:: Properties of Left-, and Right Components
:: by Artur Korni{\l}owicz
::
:: Copyright (c) 1999-2021 Association of Mizar Users

Lm1: the carrier of () = REAL 2
by EUCLID:22;

theorem :: GOBRD14:1
for f being V8() standard special_circular_sequence
for P being Subset of ()
for p being Point of () st p = 0.REAL 2 & P is_outside_component_of L~ f holds
ex r being Real st
( r > 0 & (Ball (p,r))  c= P )
proof end;

theorem :: GOBRD14:2
for n being Nat
for f being FinSequence of () st L~ f <> {} holds
2 <= len f
proof end;

theorem Th3: :: GOBRD14:3
for i, j being Nat
for G being Go-board st 1 <= i & i < len G & 1 <= j & j < width G holds
cell (G,i,j) = product ((1,2) --> ([.((G * (i,1)) 1),((G * ((i + 1),1)) 1).],[.((G * (1,j)) 2),((G * (1,(j + 1))) 2).]))
proof end;

theorem :: GOBRD14:4
for i, j being Nat
for G being Go-board st 1 <= i & i < len G & 1 <= j & j < width G holds
cell (G,i,j) is compact
proof end;

theorem :: GOBRD14:5
for i, j, n being Nat
for G being Go-board st [i,j] in Indices G & [(i + n),j] in Indices G holds
dist ((G * (i,j)),(G * ((i + n),j))) = ((G * ((i + n),j)) 1) - ((G * (i,j)) 1)
proof end;

theorem :: GOBRD14:6
for i, j, n being Nat
for G being Go-board st [i,j] in Indices G & [i,(j + n)] in Indices G holds
dist ((G * (i,j)),(G * (i,(j + n)))) = ((G * (i,(j + n))) 2) - ((G * (i,j)) 2)
proof end;

theorem :: GOBRD14:7
for n being Nat
for C being compact non horizontal non vertical Subset of () holds 3 <= (len (Gauge (C,n))) -' 1
proof end;

theorem :: GOBRD14:8
for i, j being Nat
for C being compact non horizontal non vertical Subset of () st i <= j holds
for a, b being Nat st 2 <= a & a <= (len (Gauge (C,i))) - 1 & 2 <= b & b <= (len (Gauge (C,i))) - 1 holds
ex c, d being Nat st
( 2 <= c & c <= (len (Gauge (C,j))) - 1 & 2 <= d & d <= (len (Gauge (C,j))) - 1 & [c,d] in Indices (Gauge (C,j)) & (Gauge (C,i)) * (a,b) = (Gauge (C,j)) * (c,d) & c = 2 + ((2 |^ (j -' i)) * (a -' 2)) & d = 2 + ((2 |^ (j -' i)) * (b -' 2)) )
proof end;

theorem Th9: :: GOBRD14:9
for i, j, n being Nat
for C being compact non horizontal non vertical Subset of () st [i,j] in Indices (Gauge (C,n)) & [i,(j + 1)] in Indices (Gauge (C,n)) holds
dist (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,(j + 1)))) = (() - ()) / (2 |^ n)
proof end;

theorem Th10: :: GOBRD14:10
for i, j, n being Nat
for C being compact non horizontal non vertical Subset of () st [i,j] in Indices (Gauge (C,n)) & [(i + 1),j] in Indices (Gauge (C,n)) holds
dist (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * ((i + 1),j))) = (() - ()) / (2 |^ n)
proof end;

theorem :: GOBRD14:11
for C being compact non horizontal non vertical Subset of ()
for r, t being Real st r > 0 & t > 0 holds
ex n being Nat st
( 1 < n & dist (((Gauge (C,n)) * (1,1)),((Gauge (C,n)) * (1,2))) < r & dist (((Gauge (C,n)) * (1,1)),((Gauge (C,n)) * (2,1))) < t )
proof end;

theorem Th12: :: GOBRD14:12
for f being V8() standard special_circular_sequence
for P being Subset of (() | ((L~ f) )) holds
( not P is a_component or P = RightComp f or P = LeftComp f )
proof end;

theorem :: GOBRD14:13
for f being V8() standard special_circular_sequence
for A1, A2 being Subset of () st (L~ f)  = A1 \/ A2 & A1 misses A2 & ( for C1, C2 being Subset of (() | ((L~ f) )) st C1 = A1 & C2 = A2 holds
( C1 is a_component & C2 is a_component ) ) & not ( A1 = RightComp f & A2 = LeftComp f ) holds
( A1 = LeftComp f & A2 = RightComp f )
proof end;

theorem Th14: :: GOBRD14:14
for f being V8() standard special_circular_sequence holds LeftComp f misses RightComp f
proof end;

theorem Th15: :: GOBRD14:15
for f being V8() standard special_circular_sequence holds ((L~ f) \/ ()) \/ () = the carrier of ()
proof end;

theorem Th16: :: GOBRD14:16
for f being V8() standard special_circular_sequence
for p being Point of () holds
( p in L~ f iff ( not p in LeftComp f & not p in RightComp f ) )
proof end;

theorem Th17: :: GOBRD14:17
for f being V8() standard special_circular_sequence
for p being Point of () holds
( p in LeftComp f iff ( not p in L~ f & not p in RightComp f ) )
proof end;

theorem :: GOBRD14:18
for f being V8() standard special_circular_sequence
for p being Point of () holds
( p in RightComp f iff ( not p in L~ f & not p in LeftComp f ) ) by ;

theorem Th19: :: GOBRD14:19
for f being V8() standard special_circular_sequence holds L~ f = (Cl ()) \ ()
proof end;

theorem Th20: :: GOBRD14:20
for f being V8() standard special_circular_sequence holds L~ f = (Cl ()) \ ()
proof end;

theorem Th21: :: GOBRD14:21
for f being V8() standard special_circular_sequence holds Cl () = () \/ (L~ f)
proof end;

theorem :: GOBRD14:22
for f being V8() standard special_circular_sequence holds Cl () = () \/ (L~ f)
proof end;

registration
let f be V8() standard special_circular_sequence;
cluster L~ f -> Jordan ;
coherence
L~ f is Jordan
proof end;
end;

theorem Th23: :: GOBRD14:23
for p being Point of ()
for f being V8() standard clockwise_oriented special_circular_sequence st p in RightComp f holds
W-bound (L~ f) < p 1
proof end;

theorem Th24: :: GOBRD14:24
for p being Point of ()
for f being V8() standard clockwise_oriented special_circular_sequence st p in RightComp f holds
E-bound (L~ f) > p 1
proof end;

theorem Th25: :: GOBRD14:25
for p being Point of ()
for f being V8() standard clockwise_oriented special_circular_sequence st p in RightComp f holds
N-bound (L~ f) > p 2
proof end;

theorem Th26: :: GOBRD14:26
for p being Point of ()
for f being V8() standard clockwise_oriented special_circular_sequence st p in RightComp f holds
S-bound (L~ f) < p 2
proof end;

theorem Th27: :: GOBRD14:27
for p, q being Point of ()
for f being V8() standard clockwise_oriented special_circular_sequence st p in RightComp f & q in LeftComp f holds
LSeg (p,q) meets L~ f
proof end;

theorem Th28: :: GOBRD14:28
for C being compact non horizontal non vertical Subset of () holds Cl (RightComp ()) = product ((1,2) --> ([.(W-bound (L~ ())),(E-bound (L~ ())).],[.(S-bound (L~ ())),(N-bound (L~ ())).]))
proof end;

theorem Th29: :: GOBRD14:29
for f being V8() standard clockwise_oriented special_circular_sequence holds proj1 .: (Cl ()) = proj1 .: (L~ f)
proof end;

theorem Th30: :: GOBRD14:30
for f being V8() standard clockwise_oriented special_circular_sequence holds proj2 .: (Cl ()) = proj2 .: (L~ f)
proof end;

theorem Th31: :: GOBRD14:31
for g being V8() standard clockwise_oriented special_circular_sequence holds RightComp g c= RightComp (SpStSeq (L~ g))
proof end;

theorem Th32: :: GOBRD14:32
for g being V8() standard clockwise_oriented special_circular_sequence holds Cl () is compact
proof end;

theorem Th33: :: GOBRD14:33
for g being V8() standard clockwise_oriented special_circular_sequence holds not LeftComp g is bounded
proof end;

theorem Th34: :: GOBRD14:34

theorem :: GOBRD14:35
proof end;

theorem Th36: :: GOBRD14:36
for g being V8() standard clockwise_oriented special_circular_sequence holds UBD (L~ g) = LeftComp g
proof end;

theorem Th37: :: GOBRD14:37
for g being V8() standard clockwise_oriented special_circular_sequence holds BDD (L~ g) = RightComp g
proof end;

theorem :: GOBRD14:38
for g being V8() standard clockwise_oriented special_circular_sequence
for P being Subset of () st P is_outside_component_of L~ g holds
P = LeftComp g
proof end;

theorem Th39: :: GOBRD14:39
for g being V8() standard clockwise_oriented special_circular_sequence
for P being Subset of () st P is_inside_component_of L~ g holds
P meets RightComp g
proof end;

theorem :: GOBRD14:40
for g being V8() standard clockwise_oriented special_circular_sequence
for P being Subset of () st P is_inside_component_of L~ g holds
P = BDD (L~ g)
proof end;

theorem :: GOBRD14:41
for g being V8() standard clockwise_oriented special_circular_sequence holds W-bound (L~ g) = W-bound ()
proof end;

theorem :: GOBRD14:42
for g being V8() standard clockwise_oriented special_circular_sequence holds E-bound (L~ g) = E-bound ()
proof end;

theorem :: GOBRD14:43
for g being V8() standard clockwise_oriented special_circular_sequence holds N-bound (L~ g) = N-bound ()
proof end;

theorem :: GOBRD14:44
for g being V8() standard clockwise_oriented special_circular_sequence holds S-bound (L~ g) = S-bound ()
proof end;