:: JGRAPH_6 semantic presentation
theorem Th1: :: JGRAPH_6:1
canceled;
theorem Th2: :: JGRAPH_6:2
Lemma2:
for b1, b2 being real number holds
not ( ( ( b1 >= 0 & b2 > 0 ) or ( b1 > 0 & b2 >= 0 ) ) & not b1 + b2 > 0 )
by XREAL_1:36;
theorem Th3: :: JGRAPH_6:3
canceled;
theorem Th4: :: JGRAPH_6:4
theorem Th5: :: JGRAPH_6:5
Lemma5:
for b1, b2 being real number holds
( - 1 <= b1 & b1 <= 1 & - 1 <= b2 & b2 <= 1 implies (1 + (b1 ^2 )) * (b2 ^2 ) <= 1 + (b2 ^2 ) )
theorem Th6: :: JGRAPH_6:6
theorem Th7: :: JGRAPH_6:7
Lemma8:
for b1, b2 being real number holds
( b2 <= 0 & b1 <= b2 implies b1 * (sqrt (1 + (b2 ^2 ))) <= b2 * (sqrt (1 + (b1 ^2 ))) )
Lemma9:
for b1, b2 being real number holds
( b1 <= 0 & b1 <= b2 implies b1 * (sqrt (1 + (b2 ^2 ))) <= b2 * (sqrt (1 + (b1 ^2 ))) )
Lemma10:
for b1, b2 being real number holds
( b1 >= 0 & b1 >= b2 implies b1 * (sqrt (1 + (b2 ^2 ))) >= b2 * (sqrt (1 + (b1 ^2 ))) )
theorem Th8: :: JGRAPH_6:8
theorem Th9: :: JGRAPH_6:9
theorem Th10: :: JGRAPH_6:10
theorem Th11: :: JGRAPH_6:11
theorem Th12: :: JGRAPH_6:12
theorem Th13: :: JGRAPH_6:13
theorem Th14: :: JGRAPH_6:14
theorem Th15: :: JGRAPH_6:15
theorem Th16: :: JGRAPH_6:16
theorem Th17: :: JGRAPH_6:17
theorem Th18: :: JGRAPH_6:18
theorem Th19: :: JGRAPH_6:19
theorem Th20: :: JGRAPH_6:20
theorem Th21: :: JGRAPH_6:21
theorem Th22: :: JGRAPH_6:22
Lemma25:
( |[(- 1),0]| `1 = - 1 & |[(- 1),0]| `2 = 0 & |[1,0]| `1 = 1 & |[1,0]| `2 = 0 & |[0,(- 1)]| `1 = 0 & |[0,(- 1)]| `2 = - 1 & |[0,1]| `1 = 0 & |[0,1]| `2 = 1 )
by EUCLID:56;
Lemma27:
0 in [.0,1.]
by RCOMP_1:48;
Lemma28:
1 in [.0,1.]
by RCOMP_1:48;
theorem Th23: :: JGRAPH_6:23
theorem Th24: :: JGRAPH_6:24
theorem Th25: :: JGRAPH_6:25
for b
1, b
2, b
3, b
4 being
Point of
(TOP-REAL 2)for b
5 being non
empty compact Subset of
(TOP-REAL 2)for b
6 being
Subset of
(TOP-REAL 2) holds
( b
5 = { b7 where B is Point of (TOP-REAL 2) : |.b7.| = 1 } &
LE b
1,b
2,b
5 &
LE b
2,b
3,b
5 &
LE b
3,b
4,b
5 implies for b
7, b
8 being
Function of
I[01] ,
(TOP-REAL 2) holds
not ( b
7 is
continuous & b
7 is
one-to-one & b
8 is
continuous & b
8 is
one-to-one & b
6 = { b9 where B is Point of (TOP-REAL 2) : |.b9.| <= 1 } & b
7 . 0
= b
3 & b
7 . 1
= b
1 & b
8 . 0
= b
2 & b
8 . 1
= b
4 &
rng b
7 c= b
6 &
rng b
8 c= b
6 & not
rng b
7 meets rng b
8 ) )
theorem Th26: :: JGRAPH_6:26
for b
1, b
2, b
3, b
4 being
Point of
(TOP-REAL 2)for b
5 being non
empty compact Subset of
(TOP-REAL 2)for b
6 being
Subset of
(TOP-REAL 2) holds
( b
5 = { b7 where B is Point of (TOP-REAL 2) : |.b7.| = 1 } &
LE b
1,b
2,b
5 &
LE b
2,b
3,b
5 &
LE b
3,b
4,b
5 implies for b
7, b
8 being
Function of
I[01] ,
(TOP-REAL 2) holds
not ( b
7 is
continuous & b
7 is
one-to-one & b
8 is
continuous & b
8 is
one-to-one & b
6 = { b9 where B is Point of (TOP-REAL 2) : |.b9.| <= 1 } & b
7 . 0
= b
3 & b
7 . 1
= b
1 & b
8 . 0
= b
4 & b
8 . 1
= b
2 &
rng b
7 c= b
6 &
rng b
8 c= b
6 & not
rng b
7 meets rng b
8 ) )
theorem Th27: :: JGRAPH_6:27
Lemma34:
for b1, b2, b3, b4 being real number holds
( b1 <= b2 & b3 <= b4 implies rectangle b1,b2,b3,b4 = { b5 where B is Point of (TOP-REAL 2) : not ( not ( b5 `1 = b1 & b3 <= b5 `2 & b5 `2 <= b4 ) & not ( b5 `2 = b4 & b1 <= b5 `1 & b5 `1 <= b2 ) & not ( b5 `1 = b2 & b3 <= b5 `2 & b5 `2 <= b4 ) & not ( b5 `2 = b3 & b1 <= b5 `1 & b5 `1 <= b2 ) ) } )
theorem Th28: :: JGRAPH_6:28
:: deftheorem Def1 defines inside_of_rectangle JGRAPH_6:def 1 :
:: deftheorem Def2 defines closed_inside_of_rectangle JGRAPH_6:def 2 :
:: deftheorem Def3 defines outside_of_rectangle JGRAPH_6:def 3 :
:: deftheorem Def4 defines closed_outside_of_rectangle JGRAPH_6:def 4 :
theorem Th29: :: JGRAPH_6:29
theorem Th30: :: JGRAPH_6:30
theorem Th31: :: JGRAPH_6:31
theorem Th32: :: JGRAPH_6:32
:: deftheorem Def5 defines circle JGRAPH_6:def 5 :
:: deftheorem Def6 defines inside_of_circle JGRAPH_6:def 6 :
:: deftheorem Def7 defines closed_inside_of_circle JGRAPH_6:def 7 :
:: deftheorem Def8 defines outside_of_circle JGRAPH_6:def 8 :
:: deftheorem Def9 defines closed_outside_of_circle JGRAPH_6:def 9 :
theorem Th33: :: JGRAPH_6:33
for b
1 being
real number holds
(
inside_of_circle 0,0,b
1 = { b2 where B is Point of (TOP-REAL 2) : |.b2.| < b1 } & ( b
1 > 0 implies
circle 0,0,b
1 = { b2 where B is Point of (TOP-REAL 2) : |.b2.| = b1 } ) &
outside_of_circle 0,0,b
1 = { b2 where B is Point of (TOP-REAL 2) : |.b2.| > b1 } &
closed_inside_of_circle 0,0,b
1 = { b2 where B is Point of (TOP-REAL 2) : |.b2.| <= b1 } &
closed_outside_of_circle 0,0,b
1 = { b2 where B is Point of (TOP-REAL 2) : |.b2.| >= b1 } )
theorem Th34: :: JGRAPH_6:34
theorem Th35: :: JGRAPH_6:35
theorem Th36: :: JGRAPH_6:36
theorem Th37: :: JGRAPH_6:37
theorem Th38: :: JGRAPH_6:38
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Subset of
(TOP-REAL 2)for b
9 being non
empty compact Subset of
(TOP-REAL 2)for b
10 being
Function of
(TOP-REAL 2),
(TOP-REAL 2) holds
( b
9 = circle 0,0,1 & b
1 = inside_of_circle 0,0,1 & b
2 = outside_of_circle 0,0,1 & b
3 = closed_inside_of_circle 0,0,1 & b
4 = closed_outside_of_circle 0,0,1 & b
5 = inside_of_rectangle (- 1),1,
(- 1),1 & b
6 = outside_of_rectangle (- 1),1,
(- 1),1 & b
7 = closed_inside_of_rectangle (- 1),1,
(- 1),1 & b
8 = closed_outside_of_rectangle (- 1),1,
(- 1),1 & b
10 = Sq_Circ implies ( b
10 .: (rectangle (- 1),1,(- 1),1) = b
9 &
(b10 " ) .: b
9 = rectangle (- 1),1,
(- 1),1 & b
10 .: b
5 = b
1 &
(b10 " ) .: b
1 = b
5 & b
10 .: b
6 = b
2 &
(b10 " ) .: b
2 = b
6 & b
10 .: b
7 = b
3 & b
10 .: b
8 = b
4 &
(b10 " ) .: b
3 = b
7 &
(b10 " ) .: b
4 = b
8 ) )
theorem Th39: :: JGRAPH_6:39
for b
1, b
2, b
3, b
4 being
real number holds
( b
1 <= b
2 & b
3 <= b
4 implies (
LSeg |[b1,b3]|,
|[b1,b4]| = { b5 where B is Point of (TOP-REAL 2) : ( b5 `1 = b1 & b5 `2 <= b4 & b5 `2 >= b3 ) } &
LSeg |[b1,b4]|,
|[b2,b4]| = { b5 where B is Point of (TOP-REAL 2) : ( b5 `1 <= b2 & b5 `1 >= b1 & b5 `2 = b4 ) } &
LSeg |[b1,b3]|,
|[b2,b3]| = { b5 where B is Point of (TOP-REAL 2) : ( b5 `1 <= b2 & b5 `1 >= b1 & b5 `2 = b3 ) } &
LSeg |[b2,b3]|,
|[b2,b4]| = { b5 where B is Point of (TOP-REAL 2) : ( b5 `1 = b2 & b5 `2 <= b4 & b5 `2 >= b3 ) } ) )
theorem Th40: :: JGRAPH_6:40
canceled;
theorem Th41: :: JGRAPH_6:41
for b
1, b
2, b
3, b
4 being
real number holds
( b
1 <= b
2 & b
3 <= b
4 implies
(LSeg |[b1,b3]|,|[b1,b4]|) /\ (LSeg |[b1,b3]|,|[b2,b3]|) = {|[b1,b3]|} )
theorem Th42: :: JGRAPH_6:42
for b
1, b
2, b
3, b
4 being
real number holds
( b
1 <= b
2 & b
3 <= b
4 implies
(LSeg |[b1,b3]|,|[b2,b3]|) /\ (LSeg |[b2,b3]|,|[b2,b4]|) = {|[b2,b3]|} )
theorem Th43: :: JGRAPH_6:43
for b
1, b
2, b
3, b
4 being
real number holds
( b
1 <= b
2 & b
3 <= b
4 implies
(LSeg |[b1,b4]|,|[b2,b4]|) /\ (LSeg |[b2,b3]|,|[b2,b4]|) = {|[b2,b4]|} )
theorem Th44: :: JGRAPH_6:44
for b
1, b
2, b
3, b
4 being
real number holds
( b
1 <= b
2 & b
3 <= b
4 implies
(LSeg |[b1,b3]|,|[b1,b4]|) /\ (LSeg |[b1,b4]|,|[b2,b4]|) = {|[b1,b4]|} )
theorem Th45: :: JGRAPH_6:45
theorem Th46: :: JGRAPH_6:46
theorem Th47: :: JGRAPH_6:47
theorem Th48: :: JGRAPH_6:48
theorem Th49: :: JGRAPH_6:49
theorem Th50: :: JGRAPH_6:50
theorem Th51: :: JGRAPH_6:51
theorem Th52: :: JGRAPH_6:52
theorem Th53: :: JGRAPH_6:53
theorem Th54: :: JGRAPH_6:54
theorem Th55: :: JGRAPH_6:55
theorem Th56: :: JGRAPH_6:56
for b
1, b
2, b
3, b
4 being
real number holds
( b
1 <= b
2 & b
3 <= b
4 implies (
W-min (rectangle b1,b2,b3,b4) = |[b1,b3]| &
E-max (rectangle b1,b2,b3,b4) = |[b2,b4]| ) )
theorem Th57: :: JGRAPH_6:57
for b
1, b
2, b
3, b
4 being
real number holds
( b
1 < b
2 & b
3 < b
4 implies (
(LSeg |[b1,b3]|,|[b1,b4]|) \/ (LSeg |[b1,b4]|,|[b2,b4]|) is_an_arc_of W-min (rectangle b1,b2,b3,b4),
E-max (rectangle b1,b2,b3,b4) &
(LSeg |[b1,b3]|,|[b2,b3]|) \/ (LSeg |[b2,b3]|,|[b2,b4]|) is_an_arc_of E-max (rectangle b1,b2,b3,b4),
W-min (rectangle b1,b2,b3,b4) ) )
theorem Th58: :: JGRAPH_6:58
for b
1, b
2, b
3, b
4 being
real number for b
5, b
6 being
FinSequence of
(TOP-REAL 2)for b
7, b
8, b
9, b
10 being
Point of
(TOP-REAL 2) holds
( b
1 < b
2 & b
3 < b
4 & b
7 = |[b1,b3]| & b
8 = |[b2,b4]| & b
9 = |[b1,b4]| & b
10 = |[b2,b3]| & b
5 = <*b7,b9,b8*> & b
6 = <*b7,b10,b8*> implies ( b
5 is_S-Seq &
L~ b
5 = (LSeg b7,b9) \/ (LSeg b9,b8) & b
6 is_S-Seq &
L~ b
6 = (LSeg b7,b10) \/ (LSeg b10,b8) &
rectangle b
1,b
2,b
3,b
4 = (L~ b5) \/ (L~ b6) &
(L~ b5) /\ (L~ b6) = {b7,b8} & b
5 /. 1
= b
7 & b
5 /. (len b5) = b
8 & b
6 /. 1
= b
7 & b
6 /. (len b6) = b
8 ) )
theorem Th59: :: JGRAPH_6:59
for b
1, b
2 being
Subset of
(TOP-REAL 2)for b
3, b
4, b
5, b
6 being
real number for b
7, b
8 being
FinSequence of
(TOP-REAL 2)for b
9, b
10 being
Point of
(TOP-REAL 2) holds
( b
3 < b
4 & b
5 < b
6 & b
9 = |[b3,b5]| & b
10 = |[b4,b6]| & b
7 = <*|[b3,b5]|,|[b3,b6]|,|[b4,b6]|*> & b
8 = <*|[b3,b5]|,|[b4,b5]|,|[b4,b6]|*> & b
1 = L~ b
7 & b
2 = L~ b
8 implies ( b
1 is_an_arc_of b
9,b
10 & b
2 is_an_arc_of b
9,b
10 & not b
1 is
empty & not b
2 is
empty &
rectangle b
3,b
4,b
5,b
6 = b
1 \/ b
2 & b
1 /\ b
2 = {b9,b10} ) )
theorem Th60: :: JGRAPH_6:60
theorem Th61: :: JGRAPH_6:61
for b
1, b
2, b
3, b
4 being
real number holds
( b
1 < b
2 & b
3 < b
4 implies
Upper_Arc (rectangle b1,b2,b3,b4) = (LSeg |[b1,b3]|,|[b1,b4]|) \/ (LSeg |[b1,b4]|,|[b2,b4]|) )
theorem Th62: :: JGRAPH_6:62
for b
1, b
2, b
3, b
4 being
real number holds
( b
1 < b
2 & b
3 < b
4 implies
Lower_Arc (rectangle b1,b2,b3,b4) = (LSeg |[b1,b3]|,|[b2,b3]|) \/ (LSeg |[b2,b3]|,|[b2,b4]|) )
theorem Th63: :: JGRAPH_6:63
for b
1, b
2, b
3, b
4 being
real number holds
not ( b
1 < b
2 & b
3 < b
4 & ( for b
5 being
Function of
I[01] ,
((TOP-REAL 2) | (Upper_Arc (rectangle b1,b2,b3,b4))) holds
not ( b
5 is_homeomorphism & b
5 . 0
= W-min (rectangle b1,b2,b3,b4) & b
5 . 1
= E-max (rectangle b1,b2,b3,b4) &
rng b
5 = Upper_Arc (rectangle b1,b2,b3,b4) & ( for b
6 being
Real holds
( b
6 in [.0,(1 / 2).] implies b
5 . b
6 = ((1 - (2 * b6)) * |[b1,b3]|) + ((2 * b6) * |[b1,b4]|) ) ) & ( for b
6 being
Real holds
( b
6 in [.(1 / 2),1.] implies b
5 . b
6 = ((1 - ((2 * b6) - 1)) * |[b1,b4]|) + (((2 * b6) - 1) * |[b2,b4]|) ) ) & ( for b
6 being
Point of
(TOP-REAL 2) holds
( b
6 in LSeg |[b1,b3]|,
|[b1,b4]| implies ( 0
<= (((b6 `2 ) - b3) / (b4 - b3)) / 2 &
(((b6 `2 ) - b3) / (b4 - b3)) / 2
<= 1 & b
5 . ((((b6 `2 ) - b3) / (b4 - b3)) / 2) = b
6 ) ) ) & ( for b
6 being
Point of
(TOP-REAL 2) holds
( b
6 in LSeg |[b1,b4]|,
|[b2,b4]| implies ( 0
<= ((((b6 `1 ) - b1) / (b2 - b1)) / 2) + (1 / 2) &
((((b6 `1 ) - b1) / (b2 - b1)) / 2) + (1 / 2) <= 1 & b
5 . (((((b6 `1 ) - b1) / (b2 - b1)) / 2) + (1 / 2)) = b
6 ) ) ) ) ) )
theorem Th64: :: JGRAPH_6:64
for b
1, b
2, b
3, b
4 being
real number holds
not ( b
1 < b
2 & b
3 < b
4 & ( for b
5 being
Function of
I[01] ,
((TOP-REAL 2) | (Lower_Arc (rectangle b1,b2,b3,b4))) holds
not ( b
5 is_homeomorphism & b
5 . 0
= E-max (rectangle b1,b2,b3,b4) & b
5 . 1
= W-min (rectangle b1,b2,b3,b4) &
rng b
5 = Lower_Arc (rectangle b1,b2,b3,b4) & ( for b
6 being
Real holds
( b
6 in [.0,(1 / 2).] implies b
5 . b
6 = ((1 - (2 * b6)) * |[b2,b4]|) + ((2 * b6) * |[b2,b3]|) ) ) & ( for b
6 being
Real holds
( b
6 in [.(1 / 2),1.] implies b
5 . b
6 = ((1 - ((2 * b6) - 1)) * |[b2,b3]|) + (((2 * b6) - 1) * |[b1,b3]|) ) ) & ( for b
6 being
Point of
(TOP-REAL 2) holds
( b
6 in LSeg |[b2,b4]|,
|[b2,b3]| implies ( 0
<= (((b6 `2 ) - b4) / (b3 - b4)) / 2 &
(((b6 `2 ) - b4) / (b3 - b4)) / 2
<= 1 & b
5 . ((((b6 `2 ) - b4) / (b3 - b4)) / 2) = b
6 ) ) ) & ( for b
6 being
Point of
(TOP-REAL 2) holds
( b
6 in LSeg |[b2,b3]|,
|[b1,b3]| implies ( 0
<= ((((b6 `1 ) - b2) / (b1 - b2)) / 2) + (1 / 2) &
((((b6 `1 ) - b2) / (b1 - b2)) / 2) + (1 / 2) <= 1 & b
5 . (((((b6 `1 ) - b2) / (b1 - b2)) / 2) + (1 / 2)) = b
6 ) ) ) ) ) )
theorem Th65: :: JGRAPH_6:65
for b
1, b
2, b
3, b
4 being
real number for b
5, b
6 being
Point of
(TOP-REAL 2) holds
( b
1 < b
2 & b
3 < b
4 & b
5 in LSeg |[b1,b3]|,
|[b1,b4]| & b
6 in LSeg |[b1,b3]|,
|[b1,b4]| implies (
LE b
5,b
6,
rectangle b
1,b
2,b
3,b
4 iff b
5 `2 <= b
6 `2 ) )
theorem Th66: :: JGRAPH_6:66
for b
1, b
2, b
3, b
4 being
real number for b
5, b
6 being
Point of
(TOP-REAL 2) holds
( b
1 < b
2 & b
3 < b
4 & b
5 in LSeg |[b1,b4]|,
|[b2,b4]| & b
6 in LSeg |[b1,b4]|,
|[b2,b4]| implies (
LE b
5,b
6,
rectangle b
1,b
2,b
3,b
4 iff b
5 `1 <= b
6 `1 ) )
theorem Th67: :: JGRAPH_6:67
for b
1, b
2, b
3, b
4 being
real number for b
5, b
6 being
Point of
(TOP-REAL 2) holds
( b
1 < b
2 & b
3 < b
4 & b
5 in LSeg |[b2,b3]|,
|[b2,b4]| & b
6 in LSeg |[b2,b3]|,
|[b2,b4]| implies (
LE b
5,b
6,
rectangle b
1,b
2,b
3,b
4 iff b
5 `2 >= b
6 `2 ) )
theorem Th68: :: JGRAPH_6:68
for b
1, b
2, b
3, b
4 being
real number for b
5, b
6 being
Point of
(TOP-REAL 2) holds
( b
1 < b
2 & b
3 < b
4 & b
5 in LSeg |[b1,b3]|,
|[b2,b3]| & b
6 in LSeg |[b1,b3]|,
|[b2,b3]| implies ( (
LE b
5,b
6,
rectangle b
1,b
2,b
3,b
4 & b
5 <> W-min (rectangle b1,b2,b3,b4) implies ( b
5 `1 >= b
6 `1 & b
6 <> W-min (rectangle b1,b2,b3,b4) ) ) & ( b
5 `1 >= b
6 `1 & b
6 <> W-min (rectangle b1,b2,b3,b4) implies (
LE b
5,b
6,
rectangle b
1,b
2,b
3,b
4 & b
5 <> W-min (rectangle b1,b2,b3,b4) ) ) ) )
theorem Th69: :: JGRAPH_6:69
for b
1, b
2, b
3, b
4 being
real number for b
5, b
6 being
Point of
(TOP-REAL 2) holds
( b
1 < b
2 & b
3 < b
4 & b
5 in LSeg |[b1,b3]|,
|[b1,b4]| implies (
LE b
5,b
6,
rectangle b
1,b
2,b
3,b
4 iff not ( not ( b
6 in LSeg |[b1,b3]|,
|[b1,b4]| & b
5 `2 <= b
6 `2 ) & not b
6 in LSeg |[b1,b4]|,
|[b2,b4]| & not b
6 in LSeg |[b2,b4]|,
|[b2,b3]| & not ( b
6 in LSeg |[b2,b3]|,
|[b1,b3]| & b
6 <> W-min (rectangle b1,b2,b3,b4) ) ) ) )
theorem Th70: :: JGRAPH_6:70
for b
1, b
2, b
3, b
4 being
real number for b
5, b
6 being
Point of
(TOP-REAL 2) holds
( b
1 < b
2 & b
3 < b
4 & b
5 in LSeg |[b1,b4]|,
|[b2,b4]| implies (
LE b
5,b
6,
rectangle b
1,b
2,b
3,b
4 iff not ( not ( b
6 in LSeg |[b1,b4]|,
|[b2,b4]| & b
5 `1 <= b
6 `1 ) & not b
6 in LSeg |[b2,b4]|,
|[b2,b3]| & not ( b
6 in LSeg |[b2,b3]|,
|[b1,b3]| & b
6 <> W-min (rectangle b1,b2,b3,b4) ) ) ) )
theorem Th71: :: JGRAPH_6:71
for b
1, b
2, b
3, b
4 being
real number for b
5, b
6 being
Point of
(TOP-REAL 2) holds
( b
1 < b
2 & b
3 < b
4 & b
5 in LSeg |[b2,b4]|,
|[b2,b3]| implies (
LE b
5,b
6,
rectangle b
1,b
2,b
3,b
4 iff ( ( b
6 in LSeg |[b2,b4]|,
|[b2,b3]| & b
5 `2 >= b
6 `2 ) or ( b
6 in LSeg |[b2,b3]|,
|[b1,b3]| & b
6 <> W-min (rectangle b1,b2,b3,b4) ) ) ) )
theorem Th72: :: JGRAPH_6:72
for b
1, b
2, b
3, b
4 being
real number for b
5, b
6 being
Point of
(TOP-REAL 2) holds
( b
1 < b
2 & b
3 < b
4 & b
5 in LSeg |[b2,b3]|,
|[b1,b3]| & b
5 <> W-min (rectangle b1,b2,b3,b4) implies (
LE b
5,b
6,
rectangle b
1,b
2,b
3,b
4 iff ( b
6 in LSeg |[b2,b3]|,
|[b1,b3]| & b
5 `1 >= b
6 `1 & b
6 <> W-min (rectangle b1,b2,b3,b4) ) ) )
theorem Th73: :: JGRAPH_6:73
for b
1 being
set for b
2, b
3, b
4, b
5 being
real number holds
not ( b
1 in rectangle b
2,b
3,b
4,b
5 & b
2 < b
3 & b
4 < b
5 & not b
1 in LSeg |[b2,b4]|,
|[b2,b5]| & not b
1 in LSeg |[b2,b5]|,
|[b3,b5]| & not b
1 in LSeg |[b3,b5]|,
|[b3,b4]| & not b
1 in LSeg |[b3,b4]|,
|[b2,b4]| )
theorem Th74: :: JGRAPH_6:74
for b
1, b
2 being
Point of
(TOP-REAL 2) holds
not (
LE b
1,b
2,
rectangle (- 1),1,
(- 1),1 & b
1 in LSeg |[(- 1),(- 1)]|,
|[(- 1),1]| & not ( b
2 in LSeg |[(- 1),(- 1)]|,
|[(- 1),1]| & b
2 `2 >= b
1 `2 ) & not b
2 in LSeg |[(- 1),1]|,
|[1,1]| & not b
2 in LSeg |[1,1]|,
|[1,(- 1)]| & not ( b
2 in LSeg |[1,(- 1)]|,
|[(- 1),(- 1)]| & b
2 <> |[(- 1),(- 1)]| ) )
theorem Th75: :: JGRAPH_6:75
for b
1, b
2 being
Point of
(TOP-REAL 2)for b
3 being non
empty compact Subset of
(TOP-REAL 2)for b
4 being
Function of
(TOP-REAL 2),
(TOP-REAL 2) holds
( b
3 = circle 0,0,1 & b
4 = Sq_Circ & b
1 in LSeg |[(- 1),(- 1)]|,
|[(- 1),1]| & b
1 `2 >= 0 &
LE b
1,b
2,
rectangle (- 1),1,
(- 1),1 implies
LE b
4 . b
1,b
4 . b
2,b
3 )
theorem Th76: :: JGRAPH_6:76
for b
1, b
2, b
3 being
Point of
(TOP-REAL 2)for b
4 being non
empty compact Subset of
(TOP-REAL 2)for b
5 being
Function of
(TOP-REAL 2),
(TOP-REAL 2) holds
( b
4 = circle 0,0,1 & b
5 = Sq_Circ & b
1 in LSeg |[(- 1),(- 1)]|,
|[(- 1),1]| & b
1 `2 >= 0 &
LE b
1,b
2,
rectangle (- 1),1,
(- 1),1 &
LE b
2,b
3,
rectangle (- 1),1,
(- 1),1 implies
LE b
5 . b
2,b
5 . b
3,b
4 )
theorem Th77: :: JGRAPH_6:77
theorem Th78: :: JGRAPH_6:78
theorem Th79: :: JGRAPH_6:79
theorem Th80: :: JGRAPH_6:80
theorem Th81: :: JGRAPH_6:81
theorem Th82: :: JGRAPH_6:82
for b
1, b
2, b
3, b
4 being
Point of
(TOP-REAL 2)for b
5 being non
empty compact Subset of
(TOP-REAL 2)for b
6 being
Function of
(TOP-REAL 2),
(TOP-REAL 2) holds
( b
5 = circle 0,0,1 & b
6 = Sq_Circ &
LE b
1,b
2,
rectangle (- 1),1,
(- 1),1 &
LE b
2,b
3,
rectangle (- 1),1,
(- 1),1 &
LE b
3,b
4,
rectangle (- 1),1,
(- 1),1 implies b
6 . b
1,b
6 . b
2,b
6 . b
3,b
6 . b
4 are_in_this_order_on b
5 )
theorem Th83: :: JGRAPH_6:83
theorem Th84: :: JGRAPH_6:84
for b
1, b
2, b
3 being
Point of
(TOP-REAL 2)for b
4 being non
empty compact Subset of
(TOP-REAL 2) holds
not ( b
4 is_simple_closed_curve & b
1 in b
4 & b
2 in b
4 & b
3 in b
4 & not (
LE b
1,b
2,b
4 &
LE b
2,b
3,b
4 ) & not (
LE b
1,b
3,b
4 &
LE b
3,b
2,b
4 ) & not (
LE b
2,b
1,b
4 &
LE b
1,b
3,b
4 ) & not (
LE b
2,b
3,b
4 &
LE b
3,b
1,b
4 ) & not (
LE b
3,b
1,b
4 &
LE b
1,b
2,b
4 ) & not (
LE b
3,b
2,b
4 &
LE b
2,b
1,b
4 ) )
by Th83;
theorem Th85: :: JGRAPH_6:85
for b
1, b
2, b
3 being
Point of
(TOP-REAL 2)for b
4 being non
empty compact Subset of
(TOP-REAL 2) holds
not ( b
4 is_simple_closed_curve & b
1 in b
4 & b
2 in b
4 & b
3 in b
4 &
LE b
2,b
3,b
4 & not
LE b
1,b
2,b
4 & not (
LE b
2,b
1,b
4 &
LE b
1,b
3,b
4 ) & not
LE b
3,b
1,b
4 )
by Th83;
theorem Th86: :: JGRAPH_6:86
for b
1, b
2, b
3, b
4 being
Point of
(TOP-REAL 2)for b
5 being non
empty compact Subset of
(TOP-REAL 2) holds
not ( b
5 is_simple_closed_curve & b
1 in b
5 & b
2 in b
5 & b
3 in b
5 & b
4 in b
5 &
LE b
2,b
3,b
5 &
LE b
3,b
4,b
5 & not
LE b
1,b
2,b
5 & not (
LE b
2,b
1,b
5 &
LE b
1,b
3,b
5 ) & not (
LE b
3,b
1,b
5 &
LE b
1,b
4,b
5 ) & not
LE b
4,b
1,b
5 )
by Th83;
theorem Th87: :: JGRAPH_6:87
for b
1, b
2, b
3, b
4 being
Point of
(TOP-REAL 2)for b
5 being non
empty compact Subset of
(TOP-REAL 2)for b
6 being
Function of
(TOP-REAL 2),
(TOP-REAL 2) holds
( b
5 = circle 0,0,1 & b
6 = Sq_Circ &
LE b
6 . b
1,b
6 . b
2,b
5 &
LE b
6 . b
2,b
6 . b
3,b
5 &
LE b
6 . b
3,b
6 . b
4,b
5 implies b
1,b
2,b
3,b
4 are_in_this_order_on rectangle (- 1),1,
(- 1),1 )
theorem Th88: :: JGRAPH_6:88
for b
1, b
2, b
3, b
4 being
Point of
(TOP-REAL 2)for b
5 being non
empty compact Subset of
(TOP-REAL 2)for b
6 being
Function of
(TOP-REAL 2),
(TOP-REAL 2) holds
( b
5 = circle 0,0,1 & b
6 = Sq_Circ implies ( b
1,b
2,b
3,b
4 are_in_this_order_on rectangle (- 1),1,
(- 1),1 iff b
6 . b
1,b
6 . b
2,b
6 . b
3,b
6 . b
4 are_in_this_order_on b
5 ) )
theorem Th89: :: JGRAPH_6:89
for b
1, b
2, b
3, b
4 being
Point of
(TOP-REAL 2)for b
5 being
Subset of
(TOP-REAL 2) holds
( b
1,b
2,b
3,b
4 are_in_this_order_on rectangle (- 1),1,
(- 1),1 implies for b
6, b
7 being
Function of
I[01] ,
(TOP-REAL 2) holds
not ( b
6 is
continuous & b
6 is
one-to-one & b
7 is
continuous & b
7 is
one-to-one & b
5 = closed_inside_of_rectangle (- 1),1,
(- 1),1 & b
6 . 0
= b
1 & b
6 . 1
= b
3 & b
7 . 0
= b
2 & b
7 . 1
= b
4 &
rng b
6 c= b
5 &
rng b
7 c= b
5 & not
rng b
6 meets rng b
7 ) )