:: JORDAN1F semantic presentation
theorem Th1: :: JORDAN1F:1
for b
1, b
2, b
3 being
Natfor b
4 being
FinSequence of the
carrier of
(TOP-REAL 2)for b
5 being
Go-board holds
not ( b
4 is_sequence_on b
5 &
LSeg (b5 * b1,b2),
(b5 * b1,b3) meets L~ b
4 &
[b1,b2] in Indices b
5 &
[b1,b3] in Indices b
5 & b
2 <= b
3 & ( for b
6 being
Nat holds
not ( b
2 <= b
6 & b
6 <= b
3 &
(b5 * b1,b6) `2 = inf (proj2 .: ((LSeg (b5 * b1,b2),(b5 * b1,b3)) /\ (L~ b4))) ) ) )
theorem Th2: :: JORDAN1F:2
for b
1, b
2, b
3 being
Natfor b
4 being
FinSequence of the
carrier of
(TOP-REAL 2)for b
5 being
Go-board holds
not ( b
4 is_sequence_on b
5 &
LSeg (b5 * b1,b2),
(b5 * b1,b3) meets L~ b
4 &
[b1,b2] in Indices b
5 &
[b1,b3] in Indices b
5 & b
2 <= b
3 & ( for b
6 being
Nat holds
not ( b
2 <= b
6 & b
6 <= b
3 &
(b5 * b1,b6) `2 = sup (proj2 .: ((LSeg (b5 * b1,b2),(b5 * b1,b3)) /\ (L~ b4))) ) ) )
theorem Th3: :: JORDAN1F:3
for b
1, b
2, b
3 being
Natfor b
4 being
FinSequence of the
carrier of
(TOP-REAL 2)for b
5 being
Go-board holds
not ( b
4 is_sequence_on b
5 &
LSeg (b5 * b1,b2),
(b5 * b3,b2) meets L~ b
4 &
[b1,b2] in Indices b
5 &
[b3,b2] in Indices b
5 & b
1 <= b
3 & ( for b
6 being
Nat holds
not ( b
1 <= b
6 & b
6 <= b
3 &
(b5 * b6,b2) `1 = inf (proj1 .: ((LSeg (b5 * b1,b2),(b5 * b3,b2)) /\ (L~ b4))) ) ) )
theorem Th4: :: JORDAN1F:4
for b
1, b
2, b
3 being
Natfor b
4 being
FinSequence of the
carrier of
(TOP-REAL 2)for b
5 being
Go-board holds
not ( b
4 is_sequence_on b
5 &
LSeg (b5 * b1,b2),
(b5 * b3,b2) meets L~ b
4 &
[b1,b2] in Indices b
5 &
[b3,b2] in Indices b
5 & b
1 <= b
3 & ( for b
6 being
Nat holds
not ( b
1 <= b
6 & b
6 <= b
3 &
(b5 * b6,b2) `1 = sup (proj1 .: ((LSeg (b5 * b1,b2),(b5 * b3,b2)) /\ (L~ b4))) ) ) )
theorem Th5: :: JORDAN1F:5
theorem Th6: :: JORDAN1F:6
theorem Th7: :: JORDAN1F:7
theorem Th8: :: JORDAN1F:8
theorem Th9: :: JORDAN1F:9
theorem Th10: :: JORDAN1F:10
theorem Th11: :: JORDAN1F:11
for b
1 being
Go-boardfor b
2 being
Point of
(TOP-REAL 2)for b
3 being
FinSequence of
(TOP-REAL 2) holds
( b
3 is_sequence_on b
1 & ex b
4, b
5 being
Nat st
(
[b4,b5] in Indices b
1 & b
2 = b
1 * b
4,b
5 ) & ( for b
4, b
5, b
6, b
7 being
Nat holds
(
[b4,b5] in Indices b
1 &
[b6,b7] in Indices b
1 & b
2 = b
1 * b
4,b
5 & b
3 /. 1
= b
1 * b
6,b
7 implies
(abs (b6 - b4)) + (abs (b7 - b5)) = 1 ) ) implies
<*b2*> ^ b
3 is_sequence_on b
1 )
theorem Th12: :: JORDAN1F:12
theorem Th13: :: JORDAN1F:13
for b
1 being
Natfor b
2 being non
empty being_simple_closed_curve compact non
horizontal non
vertical Subset of
(TOP-REAL 2)for b
3 being
Point of
(TOP-REAL 2) holds
not ( b
3 `1 = ((W-bound b2) + (E-bound b2)) / 2 & b
3 `2 = inf (proj2 .: ((LSeg ((Gauge b2,1) * (Center (Gauge b2,1)),1),((Gauge b2,1) * (Center (Gauge b2,1)),(width (Gauge b2,1)))) /\ (Upper_Arc (L~ (Cage b2,(b1 + 1)))))) & ( for b
4 being
Nat holds
not ( 1
<= b
4 & b
4 <= width (Gauge b2,(b1 + 1)) & b
3 = (Gauge b2,(b1 + 1)) * (Center (Gauge b2,(b1 + 1))),b
4 ) ) )