:: SPPOL_2 semantic presentation
theorem Th1: :: SPPOL_2:1
theorem Th2: :: SPPOL_2:2
theorem Th3: :: SPPOL_2:3
theorem Th4: :: SPPOL_2:4
theorem Th5: :: SPPOL_2:5
theorem Th6: :: SPPOL_2:6
theorem Th7: :: SPPOL_2:7
theorem Th8: :: SPPOL_2:8
theorem Th9: :: SPPOL_2:9
theorem Th10: :: SPPOL_2:10
theorem Th11: :: SPPOL_2:11
theorem Th12: :: SPPOL_2:12
theorem Th13: :: SPPOL_2:13
theorem Th14: :: SPPOL_2:14
theorem Th15: :: SPPOL_2:15
theorem Th16: :: SPPOL_2:16
theorem Th17: :: SPPOL_2:17
theorem Th18: :: SPPOL_2:18
theorem Th19: :: SPPOL_2:19
theorem Th20: :: SPPOL_2:20
theorem Th21: :: SPPOL_2:21
theorem Th22: :: SPPOL_2:22
theorem Th23: :: SPPOL_2:23
Lemma97:
for f being FinSequence of (TOP-REAL 2)
for n being Element of NAT holds L~ (f | n) c= L~ f
theorem Th24: :: SPPOL_2:24
canceled;
theorem Th25: :: SPPOL_2:25
theorem Th26: :: SPPOL_2:26
theorem Th27: :: SPPOL_2:27
Lemma102:
for p, q being Point of (TOP-REAL 2) holds <*p,q*> is unfolded
Lemma103:
for f being FinSequence of (TOP-REAL 2)
for n being Element of NAT st f is unfolded holds
f | n is unfolded
Lemma104:
for f being FinSequence of (TOP-REAL 2)
for n being Element of NAT st f is unfolded holds
f /^ n is unfolded
theorem Th28: :: SPPOL_2:28
Lemma106:
for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st f is unfolded holds
f -: p is unfolded
theorem Th29: :: SPPOL_2:29
theorem Th30: :: SPPOL_2:30
theorem Th31: :: SPPOL_2:31
theorem Th32: :: SPPOL_2:32
theorem Th33: :: SPPOL_2:33
theorem Th34: :: SPPOL_2:34
Lemma120:
for p, q being Point of (TOP-REAL 2) holds <*p,q*> is s.n.c.
Lemma121:
for f being FinSequence of (TOP-REAL 2)
for n being Element of NAT st f is s.n.c. holds
f | n is s.n.c.
Lemma122:
for f being FinSequence of (TOP-REAL 2)
for n being Element of NAT st f is s.n.c. holds
f /^ n is s.n.c.
Lemma123:
for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st f is s.n.c. holds
f -: p is s.n.c.
theorem Th35: :: SPPOL_2:35
theorem Th36: :: SPPOL_2:36
theorem Th37: :: SPPOL_2:37
theorem Th38: :: SPPOL_2:38
Lemma194:
<*> the carrier of (TOP-REAL 2) is special
theorem Th39: :: SPPOL_2:39
theorem Th40: :: SPPOL_2:40
Lemma197:
for f being FinSequence of (TOP-REAL 2)
for n being Element of NAT st f is special holds
f | n is special
Lemma198:
for f being FinSequence of (TOP-REAL 2)
for n being Element of NAT st f is special holds
f /^ n is special
theorem Th41: :: SPPOL_2:41
Lemma200:
for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st f is special holds
f -: p is special
theorem Th42: :: SPPOL_2:42
Lemma202:
for f, g being FinSequence of (TOP-REAL 2) st f is special & g is special & ( (f /. (len f)) `1 = (g /. 1) `1 or (f /. (len f)) `2 = (g /. 1) `2 ) holds
f ^ g is special
theorem Th43: :: SPPOL_2:43
canceled;
theorem Th44: :: SPPOL_2:44
theorem Th45: :: SPPOL_2:45
theorem Th46: :: SPPOL_2:46
theorem Th47: :: SPPOL_2:47
theorem Th48: :: SPPOL_2:48
theorem Th49: :: SPPOL_2:49
Lemma207:
for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in rng f holds
L~ (f -: p) c= L~ f
Lemma208:
for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in rng f holds
L~ (f :- p) c= L~ f
theorem Th50: :: SPPOL_2:50
theorem Th51: :: SPPOL_2:51
theorem Th52: :: SPPOL_2:52
theorem Th53: :: SPPOL_2:53
:: deftheorem Def1 defines split SPPOL_2:def 1 :
theorem Th54: :: SPPOL_2:54
Lemma214:
for P being Subset of (TOP-REAL 2)
for p1, p2, q being Point of (TOP-REAL 2)
for f1, f2 being S-Sequence_in_R2 st p1 = f1 /. 1 & p1 = f2 /. 1 & p2 = f1 /. (len f1) & p2 = f2 /. (len f2) & (L~ f1) /\ (L~ f2) = {p1,p2} & P = (L~ f1) \/ (L~ f2) & q <> p1 & q in rng f1 holds
ex g1, g2 being S-Sequence_in_R2 st
( p1 = g1 /. 1 & p1 = g2 /. 1 & q = g1 /. (len g1) & q = g2 /. (len g2) & (L~ g1) /\ (L~ g2) = {p1,q} & P = (L~ g1) \/ (L~ g2) )
theorem Th55: :: SPPOL_2:55
theorem Th56: :: SPPOL_2:56
theorem Th57: :: SPPOL_2:57
:: deftheorem Def2 defines special_polygonal SPPOL_2:def 2 :
Lemma287:
for P being Subset of (TOP-REAL 2) st P is being_special_polygon holds
ex p1, p2 being Point of (TOP-REAL 2) st
( p1 <> p2 & p1 in P & p2 in P )
definition
let r1 be
real number ;
let r2 be
real number ;
let r1' be
real number ;
let r2' be
real number ;
func [.c1,c2,c3,c4.] -> Subset of
(TOP-REAL 2) equals :: SPPOL_2:def 3
((LSeg |[r1,r1']|,|[r1,r2']|) \/ (LSeg |[r1,r2']|,|[r2,r2']|)) \/ ((LSeg |[r2,r2']|,|[r2,r1']|) \/ (LSeg |[r2,r1']|,|[r1,r1']|));
coherence
((LSeg |[r1,r1']|,|[r1,r2']|) \/ (LSeg |[r1,r2']|,|[r2,r2']|)) \/ ((LSeg |[r2,r2']|,|[r2,r1']|) \/ (LSeg |[r2,r1']|,|[r1,r1']|)) is Subset of (TOP-REAL 2)
;
end;
:: deftheorem Def3 defines [. SPPOL_2:def 3 :
for
r1,
r2,
r1',
r2' being
real number holds
[.r1,r2,r1',r2'.] = ((LSeg |[r1,r1']|,|[r1,r2']|) \/ (LSeg |[r1,r2']|,|[r2,r2']|)) \/ ((LSeg |[r2,r2']|,|[r2,r1']|) \/ (LSeg |[r2,r1']|,|[r1,r1']|));
theorem Th58: :: SPPOL_2:58
theorem Th59: :: SPPOL_2:59
theorem Th60: :: SPPOL_2:60
theorem Th61: :: SPPOL_2:61
theorem Th62: :: SPPOL_2:62
theorem Th63: :: SPPOL_2:63
theorem Th64: :: SPPOL_2:64
theorem Th65: :: SPPOL_2:65