:: JORDAN22 semantic presentation
Lemma1:
TOP-REAL 2 = TopSpaceMetr (Euclid 2)
by EUCLID:def 8;
Lemma2:
for b1 being real number
for b2 being Subset of (TOP-REAL 2) holds
not ( b1 in proj2 .: b2 & ( for b3 being Point of (TOP-REAL 2) holds
not ( b3 in b2 & proj2 . b3 = b1 ) ) )
theorem Th1: :: JORDAN22:1
theorem Th2: :: JORDAN22:2
theorem Th3: :: JORDAN22:3
theorem Th4: :: JORDAN22:4
Lemma5:
dom proj2 = the carrier of (TOP-REAL 2)
by FUNCT_2:def 1;
Lemma6:
for b1 being non empty Subset of (TOP-REAL 2)
for b2 being Nat holds 1 <= len (Gauge b1,b2)
theorem Th5: :: JORDAN22:5
theorem Th6: :: JORDAN22:6
theorem Th7: :: JORDAN22:7
theorem Th8: :: JORDAN22:8
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
2 > b
3 &
[b1,b4] in Indices (Gauge b5,b3) &
[b1,(b4 + 1)] in Indices (Gauge b5,b3) & not
dist ((Gauge b5,b2) * b1,b4),
((Gauge b5,b2) * b1,(b4 + 1)) < dist ((Gauge b5,b3) * b1,b4),
((Gauge b5,b3) * b1,(b4 + 1)) )
theorem Th9: :: JORDAN22:9
for b
1, b
2 being
Natfor b
3 being
compact non
horizontal non
vertical Subset of
(TOP-REAL 2) holds
not ( b
1 > b
2 & not
dist ((Gauge b3,b1) * 1,1),
((Gauge b3,b1) * 1,2) < dist ((Gauge b3,b2) * 1,1),
((Gauge b3,b2) * 1,2) )
theorem Th10: :: JORDAN22:10
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
2 > b
3 &
[b1,b4] in Indices (Gauge b5,b3) &
[(b1 + 1),b4] in Indices (Gauge b5,b3) & not
dist ((Gauge b5,b2) * b1,b4),
((Gauge b5,b2) * (b1 + 1),b4) < dist ((Gauge b5,b3) * b1,b4),
((Gauge b5,b3) * (b1 + 1),b4) )
theorem Th11: :: JORDAN22:11
for b
1, b
2 being
Natfor b
3 being
compact non
horizontal non
vertical Subset of
(TOP-REAL 2) holds
not ( b
1 > b
2 & not
dist ((Gauge b3,b1) * 1,1),
((Gauge b3,b1) * 2,1) < dist ((Gauge b3,b2) * 1,1),
((Gauge b3,b2) * 2,1) )
theorem Th12: :: JORDAN22:12
for b
1 being
Simple_closed_curvefor b
2 being
Natfor b
3, b
4 being
real number holds
not ( b
3 > 0 & b
4 > 0 & ( for b
5 being
Nat holds
not ( b
2 < b
5 &
dist ((Gauge b1,b5) * 1,1),
((Gauge b1,b5) * 1,2) < b
3 &
dist ((Gauge b1,b5) * 1,1),
((Gauge b1,b5) * 2,1) < b
4 ) ) )
theorem Th13: :: JORDAN22:13
for b
1 being
Simple_closed_curvefor b
2 being
Nat holds
( 0
< b
2 implies
sup (proj2 .: ((L~ (Cage b1,b2)) /\ (LSeg ((Gauge b1,b2) * (Center (Gauge b1,b2)),1),((Gauge b1,b2) * (Center (Gauge b1,b2)),(len (Gauge b1,b2)))))) = sup (proj2 .: ((L~ (Cage b1,b2)) /\ (Vertical_Line (((E-bound (L~ (Cage b1,b2))) + (W-bound (L~ (Cage b1,b2)))) / 2)))) )
theorem Th14: :: JORDAN22:14
for b
1 being
Simple_closed_curvefor b
2 being
Nat holds
( 0
< b
2 implies
inf (proj2 .: ((L~ (Cage b1,b2)) /\ (LSeg ((Gauge b1,b2) * (Center (Gauge b1,b2)),1),((Gauge b1,b2) * (Center (Gauge b1,b2)),(len (Gauge b1,b2)))))) = inf (proj2 .: ((L~ (Cage b1,b2)) /\ (Vertical_Line (((E-bound (L~ (Cage b1,b2))) + (W-bound (L~ (Cage b1,b2)))) / 2)))) )
theorem Th15: :: JORDAN22:15
for b
1 being
Simple_closed_curvefor b
2 being
Nat holds
( 0
< b
2 implies
UMP (L~ (Cage b1,b2)) = |[(((E-bound (L~ (Cage b1,b2))) + (W-bound (L~ (Cage b1,b2)))) / 2),(sup (proj2 .: ((L~ (Cage b1,b2)) /\ (LSeg ((Gauge b1,b2) * (Center (Gauge b1,b2)),1),((Gauge b1,b2) * (Center (Gauge b1,b2)),(len (Gauge b1,b2)))))))]| )
by Th13;
theorem Th16: :: JORDAN22:16
for b
1 being
Simple_closed_curvefor b
2 being
Nat holds
( 0
< b
2 implies
LMP (L~ (Cage b1,b2)) = |[(((E-bound (L~ (Cage b1,b2))) + (W-bound (L~ (Cage b1,b2)))) / 2),(inf (proj2 .: ((L~ (Cage b1,b2)) /\ (LSeg ((Gauge b1,b2) * (Center (Gauge b1,b2)),1),((Gauge b1,b2) * (Center (Gauge b1,b2)),(len (Gauge b1,b2)))))))]| )
by Th14;
theorem Th17: :: JORDAN22:17
theorem Th18: :: JORDAN22:18
theorem Th19: :: JORDAN22:19
theorem Th20: :: JORDAN22:20
theorem Th21: :: JORDAN22:21
theorem Th22: :: JORDAN22:22
theorem Th23: :: JORDAN22:23
theorem Th24: :: JORDAN22:24
theorem Th25: :: JORDAN22:25
theorem Th26: :: JORDAN22:26
theorem Th27: :: JORDAN22:27
theorem Th28: :: JORDAN22:28
theorem Th29: :: JORDAN22:29
theorem Th30: :: JORDAN22:30
theorem Th31: :: JORDAN22:31
theorem Th32: :: JORDAN22:32
theorem Th33: :: JORDAN22:33
theorem Th34: :: JORDAN22:34
theorem Th35: :: JORDAN22:35
theorem Th36: :: JORDAN22:36
theorem Th37: :: JORDAN22:37
theorem Th38: :: JORDAN22:38
theorem Th39: :: JORDAN22:39
theorem Th40: :: JORDAN22:40
theorem Th41: :: JORDAN22:41
theorem Th42: :: JORDAN22:42
theorem Th43: :: JORDAN22:43