:: JORDAN14 semantic presentation
theorem Th1: :: JORDAN14:1
theorem Th2: :: JORDAN14:2
theorem Th3: :: JORDAN14:3
theorem Th4: :: JORDAN14:4
theorem Th5: :: JORDAN14:5
:: deftheorem Def1 defines SpanStart JORDAN14:def 1 :
theorem Th6: :: JORDAN14:6
theorem Th7: :: JORDAN14:7
theorem Th8: :: JORDAN14:8
for b
1 being
Simple_closed_curvefor b
2, b
3 being
Nat holds
( b
2 is_sufficiently_large_for b
1 & 1
<= b
3 & b
3 + 1
<= len (Span b1,b2) implies (
right_cell (Span b1,b2),b
3,
(Gauge b1,b2) misses b
1 &
left_cell (Span b1,b2),b
3,
(Gauge b1,b2) meets b
1 ) )
theorem Th9: :: JORDAN14:9
theorem Th10: :: JORDAN14:10
theorem Th11: :: JORDAN14:11
theorem Th12: :: JORDAN14:12
theorem Th13: :: JORDAN14:13
theorem Th14: :: JORDAN14:14
theorem Th15: :: JORDAN14:15
theorem Th16: :: JORDAN14:16
theorem Th17: :: JORDAN14:17
theorem Th18: :: JORDAN14:18
theorem Th19: :: JORDAN14:19
theorem Th20: :: JORDAN14:20
theorem Th21: :: JORDAN14:21
theorem Th22: :: JORDAN14:22
theorem Th23: :: JORDAN14:23
theorem Th24: :: JORDAN14:24
for b
1 being
Simple_closed_curvefor b
2, b
3, b
4, b
5 being
Nat holds
not ( b
5 is_sufficiently_large_for b
1 & 1
<= b
4 & b
4 <= len (Span b1,b5) &
[b2,b3] in Indices (Gauge b1,b5) &
(Span b1,b5) /. b
4 = (Gauge b1,b5) * b
2,b
3 & not b
2 < len (Gauge b1,b5) )
theorem Th25: :: JORDAN14:25
theorem Th26: :: JORDAN14:26
for b
1 being
Simple_closed_curvefor b
2, b
3, b
4, b
5 being
Nat holds
not ( b
5 is_sufficiently_large_for b
1 & 1
<= b
4 & b
4 <= len (Span b1,b5) &
[b2,b3] in Indices (Gauge b1,b5) &
(Span b1,b5) /. b
4 = (Gauge b1,b5) * b
2,b
3 & not b
3 < width (Gauge b1,b5) )
theorem Th27: :: JORDAN14:27
theorem Th28: :: JORDAN14:28
theorem Th29: :: JORDAN14:29
theorem Th30: :: JORDAN14:30
theorem Th31: :: JORDAN14:31
theorem Th32: :: JORDAN14:32
theorem Th33: :: JORDAN14:33
theorem Th34: :: JORDAN14:34
theorem Th35: :: JORDAN14:35
theorem Th36: :: JORDAN14:36
theorem Th37: :: JORDAN14:37