:: JORDAN1G semantic presentation
theorem Th1: :: JORDAN1G:1
theorem Th2: :: JORDAN1G:2
theorem Th3: :: JORDAN1G:3
theorem Th4: :: JORDAN1G:4
theorem Th5: :: JORDAN1G:5
theorem Th6: :: JORDAN1G:6
theorem Th7: :: JORDAN1G:7
theorem Th8: :: JORDAN1G:8
canceled;
theorem Th9: :: JORDAN1G:9
canceled;
theorem Th10: :: JORDAN1G:10
canceled;
theorem Th11: :: JORDAN1G:11
canceled;
theorem Th12: :: JORDAN1G:12
canceled;
theorem Th13: :: JORDAN1G:13
canceled;
theorem Th14: :: JORDAN1G:14
canceled;
theorem Th15: :: JORDAN1G:15
canceled;
theorem Th16: :: JORDAN1G:16
theorem Th17: :: JORDAN1G:17
theorem Th18: :: JORDAN1G:18
theorem Th19: :: JORDAN1G:19
theorem Th20: :: JORDAN1G:20
theorem Th21: :: JORDAN1G:21
theorem Th22: :: JORDAN1G:22
theorem Th23: :: JORDAN1G:23
theorem Th24: :: JORDAN1G:24
theorem Th25: :: JORDAN1G:25
theorem Th26: :: JORDAN1G:26
theorem Th27: :: JORDAN1G:27
theorem Th28: :: JORDAN1G:28
theorem Th29: :: JORDAN1G:29
theorem Th30: :: JORDAN1G:30
theorem Th31: :: JORDAN1G:31
theorem Th32: :: JORDAN1G:32
theorem Th33: :: JORDAN1G:33
theorem Th34: :: JORDAN1G:34
theorem Th35: :: JORDAN1G:35
theorem Th36: :: JORDAN1G:36
theorem Th37: :: JORDAN1G:37
theorem Th38: :: JORDAN1G:38
theorem Th39: :: JORDAN1G:39
theorem Th40: :: JORDAN1G:40
theorem Th41: :: JORDAN1G:41
theorem Th42: :: JORDAN1G:42
theorem Th43: :: JORDAN1G:43
theorem Th44: :: JORDAN1G:44
theorem Th45: :: JORDAN1G:45
theorem Th46: :: JORDAN1G:46
theorem Th47: :: JORDAN1G:47
theorem Th48: :: JORDAN1G:48
theorem Th49: :: JORDAN1G:49
theorem Th50: :: JORDAN1G:50
theorem Th51: :: JORDAN1G:51
theorem Th52: :: JORDAN1G:52
theorem Th53: :: JORDAN1G:53
theorem Th54: :: JORDAN1G:54
for b
1 being
Natfor b
2 being
connected compact non
horizontal non
vertical Subset of
(TOP-REAL 2)for b
3, b
4 being
Nat holds
not ( 1
<= b
3 & b
3 <= len (Gauge b2,b1) & 1
<= b
4 & b
4 <= width (Gauge b2,b1) &
(Gauge b2,b1) * b
3,b
4 in L~ (Cage b2,b1) & not
LSeg ((Gauge b2,b1) * b3,1),
((Gauge b2,b1) * b3,b4) meets L~ (Lower_Seq b2,b1) )
theorem Th55: :: JORDAN1G:55
theorem Th56: :: JORDAN1G:56
theorem Th57: :: JORDAN1G:57
theorem Th58: :: JORDAN1G:58
for b
1 being
S-Sequence_in_R2for b
2 being
closed Subset of
(TOP-REAL 2) holds
(
L~ b
1 meets b
2 & not b
1 /. 1
in b
2 &
First_Point (L~ b1),
(b1 /. 1),
(b1 /. (len b1)),b
2 in rng b
1 implies
(L~ (mid b1,1,((First_Point (L~ b1),(b1 /. 1),(b1 /. (len b1)),b2) .. b1))) /\ b
2 = {(First_Point (L~ b1),(b1 /. 1),(b1 /. (len b1)),b2)} )
theorem Th59: :: JORDAN1G:59
for b
1 being
connected compact non
horizontal non
vertical Subset of
(TOP-REAL 2)for b
2 being
Nat holds
( b
2 > 0 implies for b
3 being
Nat holds
not ( 1
<= b
3 & b
3 < (First_Point (L~ (Upper_Seq b1,b2)),(W-min (L~ (Cage b1,b2))),(E-max (L~ (Cage b1,b2))),(Vertical_Line (((W-bound (L~ (Cage b1,b2))) + (E-bound (L~ (Cage b1,b2)))) / 2))) .. (Upper_Seq b1,b2) & not
((Upper_Seq b1,b2) /. b3) `1 < ((W-bound (L~ (Cage b1,b2))) + (E-bound (L~ (Cage b1,b2)))) / 2 ) )
theorem Th60: :: JORDAN1G:60
for b
1 being
connected compact non
horizontal non
vertical Subset of
(TOP-REAL 2)for b
2 being
Nat holds
( b
2 > 0 implies for b
3 being
Nat holds
not ( 1
<= b
3 & b
3 < (First_Point (L~ (Rev (Lower_Seq b1,b2))),(W-min (L~ (Cage b1,b2))),(E-max (L~ (Cage b1,b2))),(Vertical_Line (((W-bound (L~ (Cage b1,b2))) + (E-bound (L~ (Cage b1,b2)))) / 2))) .. (Rev (Lower_Seq b1,b2)) & not
((Rev (Lower_Seq b1,b2)) /. b3) `1 < ((W-bound (L~ (Cage b1,b2))) + (E-bound (L~ (Cage b1,b2)))) / 2 ) )
theorem Th61: :: JORDAN1G:61
for b
1 being
connected compact non
horizontal non
vertical Subset of
(TOP-REAL 2)for b
2 being
Nat holds
( b
2 > 0 implies for b
3 being
Point of
(TOP-REAL 2) holds
( b
3 in rng (mid (Upper_Seq b1,b2),2,((First_Point (L~ (Upper_Seq b1,b2)),(W-min (L~ (Cage b1,b2))),(E-max (L~ (Cage b1,b2))),(Vertical_Line (((W-bound (L~ (Cage b1,b2))) + (E-bound (L~ (Cage b1,b2)))) / 2))) .. (Upper_Seq b1,b2))) implies b
3 `1 <= ((W-bound (L~ (Cage b1,b2))) + (E-bound (L~ (Cage b1,b2)))) / 2 ) )
theorem Th62: :: JORDAN1G:62
for b
1 being
connected compact non
horizontal non
vertical Subset of
(TOP-REAL 2)for b
2 being
Nat holds
not ( b
2 > 0 & not
(First_Point (L~ (Upper_Seq b1,b2)),(W-min (L~ (Cage b1,b2))),(E-max (L~ (Cage b1,b2))),(Vertical_Line (((W-bound (L~ (Cage b1,b2))) + (E-bound (L~ (Cage b1,b2)))) / 2))) `2 > (Last_Point (L~ (Lower_Seq b1,b2)),(E-max (L~ (Cage b1,b2))),(W-min (L~ (Cage b1,b2))),(Vertical_Line (((W-bound (L~ (Cage b1,b2))) + (E-bound (L~ (Cage b1,b2)))) / 2))) `2 )
theorem Th63: :: JORDAN1G:63
theorem Th64: :: JORDAN1G:64
theorem Th65: :: JORDAN1G:65
for b
1 being
connected compact non
horizontal non
vertical Subset of
(TOP-REAL 2)for b
2 being
Nat holds
( b
2 > 0 implies for b
3, b
4 being
Nat holds
not ( 1
<= b
3 & b
3 <= len (Gauge b1,b2) & 1
<= b
4 & b
4 <= width (Gauge b1,b2) &
(Gauge b1,b2) * b
3,b
4 in L~ (Cage b1,b2) & not
LSeg ((Gauge b1,b2) * b3,1),
((Gauge b1,b2) * b3,b4) meets Lower_Arc (L~ (Cage b1,b2)) ) )