:: On the General Position of Special Polygons
:: by Mariusz Giero
::
:: Received May 27, 2002
:: Copyright (c) 2002-2021 Association of Mizar Users


Lm1: for f being FinSequence st dom f is trivial holds
len f is trivial
proof end;

Lm2: for f being FinSequence st f is trivial holds
len f is trivial
proof end;

theorem :: JORDAN12:1
for i being Nat st 1 < i holds
0 < i -' 1
proof end;

theorem Th2: :: JORDAN12:2
1 is odd
proof end;

theorem Th3: :: JORDAN12:3
for n being Nat
for f being FinSequence of (TOP-REAL n)
for i being Nat st 1 <= i & i + 1 <= len f holds
( f /. i in rng f & f /. (i + 1) in rng f )
proof end;

registration
cluster s.n.c. -> s.c.c. for FinSequence of the carrier of (TOP-REAL 2);
coherence
for b1 being FinSequence of (TOP-REAL 2) st b1 is s.n.c. holds
b1 is s.c.c. ;
end;

theorem Th4: :: JORDAN12:4
for f, g being FinSequence of (TOP-REAL 2) st f ^' g is unfolded & f ^' g is s.c.c. & len g >= 2 holds
( f is unfolded & f is s.n.c. )
proof end;

theorem Th5: :: JORDAN12:5
for g1, g2 being FinSequence of (TOP-REAL 2) holds L~ g1 c= L~ (g1 ^' g2)
proof end;

definition
let n be Nat;
let f1, f2 be FinSequence of (TOP-REAL n);
pred f1 is_in_general_position_wrt f2 means :: JORDAN12:def 1
( L~ f1 misses rng f2 & ( for i being Nat st 1 <= i & i < len f2 holds
(L~ f1) /\ (LSeg (f2,i)) is trivial ) );
end;

:: deftheorem defines is_in_general_position_wrt JORDAN12:def 1 :
for n being Nat
for f1, f2 being FinSequence of (TOP-REAL n) holds
( f1 is_in_general_position_wrt f2 iff ( L~ f1 misses rng f2 & ( for i being Nat st 1 <= i & i < len f2 holds
(L~ f1) /\ (LSeg (f2,i)) is trivial ) ) );

definition
let n be Nat;
let f1, f2 be FinSequence of (TOP-REAL n);
pred f1,f2 are_in_general_position means :: JORDAN12:def 2
( f1 is_in_general_position_wrt f2 & f2 is_in_general_position_wrt f1 );
symmetry
for f1, f2 being FinSequence of (TOP-REAL n) st f1 is_in_general_position_wrt f2 & f2 is_in_general_position_wrt f1 holds
( f2 is_in_general_position_wrt f1 & f1 is_in_general_position_wrt f2 ) ;
end;

:: deftheorem defines are_in_general_position JORDAN12:def 2 :
for n being Nat
for f1, f2 being FinSequence of (TOP-REAL n) holds
( f1,f2 are_in_general_position iff ( f1 is_in_general_position_wrt f2 & f2 is_in_general_position_wrt f1 ) );

theorem Th6: :: JORDAN12:6
for k being Nat
for f1, f2 being FinSequence of (TOP-REAL 2) st f1,f2 are_in_general_position holds
for f being FinSequence of (TOP-REAL 2) st f = f2 | (Seg k) holds
f1,f are_in_general_position
proof end;

theorem Th7: :: JORDAN12:7
for f1, f2, g1, g2 being FinSequence of (TOP-REAL 2) st f1 ^' f2,g1 ^' g2 are_in_general_position holds
f1 ^' f2,g1 are_in_general_position
proof end;

theorem Th8: :: JORDAN12:8
for k being Nat
for f, g being FinSequence of (TOP-REAL 2) st 1 <= k & k + 1 <= len g & f,g are_in_general_position holds
( g . k in (L~ f) ` & g . (k + 1) in (L~ f) ` )
proof end;

theorem Th9: :: JORDAN12:9
for f1, f2 being FinSequence of (TOP-REAL 2) st f1,f2 are_in_general_position holds
for i, j being Nat st 1 <= i & i + 1 <= len f1 & 1 <= j & j + 1 <= len f2 holds
(LSeg (f1,i)) /\ (LSeg (f2,j)) is trivial
proof end;

theorem Th10: :: JORDAN12:10
for f, g being FinSequence of (TOP-REAL 2) holds INTERSECTION ( { (LSeg (f,i)) where i is Nat : ( 1 <= i & i + 1 <= len f ) } , { (LSeg (g,j)) where j is Nat : ( 1 <= j & j + 1 <= len g ) } ) is finite
proof end;

theorem Th11: :: JORDAN12:11
for f, g being FinSequence of (TOP-REAL 2) st f,g are_in_general_position holds
(L~ f) /\ (L~ g) is finite
proof end;

theorem Th12: :: JORDAN12:12
for f, g being FinSequence of (TOP-REAL 2) st f,g are_in_general_position holds
for k being Nat holds (L~ f) /\ (LSeg (g,k)) is finite
proof end;

theorem Th13: :: JORDAN12:13
for f being non constant standard special_circular_sequence
for p1, p2 being Point of (TOP-REAL 2) st LSeg (p1,p2) misses L~ f holds
ex C being Subset of (TOP-REAL 2) st
( C is_a_component_of (L~ f) ` & p1 in C & p2 in C )
proof end;

theorem Th14: :: JORDAN12:14
for a, b being set
for f being non constant standard special_circular_sequence holds
( ( ( a in RightComp f & b in RightComp f ) or ( a in LeftComp f & b in LeftComp f ) ) iff ex C being Subset of (TOP-REAL 2) st
( C is_a_component_of (L~ f) ` & a in C & b in C ) ) by JORDAN1H:24, GOBOARD9:def 1, GOBOARD9:def 2;

theorem Th15: :: JORDAN12:15
for a, b being set
for f being non constant standard special_circular_sequence holds
( ( a in (L~ f) ` & b in (L~ f) ` & ( for C being Subset of (TOP-REAL 2) holds
( not C is_a_component_of (L~ f) ` or not a in C or not b in C ) ) ) iff ( ( a in LeftComp f & b in RightComp f ) or ( a in RightComp f & b in LeftComp f ) ) )
proof end;

theorem Th16: :: JORDAN12:16
for f being non constant standard special_circular_sequence
for a, b, c being set st ex C being Subset of (TOP-REAL 2) st
( C is_a_component_of (L~ f) ` & a in C & b in C ) & ex C being Subset of (TOP-REAL 2) st
( C is_a_component_of (L~ f) ` & b in C & c in C ) holds
ex C being Subset of (TOP-REAL 2) st
( C is_a_component_of (L~ f) ` & a in C & c in C )
proof end;

theorem Th17: :: JORDAN12:17
for f being non constant standard special_circular_sequence
for a, b, c being set st a in (L~ f) ` & b in (L~ f) ` & c in (L~ f) ` & ( for C being Subset of (TOP-REAL 2) holds
( not C is_a_component_of (L~ f) ` or not a in C or not b in C ) ) & ( for C being Subset of (TOP-REAL 2) holds
( not C is_a_component_of (L~ f) ` or not b in C or not c in C ) ) holds
ex C being Subset of (TOP-REAL 2) st
( C is_a_component_of (L~ f) ` & a in C & c in C )
proof end;

Lm3: now :: thesis: for G being Go-board
for i being Nat st i <= len G holds
for w1, w2 being Point of (TOP-REAL 2) st w1 in v_strip (G,i) & w2 in v_strip (G,i) & w1 `1 <= w2 `1 holds
LSeg (w1,w2) c= v_strip (G,i)
let G be Go-board; :: thesis: for i being Nat st i <= len G holds
for w1, w2 being Point of (TOP-REAL 2) st w1 in v_strip (G,i) & w2 in v_strip (G,i) & w1 `1 <= w2 `1 holds
LSeg (w1,w2) c= v_strip (G,i)
let i be Nat; :: thesis: ( i <= len G implies for w1, w2 being Point of (TOP-REAL 2) st w1 in v_strip (G,i) & w2 in v_strip (G,i) & w1 `1 <= w2 `1 holds
LSeg (w1,w2) c= v_strip (G,i) )
assume A1: i <= len G ; :: thesis: for w1, w2 being Point of (TOP-REAL 2) st w1 in v_strip (G,i) & w2 in v_strip (G,i) & w1 `1 <= w2 `1 holds
LSeg (w1,w2) c= v_strip (G,i)
let w1, w2 be Point of (TOP-REAL 2); :: thesis: ( w1 in v_strip (G,i) & w2 in v_strip (G,i) & w1 `1 <= w2 `1 implies LSeg (w1,w2) c= v_strip (G,i) )
assume that
A2: w1 in v_strip (G,i) and
A3: w2 in v_strip (G,i) and
A4: w1 `1 <= w2 `1 ; :: thesis: LSeg (w1,w2) c= v_strip (G,i)
thus LSeg (w1,w2) c= v_strip (G,i) :: thesis: verum
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in LSeg (w1,w2) or x in v_strip (G,i) )
assume A5: x in LSeg (w1,w2) ; :: thesis: x in v_strip (G,i)
reconsider p = x as Point of (TOP-REAL 2) by A5;
A6: w1 `1 <= p `1 by A4, A5, TOPREAL1:3;
A7: p `1 <= w2 `1 by A4, A5, TOPREAL1:3;
A8: p = |[(p `1),(p `2)]| by EUCLID:53;
per cases ( i = 0 or i = len G or ( 1 <= i & i < len G ) ) by A1, NAT_1:14, XXREAL_0:1;
suppose i = 0 ; :: thesis: x in v_strip (G,i)
then A9: v_strip (G,i) = { |[r,s]| where r, s is Real : r <= (G * (1,1)) `1 } by GOBRD11:18;
then ex r1, s1 being Real st
( w2 = |[r1,s1]| & r1 <= (G * (1,1)) `1 ) by A3;
then w2 `1 <= (G * (1,1)) `1 by EUCLID:52;
then p `1 <= (G * (1,1)) `1 by A7, XXREAL_0:2;
hence x in v_strip (G,i) by A8, A9; :: thesis: verum
end;
suppose i = len G ; :: thesis: x in v_strip (G,i)
then A10: v_strip (G,i) = { |[r,s]| where r, s is Real : (G * ((len G),1)) `1 <= r } by GOBRD11:19;
then ex r1, s1 being Real st
( w1 = |[r1,s1]| & (G * ((len G),1)) `1 <= r1 ) by A2;
then (G * ((len G),1)) `1 <= w1 `1 by EUCLID:52;
then (G * ((len G),1)) `1 <= p `1 by A6, XXREAL_0:2;
hence x in v_strip (G,i) by A8, A10; :: thesis: verum
end;
suppose ( 1 <= i & i < len G ) ; :: thesis: x in v_strip (G,i)
then A11: v_strip (G,i) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 <= r & r <= (G * ((i + 1),1)) `1 ) } by GOBRD11:20;
then ex r2, s2 being Real st
( w2 = |[r2,s2]| & (G * (i,1)) `1 <= r2 & r2 <= (G * ((i + 1),1)) `1 ) by A3;
then w2 `1 <= (G * ((i + 1),1)) `1 by EUCLID:52;
then A12: p `1 <= (G * ((i + 1),1)) `1 by A7, XXREAL_0:2;
ex r1, s1 being Real st
( w1 = |[r1,s1]| & (G * (i,1)) `1 <= r1 & r1 <= (G * ((i + 1),1)) `1 ) by A2, A11;
then (G * (i,1)) `1 <= w1 `1 by EUCLID:52;
then (G * (i,1)) `1 <= p `1 by A6, XXREAL_0:2;
hence x in v_strip (G,i) by A8, A11, A12; :: thesis: verum
end;
end;
end;
end;

theorem Th18: :: JORDAN12:18
for i being Nat
for G being Go-board st i <= len G holds
v_strip (G,i) is convex
proof end;

Lm4: now :: thesis: for G being Go-board
for j being Nat st j <= width G holds
for w1, w2 being Point of (TOP-REAL 2) st w1 in h_strip (G,j) & w2 in h_strip (G,j) & w1 `2 <= w2 `2 holds
LSeg (w1,w2) c= h_strip (G,j)
let G be Go-board; :: thesis: for j being Nat st j <= width G holds
for w1, w2 being Point of (TOP-REAL 2) st w1 in h_strip (G,j) & w2 in h_strip (G,j) & w1 `2 <= w2 `2 holds
LSeg (w1,w2) c= h_strip (G,j)
let j be Nat; :: thesis: ( j <= width G implies for w1, w2 being Point of (TOP-REAL 2) st w1 in h_strip (G,j) & w2 in h_strip (G,j) & w1 `2 <= w2 `2 holds
LSeg (w1,w2) c= h_strip (G,j) )
assume A1: j <= width G ; :: thesis: for w1, w2 being Point of (TOP-REAL 2) st w1 in h_strip (G,j) & w2 in h_strip (G,j) & w1 `2 <= w2 `2 holds
LSeg (w1,w2) c= h_strip (G,j)
let w1, w2 be Point of (TOP-REAL 2); :: thesis: ( w1 in h_strip (G,j) & w2 in h_strip (G,j) & w1 `2 <= w2 `2 implies LSeg (w1,w2) c= h_strip (G,j) )
assume that
A2: w1 in h_strip (G,j) and
A3: w2 in h_strip (G,j) and
A4: w1 `2 <= w2 `2 ; :: thesis: LSeg (w1,w2) c= h_strip (G,j)
thus LSeg (w1,w2) c= h_strip (G,j) :: thesis: verum
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in LSeg (w1,w2) or x in h_strip (G,j) )
assume A5: x in LSeg (w1,w2) ; :: thesis: x in h_strip (G,j)
then reconsider p = x as Point of (TOP-REAL 2) ;
A6: w1 `2 <= p `2 by A4, A5, TOPREAL1:4;
A7: p `2 <= w2 `2 by A4, A5, TOPREAL1:4;
A8: p = |[(p `1),(p `2)]| by EUCLID:53;
per cases ( j = 0 or j = width G or ( 1 <= j & j < width G ) ) by A1, NAT_1:14, XXREAL_0:1;
suppose j = 0 ; :: thesis: x in h_strip (G,j)
then A9: h_strip (G,j) = { |[r,s]| where r, s is Real : s <= (G * (1,1)) `2 } by GOBRD11:21;
then ex r1, s1 being Real st
( w2 = |[r1,s1]| & s1 <= (G * (1,1)) `2 ) by A3;
then w2 `2 <= (G * (1,1)) `2 by EUCLID:52;
then p `2 <= (G * (1,1)) `2 by A7, XXREAL_0:2;
hence x in h_strip (G,j) by A8, A9; :: thesis: verum
end;
suppose j = width G ; :: thesis: x in h_strip (G,j)
then A10: h_strip (G,j) = { |[r,s]| where r, s is Real : (G * (1,(width G))) `2 <= s } by GOBRD11:22;
then ex r1, s1 being Real st
( w1 = |[r1,s1]| & (G * (1,(width G))) `2 <= s1 ) by A2;
then (G * (1,(width G))) `2 <= w1 `2 by EUCLID:52;
then (G * (1,(width G))) `2 <= p `2 by A6, XXREAL_0:2;
hence x in h_strip (G,j) by A8, A10; :: thesis: verum
end;
suppose ( 1 <= j & j < width G ) ; :: thesis: x in h_strip (G,j)
then A11: h_strip (G,j) = { |[r,s]| where r, s is Real : ( (G * (1,j)) `2 <= s & s <= (G * (1,(j + 1))) `2 ) } by GOBRD11:23;
then ex r2, s2 being Real st
( w2 = |[r2,s2]| & (G * (1,j)) `2 <= s2 & s2 <= (G * (1,(j + 1))) `2 ) by A3;
then w2 `2 <= (G * (1,(j + 1))) `2 by EUCLID:52;
then A12: p `2 <= (G * (1,(j + 1))) `2 by A7, XXREAL_0:2;
ex r1, s1 being Real st
( w1 = |[r1,s1]| & (G * (1,j)) `2 <= s1 & s1 <= (G * (1,(j + 1))) `2 ) by A2, A11;
then (G * (1,j)) `2 <= w1 `2 by EUCLID:52;
then (G * (1,j)) `2 <= p `2 by A6, XXREAL_0:2;
hence x in h_strip (G,j) by A8, A11, A12; :: thesis: verum
end;
end;
end;
end;

theorem Th19: :: JORDAN12:19
for j being Nat
for G being Go-board st j <= width G holds
h_strip (G,j) is convex
proof end;

theorem Th20: :: JORDAN12:20
for i, j being Nat
for G being Go-board st i <= len G & j <= width G holds
cell (G,i,j) is convex
proof end;

theorem Th21: :: JORDAN12:21
for f being non constant standard special_circular_sequence
for k being Nat st 1 <= k & k + 1 <= len f holds
left_cell (f,k) is convex
proof end;

theorem Th22: :: JORDAN12:22
for f being non constant standard special_circular_sequence
for k being Nat st 1 <= k & k + 1 <= len f holds
( left_cell (f,k,(GoB f)) is convex & right_cell (f,k,(GoB f)) is convex )
proof end;

theorem Th23: :: JORDAN12:23
for p1, p2 being Point of (TOP-REAL 2)
for f being non constant standard special_circular_sequence
for r being Point of (TOP-REAL 2) st r in LSeg (p1,p2) & ex x being set st (L~ f) /\ (LSeg (p1,p2)) = {x} & not r in L~ f & not L~ f misses LSeg (p1,r) holds
L~ f misses LSeg (r,p2)
proof end;

Lm5: now :: thesis: for p1, p2 being Point of (TOP-REAL 2)
for f being non constant standard special_circular_sequence
for r being Point of (TOP-REAL 2) st r in LSeg (p1,p2) & ex x being set st (L~ f) /\ (LSeg (p1,p2)) = {x} & not r in L~ f & ( for C being Subset of (TOP-REAL 2) holds
( not C is_a_component_of (L~ f) ` or not r in C or not p1 in C ) ) holds
ex C being Subset of (TOP-REAL 2) st
( C is_a_component_of (L~ f) ` & r in C & p2 in C )
let p1, p2 be Point of (TOP-REAL 2); :: thesis: for f being non constant standard special_circular_sequence
for r being Point of (TOP-REAL 2) st r in LSeg (p1,p2) & ex x being set st (L~ f) /\ (LSeg (p1,p2)) = {x} & not r in L~ f & ( for C being Subset of (TOP-REAL 2) holds
( not b7 is_a_component_of (L~ b5) ` or not b6 in b7 or not C in b7 ) ) holds
ex C being Subset of (TOP-REAL 2) st
( b7 is_a_component_of (L~ b5) ` & b6 in b7 & b4 in b7 )
let f be non constant standard special_circular_sequence; :: thesis: for r being Point of (TOP-REAL 2) st r in LSeg (p1,p2) & ex x being set st (L~ f) /\ (LSeg (p1,p2)) = {x} & not r in L~ f & ( for C being Subset of (TOP-REAL 2) holds
( not b6 is_a_component_of (L~ b4) ` or not b5 in b6 or not C in b6 ) ) holds
ex C being Subset of (TOP-REAL 2) st
( b6 is_a_component_of (L~ b4) ` & b5 in b6 & b3 in b6 )
let r be Point of (TOP-REAL 2); :: thesis: ( r in LSeg (p1,p2) & ex x being set st (L~ f) /\ (LSeg (p1,p2)) = {x} & not r in L~ f & ( for C being Subset of (TOP-REAL 2) holds
( not b5 is_a_component_of (L~ b3) ` or not b4 in b5 or not C in b5 ) ) implies ex C being Subset of (TOP-REAL 2) st
( b5 is_a_component_of (L~ b3) ` & b4 in b5 & b2 in b5 ) )
assume A1: r in LSeg (p1,p2) ; :: thesis: ( ex x being set st (L~ f) /\ (LSeg (p1,p2)) = {x} & not r in L~ f & ( for C being Subset of (TOP-REAL 2) holds
( not b5 is_a_component_of (L~ b3) ` or not b4 in b5 or not C in b5 ) ) implies ex C being Subset of (TOP-REAL 2) st
( b5 is_a_component_of (L~ b3) ` & b4 in b5 & b2 in b5 ) )
assume A2: ( ex x being set st (L~ f) /\ (LSeg (p1,p2)) = {x} & not r in L~ f ) ; :: thesis: ( ex C being Subset of (TOP-REAL 2) st
( b5 is_a_component_of (L~ b3) ` & b4 in b5 & C in b5 ) or ex C being Subset of (TOP-REAL 2) st
( b5 is_a_component_of (L~ b3) ` & b4 in b5 & b2 in b5 ) )
per cases ( L~ f misses LSeg (p1,r) or L~ f misses LSeg (r,p2) ) by A1, A2, Th23;
suppose L~ f misses LSeg (p1,r) ; :: thesis: ( ex C being Subset of (TOP-REAL 2) st
( b5 is_a_component_of (L~ b3) ` & b4 in b5 & C in b5 ) or ex C being Subset of (TOP-REAL 2) st
( b5 is_a_component_of (L~ b3) ` & b4 in b5 & b2 in b5 ) )
hence ( ex C being Subset of (TOP-REAL 2) st
( C is_a_component_of (L~ f) ` & r in C & p1 in C ) or ex C being Subset of (TOP-REAL 2) st
( C is_a_component_of (L~ f) ` & r in C & p2 in C ) ) by Th13; :: thesis: verum
end;
suppose L~ f misses LSeg (r,p2) ; :: thesis: ( ex C being Subset of (TOP-REAL 2) st
( b5 is_a_component_of (L~ b3) ` & b4 in b5 & C in b5 ) or ex C being Subset of (TOP-REAL 2) st
( b5 is_a_component_of (L~ b3) ` & b4 in b5 & b2 in b5 ) )
hence ( ex C being Subset of (TOP-REAL 2) st
( C is_a_component_of (L~ f) ` & r in C & p1 in C ) or ex C being Subset of (TOP-REAL 2) st
( C is_a_component_of (L~ f) ` & r in C & p2 in C ) ) by Th13; :: thesis: verum
end;
end;
end;

theorem Th24: :: JORDAN12:24
for p, q, r, s being Point of (TOP-REAL 2) st LSeg (p,q) is vertical & LSeg (r,s) is vertical & LSeg (p,q) meets LSeg (r,s) holds
p `1 = r `1
proof end;

theorem Th25: :: JORDAN12:25
for p, p1, p2 being Point of (TOP-REAL 2) st not p in LSeg (p1,p2) & p1 `2 = p2 `2 & p2 `2 = p `2 & not p1 in LSeg (p,p2) holds
p2 in LSeg (p,p1)
proof end;

theorem Th26: :: JORDAN12:26
for p, p1, p2 being Point of (TOP-REAL 2) st not p in LSeg (p1,p2) & p1 `1 = p2 `1 & p2 `1 = p `1 & not p1 in LSeg (p,p2) holds
p2 in LSeg (p,p1)
proof end;

theorem Th27: :: JORDAN12:27
for p, p1, p2 being Point of (TOP-REAL 2) st p <> p1 & p <> p2 & p in LSeg (p1,p2) holds
not p1 in LSeg (p,p2)
proof end;

theorem Th28: :: JORDAN12:28
for p, p1, p2, q being Point of (TOP-REAL 2) st not q in LSeg (p1,p2) & p in LSeg (p1,p2) & p <> p1 & p <> p2 & ( ( p1 `1 = p2 `1 & p2 `1 = q `1 ) or ( p1 `2 = p2 `2 & p2 `2 = q `2 ) ) & not p1 in LSeg (q,p) holds
p2 in LSeg (q,p)
proof end;

theorem Th29: :: JORDAN12:29
for p1, p2, p3, p4, p being Point of (TOP-REAL 2) st ( ( p1 `1 = p2 `1 & p3 `1 = p4 `1 ) or ( p1 `2 = p2 `2 & p3 `2 = p4 `2 ) ) & (LSeg (p1,p2)) /\ (LSeg (p3,p4)) = {p} & not p = p1 & not p = p2 holds
p = p3
proof end;

theorem Th30: :: JORDAN12:30
for p, p1, p2 being Point of (TOP-REAL 2)
for f being non constant standard special_circular_sequence st (L~ f) /\ (LSeg (p1,p2)) = {p} holds
for r being Point of (TOP-REAL 2) st not r in LSeg (p1,p2) & not p1 in L~ f & not p2 in L~ f & ( ( p1 `1 = p2 `1 & p1 `1 = r `1 ) or ( p1 `2 = p2 `2 & p1 `2 = r `2 ) ) & ex i being Nat st
( 1 <= i & i + 1 <= len f & ( r in right_cell (f,i,(GoB f)) or r in left_cell (f,i,(GoB f)) ) & p in LSeg (f,i) ) & not r in L~ f & ( for C being Subset of (TOP-REAL 2) holds
( not C is_a_component_of (L~ f) ` or not r in C or not p1 in C ) ) holds
ex C being Subset of (TOP-REAL 2) st
( C is_a_component_of (L~ f) ` & r in C & p2 in C )
proof end;

theorem Th31: :: JORDAN12:31
for f being non constant standard special_circular_sequence
for p1, p2, p being Point of (TOP-REAL 2) st (L~ f) /\ (LSeg (p1,p2)) = {p} holds
for rl, rp being Point of (TOP-REAL 2) st not p1 in L~ f & not p2 in L~ f & ( ( p1 `1 = p2 `1 & p1 `1 = rl `1 & rl `1 = rp `1 ) or ( p1 `2 = p2 `2 & p1 `2 = rl `2 & rl `2 = rp `2 ) ) & ex i being Nat st
( 1 <= i & i + 1 <= len f & rl in left_cell (f,i,(GoB f)) & rp in right_cell (f,i,(GoB f)) & p in LSeg (f,i) ) & not rl in L~ f & not rp in L~ f holds
for C being Subset of (TOP-REAL 2) holds
( not C is_a_component_of (L~ f) ` or not p1 in C or not p2 in C )
proof end;

theorem Th32: :: JORDAN12:32
for p being Point of (TOP-REAL 2)
for f being non constant standard special_circular_sequence
for p1, p2 being Point of (TOP-REAL 2) st (L~ f) /\ (LSeg (p1,p2)) = {p} & ( p1 `1 = p2 `1 or p1 `2 = p2 `2 ) & not p1 in L~ f & not p2 in L~ f & rng f misses LSeg (p1,p2) holds
for C being Subset of (TOP-REAL 2) holds
( not C is_a_component_of (L~ f) ` or not p1 in C or not p2 in C )
proof end;

theorem Th33: :: JORDAN12:33
for f being non constant standard special_circular_sequence
for g being special FinSequence of (TOP-REAL 2) st f,g are_in_general_position holds
for k being Nat st 1 <= k & k + 1 <= len g holds
( card ((L~ f) /\ (LSeg (g,k))) is even Element of NAT iff ex C being Subset of (TOP-REAL 2) st
( C is_a_component_of (L~ f) ` & g . k in C & g . (k + 1) in C ) )
proof end;

theorem Th34: :: JORDAN12:34
for f1, f2, g1 being special FinSequence of (TOP-REAL 2) st f1 ^' f2 is non constant standard special_circular_sequence & f1 ^' f2,g1 are_in_general_position & len g1 >= 2 & g1 is unfolded & g1 is s.n.c. holds
( card ((L~ (f1 ^' f2)) /\ (L~ g1)) is even Element of NAT iff ex C being Subset of (TOP-REAL 2) st
( C is_a_component_of (L~ (f1 ^' f2)) ` & g1 . 1 in C & g1 . (len g1) in C ) )
proof end;

theorem :: JORDAN12:35
for f1, f2, g1, g2 being special FinSequence of (TOP-REAL 2) st f1 ^' f2 is non constant standard special_circular_sequence & g1 ^' g2 is non constant standard special_circular_sequence & L~ f1 misses L~ g2 & L~ f2 misses L~ g1 & f1 ^' f2,g1 ^' g2 are_in_general_position holds
for p1, p2, q1, q2 being Point of (TOP-REAL 2) st f1 . 1 = p1 & f1 . (len f1) = p2 & g1 . 1 = q1 & g1 . (len g1) = q2 & f1 /. (len f1) = f2 /. 1 & g1 /. (len g1) = g2 /. 1 & p1 in (L~ f1) /\ (L~ f2) & q1 in (L~ g1) /\ (L~ g2) & ex C being Subset of (TOP-REAL 2) st
( C is_a_component_of (L~ (f1 ^' f2)) ` & q1 in C & q2 in C ) holds
ex C being Subset of (TOP-REAL 2) st
( C is_a_component_of (L~ (g1 ^' g2)) ` & p1 in C & p2 in C )
proof end;