:: Some Topological Properties of Cells in $R^2$
:: by Yatsuka Nakamura and Andrzej Trybulec
::
:: Copyright (c) 1996-2021 Association of Mizar Users

Lm1:
by SQUARE_1:25;

theorem Th1: :: GOBRD11:1
for GX being non empty TopSpace
for A being Subset of GX
for p being Point of GX st p in A & A is connected holds
A c= Component_of p
proof end;

theorem :: GOBRD11:2
for GX being non empty TopSpace
for A, B, C being Subset of GX st C is a_component & A c= C & B is connected & Cl A meets Cl B holds
B c= C
proof end;

theorem :: GOBRD11:3
for GZ being non empty TopSpace
for A, B being Subset of GZ st A is a_component & B is a_component holds
A \/ B is a_union_of_components of GZ
proof end;

theorem :: GOBRD11:4
for GX being non empty TopSpace
for B1, B2, V being Subset of GX holds Down ((B1 \/ B2),V) = (Down (B1,V)) \/ (Down (B2,V))
proof end;

theorem :: GOBRD11:5
for GX being non empty TopSpace
for B1, B2, V being Subset of GX holds Down ((B1 /\ B2),V) = (Down (B1,V)) /\ (Down (B2,V))
proof end;

theorem Th6: :: GOBRD11:6
for f being non constant standard special_circular_sequence holds (L~ f)  <> {}
proof end;

registration
let f be non constant standard special_circular_sequence;
cluster (L~ f)  -> non empty ;
coherence
not (L~ f)  is empty
by Th6;
end;

Lm2: the carrier of () = REAL 2
by EUCLID:22;

theorem :: GOBRD11:7
for f being non constant standard special_circular_sequence holds the carrier of () = union { (cell ((GoB f),i,j)) where i, j is Nat : ( i <= len (GoB f) & j <= width (GoB f) ) }
proof end;

Lm3: for s1 being Real holds { |[tb,sb]| where tb, sb is Real : sb >= s1 } is Subset of ()
proof end;

Lm4: for s1 being Real holds { |[tb,sb]| where tb, sb is Real : sb > s1 } is Subset of ()
proof end;

Lm5: for s1 being Real holds { |[tb,sb]| where tb, sb is Real : sb <= s1 } is Subset of ()
proof end;

Lm6: for s1 being Real holds { |[tb,sb]| where tb, sb is Real : sb < s1 } is Subset of ()
proof end;

Lm7: for s1 being Real holds { |[sb,tb]| where sb, tb is Real : sb <= s1 } is Subset of ()
proof end;

Lm8: for s1 being Real holds { |[sb,tb]| where sb, tb is Real : sb < s1 } is Subset of ()
proof end;

Lm9: for s1 being Real holds { |[sb,tb]| where sb, tb is Real : sb >= s1 } is Subset of ()
proof end;

Lm10: for s1 being Real holds { |[sb,tb]| where sb, tb is Real : sb > s1 } is Subset of ()
proof end;

theorem Th8: :: GOBRD11:8
for s1 being Real
for P1, P2 being Subset of () st P1 = { |[r,s]| where r, s is Real : s <= s1 } & P2 = { |[r2,s2]| where r2, s2 is Real : s2 > s1 } holds
P1 = P2
proof end;

theorem Th9: :: GOBRD11:9
for s1 being Real
for P1, P2 being Subset of () st P1 = { |[r,s]| where r, s is Real : s >= s1 } & P2 = { |[r2,s2]| where r2, s2 is Real : s2 < s1 } holds
P1 = P2 
proof end;

theorem Th10: :: GOBRD11:10
for s1 being Real
for P1, P2 being Subset of () st P1 = { |[s,r]| where s, r is Real : s >= s1 } & P2 = { |[s2,r2]| where s2, r2 is Real : s2 < s1 } holds
P1 = P2
proof end;

theorem Th11: :: GOBRD11:11
for s1 being Real
for P1, P2 being Subset of () st P1 = { |[s,r]| where s, r is Real : s <= s1 } & P2 = { |[s2,r2]| where s2, r2 is Real : s2 > s1 } holds
P1 = P2 
proof end;

theorem Th12: :: GOBRD11:12
for P being Subset of ()
for s1 being Real st P = { |[r,s]| where r, s is Real : s <= s1 } holds
P is closed
proof end;

theorem Th13: :: GOBRD11:13
for P being Subset of ()
for s1 being Real st P = { |[r,s]| where r, s is Real : s1 <= s } holds
P is closed
proof end;

theorem Th14: :: GOBRD11:14
for P being Subset of ()
for s1 being Real st P = { |[s,r]| where s, r is Real : s <= s1 } holds
P is closed
proof end;

theorem Th15: :: GOBRD11:15
for P being Subset of ()
for s1 being Real st P = { |[s,r]| where s, r is Real : s1 <= s } holds
P is closed
proof end;

theorem Th16: :: GOBRD11:16
for j being Nat
for G being Matrix of () holds h_strip (G,j) is closed
proof end;

theorem Th17: :: GOBRD11:17
for j being Nat
for G being Matrix of () holds v_strip (G,j) is closed
proof end;

theorem Th18: :: GOBRD11:18
for G being V9() Matrix of () st G is X_equal-in-line holds
v_strip (G,0) = { |[r,s]| where r, s is Real : r <= (G * (1,1)) 1 }
proof end;

theorem Th19: :: GOBRD11:19
for G being V9() Matrix of () st G is X_equal-in-line holds
v_strip (G,(len G)) = { |[r,s]| where r, s is Real : (G * ((len G),1)) 1 <= r }
proof end;

theorem Th20: :: GOBRD11:20
for i being Nat
for G being V9() Matrix of () st G is X_equal-in-line & 1 <= i & i < len G holds
v_strip (G,i) = { |[r,s]| where r, s is Real : ( (G * (i,1)) 1 <= r & r <= (G * ((i + 1),1)) 1 ) }
proof end;

theorem Th21: :: GOBRD11:21
for G being V9() Matrix of () st G is Y_equal-in-column holds
h_strip (G,0) = { |[r,s]| where r, s is Real : s <= (G * (1,1)) 2 }
proof end;

theorem Th22: :: GOBRD11:22
for G being V9() Matrix of () st G is Y_equal-in-column holds
h_strip (G,()) = { |[r,s]| where r, s is Real : (G * (1,())) 2 <= s }
proof end;

theorem Th23: :: GOBRD11:23
for j being Nat
for G being V9() Matrix of () st G is Y_equal-in-column & 1 <= j & j < width G holds
h_strip (G,j) = { |[r,s]| where r, s is Real : ( (G * (1,j)) 2 <= s & s <= (G * (1,(j + 1))) 2 ) }
proof end;

theorem Th24: :: GOBRD11:24
for G being V9() X_equal-in-line Y_equal-in-column Matrix of () holds cell (G,0,0) = { |[r,s]| where r, s is Real : ( r <= (G * (1,1)) 1 & s <= (G * (1,1)) 2 ) }
proof end;

theorem Th25: :: GOBRD11:25
for G being V9() X_equal-in-line Y_equal-in-column Matrix of () holds cell (G,0,()) = { |[r,s]| where r, s is Real : ( r <= (G * (1,1)) 1 & (G * (1,())) 2 <= s ) }
proof end;

theorem Th26: :: GOBRD11:26
for j being Nat
for G being V9() X_equal-in-line Y_equal-in-column Matrix of () st 1 <= j & j < width G holds
cell (G,0,j) = { |[r,s]| where r, s is Real : ( r <= (G * (1,1)) 1 & (G * (1,j)) 2 <= s & s <= (G * (1,(j + 1))) 2 ) }
proof end;

theorem Th27: :: GOBRD11:27
for G being V9() X_equal-in-line Y_equal-in-column Matrix of () holds cell (G,(len G),0) = { |[r,s]| where r, s is Real : ( (G * ((len G),1)) 1 <= r & s <= (G * (1,1)) 2 ) }
proof end;

theorem Th28: :: GOBRD11:28
for G being V9() X_equal-in-line Y_equal-in-column Matrix of () holds cell (G,(len G),()) = { |[r,s]| where r, s is Real : ( (G * ((len G),1)) 1 <= r & (G * (1,())) 2 <= s ) }
proof end;

theorem Th29: :: GOBRD11:29
for j being Nat
for G being V9() X_equal-in-line Y_equal-in-column Matrix of () st 1 <= j & j < width G holds
cell (G,(len G),j) = { |[r,s]| where r, s is Real : ( (G * ((len G),1)) 1 <= r & (G * (1,j)) 2 <= s & s <= (G * (1,(j + 1))) 2 ) }
proof end;

theorem Th30: :: GOBRD11:30
for i being Nat
for G being V9() X_equal-in-line Y_equal-in-column Matrix of () st 1 <= i & i < len G holds
cell (G,i,0) = { |[r,s]| where r, s is Real : ( (G * (i,1)) 1 <= r & r <= (G * ((i + 1),1)) 1 & s <= (G * (1,1)) 2 ) }
proof end;

theorem Th31: :: GOBRD11:31
for i being Nat
for G being V9() X_equal-in-line Y_equal-in-column Matrix of () st 1 <= i & i < len G holds
cell (G,i,()) = { |[r,s]| where r, s is Real : ( (G * (i,1)) 1 <= r & r <= (G * ((i + 1),1)) 1 & (G * (1,())) 2 <= s ) }
proof end;

theorem Th32: :: GOBRD11:32
for i, j being Nat
for G being V9() X_equal-in-line Y_equal-in-column Matrix of () st 1 <= i & i < len G & 1 <= j & j < width G holds
cell (G,i,j) = { |[r,s]| where r, s is Real : ( (G * (i,1)) 1 <= r & r <= (G * ((i + 1),1)) 1 & (G * (1,j)) 2 <= s & s <= (G * (1,(j + 1))) `2 ) }
proof end;

theorem Th33: :: GOBRD11:33
for i, j being Nat
for G being Matrix of () holds cell (G,i,j) is closed
proof end;

theorem Th34: :: GOBRD11:34
for G being V9() Matrix of () holds
( 1 <= len G & 1 <= width G )
proof end;

theorem :: GOBRD11:35
for i, j being Nat
for G being Go-board st i <= len G & j <= width G holds
cell (G,i,j) = Cl (Int (cell (G,i,j)))
proof end;