:: JORDAN1C semantic presentation
Lemma2:
for b1, b2, b3 being real number holds
( b1 <> 0 & b2 <> 0 implies (b1 / b2) * ((b3 / b1) * b2) = b3 )
Lemma3:
for b1 being Point of (TOP-REAL 2) holds
b1 is Point of (Euclid 2)
Lemma4:
for b1 being real number holds
not ( b1 > 0 & not 2 * (b1 / 4) < b1 )
theorem Th1: :: JORDAN1C:1
canceled;
theorem Th2: :: JORDAN1C:2
for b
1 being
Simple_closed_curvefor b
2, b
3, b
4 being
Nat holds
(
[b2,b3] in Indices (Gauge b1,b4) &
[(b2 + 1),b3] in Indices (Gauge b1,b4) implies
dist ((Gauge b1,b4) * 1,1),
((Gauge b1,b4) * 2,1) = (((Gauge b1,b4) * (b2 + 1),b3) `1 ) - (((Gauge b1,b4) * b2,b3) `1 ) )
theorem Th3: :: JORDAN1C:3
for b
1 being
Simple_closed_curvefor b
2, b
3, b
4 being
Nat holds
(
[b2,b3] in Indices (Gauge b1,b4) &
[b2,(b3 + 1)] in Indices (Gauge b1,b4) implies
dist ((Gauge b1,b4) * 1,1),
((Gauge b1,b4) * 1,2) = (((Gauge b1,b4) * b2,(b3 + 1)) `2 ) - (((Gauge b1,b4) * b2,b3) `2 ) )
TopSpaceMetr (Euclid 2) = TOP-REAL 2
by EUCLID:def 8;
then Lemma7:
TOP-REAL 2 = TopStruct(# the carrier of (Euclid 2),(Family_open_set (Euclid 2)) #)
by PCOMPS_1:def 6;
Lemma8:
for b1 being Simple_closed_curve
for b2, b3, b4 being Nat
for b5 being Point of (TOP-REAL 2)
for b6 being real number
for b7 being Point of (Euclid 2) holds
( 1 <= b2 & b2 + 1 <= len (Gauge b1,b3) & 1 <= b4 & b4 + 1 <= width (Gauge b1,b3) & b6 > 0 & b5 = b7 & dist ((Gauge b1,b3) * 1,1),((Gauge b1,b3) * 1,2) < b6 / 4 & dist ((Gauge b1,b3) * 1,1),((Gauge b1,b3) * 2,1) < b6 / 4 & b5 in cell (Gauge b1,b3),b2,b4 & (Gauge b1,b3) * b2,b4 in Ball b7,b6 & (Gauge b1,b3) * (b2 + 1),b4 in Ball b7,b6 & (Gauge b1,b3) * b2,(b4 + 1) in Ball b7,b6 & (Gauge b1,b3) * (b2 + 1),(b4 + 1) in Ball b7,b6 implies cell (Gauge b1,b3),b2,b4 c= Ball b7,b6 )
theorem Th4: :: JORDAN1C:4
theorem Th5: :: JORDAN1C:5
Lemma10:
for b1 being Point of (TOP-REAL 2)
for b2 being Subset of (TOP-REAL 2) holds
( b1 in b2 & b2 is Bounded implies ( inf (proj1 || b2) <= b1 `1 & b1 `1 <= sup (proj1 || b2) ) )
Lemma11:
for b1 being Point of (TOP-REAL 2)
for b2 being Subset of (TOP-REAL 2) holds
( b1 in b2 & b2 is Bounded implies ( inf (proj2 || b2) <= b1 `2 & b1 `2 <= sup (proj2 || b2) ) )
theorem Th6: :: JORDAN1C:6
theorem Th7: :: JORDAN1C:7
Lemma13:
for b1 being Subset of (TOP-REAL 2) holds
( b1 is Bounded implies for b2 being real number holds
not ( BDD b1 <> {} & b2 > W-bound (BDD b1) & ( for b3 being Point of (TOP-REAL 2) holds
( b3 in BDD b1 implies b3 `1 >= b2 ) ) ) )
Lemma14:
for b1 being Subset of (TOP-REAL 2) holds
( b1 is Bounded implies for b2 being real number holds
not ( BDD b1 <> {} & b2 < E-bound (BDD b1) & ( for b3 being Point of (TOP-REAL 2) holds
( b3 in BDD b1 implies b3 `1 <= b2 ) ) ) )
Lemma15:
for b1 being Subset of (TOP-REAL 2) holds
( b1 is Bounded implies for b2 being real number holds
not ( BDD b1 <> {} & b2 < N-bound (BDD b1) & ( for b3 being Point of (TOP-REAL 2) holds
( b3 in BDD b1 implies b3 `2 <= b2 ) ) ) )
Lemma16:
for b1 being Subset of (TOP-REAL 2) holds
( b1 is Bounded implies for b2 being real number holds
not ( BDD b1 <> {} & b2 > S-bound (BDD b1) & ( for b3 being Point of (TOP-REAL 2) holds
( b3 in BDD b1 implies b3 `2 >= b2 ) ) ) )
theorem Th8: :: JORDAN1C:8
theorem Th9: :: JORDAN1C:9
theorem Th10: :: JORDAN1C:10
theorem Th11: :: JORDAN1C:11
theorem Th12: :: JORDAN1C:12
theorem Th13: :: JORDAN1C:13
theorem Th14: :: JORDAN1C:14
theorem Th15: :: JORDAN1C:15
theorem Th16: :: JORDAN1C:16
theorem Th17: :: JORDAN1C:17
theorem Th18: :: JORDAN1C:18
theorem Th19: :: JORDAN1C:19
theorem Th20: :: JORDAN1C:20
theorem Th21: :: JORDAN1C:21
theorem Th22: :: JORDAN1C:22
theorem Th23: :: JORDAN1C:23
theorem Th24: :: JORDAN1C:24
theorem Th25: :: JORDAN1C:25
theorem Th26: :: JORDAN1C:26
theorem Th27: :: JORDAN1C:27
theorem Th28: :: JORDAN1C:28
theorem Th29: :: JORDAN1C:29
theorem Th30: :: JORDAN1C:30
for b
1 being
Simple_closed_curvefor b
2, b
3, b
4 being
Natfor b
5, b
6 being
Point of
(TOP-REAL 2)for b
7 being
real number holds
not (
dist ((Gauge b1,b2) * 1,1),
((Gauge b1,b2) * 1,2) < b
7 &
dist ((Gauge b1,b2) * 1,1),
((Gauge b1,b2) * 2,1) < b
7 & b
5 in cell (Gauge b1,b2),b
3,b
4 & b
6 in cell (Gauge b1,b2),b
3,b
4 & 1
<= b
3 & b
3 + 1
<= len (Gauge b1,b2) & 1
<= b
4 & b
4 + 1
<= width (Gauge b1,b2) & not
dist b
5,b
6 < 2
* b
7 )
theorem Th31: :: JORDAN1C:31
theorem Th32: :: JORDAN1C:32
theorem Th33: :: JORDAN1C:33
theorem Th34: :: JORDAN1C:34
theorem Th35: :: JORDAN1C:35
for b
1 being
Simple_closed_curvefor b
2 being
Point of
(TOP-REAL 2) holds
not ( b
2 in BDD b
1 & ( for b
3, b
4, b
5 being
Nat holds
not ( 1
< b
4 & b
4 < len (Gauge b1,b3) & 1
< b
5 & b
5 < width (Gauge b1,b3) & b
2 `1 <> ((Gauge b1,b3) * b4,b5) `1 & b
2 in cell (Gauge b1,b3),b
4,b
5 &
cell (Gauge b1,b3),b
4,b
5 c= BDD b
1 ) ) )
theorem Th36: :: JORDAN1C:36