:: JORDAN17 semantic presentation
theorem Th1: :: JORDAN17:1
theorem Th2: :: JORDAN17:2
theorem Th3: :: JORDAN17:3
theorem Th4: :: JORDAN17:4
theorem Th5: :: JORDAN17:5
theorem Th6: :: JORDAN17:6
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
not ( b
1 <> b
2 & b
5 is_an_arc_of b
3,b
4 &
LE b
1,b
2,b
5,b
3,b
4 & ( for b
6 being
Point of
(TOP-REAL 2) holds
not ( b
1 <> b
6 & b
2 <> b
6 &
LE b
1,b
6,b
5,b
3,b
4 &
LE b
6,b
2,b
5,b
3,b
4 ) ) )
theorem Th7: :: JORDAN17:7
theorem Th8: :: JORDAN17:8
definition
let c
1 be
Subset of
(TOP-REAL 2);
let c
2, c
3, c
4, c
5 be
Point of
(TOP-REAL 2);
pred c
2,c
3,c
4,c
5 are_in_this_order_on c
1 means :
Def1:
:: JORDAN17:def 1
not ( not (
LE a
2,a
3,a
1 &
LE a
3,a
4,a
1 &
LE a
4,a
5,a
1 ) & not (
LE a
3,a
4,a
1 &
LE a
4,a
5,a
1 &
LE a
5,a
2,a
1 ) & not (
LE a
4,a
5,a
1 &
LE a
5,a
2,a
1 &
LE a
2,a
3,a
1 ) & not (
LE a
5,a
2,a
1 &
LE a
2,a
3,a
1 &
LE a
3,a
4,a
1 ) );
end;
:: deftheorem Def1 defines are_in_this_order_on JORDAN17:def 1 :
for b
1 being
Subset of
(TOP-REAL 2)for b
2, b
3, b
4, b
5 being
Point of
(TOP-REAL 2) holds
( b
2,b
3,b
4,b
5 are_in_this_order_on b
1 iff not ( not (
LE b
2,b
3,b
1 &
LE b
3,b
4,b
1 &
LE b
4,b
5,b
1 ) & not (
LE b
3,b
4,b
1 &
LE b
4,b
5,b
1 &
LE b
5,b
2,b
1 ) & not (
LE b
4,b
5,b
1 &
LE b
5,b
2,b
1 &
LE b
2,b
3,b
1 ) & not (
LE b
5,b
2,b
1 &
LE b
2,b
3,b
1 &
LE b
3,b
4,b
1 ) ) );
theorem Th9: :: JORDAN17:9
theorem Th10: :: JORDAN17:10
theorem Th11: :: JORDAN17:11
theorem Th12: :: JORDAN17:12
theorem Th13: :: JORDAN17:13
theorem Th14: :: JORDAN17:14
theorem Th15: :: JORDAN17:15
theorem Th16: :: JORDAN17:16
theorem Th17: :: JORDAN17:17
theorem Th18: :: JORDAN17:18
theorem Th19: :: JORDAN17:19
theorem Th20: :: JORDAN17:20
theorem Th21: :: JORDAN17:21
theorem Th22: :: JORDAN17:22
theorem Th23: :: JORDAN17:23
theorem Th24: :: JORDAN17:24
theorem Th25: :: JORDAN17:25
theorem Th26: :: JORDAN17:26
theorem Th27: :: JORDAN17:27
for b
1 being
Simple_closed_curvefor b
2, b
3, b
4, b
5 being
Point of
(TOP-REAL 2) holds
not ( b
2 in b
1 & b
3 in b
1 & b
4 in b
1 & b
5 in b
1 & not b
2,b
3,b
4,b
5 are_in_this_order_on b
1 & not b
2,b
3,b
5,b
4 are_in_this_order_on b
1 & not b
2,b
4,b
3,b
5 are_in_this_order_on b
1 & not b
2,b
4,b
5,b
3 are_in_this_order_on b
1 & not b
2,b
5,b
3,b
4 are_in_this_order_on b
1 & not b
2,b
5,b
4,b
3 are_in_this_order_on b
1 )