:: JORDAN1J semantic presentation
theorem Th1: :: JORDAN1J:1
theorem Th2: :: JORDAN1J:2
theorem Th3: :: JORDAN1J:3
theorem Th4: :: JORDAN1J:4
theorem Th5: :: JORDAN1J:5
theorem Th6: :: JORDAN1J:6
theorem Th7: :: JORDAN1J:7
theorem Th8: :: JORDAN1J:8
theorem Th9: :: JORDAN1J:9
theorem Th10: :: JORDAN1J:10
theorem Th11: :: JORDAN1J:11
theorem Th12: :: JORDAN1J:12
theorem Th13: :: JORDAN1J:13
theorem Th14: :: JORDAN1J:14
theorem Th15: :: JORDAN1J:15
theorem Th16: :: JORDAN1J:16
theorem Th17: :: JORDAN1J:17
theorem Th18: :: JORDAN1J:18
theorem Th19: :: JORDAN1J:19
theorem Th20: :: JORDAN1J:20
theorem Th21: :: JORDAN1J:21
theorem Th22: :: JORDAN1J:22
theorem Th23: :: JORDAN1J:23
theorem Th24: :: JORDAN1J:24
theorem Th25: :: JORDAN1J:25
theorem Th26: :: JORDAN1J:26
theorem Th27: :: JORDAN1J:27
theorem Th28: :: JORDAN1J:28
theorem Th29: :: JORDAN1J:29
theorem Th30: :: JORDAN1J:30
theorem Th31: :: JORDAN1J:31
theorem Th32: :: JORDAN1J:32
theorem Th33: :: JORDAN1J:33
theorem Th34: :: JORDAN1J:34
theorem Th35: :: JORDAN1J:35
theorem Th36: :: JORDAN1J:36
theorem Th37: :: JORDAN1J:37
theorem Th38: :: JORDAN1J:38
theorem Th39: :: JORDAN1J:39
theorem Th40: :: JORDAN1J:40
theorem Th41: :: JORDAN1J:41
theorem Th42: :: JORDAN1J:42
theorem Th43: :: JORDAN1J:43
theorem Th44: :: JORDAN1J:44
theorem Th45: :: JORDAN1J:45
theorem Th46: :: JORDAN1J:46
theorem Th47: :: JORDAN1J:47
theorem Th48: :: JORDAN1J:48
theorem Th49: :: JORDAN1J:49
theorem Th50: :: JORDAN1J:50
theorem Th51: :: JORDAN1J:51
theorem Th52: :: JORDAN1J:52
for b
1 being
Go-boardfor b
2 being
S-Sequence_in_R2for b
3 being
Point of
(TOP-REAL 2)for b
4 being
Nat holds
( 1
<= b
4 & b
4 < b
3 .. b
2 & b
2 is_sequence_on b
1 & b
3 in rng b
2 implies (
left_cell (R_Cut b2,b3),b
4,b
1 = left_cell b
2,b
4,b
1 &
right_cell (R_Cut b2,b3),b
4,b
1 = right_cell b
2,b
4,b
1 ) )
theorem Th53: :: JORDAN1J:53
theorem Th54: :: JORDAN1J:54
theorem Th55: :: JORDAN1J:55
theorem Th56: :: JORDAN1J:56
theorem Th57: :: JORDAN1J:57
for b
1 being
Natfor b
2 being
connected compact non
horizontal non
vertical Subset of
(TOP-REAL 2)for b
3, b
4, b
5 being
Nat holds
not ( 1
< b
3 & b
3 < len (Gauge b2,b1) & 1
<= b
4 & b
5 <= width (Gauge b2,b1) &
(Gauge b2,b1) * b
3,b
5 in L~ (Upper_Seq b2,b1) &
(Gauge b2,b1) * b
3,b
4 in L~ (Lower_Seq b2,b1) & not b
4 <> b
5 )
theorem Th58: :: JORDAN1J:58
for b
1 being
Natfor b
2 being
Simple_closed_curvefor b
3, b
4, b
5 being
Nat holds
not ( 1
< b
3 & b
3 < len (Gauge b2,b1) & 1
<= b
4 & b
4 <= b
5 & b
5 <= width (Gauge b2,b1) &
(LSeg ((Gauge b2,b1) * b3,b4),((Gauge b2,b1) * b3,b5)) /\ (L~ (Upper_Seq b2,b1)) = {((Gauge b2,b1) * b3,b5)} &
(LSeg ((Gauge b2,b1) * b3,b4),((Gauge b2,b1) * b3,b5)) /\ (L~ (Lower_Seq b2,b1)) = {((Gauge b2,b1) * b3,b4)} & not
LSeg ((Gauge b2,b1) * b3,b4),
((Gauge b2,b1) * b3,b5) meets Lower_Arc b
2 )
theorem Th59: :: JORDAN1J:59
for b
1 being
Natfor b
2 being
Simple_closed_curvefor b
3, b
4, b
5 being
Nat holds
not ( 1
< b
3 & b
3 < len (Gauge b2,b1) & 1
<= b
4 & b
4 <= b
5 & b
5 <= width (Gauge b2,b1) &
(LSeg ((Gauge b2,b1) * b3,b4),((Gauge b2,b1) * b3,b5)) /\ (L~ (Upper_Seq b2,b1)) = {((Gauge b2,b1) * b3,b5)} &
(LSeg ((Gauge b2,b1) * b3,b4),((Gauge b2,b1) * b3,b5)) /\ (L~ (Lower_Seq b2,b1)) = {((Gauge b2,b1) * b3,b4)} & not
LSeg ((Gauge b2,b1) * b3,b4),
((Gauge b2,b1) * b3,b5) meets Upper_Arc b
2 )
theorem Th60: :: JORDAN1J:60
for b
1 being
Natfor b
2 being
Simple_closed_curvefor b
3, b
4, b
5 being
Nat holds
not ( 1
< b
3 & b
3 < len (Gauge b2,b1) & 1
<= b
4 & b
4 <= b
5 & b
5 <= width (Gauge b2,b1) & b
1 > 0 &
(LSeg ((Gauge b2,b1) * b3,b4),((Gauge b2,b1) * b3,b5)) /\ (Upper_Arc (L~ (Cage b2,b1))) = {((Gauge b2,b1) * b3,b5)} &
(LSeg ((Gauge b2,b1) * b3,b4),((Gauge b2,b1) * b3,b5)) /\ (Lower_Arc (L~ (Cage b2,b1))) = {((Gauge b2,b1) * b3,b4)} & not
LSeg ((Gauge b2,b1) * b3,b4),
((Gauge b2,b1) * b3,b5) meets Lower_Arc b
2 )
theorem Th61: :: JORDAN1J:61
for b
1 being
Natfor b
2 being
Simple_closed_curvefor b
3, b
4, b
5 being
Nat holds
not ( 1
< b
3 & b
3 < len (Gauge b2,b1) & 1
<= b
4 & b
4 <= b
5 & b
5 <= width (Gauge b2,b1) & b
1 > 0 &
(LSeg ((Gauge b2,b1) * b3,b4),((Gauge b2,b1) * b3,b5)) /\ (Upper_Arc (L~ (Cage b2,b1))) = {((Gauge b2,b1) * b3,b5)} &
(LSeg ((Gauge b2,b1) * b3,b4),((Gauge b2,b1) * b3,b5)) /\ (Lower_Arc (L~ (Cage b2,b1))) = {((Gauge b2,b1) * b3,b4)} & not
LSeg ((Gauge b2,b1) * b3,b4),
((Gauge b2,b1) * b3,b5) meets Upper_Arc b
2 )
theorem Th62: :: JORDAN1J:62
for b
1 being
Natfor b
2 being
connected compact non
horizontal non
vertical Subset of
(TOP-REAL 2)for b
3 being
Nat holds
not (
(Gauge b2,(b1 + 1)) * (Center (Gauge b2,(b1 + 1))),b
3 in Upper_Arc (L~ (Cage b2,(b1 + 1))) & 1
<= b
3 & b
3 <= width (Gauge b2,(b1 + 1)) & not
LSeg ((Gauge b2,1) * (Center (Gauge b2,1)),1),
((Gauge b2,(b1 + 1)) * (Center (Gauge b2,(b1 + 1))),b3) meets Lower_Arc (L~ (Cage b2,(b1 + 1))) )
theorem Th63: :: JORDAN1J:63
for b
1 being
Natfor b
2 being
Simple_closed_curvefor b
3, b
4 being
Nat holds
not ( 1
<= b
3 & b
3 <= b
4 & b
4 <= width (Gauge b2,(b1 + 1)) &
(LSeg ((Gauge b2,(b1 + 1)) * (Center (Gauge b2,(b1 + 1))),b3),((Gauge b2,(b1 + 1)) * (Center (Gauge b2,(b1 + 1))),b4)) /\ (Upper_Arc (L~ (Cage b2,(b1 + 1)))) = {((Gauge b2,(b1 + 1)) * (Center (Gauge b2,(b1 + 1))),b4)} &
(LSeg ((Gauge b2,(b1 + 1)) * (Center (Gauge b2,(b1 + 1))),b3),((Gauge b2,(b1 + 1)) * (Center (Gauge b2,(b1 + 1))),b4)) /\ (Lower_Arc (L~ (Cage b2,(b1 + 1)))) = {((Gauge b2,(b1 + 1)) * (Center (Gauge b2,(b1 + 1))),b3)} & not
LSeg ((Gauge b2,(b1 + 1)) * (Center (Gauge b2,(b1 + 1))),b3),
((Gauge b2,(b1 + 1)) * (Center (Gauge b2,(b1 + 1))),b4) meets Lower_Arc b
2 )
theorem Th64: :: JORDAN1J:64
for b
1 being
Natfor b
2 being
Simple_closed_curvefor b
3, b
4 being
Nat holds
not ( 1
<= b
3 & b
3 <= b
4 & b
4 <= width (Gauge b2,(b1 + 1)) &
(LSeg ((Gauge b2,(b1 + 1)) * (Center (Gauge b2,(b1 + 1))),b3),((Gauge b2,(b1 + 1)) * (Center (Gauge b2,(b1 + 1))),b4)) /\ (Upper_Arc (L~ (Cage b2,(b1 + 1)))) = {((Gauge b2,(b1 + 1)) * (Center (Gauge b2,(b1 + 1))),b4)} &
(LSeg ((Gauge b2,(b1 + 1)) * (Center (Gauge b2,(b1 + 1))),b3),((Gauge b2,(b1 + 1)) * (Center (Gauge b2,(b1 + 1))),b4)) /\ (Lower_Arc (L~ (Cage b2,(b1 + 1)))) = {((Gauge b2,(b1 + 1)) * (Center (Gauge b2,(b1 + 1))),b3)} & not
LSeg ((Gauge b2,(b1 + 1)) * (Center (Gauge b2,(b1 + 1))),b3),
((Gauge b2,(b1 + 1)) * (Center (Gauge b2,(b1 + 1))),b4) meets Upper_Arc b
2 )
theorem Th65: :: JORDAN1J:65
theorem Th66: :: JORDAN1J:66
theorem Th67: :: JORDAN1J:67
theorem Th68: :: JORDAN1J:68