:: SPPOL_1 semantic presentation
theorem Th1: :: SPPOL_1:1
canceled;
theorem Th2: :: SPPOL_1:2
canceled;
theorem Th3: :: SPPOL_1:3
canceled;
theorem Th4: :: SPPOL_1:4
canceled;
theorem Th5: :: SPPOL_1:5
theorem Th6: :: SPPOL_1:6
for b
1, b
2 being
Nat holds
( b
1 < b
2 implies b
1 <= b
2 - 1 )
theorem Th7: :: SPPOL_1:7
for b
1, b
2, b
3 being
Nat holds
( 1
<= b
1 - b
2 & b
1 - b
2 <= b
3 implies ( b
1 - b
2 in Seg b
3 & b
1 - b
2 is
Nat ) )
Lemma1:
for b1, b2, b3, b4 being real number holds
( b1 >= 0 & b2 >= 0 & b3 >= 0 & b4 >= 0 & (b1 * b3) + (b2 * b4) = 0 implies ( ( b1 = 0 or b3 = 0 ) & ( b2 = 0 or b4 = 0 ) ) )
by XREAL_1:73;
theorem Th8: :: SPPOL_1:8
canceled;
theorem Th9: :: SPPOL_1:9
canceled;
theorem Th10: :: SPPOL_1:10
canceled;
theorem Th11: :: SPPOL_1:11
canceled;
theorem Th12: :: SPPOL_1:12
for b
1, b
2, b
3 being
real number holds
not ( 0
<= b
1 & b
1 <= 1 & b
2 >= 0 & b
3 >= 0 &
(b1 * b2) + ((1 - b1) * b3) = 0 & not ( b
1 = 0 & b
3 = 0 ) & not ( b
1 = 1 & b
2 = 0 ) & not ( b
2 = 0 & b
3 = 0 ) )
theorem Th13: :: SPPOL_1:13
theorem Th14: :: SPPOL_1:14
theorem Th15: :: SPPOL_1:15
theorem Th16: :: SPPOL_1:16
theorem Th17: :: SPPOL_1:17
theorem Th18: :: SPPOL_1:18
theorem Th19: :: SPPOL_1:19
theorem Th20: :: SPPOL_1:20
theorem Th21: :: SPPOL_1:21
for b
1 being
Natfor b
2, b
3, b
4 being
Point of
(TOP-REAL b1) holds
not ( b
2 in LSeg b
3,b
4 & ( for b
5 being
Real holds
not ( 0
<= b
5 & b
5 <= 1 & b
2 = ((1 - b5) * b3) + (b5 * b4) ) ) )
theorem Th22: :: SPPOL_1:22
theorem Th23: :: SPPOL_1:23
theorem Th24: :: SPPOL_1:24
theorem Th25: :: SPPOL_1:25
for b
1 being
Natfor b
2, b
3, b
4, b
5 being
Point of
(TOP-REAL b1) holds
not (
LSeg b
2,b
3 = LSeg b
4,b
5 & not ( b
2 = b
4 & b
3 = b
5 ) & not ( b
2 = b
5 & b
3 = b
4 ) )
theorem Th26: :: SPPOL_1:26
theorem Th27: :: SPPOL_1:27
theorem Th28: :: SPPOL_1:28
theorem Th29: :: SPPOL_1:29
:: deftheorem Def1 defines is_extremal_in SPPOL_1:def 1 :
theorem Th30: :: SPPOL_1:30
theorem Th31: :: SPPOL_1:31
theorem Th32: :: SPPOL_1:32
theorem Th33: :: SPPOL_1:33
theorem Th34: :: SPPOL_1:34
theorem Th35: :: SPPOL_1:35
:: deftheorem Def2 defines horizontal SPPOL_1:def 2 :
:: deftheorem Def3 defines vertical SPPOL_1:def 3 :
Lemma15:
for b1 being Subset of (TOP-REAL 2) holds
not ( not b1 is trivial & b1 is horizontal & b1 is vertical )
theorem Th36: :: SPPOL_1:36
theorem Th37: :: SPPOL_1:37
theorem Th38: :: SPPOL_1:38
theorem Th39: :: SPPOL_1:39
theorem Th40: :: SPPOL_1:40
theorem Th41: :: SPPOL_1:41
theorem Th42: :: SPPOL_1:42
theorem Th43: :: SPPOL_1:43
theorem Th44: :: SPPOL_1:44
theorem Th45: :: SPPOL_1:45
Lemma23:
for b1 being FinSequence of the carrier of (TOP-REAL 2)
for b2 being Nat holds { (LSeg b1,b3) where B is Nat : ( 1 <= b3 & b3 + 1 <= len b1 & b3 <> b2 & b3 <> b2 + 1 ) } is finite
theorem Th46: :: SPPOL_1:46
theorem Th47: :: SPPOL_1:47
Lemma25:
for b1 being FinSequence of the carrier of (TOP-REAL 2)
for b2 being Nat holds
{ (LSeg b1,b3) where B is Nat : ( 1 <= b3 & b3 + 1 <= len b1 & b3 <> b2 & b3 <> b2 + 1 ) } is Subset-Family of (TOP-REAL 2)
theorem Th48: :: SPPOL_1:48
theorem Th49: :: SPPOL_1:49
Lemma27:
for b1 being FinSequence of the carrier of (TOP-REAL 2)
for b2 being Subset of (TOP-REAL 2)
for b3 being Nat holds
( b2 = union { (LSeg b1,b4) where B is Nat : ( 1 <= b4 & b4 + 1 <= len b1 & b4 <> b3 & b4 <> b3 + 1 ) } implies b2 is closed )
:: deftheorem Def4 defines alternating SPPOL_1:def 4 :
theorem Th50: :: SPPOL_1:50
theorem Th51: :: SPPOL_1:51
theorem Th52: :: SPPOL_1:52
theorem Th53: :: SPPOL_1:53
Lemma32:
for b1 being FinSequence of the carrier of (TOP-REAL 2)
for b2 being Nat
for b3, b4 being Point of (TOP-REAL 2) holds
not ( b1 is alternating & 1 <= b2 & b2 + 2 <= len b1 & b3 = b1 /. b2 & b4 = b1 /. (b2 + 2) & ( for b5 being Point of (TOP-REAL 2) holds
not ( b5 in LSeg b3,b4 & b5 `1 <> b3 `1 & b5 `1 <> b4 `1 & b5 `2 <> b3 `2 & b5 `2 <> b4 `2 ) ) )
theorem Th54: :: SPPOL_1:54
theorem Th55: :: SPPOL_1:55
theorem Th56: :: SPPOL_1:56
theorem Th57: :: SPPOL_1:57
theorem Th58: :: SPPOL_1:58
theorem Th59: :: SPPOL_1:59
theorem Th60: :: SPPOL_1:60
:: deftheorem Def5 defines are_generators_of SPPOL_1:def 5 :
theorem Th61: :: SPPOL_1:61
theorem Th62: :: SPPOL_1:62
theorem Th63: :: SPPOL_1:63
theorem Th64: :: SPPOL_1:64