:: SPPOL_1 semantic presentation
:: Showing IDV graph ... (Click the Palm Trees again to close it)
theorem :: SPPOL_1:1
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
canceled;
theorem :: SPPOL_1:2
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
canceled;
theorem :: SPPOL_1:3
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
canceled;
theorem :: SPPOL_1:4
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
canceled;
theorem :: SPPOL_1:5
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
theorem :: SPPOL_1:6
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
for
n,
k being
Nat st
n < k holds
n <= k - 1
theorem :: SPPOL_1:7
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
Lm1:
for r1, r2, s1, s2 being real number st r1 >= 0 & r2 >= 0 & s1 >= 0 & s2 >= 0 & (r1 * s1) + (r2 * s2) = 0 holds
( ( r1 = 0 or s1 = 0 ) & ( r2 = 0 or s2 = 0 ) )
by XREAL_1:73;
theorem :: SPPOL_1:8
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
canceled;
theorem :: SPPOL_1:9
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
canceled;
theorem :: SPPOL_1:10
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
canceled;
theorem :: SPPOL_1:11
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
canceled;
theorem Th12: :: SPPOL_1:12
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
for
r,
s1,
s2 being
real number st 0
<= r &
r <= 1 &
s1 >= 0 &
s2 >= 0 &
(r * s1) + ((1 - r) * s2) = 0 & not (
r = 0 &
s2 = 0 ) & not (
r = 1 &
s1 = 0 ) holds
(
s1 = 0 &
s2 = 0 )
theorem Th13: :: SPPOL_1:13
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
theorem Th14: :: SPPOL_1:14
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
theorem Th15: :: SPPOL_1:15
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
theorem :: SPPOL_1:16
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
theorem Th17: :: SPPOL_1:17
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
theorem :: SPPOL_1:18
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
theorem :: SPPOL_1:19
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
theorem Th20: :: SPPOL_1:20
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
theorem Th21: :: SPPOL_1:21
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
theorem :: SPPOL_1:22
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
theorem :: SPPOL_1:23
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
theorem Th24: :: SPPOL_1:24
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
theorem :: SPPOL_1:25
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
for
n being
Nat for
p1,
p2,
q1,
q2 being
Point of
(TOP-REAL n) holds
( not
LSeg p1,
p2 = LSeg q1,
q2 or (
p1 = q1 &
p2 = q2 ) or (
p1 = q2 &
p2 = q1 ) )
theorem :: SPPOL_1:26
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
theorem :: SPPOL_1:27
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
theorem Th28: :: SPPOL_1:28
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
theorem Th29: :: SPPOL_1:29
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
:: deftheorem Def1 defines is_extremal_in SPPOL_1:def 1 :
theorem :: SPPOL_1:30
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
theorem :: SPPOL_1:31
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
theorem Th32: :: SPPOL_1:32
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
theorem :: SPPOL_1:33
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
theorem :: SPPOL_1:34
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
theorem Th35: :: SPPOL_1:35
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
:: deftheorem Def2 defines horizontal SPPOL_1:def 2 :
:: deftheorem Def3 defines vertical SPPOL_1:def 3 :
Lm2:
for P being Subset of (TOP-REAL 2) st not P is trivial & P is horizontal holds
not P is vertical
theorem Th36: :: SPPOL_1:36
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
theorem Th37: :: SPPOL_1:37
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
theorem :: SPPOL_1:38
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
theorem :: SPPOL_1:39
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
theorem Th40: :: SPPOL_1:40
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
theorem Th41: :: SPPOL_1:41
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
theorem Th42: :: SPPOL_1:42
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
theorem :: SPPOL_1:43
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
theorem Th44: :: SPPOL_1:44
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
theorem Th45: :: SPPOL_1:45
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
Lm3:
for f being FinSequence of the carrier of (TOP-REAL 2)
for k being Nat holds { (LSeg f,i) where i is Nat : ( 1 <= i & i + 1 <= len f & i <> k & i <> k + 1 ) } is finite
theorem :: SPPOL_1:46
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
theorem Th47: :: SPPOL_1:47
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
Lm4:
for f being FinSequence of the carrier of (TOP-REAL 2)
for k being Nat holds { (LSeg f,i) where i is Nat : ( 1 <= i & i + 1 <= len f & i <> k & i <> k + 1 ) } is Subset-Family of (TOP-REAL 2)
theorem Th48: :: SPPOL_1:48
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
theorem :: SPPOL_1:49
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
Lm5:
for f being FinSequence of the carrier of (TOP-REAL 2)
for Q being Subset of (TOP-REAL 2)
for k being Nat st Q = union { (LSeg f,i) where i is Nat : ( 1 <= i & i + 1 <= len f & i <> k & i <> k + 1 ) } holds
Q is closed
:: deftheorem Def4 defines alternating SPPOL_1:def 4 :
theorem Th50: :: SPPOL_1:50
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
theorem Th51: :: SPPOL_1:51
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
theorem Th52: :: SPPOL_1:52
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
theorem :: SPPOL_1:53
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
Lm6:
for f being FinSequence of the carrier of (TOP-REAL 2)
for i being Nat
for p1, p2 being Point of (TOP-REAL 2) st f is alternating & 1 <= i & i + 2 <= len f & p1 = f /. i & p2 = f /. (i + 2) holds
ex p being Point of (TOP-REAL 2) st
( p in LSeg p1,p2 & p `1 <> p1 `1 & p `1 <> p2 `1 & p `2 <> p1 `2 & p `2 <> p2 `2 )
theorem :: SPPOL_1:54
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
theorem Th55: :: SPPOL_1:55
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
theorem Th56: :: SPPOL_1:56
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
theorem :: SPPOL_1:57
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
theorem Th58: :: SPPOL_1:58
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
theorem Th59: :: SPPOL_1:59
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
theorem Th60: :: SPPOL_1:60
:: Showing IDV graph ... (Click the Palm Tree again to close it) 
:: deftheorem Def5 defines are_generators_of SPPOL_1:def 5 :
theorem :: SPPOL_1:61
:: Showing IDV graph ... (Click the Palm Tree again to close it) 