:: JORDAN15 semantic presentation
theorem Th1: :: JORDAN15:1
theorem Th2: :: JORDAN15:2
theorem Th3: :: JORDAN15:3
theorem Th4: :: JORDAN15:4
theorem Th5: :: JORDAN15:5
theorem Th6: :: JORDAN15:6
theorem Th7: :: JORDAN15:7
for b
1 being
Go-boardfor b
2, b
3, b
4, b
5, b
6 being
Nat holds
( 1
<= b
2 & b
2 <= len b
1 & 1
<= b
3 & b
3 <= b
5 & b
5 <= b
6 & b
6 <= b
4 & b
4 <= width b
1 implies
LSeg (b1 * b2,b5),
(b1 * b2,b6) c= LSeg (b1 * b2,b3),
(b1 * b2,b4) )
theorem Th8: :: JORDAN15:8
for b
1 being
Go-boardfor b
2, b
3, b
4, b
5, b
6 being
Nat holds
( 1
<= b
2 & b
2 <= width b
1 & 1
<= b
3 & b
3 <= b
5 & b
5 <= b
6 & b
6 <= b
4 & b
4 <= len b
1 implies
LSeg (b1 * b5,b2),
(b1 * b6,b2) c= LSeg (b1 * b3,b2),
(b1 * b4,b2) )
theorem Th9: :: JORDAN15:9
theorem Th10: :: JORDAN15:10
theorem Th11: :: JORDAN15:11
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
4 <= b
5 & b
5 <= width (Gauge b2,b1) &
(Gauge b2,b1) * b
3,b
4 in L~ (Lower_Seq b2,b1) & ( for b
6 being
Nat holds
not ( b
4 <= b
6 & b
6 <= b
5 &
(LSeg ((Gauge b2,b1) * b3,b6),((Gauge b2,b1) * b3,b5)) /\ (L~ (Lower_Seq b2,b1)) = {((Gauge b2,b1) * b3,b6)} ) ) )
theorem Th12: :: JORDAN15:12
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
4 <= b
5 & b
5 <= width (Gauge b2,b1) &
(Gauge b2,b1) * b
3,b
5 in L~ (Upper_Seq b2,b1) & ( for b
6 being
Nat holds
not ( b
4 <= b
6 & b
6 <= b
5 &
(LSeg ((Gauge b2,b1) * b3,b4),((Gauge b2,b1) * b3,b6)) /\ (L~ (Upper_Seq b2,b1)) = {((Gauge b2,b1) * b3,b6)} ) ) )
theorem Th13: :: JORDAN15:13
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
4 <= b
5 & b
5 <= width (Gauge b2,b1) &
(Gauge b2,b1) * b
3,b
4 in L~ (Lower_Seq b2,b1) &
(Gauge b2,b1) * b
3,b
5 in L~ (Upper_Seq b2,b1) & ( for b
6, b
7 being
Nat holds
not ( b
4 <= b
6 & b
6 <= b
7 & b
7 <= b
5 &
(LSeg ((Gauge b2,b1) * b3,b6),((Gauge b2,b1) * b3,b7)) /\ (L~ (Lower_Seq b2,b1)) = {((Gauge b2,b1) * b3,b6)} &
(LSeg ((Gauge b2,b1) * b3,b6),((Gauge b2,b1) * b3,b7)) /\ (L~ (Upper_Seq b2,b1)) = {((Gauge b2,b1) * b3,b7)} ) ) )
theorem Th14: :: JORDAN15:14
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
4 & b
4 <= b
5 & b
5 <= len (Gauge b2,b1) & 1
<= b
3 & b
3 <= width (Gauge b2,b1) &
(Gauge b2,b1) * b
4,b
3 in L~ (Lower_Seq b2,b1) & ( for b
6 being
Nat holds
not ( b
4 <= b
6 & b
6 <= b
5 &
(LSeg ((Gauge b2,b1) * b6,b3),((Gauge b2,b1) * b5,b3)) /\ (L~ (Lower_Seq b2,b1)) = {((Gauge b2,b1) * b6,b3)} ) ) )
theorem Th15: :: JORDAN15:15
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
4 & b
4 <= b
5 & b
5 <= len (Gauge b2,b1) & 1
<= b
3 & b
3 <= width (Gauge b2,b1) &
(Gauge b2,b1) * b
5,b
3 in L~ (Upper_Seq b2,b1) & ( for b
6 being
Nat holds
not ( b
4 <= b
6 & b
6 <= b
5 &
(LSeg ((Gauge b2,b1) * b4,b3),((Gauge b2,b1) * b6,b3)) /\ (L~ (Upper_Seq b2,b1)) = {((Gauge b2,b1) * b6,b3)} ) ) )
theorem Th16: :: JORDAN15:16
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
4 & b
4 <= b
5 & b
5 <= len (Gauge b2,b1) & 1
<= b
3 & b
3 <= width (Gauge b2,b1) &
(Gauge b2,b1) * b
4,b
3 in L~ (Lower_Seq b2,b1) &
(Gauge b2,b1) * b
5,b
3 in L~ (Upper_Seq b2,b1) & ( for b
6, b
7 being
Nat holds
not ( b
4 <= b
6 & b
6 <= b
7 & b
7 <= b
5 &
(LSeg ((Gauge b2,b1) * b6,b3),((Gauge b2,b1) * b7,b3)) /\ (L~ (Lower_Seq b2,b1)) = {((Gauge b2,b1) * b6,b3)} &
(LSeg ((Gauge b2,b1) * b6,b3),((Gauge b2,b1) * b7,b3)) /\ (L~ (Upper_Seq b2,b1)) = {((Gauge b2,b1) * b7,b3)} ) ) )
theorem Th17: :: JORDAN15:17
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
4 <= b
5 & b
5 <= width (Gauge b2,b1) &
(Gauge b2,b1) * b
3,b
4 in L~ (Upper_Seq b2,b1) & ( for b
6 being
Nat holds
not ( b
4 <= b
6 & b
6 <= b
5 &
(LSeg ((Gauge b2,b1) * b3,b6),((Gauge b2,b1) * b3,b5)) /\ (L~ (Upper_Seq b2,b1)) = {((Gauge b2,b1) * b3,b6)} ) ) )
theorem Th18: :: JORDAN15:18
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
4 <= b
5 & b
5 <= width (Gauge b2,b1) &
(Gauge b2,b1) * b
3,b
5 in L~ (Lower_Seq b2,b1) & ( for b
6 being
Nat holds
not ( b
4 <= b
6 & b
6 <= b
5 &
(LSeg ((Gauge b2,b1) * b3,b4),((Gauge b2,b1) * b3,b6)) /\ (L~ (Lower_Seq b2,b1)) = {((Gauge b2,b1) * b3,b6)} ) ) )
theorem Th19: :: JORDAN15:19
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
4 <= b
5 & b
5 <= width (Gauge b2,b1) &
(Gauge b2,b1) * b
3,b
4 in L~ (Upper_Seq b2,b1) &
(Gauge b2,b1) * b
3,b
5 in L~ (Lower_Seq b2,b1) & ( for b
6, b
7 being
Nat holds
not ( b
4 <= b
6 & b
6 <= b
7 & b
7 <= b
5 &
(LSeg ((Gauge b2,b1) * b3,b6),((Gauge b2,b1) * b3,b7)) /\ (L~ (Upper_Seq b2,b1)) = {((Gauge b2,b1) * b3,b6)} &
(LSeg ((Gauge b2,b1) * b3,b6),((Gauge b2,b1) * b3,b7)) /\ (L~ (Lower_Seq b2,b1)) = {((Gauge b2,b1) * b3,b7)} ) ) )
theorem Th20: :: JORDAN15:20
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
4 & b
4 <= b
5 & b
5 <= len (Gauge b2,b1) & 1
<= b
3 & b
3 <= width (Gauge b2,b1) &
(Gauge b2,b1) * b
4,b
3 in L~ (Upper_Seq b2,b1) & ( for b
6 being
Nat holds
not ( b
4 <= b
6 & b
6 <= b
5 &
(LSeg ((Gauge b2,b1) * b6,b3),((Gauge b2,b1) * b5,b3)) /\ (L~ (Upper_Seq b2,b1)) = {((Gauge b2,b1) * b6,b3)} ) ) )
theorem Th21: :: JORDAN15:21
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
4 & b
4 <= b
5 & b
5 <= len (Gauge b2,b1) & 1
<= b
3 & b
3 <= width (Gauge b2,b1) &
(Gauge b2,b1) * b
5,b
3 in L~ (Lower_Seq b2,b1) & ( for b
6 being
Nat holds
not ( b
4 <= b
6 & b
6 <= b
5 &
(LSeg ((Gauge b2,b1) * b4,b3),((Gauge b2,b1) * b6,b3)) /\ (L~ (Lower_Seq b2,b1)) = {((Gauge b2,b1) * b6,b3)} ) ) )
theorem Th22: :: JORDAN15:22
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
4 & b
4 <= b
5 & b
5 <= len (Gauge b2,b1) & 1
<= b
3 & b
3 <= width (Gauge b2,b1) &
(Gauge b2,b1) * b
4,b
3 in L~ (Upper_Seq b2,b1) &
(Gauge b2,b1) * b
5,b
3 in L~ (Lower_Seq b2,b1) & ( for b
6, b
7 being
Nat holds
not ( b
4 <= b
6 & b
6 <= b
7 & b
7 <= b
5 &
(LSeg ((Gauge b2,b1) * b6,b3),((Gauge b2,b1) * b7,b3)) /\ (L~ (Upper_Seq b2,b1)) = {((Gauge b2,b1) * b6,b3)} &
(LSeg ((Gauge b2,b1) * b6,b3),((Gauge b2,b1) * b7,b3)) /\ (L~ (Lower_Seq b2,b1)) = {((Gauge b2,b1) * b7,b3)} ) ) )
theorem Th23: :: JORDAN15:23
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) &
(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
LSeg ((Gauge b2,b1) * b3,b4),
((Gauge b2,b1) * b3,b5) meets Lower_Arc b
2 )
theorem Th24: :: JORDAN15:24
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) &
(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
LSeg ((Gauge b2,b1) * b3,b4),
((Gauge b2,b1) * b3,b5) meets Upper_Arc b
2 )
theorem Th25: :: JORDAN15:25
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 &
(Gauge b2,b1) * b
3,b
5 in Upper_Arc (L~ (Cage b2,b1)) &
(Gauge b2,b1) * b
3,b
4 in Lower_Arc (L~ (Cage b2,b1)) & not
LSeg ((Gauge b2,b1) * b3,b4),
((Gauge b2,b1) * b3,b5) meets Lower_Arc b
2 )
theorem Th26: :: JORDAN15:26
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 &
(Gauge b2,b1) * b
3,b
5 in Upper_Arc (L~ (Cage b2,b1)) &
(Gauge b2,b1) * b
3,b
4 in Lower_Arc (L~ (Cage b2,b1)) & not
LSeg ((Gauge b2,b1) * b3,b4),
((Gauge b2,b1) * b3,b5) meets Upper_Arc b
2 )
theorem Th27: :: JORDAN15:27
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)) &
(Gauge b2,(b1 + 1)) * (Center (Gauge b2,(b1 + 1))),b
4 in Upper_Arc (L~ (Cage b2,(b1 + 1))) &
(Gauge b2,(b1 + 1)) * (Center (Gauge b2,(b1 + 1))),b
3 in Lower_Arc (L~ (Cage b2,(b1 + 1))) & 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 Th28: :: JORDAN15:28
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)) &
(Gauge b2,(b1 + 1)) * (Center (Gauge b2,(b1 + 1))),b
4 in Upper_Arc (L~ (Cage b2,(b1 + 1))) &
(Gauge b2,(b1 + 1)) * (Center (Gauge b2,(b1 + 1))),b
3 in Lower_Arc (L~ (Cage b2,(b1 + 1))) & 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 Th29: :: JORDAN15:29
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
4 & b
5 < len (Gauge b2,b1) & 1
<= b
3 & b
3 <= width (Gauge b2,b1) &
(Gauge b2,b1) * b
5,b
3 in L~ (Upper_Seq b2,b1) &
(Gauge b2,b1) * b
4,b
3 in L~ (Lower_Seq b2,b1) & not b
4 <> b
5 )
theorem Th30: :: JORDAN15:30
for b
1 being
Natfor b
2 being
Simple_closed_curvefor b
3, b
4, b
5 being
Nat holds
not ( 1
< b
4 & b
4 <= b
5 & b
5 < len (Gauge b2,b1) & 1
<= b
3 & b
3 <= width (Gauge b2,b1) &
(LSeg ((Gauge b2,b1) * b4,b3),((Gauge b2,b1) * b5,b3)) /\ (L~ (Upper_Seq b2,b1)) = {((Gauge b2,b1) * b5,b3)} &
(LSeg ((Gauge b2,b1) * b4,b3),((Gauge b2,b1) * b5,b3)) /\ (L~ (Lower_Seq b2,b1)) = {((Gauge b2,b1) * b4,b3)} & not
LSeg ((Gauge b2,b1) * b4,b3),
((Gauge b2,b1) * b5,b3) meets Lower_Arc b
2 )
theorem Th31: :: JORDAN15:31
for b
1 being
Natfor b
2 being
Simple_closed_curvefor b
3, b
4, b
5 being
Nat holds
not ( 1
< b
4 & b
4 <= b
5 & b
5 < len (Gauge b2,b1) & 1
<= b
3 & b
3 <= width (Gauge b2,b1) &
(LSeg ((Gauge b2,b1) * b4,b3),((Gauge b2,b1) * b5,b3)) /\ (L~ (Upper_Seq b2,b1)) = {((Gauge b2,b1) * b5,b3)} &
(LSeg ((Gauge b2,b1) * b4,b3),((Gauge b2,b1) * b5,b3)) /\ (L~ (Lower_Seq b2,b1)) = {((Gauge b2,b1) * b4,b3)} & not
LSeg ((Gauge b2,b1) * b4,b3),
((Gauge b2,b1) * b5,b3) meets Upper_Arc b
2 )
theorem Th32: :: JORDAN15:32
for b
1 being
Natfor b
2 being
Simple_closed_curvefor b
3, b
4, b
5 being
Nat holds
not ( 1
< b
4 & b
4 <= b
5 & b
5 < len (Gauge b2,b1) & 1
<= b
3 & b
3 <= width (Gauge b2,b1) &
(Gauge b2,b1) * b
5,b
3 in L~ (Upper_Seq b2,b1) &
(Gauge b2,b1) * b
4,b
3 in L~ (Lower_Seq b2,b1) & not
LSeg ((Gauge b2,b1) * b4,b3),
((Gauge b2,b1) * b5,b3) meets Lower_Arc b
2 )
theorem Th33: :: JORDAN15:33
for b
1 being
Natfor b
2 being
Simple_closed_curvefor b
3, b
4, b
5 being
Nat holds
not ( 1
< b
4 & b
4 <= b
5 & b
5 < len (Gauge b2,b1) & 1
<= b
3 & b
3 <= width (Gauge b2,b1) &
(Gauge b2,b1) * b
5,b
3 in L~ (Upper_Seq b2,b1) &
(Gauge b2,b1) * b
4,b
3 in L~ (Lower_Seq b2,b1) & not
LSeg ((Gauge b2,b1) * b4,b3),
((Gauge b2,b1) * b5,b3) meets Upper_Arc b
2 )
theorem Th34: :: JORDAN15:34
for b
1 being
Natfor b
2 being
Simple_closed_curvefor b
3, b
4, b
5 being
Nat holds
not ( 1
< b
4 & b
4 <= b
5 & b
5 < len (Gauge b2,b1) & 1
<= b
3 & b
3 <= width (Gauge b2,b1) & b
1 > 0 &
(Gauge b2,b1) * b
5,b
3 in Upper_Arc (L~ (Cage b2,b1)) &
(Gauge b2,b1) * b
4,b
3 in Lower_Arc (L~ (Cage b2,b1)) & not
LSeg ((Gauge b2,b1) * b4,b3),
((Gauge b2,b1) * b5,b3) meets Lower_Arc b
2 )
theorem Th35: :: JORDAN15:35
for b
1 being
Natfor b
2 being
Simple_closed_curvefor b
3, b
4, b
5 being
Nat holds
not ( 1
< b
4 & b
4 <= b
5 & b
5 < len (Gauge b2,b1) & 1
<= b
3 & b
3 <= width (Gauge b2,b1) & b
1 > 0 &
(Gauge b2,b1) * b
5,b
3 in Upper_Arc (L~ (Cage b2,b1)) &
(Gauge b2,b1) * b
4,b
3 in Lower_Arc (L~ (Cage b2,b1)) & not
LSeg ((Gauge b2,b1) * b4,b3),
((Gauge b2,b1) * b5,b3) meets Upper_Arc b
2 )
theorem Th36: :: JORDAN15:36
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 < len (Gauge b2,(b1 + 1)) &
(Gauge b2,(b1 + 1)) * b
4,
(Center (Gauge b2,(b1 + 1))) in Upper_Arc (L~ (Cage b2,(b1 + 1))) &
(Gauge b2,(b1 + 1)) * b
3,
(Center (Gauge b2,(b1 + 1))) in Lower_Arc (L~ (Cage b2,(b1 + 1))) & not
LSeg ((Gauge b2,(b1 + 1)) * b3,(Center (Gauge b2,(b1 + 1)))),
((Gauge b2,(b1 + 1)) * b4,(Center (Gauge b2,(b1 + 1)))) meets Lower_Arc b
2 )
theorem Th37: :: JORDAN15:37
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 < len (Gauge b2,(b1 + 1)) &
(Gauge b2,(b1 + 1)) * b
4,
(Center (Gauge b2,(b1 + 1))) in Upper_Arc (L~ (Cage b2,(b1 + 1))) &
(Gauge b2,(b1 + 1)) * b
3,
(Center (Gauge b2,(b1 + 1))) in Lower_Arc (L~ (Cage b2,(b1 + 1))) & not
LSeg ((Gauge b2,(b1 + 1)) * b3,(Center (Gauge b2,(b1 + 1)))),
((Gauge b2,(b1 + 1)) * b4,(Center (Gauge b2,(b1 + 1)))) meets Upper_Arc b
2 )
theorem Th38: :: JORDAN15:38
for b
1 being
Natfor b
2 being
Simple_closed_curvefor b
3, b
4, b
5 being
Nat holds
not ( 1
< b
4 & b
4 <= b
5 & b
5 < len (Gauge b2,b1) & 1
<= b
3 & b
3 <= width (Gauge b2,b1) &
(LSeg ((Gauge b2,b1) * b4,b3),((Gauge b2,b1) * b5,b3)) /\ (L~ (Upper_Seq b2,b1)) = {((Gauge b2,b1) * b4,b3)} &
(LSeg ((Gauge b2,b1) * b4,b3),((Gauge b2,b1) * b5,b3)) /\ (L~ (Lower_Seq b2,b1)) = {((Gauge b2,b1) * b5,b3)} & not
LSeg ((Gauge b2,b1) * b4,b3),
((Gauge b2,b1) * b5,b3) meets Lower_Arc b
2 )
theorem Th39: :: JORDAN15:39
for b
1 being
Natfor b
2 being
Simple_closed_curvefor b
3, b
4, b
5 being
Nat holds
not ( 1
< b
4 & b
4 <= b
5 & b
5 < len (Gauge b2,b1) & 1
<= b
3 & b
3 <= width (Gauge b2,b1) &
(LSeg ((Gauge b2,b1) * b4,b3),((Gauge b2,b1) * b5,b3)) /\ (L~ (Upper_Seq b2,b1)) = {((Gauge b2,b1) * b4,b3)} &
(LSeg ((Gauge b2,b1) * b4,b3),((Gauge b2,b1) * b5,b3)) /\ (L~ (Lower_Seq b2,b1)) = {((Gauge b2,b1) * b5,b3)} & not
LSeg ((Gauge b2,b1) * b4,b3),
((Gauge b2,b1) * b5,b3) meets Upper_Arc b
2 )
theorem Th40: :: JORDAN15:40
for b
1 being
Natfor b
2 being
Simple_closed_curvefor b
3, b
4, b
5 being
Nat holds
not ( 1
< b
4 & b
4 <= b
5 & b
5 < len (Gauge b2,b1) & 1
<= b
3 & b
3 <= width (Gauge b2,b1) &
(Gauge b2,b1) * b
4,b
3 in L~ (Upper_Seq b2,b1) &
(Gauge b2,b1) * b
5,b
3 in L~ (Lower_Seq b2,b1) & not
LSeg ((Gauge b2,b1) * b4,b3),
((Gauge b2,b1) * b5,b3) meets Lower_Arc b
2 )
theorem Th41: :: JORDAN15:41
for b
1 being
Natfor b
2 being
Simple_closed_curvefor b
3, b
4, b
5 being
Nat holds
not ( 1
< b
4 & b
4 <= b
5 & b
5 < len (Gauge b2,b1) & 1
<= b
3 & b
3 <= width (Gauge b2,b1) &
(Gauge b2,b1) * b
4,b
3 in L~ (Upper_Seq b2,b1) &
(Gauge b2,b1) * b
5,b
3 in L~ (Lower_Seq b2,b1) & not
LSeg ((Gauge b2,b1) * b4,b3),
((Gauge b2,b1) * b5,b3) meets Upper_Arc b
2 )
theorem Th42: :: JORDAN15:42
for b
1 being
Natfor b
2 being
Simple_closed_curvefor b
3, b
4, b
5 being
Nat holds
not ( 1
< b
4 & b
4 <= b
5 & b
5 < len (Gauge b2,b1) & 1
<= b
3 & b
3 <= width (Gauge b2,b1) & b
1 > 0 &
(Gauge b2,b1) * b
4,b
3 in Upper_Arc (L~ (Cage b2,b1)) &
(Gauge b2,b1) * b
5,b
3 in Lower_Arc (L~ (Cage b2,b1)) & not
LSeg ((Gauge b2,b1) * b4,b3),
((Gauge b2,b1) * b5,b3) meets Lower_Arc b
2 )
theorem Th43: :: JORDAN15:43
for b
1 being
Natfor b
2 being
Simple_closed_curvefor b
3, b
4, b
5 being
Nat holds
not ( 1
< b
4 & b
4 <= b
5 & b
5 < len (Gauge b2,b1) & 1
<= b
3 & b
3 <= width (Gauge b2,b1) & b
1 > 0 &
(Gauge b2,b1) * b
4,b
3 in Upper_Arc (L~ (Cage b2,b1)) &
(Gauge b2,b1) * b
5,b
3 in Lower_Arc (L~ (Cage b2,b1)) & not
LSeg ((Gauge b2,b1) * b4,b3),
((Gauge b2,b1) * b5,b3) meets Upper_Arc b
2 )
theorem Th44: :: JORDAN15:44
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 < len (Gauge b2,(b1 + 1)) &
(Gauge b2,(b1 + 1)) * b
3,
(Center (Gauge b2,(b1 + 1))) in Upper_Arc (L~ (Cage b2,(b1 + 1))) &
(Gauge b2,(b1 + 1)) * b
4,
(Center (Gauge b2,(b1 + 1))) in Lower_Arc (L~ (Cage b2,(b1 + 1))) & not
LSeg ((Gauge b2,(b1 + 1)) * b3,(Center (Gauge b2,(b1 + 1)))),
((Gauge b2,(b1 + 1)) * b4,(Center (Gauge b2,(b1 + 1)))) meets Lower_Arc b
2 )
theorem Th45: :: JORDAN15:45
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 < len (Gauge b2,(b1 + 1)) &
(Gauge b2,(b1 + 1)) * b
3,
(Center (Gauge b2,(b1 + 1))) in Upper_Arc (L~ (Cage b2,(b1 + 1))) &
(Gauge b2,(b1 + 1)) * b
4,
(Center (Gauge b2,(b1 + 1))) in Lower_Arc (L~ (Cage b2,(b1 + 1))) & not
LSeg ((Gauge b2,(b1 + 1)) * b3,(Center (Gauge b2,(b1 + 1)))),
((Gauge b2,(b1 + 1)) * b4,(Center (Gauge b2,(b1 + 1)))) meets Upper_Arc b
2 )
theorem Th46: :: JORDAN15:46
for b
1 being
Natfor b
2 being
Simple_closed_curvefor b
3, b
4, b
5, b
6 being
Nat holds
not ( 1
< b
3 & b
3 <= b
4 & b
4 < len (Gauge b2,b1) & 1
<= b
5 & b
5 <= b
6 & b
6 <= width (Gauge b2,b1) &
((LSeg ((Gauge b2,b1) * b3,b5),((Gauge b2,b1) * b3,b6)) \/ (LSeg ((Gauge b2,b1) * b3,b6),((Gauge b2,b1) * b4,b6))) /\ (L~ (Upper_Seq b2,b1)) = {((Gauge b2,b1) * b4,b6)} &
((LSeg ((Gauge b2,b1) * b3,b5),((Gauge b2,b1) * b3,b6)) \/ (LSeg ((Gauge b2,b1) * b3,b6),((Gauge b2,b1) * b4,b6))) /\ (L~ (Lower_Seq b2,b1)) = {((Gauge b2,b1) * b3,b5)} & not
(LSeg ((Gauge b2,b1) * b3,b5),((Gauge b2,b1) * b3,b6)) \/ (LSeg ((Gauge b2,b1) * b3,b6),((Gauge b2,b1) * b4,b6)) meets Upper_Arc b
2 )
theorem Th47: :: JORDAN15:47
for b
1 being
Natfor b
2 being
Simple_closed_curvefor b
3, b
4, b
5, b
6 being
Nat holds
not ( 1
< b
3 & b
3 <= b
4 & b
4 < len (Gauge b2,b1) & 1
<= b
5 & b
5 <= b
6 & b
6 <= width (Gauge b2,b1) &
((LSeg ((Gauge b2,b1) * b3,b5),((Gauge b2,b1) * b3,b6)) \/ (LSeg ((Gauge b2,b1) * b3,b6),((Gauge b2,b1) * b4,b6))) /\ (L~ (Upper_Seq b2,b1)) = {((Gauge b2,b1) * b4,b6)} &
((LSeg ((Gauge b2,b1) * b3,b5),((Gauge b2,b1) * b3,b6)) \/ (LSeg ((Gauge b2,b1) * b3,b6),((Gauge b2,b1) * b4,b6))) /\ (L~ (Lower_Seq b2,b1)) = {((Gauge b2,b1) * b3,b5)} & not
(LSeg ((Gauge b2,b1) * b3,b5),((Gauge b2,b1) * b3,b6)) \/ (LSeg ((Gauge b2,b1) * b3,b6),((Gauge b2,b1) * b4,b6)) meets Lower_Arc b
2 )
theorem Th48: :: JORDAN15:48
for b
1 being
Natfor b
2 being
Simple_closed_curvefor b
3, b
4, b
5, b
6 being
Nat holds
not ( 1
< b
4 & b
4 <= b
3 & b
3 < len (Gauge b2,b1) & 1
<= b
5 & b
5 <= b
6 & b
6 <= width (Gauge b2,b1) &
((LSeg ((Gauge b2,b1) * b3,b5),((Gauge b2,b1) * b3,b6)) \/ (LSeg ((Gauge b2,b1) * b3,b6),((Gauge b2,b1) * b4,b6))) /\ (L~ (Upper_Seq b2,b1)) = {((Gauge b2,b1) * b4,b6)} &
((LSeg ((Gauge b2,b1) * b3,b5),((Gauge b2,b1) * b3,b6)) \/ (LSeg ((Gauge b2,b1) * b3,b6),((Gauge b2,b1) * b4,b6))) /\ (L~ (Lower_Seq b2,b1)) = {((Gauge b2,b1) * b3,b5)} & not
(LSeg ((Gauge b2,b1) * b3,b5),((Gauge b2,b1) * b3,b6)) \/ (LSeg ((Gauge b2,b1) * b3,b6),((Gauge b2,b1) * b4,b6)) meets Upper_Arc b
2 )
theorem Th49: :: JORDAN15:49
for b
1 being
Natfor b
2 being
Simple_closed_curvefor b
3, b
4, b
5, b
6 being
Nat holds
not ( 1
< b
4 & b
4 <= b
3 & b
3 < len (Gauge b2,b1) & 1
<= b
5 & b
5 <= b
6 & b
6 <= width (Gauge b2,b1) &
((LSeg ((Gauge b2,b1) * b3,b5),((Gauge b2,b1) * b3,b6)) \/ (LSeg ((Gauge b2,b1) * b3,b6),((Gauge b2,b1) * b4,b6))) /\ (L~ (Upper_Seq b2,b1)) = {((Gauge b2,b1) * b4,b6)} &
((LSeg ((Gauge b2,b1) * b3,b5),((Gauge b2,b1) * b3,b6)) \/ (LSeg ((Gauge b2,b1) * b3,b6),((Gauge b2,b1) * b4,b6))) /\ (L~ (Lower_Seq b2,b1)) = {((Gauge b2,b1) * b3,b5)} & not
(LSeg ((Gauge b2,b1) * b3,b5),((Gauge b2,b1) * b3,b6)) \/ (LSeg ((Gauge b2,b1) * b3,b6),((Gauge b2,b1) * b4,b6)) meets Lower_Arc b
2 )
theorem Th50: :: JORDAN15:50
for b
1 being
Natfor b
2 being
Simple_closed_curvefor b
3, b
4, b
5, b
6 being
Nat holds
not ( 1
< b
3 & b
3 < len (Gauge b2,(b1 + 1)) & 1
< b
4 & b
4 < len (Gauge b2,(b1 + 1)) & 1
<= b
5 & b
5 <= b
6 & b
6 <= width (Gauge b2,(b1 + 1)) &
(Gauge b2,(b1 + 1)) * b
3,b
6 in Upper_Arc (L~ (Cage b2,(b1 + 1))) &
(Gauge b2,(b1 + 1)) * b
4,b
5 in Lower_Arc (L~ (Cage b2,(b1 + 1))) & not
(LSeg ((Gauge b2,(b1 + 1)) * b4,b5),((Gauge b2,(b1 + 1)) * b4,b6)) \/ (LSeg ((Gauge b2,(b1 + 1)) * b4,b6),((Gauge b2,(b1 + 1)) * b3,b6)) meets Upper_Arc b
2 )
theorem Th51: :: JORDAN15:51
for b
1 being
Natfor b
2 being
Simple_closed_curvefor b
3, b
4, b
5, b
6 being
Nat holds
not ( 1
< b
3 & b
3 < len (Gauge b2,(b1 + 1)) & 1
< b
4 & b
4 < len (Gauge b2,(b1 + 1)) & 1
<= b
5 & b
5 <= b
6 & b
6 <= width (Gauge b2,(b1 + 1)) &
(Gauge b2,(b1 + 1)) * b
3,b
6 in Upper_Arc (L~ (Cage b2,(b1 + 1))) &
(Gauge b2,(b1 + 1)) * b
4,b
5 in Lower_Arc (L~ (Cage b2,(b1 + 1))) & not
(LSeg ((Gauge b2,(b1 + 1)) * b4,b5),((Gauge b2,(b1 + 1)) * b4,b6)) \/ (LSeg ((Gauge b2,(b1 + 1)) * b4,b6),((Gauge b2,(b1 + 1)) * b3,b6)) meets Lower_Arc b
2 )
theorem Th52: :: JORDAN15:52
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)) & 1
<= b
4 & b
4 <= b
5 & b
5 <= width (Gauge b2,(b1 + 1)) &
(Gauge b2,(b1 + 1)) * b
3,b
5 in Upper_Arc (L~ (Cage b2,(b1 + 1))) &
(Gauge b2,(b1 + 1)) * (Center (Gauge b2,(b1 + 1))),b
4 in Lower_Arc (L~ (Cage b2,(b1 + 1))) & not
(LSeg ((Gauge b2,(b1 + 1)) * (Center (Gauge b2,(b1 + 1))),b4),((Gauge b2,(b1 + 1)) * (Center (Gauge b2,(b1 + 1))),b5)) \/ (LSeg ((Gauge b2,(b1 + 1)) * (Center (Gauge b2,(b1 + 1))),b5),((Gauge b2,(b1 + 1)) * b3,b5)) meets Upper_Arc b
2 )
theorem Th53: :: JORDAN15:53
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)) & 1
<= b
4 & b
4 <= b
5 & b
5 <= width (Gauge b2,(b1 + 1)) &
(Gauge b2,(b1 + 1)) * b
3,b
5 in Upper_Arc (L~ (Cage b2,(b1 + 1))) &
(Gauge b2,(b1 + 1)) * (Center (Gauge b2,(b1 + 1))),b
4 in Lower_Arc (L~ (Cage b2,(b1 + 1))) & not
(LSeg ((Gauge b2,(b1 + 1)) * (Center (Gauge b2,(b1 + 1))),b4),((Gauge b2,(b1 + 1)) * (Center (Gauge b2,(b1 + 1))),b5)) \/ (LSeg ((Gauge b2,(b1 + 1)) * (Center (Gauge b2,(b1 + 1))),b5),((Gauge b2,(b1 + 1)) * b3,b5)) meets Lower_Arc b
2 )