compact with_the_max_arc Subset of (TOP-REAL 2) holds UMP D in D

proof end;

theorem Th31: :: JORDAN21:31

for D being compact with_the_max_arc Subset of (TOP-REAL 2) holds LMP D in D

proof end;

theorem Th32: :: JORDAN21:32

for P being Subset of (TOP-REAL 2) holds LSeg ((UMP P),|[(((W-bound P) + (E-bound P)) / 2),(N-bound P)]|) is vertical

proof end;

theorem Th33: :: JORDAN21:33

for P being Subset of (TOP-REAL 2) holds LSeg ((LMP P),|[(((W-bound P) + (E-bound P)) / 2),(S-bound P)]|) is vertical

proof end;

theorem Th34: :: JORDAN21:34

for D being compact with_the_max_arc Subset of (TOP-REAL 2) holds (LSeg ((UMP D),|[(((W-bound D) + (E-bound D)) / 2),(N-bound D)]|)) /\ D = {(UMP D)}

proof end;

theorem Th35: :: JORDAN21:35

for D being compact with_the_max_arc Subset of (TOP-REAL 2) holds (LSeg ((LMP D),|[(((W-bound D) + (E-bound D)) / 2),(S-bound D)]|)) /\ D = {(LMP D)}

proof end;

theorem Th36: :: JORDAN21:36

for C being Simple_closed_curve holds (LMP C) `2 < (UMP C) `2

proof end;

theorem Th37: :: JORDAN21:37

for C being Simple_closed_curve holds UMP C <> LMP C

proof end;

theorem Th38: :: JORDAN21:38

for C being Simple_closed_curve holds S-bound C < (UMP C) `2

proof end;

theorem Th39: :: JORDAN21:39

for C being Simple_closed_curve holds (UMP C) `2 <= N-bound C

proof end;

theorem Th40: :: JORDAN21:40

for C being Simple_closed_curve holds S-bound C <= (LMP C) `2

proof end;

theorem Th41: :: JORDAN21:41

for C being Simple_closed_curve holds (LMP C) `2 < N-bound C

proof end;

theorem Th42: :: JORDAN21:42

for C being Simple_closed_curve holds LSeg ((UMP C),|[(((W-bound C) + (E-bound C)) / 2),(N-bound C)]|) misses LSeg ((LMP C),|[(((W-bound C) + (E-bound C)) / 2),(S-bound C)]|)

proof end;

theorem :: JORDAN21:43

for A, B being Subset of (TOP-REAL 2) st A c= B & (W-bound A) + (E-bound A) = (W-bound B) + (E-bound B) & not A /\ (Vertical_Line (((W-bound A) + (E-bound A)) / 2)) is empty & proj2 .: (B /\ (Vertical_Line (((W-bound A) + (E-bound A)) / 2))) is bounded_above holds
(UMP A) `2 <= (UMP B) `2

proof end;

theorem :: JORDAN21:44

for A, B being Subset of (TOP-REAL 2) st A c= B & (W-bound A) + (E-bound A) = (W-bound B) + (E-bound B) & not A /\ (Vertical_Line (((W-bound A) + (E-bound A)) / 2)) is empty & proj2 .: (B /\ (Vertical_Line (((W-bound A) + (E-bound A)) / 2))) is bounded_below holds
(LMP B) `2 <= (LMP A) `2

proof end;

theorem :: JORDAN21:45

for A, B being Subset of (TOP-REAL 2) st A c= B & UMP B in A & not A /\ (Vertical_Line (((W-bound A) + (E-bound A)) / 2)) is empty & proj2 .: (B /\ (Vertical_Line (((W-bound B) + (E-bound B)) / 2))) is bounded_above & (W-bound A) + (E-bound A) = (W-bound B) + (E-bound B) holds
UMP A = UMP B

proof end;

theorem :: JORDAN21:46

for A, B being Subset of (TOP-REAL 2) st A c= B & LMP B in A & not A /\ (Vertical_Line (((W-bound A) + (E-bound A)) / 2)) is empty & proj2 .: (B /\ (Vertical_Line (((W-bound B) + (E-bound B)) / 2))) is bounded_below & (W-bound A) + (E-bound A) = (W-bound B) + (E-bound B) holds
LMP A = LMP B

proof end;

theorem :: JORDAN21:47

for C being Simple_closed_curve holds (UMP (Upper_Arc C)) `2 <= N-bound C

proof end;

theorem :: JORDAN21:48

for C being Simple_closed_curve holds S-bound C <= (LMP (Lower_Arc C)) `2

proof end;

theorem :: JORDAN21:49

for C being Simple_closed_curve holds
( not LMP C in Lower_Arc C or not UMP C in Lower_Arc C )

proof end;

theorem :: JORDAN21:50

for C being Simple_closed_curve holds
( not LMP C in Upper_Arc C or not UMP C in Upper_Arc C )

proof end;