:: On Rectangular Finite Sequences of the Points of the Plane
:: by Andrzej Trybulec and Yatsuka Nakamura
::
:: Received November 30, 1997
:: Copyright (c) 1997-2021 Association of Mizar Users

registration

cluster V1() V4( NAT ) non empty Function-like constant V35() FinSequence-like FinSubsequence-like for set ;

existence
ex b₁ being FinSequence st
( not b₁ is empty & b₁ is constant )

proof end;

end;

theorem Th1: :: SPRECT_1:1

for f, g being FinSequence st f ^ g is constant holds
( f is constant & g is constant )

proof end;

theorem Th2: :: SPRECT_1:2

for x, y being set st <*x,y*> is constant holds
x = y

proof end;

theorem Th3: :: SPRECT_1:3

for x, y, z being set st <*x,y,z*> is constant holds
( x = y & y = z & z = x )

proof end;

theorem Th4: :: SPRECT_1:4

for GX being non empty TopSpace
for A being Subset of GX
for B being non empty Subset of GX st A is_a_component_of B holds
A <> {}

proof end;

theorem Th5: :: SPRECT_1:5

for GX being TopStruct
for A, B being Subset of GX st A is_a_component_of B holds
A c= B

proof end;

theorem Th6: :: SPRECT_1:6

for T being non empty TopSpace
for A being non empty Subset of T
for B1, B2, S being Subset of T st B1 is_a_component_of A & B2 is_a_component_of A & S is_a_component_of A & B1 \/ B2 = A & not S = B1 holds
S = B2

proof end;

theorem Th7: :: SPRECT_1:7

for T being non empty TopSpace
for A being non empty Subset of T
for B1, B2, C1, C2 being Subset of T st B1 is_a_component_of A & B2 is_a_component_of A & C1 is_a_component_of A & C2 is_a_component_of A & B1 \/ B2 = A & C1 \/ C2 = A holds
{B1,B2} = {C1,C2}

proof end;

theorem Th8: :: SPRECT_1:8

for p, q, r being Point of (TOP-REAL 2) holds L~ <*p,q,r*> = (LSeg (p,q)) \/ (LSeg (q,r))

proof end;

registration

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

cluster L~ f -> non empty ;

coherence
not L~ f is empty

proof end;

end;

registration

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

cluster L~ f -> compact ;

coherence
L~ f is compact

proof end;

end;

theorem Th9: :: SPRECT_1:9

for A, B being Subset of (TOP-REAL 2) st A c= B & B is horizontal holds
A is horizontal ;

theorem Th10: :: SPRECT_1:10

for A, B being Subset of (TOP-REAL 2) st A c= B & B is vertical holds
A is vertical ;

registration

cluster R^2-unit_square -> special_polygonal non horizontal non vertical ;

coherence
( R^2-unit_square is special_polygonal & not R^2-unit_square is horizontal & not R^2-unit_square is vertical )

proof end;

end;

registration

cluster non empty compact non horizontal non vertical for Element of bool the carrier of (TOP-REAL 2);

existence
ex b₁ being Subset of (TOP-REAL 2) st
( not b₁ is vertical & not b₁ is horizontal & not b₁ is empty & b₁ is compact )

proof end;

end;

theorem Th11: :: SPRECT_1:11

for C being non empty compact Subset of (TOP-REAL 2) holds
( N-min C in C & N-max C in C )

proof end;

theorem Th12: :: SPRECT_1:12

for C being non empty compact Subset of (TOP-REAL 2) holds
( S-min C in C & S-max C in C )

proof end;

theorem Th13: :: SPRECT_1:13

for C being non empty compact Subset of (TOP-REAL 2) holds
( W-min C in C & W-max C in C )

proof end;

theorem Th14: :: SPRECT_1:14

for C being non empty compact Subset of (TOP-REAL 2) holds
( E-min C in C & E-max C in C )

proof end;

theorem Th15: :: SPRECT_1:15

for C being non empty compact Subset of (TOP-REAL 2) holds
( C is vertical iff W-bound C = E-bound C )

proof end;

theorem Th16: :: SPRECT_1:16

for C being non empty compact Subset of (TOP-REAL 2) holds
( C is horizontal iff S-bound C = N-bound C )

proof end;

theorem :: SPRECT_1:17

for C being non empty compact Subset of (TOP-REAL 2) st NW-corner C = NE-corner C holds
C is vertical

proof end;

theorem :: SPRECT_1:18

for C being non empty compact Subset of (TOP-REAL 2) st SW-corner C = SE-corner C holds
C is vertical

proof end;

theorem :: SPRECT_1:19

for C being non empty compact Subset of (TOP-REAL 2) st NW-corner C = SW-corner C holds
C is horizontal

proof end;

theorem :: SPRECT_1:20

for C being non empty compact Subset of (TOP-REAL 2) st NE-corner C = SE-corner C holds
C is horizontal

proof end;

theorem Th21: :: SPRECT_1:21

for C being non empty compact Subset of (TOP-REAL 2) holds W-bound C <= E-bound C

proof end;

theorem Th22: :: SPRECT_1:22

for C being non empty compact Subset of (TOP-REAL 2) holds S-bound C <= N-bound C

proof end;

theorem Th23: :: SPRECT_1:23

for C being non empty compact Subset of (TOP-REAL 2) holds LSeg ((SE-corner C),(NE-corner C)) = { p where p is Point of (TOP-REAL 2) : ( p `1 = E-bound C & p `2 <= N-bound C & p `2 >= S-bound C ) }

proof end;

theorem Th24: :: SPRECT_1:24

for C being non empty compact Subset of (TOP-REAL 2) holds LSeg ((SW-corner C),(SE-corner C)) = { p where p is Point of (TOP-REAL 2) : ( p `1 <= E-bound C & p `1 >= W-bound C & p `2 = S-bound C ) }

proof end;

theorem Th25: :: SPRECT_1:25

for C being non empty compact Subset of (TOP-REAL 2) holds LSeg ((NW-corner C),(NE-corner C)) = { p where p is Point of (TOP-REAL 2) : ( p `1 <= E-bound C & p `1 >= W-bound C & p `2 = N-bound C ) }

proof end;

theorem Th26: :: SPRECT_1:26

for C being non empty compact Subset of (TOP-REAL 2) holds LSeg ((SW-corner C),(NW-corner C)) = { p where p is Point of (TOP-REAL 2) : ( p `1 = W-bound C & p `2 <= N-bound C & p `2 >= S-bound C ) }

proof end;

theorem :: SPRECT_1:27

for C being non empty compact Subset of (TOP-REAL 2) holds (LSeg ((SW-corner C),(NW-corner C))) /\ (LSeg ((NW-corner C),(NE-corner C))) = {(NW-corner C)}

proof end;

theorem Th28: :: SPRECT_1:28

for C being non empty compact Subset of (TOP-REAL 2) holds (LSeg ((NW-corner C),(NE-corner C))) /\ (LSeg ((NE-corner C),(SE-corner C))) = {(NE-corner C)}

proof end;

theorem Th29: :: SPRECT_1:29

for C being non empty compact Subset of (TOP-REAL 2) holds (LSeg ((SE-corner C),(NE-corner C))) /\ (LSeg ((SW-corner C),(SE-corner C))) = {(SE-corner C)}

proof end;

theorem Th30: :: SPRECT_1:30

for C being non empty compact Subset of (TOP-REAL 2) holds (LSeg ((NW-corner C),(SW-corner C))) /\ (LSeg ((SW-corner C),(SE-corner C))) = {(SW-corner C)}

proof end;

theorem Th31: :: SPRECT_1:31

for D1 being non empty compact non vertical Subset of (TOP-REAL 2) holds W-bound D1 < E-bound D1

proof end;

theorem Th32: :: SPRECT_1:32

for D2 being non empty compact non horizontal Subset of (TOP-REAL 2) holds S-bound D2 < N-bound D2

proof end;

theorem Th33: :: SPRECT_1:33

for D1 being non empty compact non vertical Subset of (TOP-REAL 2) holds LSeg ((SW-corner D1),(NW-corner D1)) misses LSeg ((SE-corner D1),(NE-corner D1))

proof end;

theorem Th34: :: SPRECT_1:34

for D2 being non empty compact non horizontal Subset of (TOP-REAL 2) holds LSeg ((SW-corner D2),(SE-corner D2)) misses LSeg ((NW-corner D2),(NE-corner D2))

proof end;

definition

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

func SpStSeq C -> FinSequence of (TOP-REAL 2) equals :: SPRECT_1:def 1
<*(NW-corner C),(NE-corner C),(SE-corner C)*> ^ <*(SW-corner C),(NW-corner C)*>;

coherence
<*(NW-corner C),(NE-corner C),(SE-corner C)*> ^ <*(SW-corner C),(NW-corner C)*> is FinSequence of (TOP-REAL 2) ;

end;

:: deftheorem defines SpStSeq SPRECT_1:def 1 :
for C being Subset of (TOP-REAL 2) holds SpStSeq C = <*(NW-corner C),(NE-corner C),(SE-corner C)*> ^ <*(SW-corner C),(NW-corner C)*>;

theorem Th35: :: SPRECT_1:35

for S being Subset of (TOP-REAL 2) holds (SpStSeq S) /. 1 = NW-corner S

proof end;

theorem Th36: :: SPRECT_1:36

for S being Subset of (TOP-REAL 2) holds (SpStSeq S) /. 2 = NE-corner S

proof end;

theorem Th37: :: SPRECT_1:37

for S being Subset of (TOP-REAL 2) holds (SpStSeq S) /. 3 = SE-corner S

proof end;

theorem Th38: :: SPRECT_1:38

for S being Subset of (TOP-REAL 2) holds (SpStSeq S) /. 4 = SW-corner S

proof end;

theorem :: SPRECT_1:39

for S being Subset of (TOP-REAL 2) holds (SpStSeq S) /. 5 = NW-corner S

proof end;

theorem Th40: :: SPRECT_1:40

for S being Subset of (TOP-REAL 2) holds len (SpStSeq S) = 5

proof end;

theorem Th41: :: SPRECT_1:41

for S being Subset of (TOP-REAL 2) holds L~ (SpStSeq S) = ((LSeg ((NW-corner S),(NE-corner S))) \/ (LSeg ((NE-corner S),(SE-corner S)))) \/ ((LSeg ((SE-corner S),(SW-corner S))) \/ (LSeg ((SW-corner S),(NW-corner S))))

proof end;

registration

let D be non empty compact non vertical Subset of (TOP-REAL 2);

cluster SpStSeq D -> non constant ;

coherence
not SpStSeq D is constant

proof end;

end;

registration

let D be non empty compact non horizontal Subset of (TOP-REAL 2);

cluster SpStSeq D -> non constant ;

coherence
not SpStSeq D is constant

proof end;

end;

registration

let D be non empty compact non horizontal non vertical Subset of (TOP-REAL 2);

cluster SpStSeq D -> circular special unfolded s.c.c. standard ;

coherence
( SpStSeq D is special & SpStSeq D is unfolded & SpStSeq D is circular & SpStSeq D is s.c.c. & SpStSeq D is standard )

proof end;

end;

theorem Th42: :: SPRECT_1:42

for D being non empty compact non horizontal non vertical Subset of (TOP-REAL 2) holds L~ (SpStSeq D) = [.(W-bound D),(E-bound D),(S-bound D),(N-bound D).]

proof end;

registration

let T be non empty TopSpace;
let X be non empty compact Subset of T;
let f be continuous RealMap of T;

cluster f .: X -> bounded_below ;

coherence
f .: X is bounded_below

proof end;

cluster f .: X -> bounded_above ;

coherence
f .: X is bounded_above

proof end;

end;

theorem Th43: :: SPRECT_1:43

for S being Subset of (TOP-REAL 2) holds W-bound S = lower_bound (proj1 .: S)

proof end;

theorem Th44: :: SPRECT_1:44

for S being Subset of (TOP-REAL 2) holds S-bound S = lower_bound (proj2 .: S)

proof end;

theorem Th45: :: SPRECT_1:45

for S being Subset of (TOP-REAL 2) holds N-bound S = upper_bound (proj2 .: S)

proof end;

theorem Th46: :: SPRECT_1:46

for S being Subset of (TOP-REAL 2) holds E-bound S = upper_bound (proj1 .: S)

proof end;

theorem Th47: :: SPRECT_1:47

for S being Subset of (TOP-REAL 2)
for C1, C2 being non empty compact Subset of (TOP-REAL 2) st S = C1 \/ C2 holds
W-bound S = min ((W-bound C1),(W-bound C2))

proof end;

theorem Th48: :: SPRECT_1:48

for S being Subset of (TOP-REAL 2)
for C1, C2 being non empty compact Subset of (TOP-REAL 2) st S = C1 \/ C2 holds
S-bound S = min ((S-bound C1),(S-bound C2))

proof end;

theorem Th49: :: SPRECT_1:49

for S being Subset of (TOP-REAL 2)
for C1, C2 being non empty compact Subset of (TOP-REAL 2) st S = C1 \/ C2 holds
N-bound S = max ((N-bound C1),(N-bound C2))

proof end;

theorem Th50: :: SPRECT_1:50

for S being Subset of (TOP-REAL 2)
for C1, C2 being non empty compact Subset of (TOP-REAL 2) st S = C1 \/ C2 holds
E-bound S = max ((E-bound C1),(E-bound C2))

proof end;

registration

let r1, r2 be Real;

cluster K74(r1,r2) -> real-bounded for Subset of REAL;

coherence
for b₁ being Subset of REAL st b₁ = [.r1,r2.] holds
b₁ is real-bounded

proof end;

end;

theorem Th51: :: SPRECT_1:51

for t, r1, r2 being Real st r1 <= r2 holds
( t in [.r1,r2.] iff ex s1 being Real st
( 0 <= s1 & s1 <= 1 & t = (s1 * r1) + ((1 - s1) * r2) ) )

proof end;

theorem Th52: :: SPRECT_1:52

for p, q being Point of (TOP-REAL 2) st p `1 <= q `1 holds
proj1 .: (LSeg (p,q)) = [.(p `1),(q `1).]

proof end;

theorem Th53: :: SPRECT_1:53

for p, q being Point of (TOP-REAL 2) st p `2 <= q `2 holds
proj2 .: (LSeg (p,q)) = [.(p `2),(q `2).]

proof end;

theorem Th54: :: SPRECT_1:54

for p, q being Point of (TOP-REAL 2) st p `1 <= q `1 holds
W-bound (LSeg (p,q)) = p `1

proof end;

theorem Th55: :: SPRECT_1:55

for p, q being Point of (TOP-REAL 2) st p `2 <= q `2 holds
S-bound (LSeg (p,q)) = p `2

proof end;

theorem Th56: :: SPRECT_1:56

for p, q being Point of (TOP-REAL 2) st p `2 <= q `2 holds
N-bound (LSeg (p,q)) = q `2

proof end;

theorem Th57: :: SPRECT_1:57

for p, q being Point of (TOP-REAL 2) st p `1 <= q `1 holds
E-bound (LSeg (p,q)) = q `1

proof end;

theorem Th58: :: SPRECT_1:58

for C being non empty compact Subset of (TOP-REAL 2) holds W-bound (L~ (SpStSeq C)) = W-bound C

proof end;

theorem Th59: :: SPRECT_1:59

for C being non empty compact Subset of (TOP-REAL 2) holds S-bound (L~ (SpStSeq C)) = S-bound C

proof end;

theorem Th60: :: SPRECT_1:60

for C being non empty compact Subset of (TOP-REAL 2) holds N-bound (L~ (SpStSeq C)) = N-bound C

proof end;

theorem Th61: :: SPRECT_1:61

for C being non empty compact Subset of (TOP-REAL 2) holds E-bound (L~ (SpStSeq C)) = E-bound C

proof end;

theorem Th62: :: SPRECT_1:62

for C being non empty compact Subset of (TOP-REAL 2) holds NW-corner (L~ (SpStSeq C)) = NW-corner C

proof end;

theorem Th63: :: SPRECT_1:63

for C being non empty compact Subset of (TOP-REAL 2) holds NE-corner (L~ (SpStSeq C)) = NE-corner C

proof end;

theorem Th64: :: SPRECT_1:64

for C being non empty compact Subset of (TOP-REAL 2) holds SW-corner (L~ (SpStSeq C)) = SW-corner C

proof end;

theorem Th65: :: SPRECT_1:65

for C being non empty compact Subset of (TOP-REAL 2) holds SE-corner (L~ (SpStSeq C)) = SE-corner C

proof end;

theorem Th66: :: SPRECT_1:66

for C being non empty compact Subset of (TOP-REAL 2) holds W-most (L~ (SpStSeq C)) = LSeg ((SW-corner C),(NW-corner C))

proof end;

theorem Th67: :: SPRECT_1:67

for C being non empty compact Subset of (TOP-REAL 2) holds N-most (L~ (SpStSeq C)) = LSeg ((NW-corner C),(NE-corner C))

proof end;

theorem Th68: :: SPRECT_1:68

for C being non empty compact Subset of (TOP-REAL 2) holds S-most (L~ (SpStSeq C)) = LSeg ((SW-corner C),(SE-corner C))

proof end;

theorem Th69: :: SPRECT_1:69

for C being non empty compact Subset of (TOP-REAL 2) holds E-most (L~ (SpStSeq C)) = LSeg ((SE-corner C),(NE-corner C))

proof end;

theorem Th70: :: SPRECT_1:70

for C being non empty compact Subset of (TOP-REAL 2) holds proj2 .: (LSeg ((SW-corner C),(NW-corner C))) = [.(S-bound C),(N-bound C).]

proof end;

theorem Th71: :: SPRECT_1:71

for C being non empty compact Subset of (TOP-REAL 2) holds proj1 .: (LSeg ((NW-corner C),(NE-corner C))) = [.(W-bound C),(E-bound C).]

proof end;

theorem Th72: :: SPRECT_1:72

for C being non empty compact Subset of (TOP-REAL 2) holds proj2 .: (LSeg ((NE-corner C),(SE-corner C))) = [.(S-bound C),(N-bound C).]

proof end;

theorem Th73: :: SPRECT_1:73

for C being non empty compact Subset of (TOP-REAL 2) holds proj1 .: (LSeg ((SE-corner C),(SW-corner C))) = [.(W-bound C),(E-bound C).]

proof end;

theorem Th74: :: SPRECT_1:74

for C being non empty compact Subset of (TOP-REAL 2) holds W-min (L~ (SpStSeq C)) = SW-corner C

proof end;

theorem Th75: :: SPRECT_1:75

for C being non empty compact Subset of (TOP-REAL 2) holds W-max (L~ (SpStSeq C)) = NW-corner C

proof end;

theorem Th76: :: SPRECT_1:76

for C being non empty compact Subset of (TOP-REAL 2) holds N-min (L~ (SpStSeq C)) = NW-corner C

proof end;

theorem Th77: :: SPRECT_1:77

for C being non empty compact Subset of (TOP-REAL 2) holds N-max (L~ (SpStSeq C)) = NE-corner C

proof end;

theorem Th78: :: SPRECT_1:78

for C being non empty compact Subset of (TOP-REAL 2) holds E-min (L~ (SpStSeq C)) = SE-corner C

proof end;

theorem Th79: :: SPRECT_1:79

for C being non empty compact Subset of (TOP-REAL 2) holds E-max (L~ (SpStSeq C)) = NE-corner C

proof end;

theorem Th80: :: SPRECT_1:80

for C being non empty compact Subset of (TOP-REAL 2) holds S-min (L~ (SpStSeq C)) = SW-corner C

proof end;

theorem Th81: :: SPRECT_1:81

for C being non empty compact Subset of (TOP-REAL 2) holds S-max (L~ (SpStSeq C)) = SE-corner C

proof end;

definition

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

attr f is rectangular means :Def2: :: SPRECT_1:def 2
ex D being non empty compact non horizontal non vertical Subset of (TOP-REAL 2) st f = SpStSeq D;

end;

:: deftheorem Def2 defines rectangular SPRECT_1:def 2 :
for f being FinSequence of (TOP-REAL 2) holds
( f is rectangular iff ex D being non empty compact non horizontal non vertical Subset of (TOP-REAL 2) st f = SpStSeq D );

registration

let D be non empty compact non horizontal non vertical Subset of (TOP-REAL 2);

cluster SpStSeq D -> rectangular ;

coherence
SpStSeq D is rectangular ;

end;

registration

cluster V1() V4( NAT ) V5( the carrier of (TOP-REAL 2)) Function-like V35() FinSequence-like FinSubsequence-like rectangular for FinSequence of the carrier of (TOP-REAL 2);

existence
ex b₁ being FinSequence of (TOP-REAL 2) st b₁ is rectangular

proof end;

end;

theorem :: SPRECT_1:82

for s being rectangular FinSequence of (TOP-REAL 2) holds len s = 5

proof end;

registration

cluster rectangular -> non constant for FinSequence of the carrier of (TOP-REAL 2);

coherence
for b₁ being FinSequence of (TOP-REAL 2) st b₁ is rectangular holds
not b₁ is constant ;

end;

registration

cluster non empty rectangular -> non empty circular special unfolded s.c.c. standard for FinSequence of the carrier of (TOP-REAL 2);

coherence
for b₁ being non empty FinSequence of (TOP-REAL 2) st b₁ is rectangular holds
( b₁ is standard & b₁ is special & b₁ is unfolded & b₁ is circular & b₁ is s.c.c. ) ;

end;

:: Special points of L~f, f - rectangular

theorem :: SPRECT_1:83

for s being rectangular FinSequence of (TOP-REAL 2) holds
( s /. 1 = N-min (L~ s) & s /. 1 = W-max (L~ s) )

proof end;

theorem :: SPRECT_1:84

for s being rectangular FinSequence of (TOP-REAL 2) holds
( s /. 2 = N-max (L~ s) & s /. 2 = E-max (L~ s) )

proof end;

theorem :: SPRECT_1:85

for s being rectangular FinSequence of (TOP-REAL 2) holds
( s /. 3 = S-max (L~ s) & s /. 3 = E-min (L~ s) )

proof end;

theorem :: SPRECT_1:86

for s being rectangular FinSequence of (TOP-REAL 2) holds
( s /. 4 = S-min (L~ s) & s /. 4 = W-min (L~ s) )

proof end;

theorem Th87: :: SPRECT_1:87

for r1, r2, s1, s2 being Real st r1 < r2 & s1 < s2 holds
[.r1,r2,s1,s2.] is Jordan

proof end;

registration

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

cluster L~ f -> Jordan ;

coherence
L~ f is Jordan

proof end;

end;

definition

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

redefine attr S is Jordan means :Def3: :: SPRECT_1:def 3
( S ` <> {} & ex A1, A2 being Subset of (TOP-REAL 2) st
( S ` = A1 \/ A2 & A1 misses A2 & (Cl A1) \ A1 = (Cl A2) \ A2 & A1 is_a_component_of S ` & A2 is_a_component_of S ` ) );

compatibility
( S is Jordan iff ( S ` <> {} & ex A1, A2 being Subset of (TOP-REAL 2) st
( S ` = A1 \/ A2 & A1 misses A2 & (Cl A1) \ A1 = (Cl A2) \ A2 & A1 is_a_component_of S ` & A2 is_a_component_of S ` ) ) )

proof end;

end;

:: deftheorem Def3 defines Jordan SPRECT_1:def 3 :
for S being Subset of (TOP-REAL 2) holds
( S is Jordan iff ( S ` <> {} & ex A1, A2 being Subset of (TOP-REAL 2) st
( S ` = A1 \/ A2 & A1 misses A2 & (Cl A1) \ A1 = (Cl A2) \ A2 & A1 is_a_component_of S ` & A2 is_a_component_of S ` ) ) );

theorem Th88: :: SPRECT_1:88

for f being rectangular FinSequence of (TOP-REAL 2) holds LeftComp f misses RightComp f

proof end;

registration

let f be non constant standard special_circular_sequence;

cluster LeftComp f -> non empty ;

coherence
not LeftComp f is empty

proof end;

cluster RightComp f -> non empty ;

coherence
not RightComp f is empty

proof end;

end;

theorem :: SPRECT_1:89

for f being rectangular FinSequence of (TOP-REAL 2) holds LeftComp f <> RightComp f

proof end;