:: JORDAN12 semantic presentation
Lemma1:
for b1 being Relation holds
( b1 is trivial implies dom b1 is trivial )
Lemma2:
for b1 being FinSequence holds
( dom b1 is trivial implies len b1 is trivial )
Lemma3:
for b1 being FinSequence holds
( b1 is trivial implies len b1 is trivial )
theorem Th1: :: JORDAN12:1
for b
1 being
Nat holds
not ( 1
< b
1 & not 0
< b
1 -' 1 )
theorem Th2: :: JORDAN12:2
canceled;
theorem Th3: :: JORDAN12:3
theorem Th4: :: JORDAN12:4
theorem Th5: :: JORDAN12:5
theorem Th6: :: JORDAN12:6
:: deftheorem Def1 defines is_in_general_position_wrt JORDAN12:def 1 :
:: deftheorem Def2 defines are_in_general_position JORDAN12:def 2 :
theorem Th7: :: JORDAN12:7
theorem Th8: :: JORDAN12:8
theorem Th9: :: JORDAN12:9
theorem Th10: :: JORDAN12:10
theorem Th11: :: JORDAN12:11
theorem Th12: :: JORDAN12:12
theorem Th13: :: JORDAN12:13
theorem Th14: :: JORDAN12:14
theorem Th15: :: JORDAN12:15
theorem Th16: :: JORDAN12:16
theorem Th17: :: JORDAN12:17
theorem Th18: :: JORDAN12:18
E22:
now
let c
1 be
Go-board;
let c
2 be
Nat;
assume E23:
c
2 <= len c
1
;
let c
3, c
4 be
Point of
(TOP-REAL 2);
assume that E24:
( c
3 in v_strip c
1,c
2 & c
4 in v_strip c
1,c
2 )
and E25:
c
3 `1 <= c
4 `1
;
thus
LSeg c
3,c
4 c= v_strip c
1,c
2
proof
let c
5 be
set ;
:: according to TARSKI:def 3
assume E26:
c
5 in LSeg c
3,c
4
;
reconsider c
6 = c
5 as
Point of
(TOP-REAL 2) by E26;
E27:
c
6 = |[(c6 `1 ),(c6 `2 )]|
by EUCLID:57;
E28:
( c
3 `1 <= c
6 `1 & c
6 `1 <= c
4 `1 )
by E25, E26, TOPREAL1:9;
per cases
not ( not c2 = 0 & not c2 = len c1 & not ( 1 <= c2 & c2 < len c1 ) )
by E23, XXREAL_0:1, NAT_1:39;
suppose
c
2 = 0
;
then E29:
v_strip c
1,c
2 = { |[b1,b2]| where B is Real, B is Real : b1 <= (c1 * 1,1) `1 }
by GOBRD11:18;
then consider c
7, c
8 being
Real such that E30:
( c
4 = |[c7,c8]| & c
7 <= (c1 * 1,1) `1 )
by E24;
c
4 `1 <= (c1 * 1,1) `1
by E30, EUCLID:56;
then
c
6 `1 <= (c1 * 1,1) `1
by E28, XXREAL_0:2;
hence
c
5 in v_strip c
1,c
2
by E27, E29;
end;
suppose
c
2 = len c
1
;
then E29:
v_strip c
1,c
2 = { |[b1,b2]| where B is Real, B is Real : (c1 * (len c1),1) `1 <= b1 }
by GOBRD11:19;
then consider c
7, c
8 being
Real such that E30:
( c
3 = |[c7,c8]| &
(c1 * (len c1),1) `1 <= c
7 )
by E24;
(c1 * (len c1),1) `1 <= c
3 `1
by E30, EUCLID:56;
then
(c1 * (len c1),1) `1 <= c
6 `1
by E28, XXREAL_0:2;
hence
c
5 in v_strip c
1,c
2
by E27, E29;
end;
suppose
( 1
<= c
2 & c
2 < len c
1 )
;
then E29:
v_strip c
1,c
2 = { |[b1,b2]| where B is Real, B is Real : ( (c1 * c2,1) `1 <= b1 & b1 <= (c1 * (c2 + 1),1) `1 ) }
by GOBRD11:20;
then consider c
7, c
8 being
Real such that E30:
( c
3 = |[c7,c8]| &
(c1 * c2,1) `1 <= c
7 & c
7 <= (c1 * (c2 + 1),1) `1 )
by E24;
(
(c1 * c2,1) `1 <= c
3 `1 & c
3 `1 <= (c1 * (c2 + 1),1) `1 )
by E30, EUCLID:56;
then E31:
(c1 * c2,1) `1 <= c
6 `1
by E28, XXREAL_0:2;
consider c
9, c
10 being
Real such that E32:
( c
4 = |[c9,c10]| &
(c1 * c2,1) `1 <= c
9 & c
9 <= (c1 * (c2 + 1),1) `1 )
by E24, E29;
(
(c1 * c2,1) `1 <= c
4 `1 & c
4 `1 <= (c1 * (c2 + 1),1) `1 )
by E32, EUCLID:56;
then
c
6 `1 <= (c1 * (c2 + 1),1) `1
by E28, XXREAL_0:2;
hence
c
5 in v_strip c
1,c
2
by E27, E29, E31;
end;
end;
end;
end;
theorem Th19: :: JORDAN12:19
E24:
now
let c
1 be
Go-board;
let c
2 be
Nat;
assume E25:
c
2 <= width c
1
;
let c
3, c
4 be
Point of
(TOP-REAL 2);
assume that E26:
( c
3 in h_strip c
1,c
2 & c
4 in h_strip c
1,c
2 )
and E27:
c
3 `2 <= c
4 `2
;
thus
LSeg c
3,c
4 c= h_strip c
1,c
2
proof
let c
5 be
set ;
:: according to TARSKI:def 3
assume E28:
c
5 in LSeg c
3,c
4
;
then reconsider c
6 = c
5 as
Point of
(TOP-REAL 2) ;
E29:
c
6 = |[(c6 `1 ),(c6 `2 )]|
by EUCLID:57;
E30:
( c
3 `2 <= c
6 `2 & c
6 `2 <= c
4 `2 )
by E27, E28, TOPREAL1:10;
per cases
not ( not c2 = 0 & not c2 = width c1 & not ( 1 <= c2 & c2 < width c1 ) )
by E25, XXREAL_0:1, NAT_1:39;
suppose
c
2 = 0
;
then E31:
h_strip c
1,c
2 = { |[b1,b2]| where B is Real, B is Real : b2 <= (c1 * 1,1) `2 }
by GOBRD11:21;
then consider c
7, c
8 being
Real such that E32:
( c
4 = |[c7,c8]| & c
8 <= (c1 * 1,1) `2 )
by E26;
c
4 `2 <= (c1 * 1,1) `2
by E32, EUCLID:56;
then
c
6 `2 <= (c1 * 1,1) `2
by E30, XXREAL_0:2;
hence
c
5 in h_strip c
1,c
2
by E29, E31;
end;
suppose
c
2 = width c
1
;
then E31:
h_strip c
1,c
2 = { |[b1,b2]| where B is Real, B is Real : (c1 * 1,(width c1)) `2 <= b2 }
by GOBRD11:22;
then consider c
7, c
8 being
Real such that E32:
( c
3 = |[c7,c8]| &
(c1 * 1,(width c1)) `2 <= c
8 )
by E26;
(c1 * 1,(width c1)) `2 <= c
3 `2
by E32, EUCLID:56;
then
(c1 * 1,(width c1)) `2 <= c
6 `2
by E30, XXREAL_0:2;
hence
c
5 in h_strip c
1,c
2
by E29, E31;
end;
suppose
( 1
<= c
2 & c
2 < width c
1 )
;
then E31:
h_strip c
1,c
2 = { |[b1,b2]| where B is Real, B is Real : ( (c1 * 1,c2) `2 <= b2 & b2 <= (c1 * 1,(c2 + 1)) `2 ) }
by GOBRD11:23;
then consider c
7, c
8 being
Real such that E32:
( c
3 = |[c7,c8]| &
(c1 * 1,c2) `2 <= c
8 & c
8 <= (c1 * 1,(c2 + 1)) `2 )
by E26;
(
(c1 * 1,c2) `2 <= c
3 `2 & c
3 `2 <= (c1 * 1,(c2 + 1)) `2 )
by E32, EUCLID:56;
then E33:
(c1 * 1,c2) `2 <= c
6 `2
by E30, XXREAL_0:2;
consider c
9, c
10 being
Real such that E34:
( c
4 = |[c9,c10]| &
(c1 * 1,c2) `2 <= c
10 & c
10 <= (c1 * 1,(c2 + 1)) `2 )
by E26, E31;
(
(c1 * 1,c2) `2 <= c
4 `2 & c
4 `2 <= (c1 * 1,(c2 + 1)) `2 )
by E34, EUCLID:56;
then
c
6 `2 <= (c1 * 1,(c2 + 1)) `2
by E30, XXREAL_0:2;
hence
c
5 in h_strip c
1,c
2
by E29, E31, E33;
end;
end;
end;
end;
theorem Th20: :: JORDAN12:20
theorem Th21: :: JORDAN12:21
theorem Th22: :: JORDAN12:22
theorem Th23: :: JORDAN12:23
theorem Th24: :: JORDAN12:24
theorem Th25: :: JORDAN12:25
theorem Th26: :: JORDAN12:26
theorem Th27: :: JORDAN12:27
theorem Th28: :: JORDAN12:28
theorem Th29: :: JORDAN12:29
theorem Th30: :: JORDAN12:30
theorem Th31: :: JORDAN12:31
theorem Th32: :: JORDAN12:32
theorem Th33: :: JORDAN12:33
theorem Th34: :: JORDAN12:34
theorem Th35: :: JORDAN12:35
theorem Th36: :: JORDAN12:36