:: TOPREAL4 semantic presentation
:: deftheorem Def1 defines is_S-P_arc_joining TOPREAL4:def 1 :
:: deftheorem Def2 defines being_special_polygon TOPREAL4:def 2 :
:: deftheorem Def3 defines being_Region TOPREAL4:def 3 :
theorem Th1: :: TOPREAL4:1
canceled;
theorem Th2: :: TOPREAL4:2
theorem Th3: :: TOPREAL4:3
theorem Th4: :: TOPREAL4:4
theorem Th5: :: TOPREAL4:5
theorem Th6: :: TOPREAL4:6
theorem Th7: :: TOPREAL4:7
theorem Th8: :: TOPREAL4:8
theorem Th9: :: TOPREAL4:9
for b
1, b
2 being
Point of
(TOP-REAL 2)for b
3 being
FinSequence of
(TOP-REAL 2)for b
4 being
Realfor b
5 being
Point of
(Euclid 2) holds
( b
1 `1 <> b
2 `1 & b
1 `2 <> b
2 `2 & b
1 in Ball b
5,b
4 & b
2 in Ball b
5,b
4 &
|[(b1 `1 ),(b2 `2 )]| in Ball b
5,b
4 & b
3 = <*b1,|[(b1 `1 ),(b2 `2 )]|,b2*> implies ( b
3 is_S-Seq & b
3 /. 1
= b
1 & b
3 /. (len b3) = b
2 &
L~ b
3 is_S-P_arc_joining b
1,b
2 &
L~ b
3 c= Ball b
5,b
4 ) )
theorem Th10: :: TOPREAL4:10
for b
1, b
2 being
Point of
(TOP-REAL 2)for b
3 being
FinSequence of
(TOP-REAL 2)for b
4 being
Realfor b
5 being
Point of
(Euclid 2) holds
( b
1 `1 <> b
2 `1 & b
1 `2 <> b
2 `2 & b
1 in Ball b
5,b
4 & b
2 in Ball b
5,b
4 &
|[(b2 `1 ),(b1 `2 )]| in Ball b
5,b
4 & b
3 = <*b1,|[(b2 `1 ),(b1 `2 )]|,b2*> implies ( b
3 is_S-Seq & b
3 /. 1
= b
1 & b
3 /. (len b3) = b
2 &
L~ b
3 is_S-P_arc_joining b
1,b
2 &
L~ b
3 c= Ball b
5,b
4 ) )
theorem Th11: :: TOPREAL4:11
theorem Th12: :: TOPREAL4:12
theorem Th13: :: TOPREAL4:13
theorem Th14: :: TOPREAL4:14
theorem Th15: :: TOPREAL4:15
theorem Th16: :: TOPREAL4:16
theorem Th17: :: TOPREAL4:17
theorem Th18: :: TOPREAL4:18
theorem Th19: :: TOPREAL4:19
theorem Th20: :: TOPREAL4:20
theorem Th21: :: TOPREAL4:21
for b
1 being
Point of
(TOP-REAL 2)for b
2, b
3 being
FinSequence of
(TOP-REAL 2)for b
4 being
Realfor b
5 being
Point of
(Euclid 2) holds
( not b
2 /. 1
in Ball b
5,b
4 & b
2 /. (len b2) in Ball b
5,b
4 & b
1 in Ball b
5,b
4 &
|[(b1 `1 ),((b2 /. (len b2)) `2 )]| in Ball b
5,b
4 & b
2 is_S-Seq & b
1 `1 <> (b2 /. (len b2)) `1 & b
1 `2 <> (b2 /. (len b2)) `2 & b
3 = b
2 ^ <*|[(b1 `1 ),((b2 /. (len b2)) `2 )]|,b1*> &
((LSeg (b2 /. (len b2)),|[(b1 `1 ),((b2 /. (len b2)) `2 )]|) \/ (LSeg |[(b1 `1 ),((b2 /. (len b2)) `2 )]|,b1)) /\ (L~ b2) = {(b2 /. (len b2))} implies (
L~ b
3 is_S-P_arc_joining b
2 /. 1,b
1 &
L~ b
3 c= (L~ b2) \/ (Ball b5,b4) ) )
theorem Th22: :: TOPREAL4:22
for b
1 being
Point of
(TOP-REAL 2)for b
2, b
3 being
FinSequence of
(TOP-REAL 2)for b
4 being
Realfor b
5 being
Point of
(Euclid 2) holds
( not b
2 /. 1
in Ball b
5,b
4 & b
2 /. (len b2) in Ball b
5,b
4 & b
1 in Ball b
5,b
4 &
|[((b2 /. (len b2)) `1 ),(b1 `2 )]| in Ball b
5,b
4 & b
2 is_S-Seq & b
1 `1 <> (b2 /. (len b2)) `1 & b
1 `2 <> (b2 /. (len b2)) `2 & b
3 = b
2 ^ <*|[((b2 /. (len b2)) `1 ),(b1 `2 )]|,b1*> &
((LSeg (b2 /. (len b2)),|[((b2 /. (len b2)) `1 ),(b1 `2 )]|) \/ (LSeg |[((b2 /. (len b2)) `1 ),(b1 `2 )]|,b1)) /\ (L~ b2) = {(b2 /. (len b2))} implies (
L~ b
3 is_S-P_arc_joining b
2 /. 1,b
1 &
L~ b
3 c= (L~ b2) \/ (Ball b5,b4) ) )
theorem Th23: :: TOPREAL4:23
theorem Th24: :: TOPREAL4:24
Lemma24:
TopSpaceMetr (Euclid 2) = TOP-REAL 2
by EUCLID:def 8;
theorem Th25: :: TOPREAL4:25
theorem Th26: :: TOPREAL4:26
theorem Th27: :: TOPREAL4:27
theorem Th28: :: TOPREAL4:28
theorem Th29: :: TOPREAL4:29
theorem Th30: :: TOPREAL4:30