:: 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
a,
b,
c,
d being
Point of
(TOP-REAL 2) for
P being
Subset of
(TOP-REAL 2) st
a <> b &
P is_an_arc_of c,
d &
LE a,
b,
P,
c,
d holds
ex
e being
Point of
(TOP-REAL 2) st
(
a <> e &
b <> e &
LE a,
e,
P,
c,
d &
LE e,
b,
P,
c,
d )
theorem Th7: :: JORDAN17:7
theorem Th8: :: JORDAN17:8
definition
let P be
Subset of
(TOP-REAL 2);
let a be
Point of
(TOP-REAL 2),
b be
Point of
(TOP-REAL 2),
c be
Point of
(TOP-REAL 2),
d be
Point of
(TOP-REAL 2);
pred c2,
c3,
c4,
c5 are_in_this_order_on c1 means :
Def1:
:: JORDAN17:def 1
( (
LE a,
b,
P &
LE b,
c,
P &
LE c,
d,
P ) or (
LE b,
c,
P &
LE c,
d,
P &
LE d,
a,
P ) or (
LE c,
d,
P &
LE d,
a,
P &
LE a,
b,
P ) or (
LE d,
a,
P &
LE a,
b,
P &
LE b,
c,
P ) );
end;
:: deftheorem Def1 defines are_in_this_order_on JORDAN17:def 1 :
for
P being
Subset of
(TOP-REAL 2) for
a,
b,
c,
d being
Point of
(TOP-REAL 2) holds
(
a,
b,
c,
d are_in_this_order_on P iff ( (
LE a,
b,
P &
LE b,
c,
P &
LE c,
d,
P ) or (
LE b,
c,
P &
LE c,
d,
P &
LE d,
a,
P ) or (
LE c,
d,
P &
LE d,
a,
P &
LE a,
b,
P ) or (
LE d,
a,
P &
LE a,
b,
P &
LE b,
c,
P ) ) );
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
C being
Simple_closed_curve for
a,
b,
c,
d being
Point of
(TOP-REAL 2) st
a in C &
b in C &
c in C &
d in C & not
a,
b,
c,
d are_in_this_order_on C & not
a,
b,
d,
c are_in_this_order_on C & not
a,
c,
b,
d are_in_this_order_on C & not
a,
c,
d,
b are_in_this_order_on C & not
a,
d,
b,
c are_in_this_order_on C holds
a,
d,
c,
b are_in_this_order_on C