:: JORDAN1H semantic presentation
theorem Th1: :: JORDAN1H:1
canceled;
theorem Th2: :: JORDAN1H:2
canceled;
theorem Th3: :: JORDAN1H:3
canceled;
theorem Th4: :: JORDAN1H:4
theorem Th5: :: JORDAN1H:5
theorem Th6: :: JORDAN1H:6
theorem Th7: :: JORDAN1H:7
theorem Th8: :: JORDAN1H:8
:: deftheorem Def1 defines RealOrd JORDAN1H:def 1 :
theorem Th9: :: JORDAN1H:9
Lemma4:
RealOrd is_reflexive_in REAL
Lemma5:
RealOrd is_antisymmetric_in REAL
Lemma6:
RealOrd is_transitive_in REAL
Lemma7:
RealOrd is_connected_in REAL
theorem Th10: :: JORDAN1H:10
theorem Th11: :: JORDAN1H:11
theorem Th12: :: JORDAN1H:12
theorem Th13: :: JORDAN1H:13
theorem Th14: :: JORDAN1H:14
canceled;
theorem Th15: :: JORDAN1H:15
theorem Th16: :: JORDAN1H:16
theorem Th17: :: JORDAN1H:17
theorem Th18: :: JORDAN1H:18
theorem Th19: :: JORDAN1H:19
theorem Th20: :: JORDAN1H:20
theorem Th21: :: JORDAN1H:21
theorem Th22: :: JORDAN1H:22
theorem Th23: :: JORDAN1H:23
theorem Th24: :: JORDAN1H:24
theorem Th25: :: JORDAN1H:25
theorem Th26: :: JORDAN1H:26
theorem Th27: :: JORDAN1H:27
theorem Th28: :: JORDAN1H:28
theorem Th29: :: JORDAN1H:29
theorem Th30: :: JORDAN1H:30
theorem Th31: :: JORDAN1H:31
theorem Th32: :: JORDAN1H:32
theorem Th33: :: JORDAN1H:33
theorem Th34: :: JORDAN1H:34
theorem Th35: :: JORDAN1H:35
theorem Th36: :: JORDAN1H:36
theorem Th37: :: JORDAN1H:37
theorem Th38: :: JORDAN1H:38
theorem Th39: :: JORDAN1H:39
theorem Th40: :: JORDAN1H:40
theorem Th41: :: JORDAN1H:41
theorem Th42: :: JORDAN1H:42
theorem Th43: :: JORDAN1H:43
for b
1, b
2, b
3, b
4 being
Natfor b
5 being
compact non
horizontal non
vertical Subset of
(TOP-REAL 2) holds
not ( b
1 <= b
2 & 1
<= b
3 & b
3 + 1
<= len (Gauge b5,b2) & 1
<= b
4 & b
4 + 1
<= width (Gauge b5,b2) & ( for b
6, b
7 being
Nat holds
not ( b
6 = [\(((b3 - 2) / (2 |^ (b2 -' b1))) + 2)/] & b
7 = [\(((b4 - 2) / (2 |^ (b2 -' b1))) + 2)/] &
cell (Gauge b5,b2),b
3,b
4 c= cell (Gauge b5,b1),b
6,b
7 ) ) )
theorem Th44: :: JORDAN1H:44
for b
1, b
2, b
3, b
4 being
Natfor b
5 being
compact non
horizontal non
vertical Subset of
(TOP-REAL 2) holds
not ( b
1 <= b
2 & 1
<= b
3 & b
3 + 1
<= len (Gauge b5,b2) & 1
<= b
4 & b
4 + 1
<= width (Gauge b5,b2) & ( for b
6, b
7 being
Nat holds
not ( 1
<= b
6 & b
6 + 1
<= len (Gauge b5,b1) & 1
<= b
7 & b
7 + 1
<= width (Gauge b5,b1) &
cell (Gauge b5,b2),b
3,b
4 c= cell (Gauge b5,b1),b
6,b
7 ) ) )
theorem Th45: :: JORDAN1H:45
canceled;
theorem Th46: :: JORDAN1H:46
canceled;
theorem Th47: :: JORDAN1H:47
theorem Th48: :: JORDAN1H:48
theorem Th49: :: JORDAN1H:49
theorem Th50: :: JORDAN1H:50
theorem Th51: :: JORDAN1H:51
theorem Th52: :: JORDAN1H:52
theorem Th53: :: JORDAN1H:53
theorem Th54: :: JORDAN1H:54
theorem Th55: :: JORDAN1H:55
theorem Th56: :: JORDAN1H:56
:: deftheorem Def2 defines X-SpanStart JORDAN1H:def 2 :
theorem Th57: :: JORDAN1H:57
theorem Th58: :: JORDAN1H:58
theorem Th59: :: JORDAN1H:59
:: deftheorem Def3 defines is_sufficiently_large_for JORDAN1H:def 3 :
theorem Th60: :: JORDAN1H:60
theorem Th61: :: JORDAN1H:61
for b
1 being non
empty compact non
horizontal non
vertical Subset of
(TOP-REAL 2)for b
2 being
Natfor b
3 being
FinSequence of
(TOP-REAL 2) holds
( b
3 is_sequence_on Gauge b
1,b
2 &
len b
3 > 1 implies for b
4, b
5 being
Nat holds
(
left_cell b
3,
((len b3) -' 1),
(Gauge b1,b2) meets b
1 &
[b4,b5] in Indices (Gauge b1,b2) & b
3 /. ((len b3) -' 1) = (Gauge b1,b2) * b
4,b
5 &
[b4,(b5 + 1)] in Indices (Gauge b1,b2) & b
3 /. (len b3) = (Gauge b1,b2) * b
4,
(b5 + 1) implies
[(b4 -' 1),(b5 + 1)] in Indices (Gauge b1,b2) ) )
theorem Th62: :: JORDAN1H:62
for b
1 being non
empty compact non
horizontal non
vertical Subset of
(TOP-REAL 2)for b
2 being
Natfor b
3 being
FinSequence of
(TOP-REAL 2) holds
( b
3 is_sequence_on Gauge b
1,b
2 &
len b
3 > 1 implies for b
4, b
5 being
Nat holds
(
left_cell b
3,
((len b3) -' 1),
(Gauge b1,b2) meets b
1 &
[b4,b5] in Indices (Gauge b1,b2) & b
3 /. ((len b3) -' 1) = (Gauge b1,b2) * b
4,b
5 &
[(b4 + 1),b5] in Indices (Gauge b1,b2) & b
3 /. (len b3) = (Gauge b1,b2) * (b4 + 1),b
5 implies
[(b4 + 1),(b5 + 1)] in Indices (Gauge b1,b2) ) )
theorem Th63: :: JORDAN1H:63
for b
1 being non
empty compact non
horizontal non
vertical Subset of
(TOP-REAL 2)for b
2 being
Natfor b
3 being
FinSequence of
(TOP-REAL 2) holds
( b
3 is_sequence_on Gauge b
1,b
2 &
len b
3 > 1 implies for b
4, b
5 being
Nat holds
(
left_cell b
3,
((len b3) -' 1),
(Gauge b1,b2) meets b
1 &
[(b5 + 1),b4] in Indices (Gauge b1,b2) & b
3 /. ((len b3) -' 1) = (Gauge b1,b2) * (b5 + 1),b
4 &
[b5,b4] in Indices (Gauge b1,b2) & b
3 /. (len b3) = (Gauge b1,b2) * b
5,b
4 implies
[b5,(b4 -' 1)] in Indices (Gauge b1,b2) ) )
theorem Th64: :: JORDAN1H:64
for b
1 being non
empty compact non
horizontal non
vertical Subset of
(TOP-REAL 2)for b
2 being
Natfor b
3 being
FinSequence of
(TOP-REAL 2) holds
( b
3 is_sequence_on Gauge b
1,b
2 &
len b
3 > 1 implies for b
4, b
5 being
Nat holds
(
left_cell b
3,
((len b3) -' 1),
(Gauge b1,b2) meets b
1 &
[b4,(b5 + 1)] in Indices (Gauge b1,b2) & b
3 /. ((len b3) -' 1) = (Gauge b1,b2) * b
4,
(b5 + 1) &
[b4,b5] in Indices (Gauge b1,b2) & b
3 /. (len b3) = (Gauge b1,b2) * b
4,b
5 implies
[(b4 + 1),b5] in Indices (Gauge b1,b2) ) )
theorem Th65: :: JORDAN1H:65
for b
1 being non
empty compact non
horizontal non
vertical Subset of
(TOP-REAL 2)for b
2 being
Natfor b
3 being
FinSequence of
(TOP-REAL 2) holds
( b
3 is_sequence_on Gauge b
1,b
2 &
len b
3 > 1 implies for b
4, b
5 being
Nat holds
(
front_left_cell b
3,
((len b3) -' 1),
(Gauge b1,b2) meets b
1 &
[b4,b5] in Indices (Gauge b1,b2) & b
3 /. ((len b3) -' 1) = (Gauge b1,b2) * b
4,b
5 &
[b4,(b5 + 1)] in Indices (Gauge b1,b2) & b
3 /. (len b3) = (Gauge b1,b2) * b
4,
(b5 + 1) implies
[b4,(b5 + 2)] in Indices (Gauge b1,b2) ) )
theorem Th66: :: JORDAN1H:66
for b
1 being non
empty compact non
horizontal non
vertical Subset of
(TOP-REAL 2)for b
2 being
Natfor b
3 being
FinSequence of
(TOP-REAL 2) holds
( b
3 is_sequence_on Gauge b
1,b
2 &
len b
3 > 1 implies for b
4, b
5 being
Nat holds
(
front_left_cell b
3,
((len b3) -' 1),
(Gauge b1,b2) meets b
1 &
[b4,b5] in Indices (Gauge b1,b2) & b
3 /. ((len b3) -' 1) = (Gauge b1,b2) * b
4,b
5 &
[(b4 + 1),b5] in Indices (Gauge b1,b2) & b
3 /. (len b3) = (Gauge b1,b2) * (b4 + 1),b
5 implies
[(b4 + 2),b5] in Indices (Gauge b1,b2) ) )
theorem Th67: :: JORDAN1H:67
for b
1 being non
empty compact non
horizontal non
vertical Subset of
(TOP-REAL 2)for b
2 being
Natfor b
3 being
FinSequence of
(TOP-REAL 2) holds
( b
3 is_sequence_on Gauge b
1,b
2 &
len b
3 > 1 implies for b
4, b
5 being
Nat holds
(
front_left_cell b
3,
((len b3) -' 1),
(Gauge b1,b2) meets b
1 &
[(b5 + 1),b4] in Indices (Gauge b1,b2) & b
3 /. ((len b3) -' 1) = (Gauge b1,b2) * (b5 + 1),b
4 &
[b5,b4] in Indices (Gauge b1,b2) & b
3 /. (len b3) = (Gauge b1,b2) * b
5,b
4 implies
[(b5 -' 1),b4] in Indices (Gauge b1,b2) ) )
theorem Th68: :: JORDAN1H:68
for b
1 being non
empty compact non
horizontal non
vertical Subset of
(TOP-REAL 2)for b
2 being
Natfor b
3 being
FinSequence of
(TOP-REAL 2) holds
( b
3 is_sequence_on Gauge b
1,b
2 &
len b
3 > 1 implies for b
4, b
5 being
Nat holds
(
front_left_cell b
3,
((len b3) -' 1),
(Gauge b1,b2) meets b
1 &
[b4,(b5 + 1)] in Indices (Gauge b1,b2) & b
3 /. ((len b3) -' 1) = (Gauge b1,b2) * b
4,
(b5 + 1) &
[b4,b5] in Indices (Gauge b1,b2) & b
3 /. (len b3) = (Gauge b1,b2) * b
4,b
5 implies
[b4,(b5 -' 1)] in Indices (Gauge b1,b2) ) )
theorem Th69: :: JORDAN1H:69
for b
1 being non
empty compact non
horizontal non
vertical Subset of
(TOP-REAL 2)for b
2 being
Natfor b
3 being
FinSequence of
(TOP-REAL 2) holds
( b
3 is_sequence_on Gauge b
1,b
2 &
len b
3 > 1 implies for b
4, b
5 being
Nat holds
(
front_right_cell b
3,
((len b3) -' 1),
(Gauge b1,b2) meets b
1 &
[b4,b5] in Indices (Gauge b1,b2) & b
3 /. ((len b3) -' 1) = (Gauge b1,b2) * b
4,b
5 &
[b4,(b5 + 1)] in Indices (Gauge b1,b2) & b
3 /. (len b3) = (Gauge b1,b2) * b
4,
(b5 + 1) implies
[(b4 + 1),(b5 + 1)] in Indices (Gauge b1,b2) ) )
theorem Th70: :: JORDAN1H:70
for b
1 being non
empty compact non
horizontal non
vertical Subset of
(TOP-REAL 2)for b
2 being
Natfor b
3 being
FinSequence of
(TOP-REAL 2) holds
( b
3 is_sequence_on Gauge b
1,b
2 &
len b
3 > 1 implies for b
4, b
5 being
Nat holds
(
front_right_cell b
3,
((len b3) -' 1),
(Gauge b1,b2) meets b
1 &
[b4,b5] in Indices (Gauge b1,b2) & b
3 /. ((len b3) -' 1) = (Gauge b1,b2) * b
4,b
5 &
[(b4 + 1),b5] in Indices (Gauge b1,b2) & b
3 /. (len b3) = (Gauge b1,b2) * (b4 + 1),b
5 implies
[(b4 + 1),(b5 -' 1)] in Indices (Gauge b1,b2) ) )
theorem Th71: :: JORDAN1H:71
for b
1 being non
empty compact non
horizontal non
vertical Subset of
(TOP-REAL 2)for b
2 being
Natfor b
3 being
FinSequence of
(TOP-REAL 2) holds
( b
3 is_sequence_on Gauge b
1,b
2 &
len b
3 > 1 implies for b
4, b
5 being
Nat holds
(
front_right_cell b
3,
((len b3) -' 1),
(Gauge b1,b2) meets b
1 &
[(b5 + 1),b4] in Indices (Gauge b1,b2) & b
3 /. ((len b3) -' 1) = (Gauge b1,b2) * (b5 + 1),b
4 &
[b5,b4] in Indices (Gauge b1,b2) & b
3 /. (len b3) = (Gauge b1,b2) * b
5,b
4 implies
[b5,(b4 + 1)] in Indices (Gauge b1,b2) ) )
theorem Th72: :: JORDAN1H:72
for b
1 being non
empty compact non
horizontal non
vertical Subset of
(TOP-REAL 2)for b
2 being
Natfor b
3 being
FinSequence of
(TOP-REAL 2) holds
( b
3 is_sequence_on Gauge b
1,b
2 &
len b
3 > 1 implies for b
4, b
5 being
Nat holds
(
front_right_cell b
3,
((len b3) -' 1),
(Gauge b1,b2) meets b
1 &
[b4,(b5 + 1)] in Indices (Gauge b1,b2) & b
3 /. ((len b3) -' 1) = (Gauge b1,b2) * b
4,
(b5 + 1) &
[b4,b5] in Indices (Gauge b1,b2) & b
3 /. (len b3) = (Gauge b1,b2) * b
4,b
5 implies
[(b4 -' 1),b5] in Indices (Gauge b1,b2) ) )