:: JORDAN7 semantic presentation
E1: 2 -' 1 =
2 - 1
by BINARITH:50
.=
1
;
Lemma2:
for b1, b2, b3 being Nat holds
( b1 -' b3 <= b2 implies b1 <= b2 + b3 )
Lemma3:
for b1, b2, b3 being Nat holds
( b2 + b3 <= b1 implies b3 <= b1 -' b2 )
theorem Th1: :: JORDAN7:1
theorem Th2: :: JORDAN7:2
theorem Th3: :: JORDAN7:3
definition
let c
1 be non
empty compact Subset of
(TOP-REAL 2);
let c
2, c
3 be
Point of
(TOP-REAL 2);
func Segment c
2,c
3,c
1 -> Subset of
(TOP-REAL 2) equals :
Def1:
:: JORDAN7:def 1
{ b1 where B is Point of (TOP-REAL 2) : ( LE a2,b1,a1 & LE b1,a3,a1 ) } if a
3 <> W-min a
1 otherwise { b1 where B is Point of (TOP-REAL 2) : ( LE a2,b1,a1 or ( a2 in a1 & b1 = W-min a1 ) ) } ;
correctness
coherence
( ( c3 <> W-min c1 implies { b1 where B is Point of (TOP-REAL 2) : ( LE c2,b1,c1 & LE b1,c3,c1 ) } is Subset of (TOP-REAL 2) ) & ( not c3 <> W-min c1 implies { b1 where B is Point of (TOP-REAL 2) : ( LE c2,b1,c1 or ( c2 in c1 & b1 = W-min c1 ) ) } is Subset of (TOP-REAL 2) ) );
consistency
for b1 being Subset of (TOP-REAL 2) holds
verum;
end;
:: deftheorem Def1 defines Segment JORDAN7:def 1 :
for b
1 being non
empty compact Subset of
(TOP-REAL 2)for b
2, b
3 being
Point of
(TOP-REAL 2) holds
( ( b
3 <> W-min b
1 implies
Segment b
2,b
3,b
1 = { b4 where B is Point of (TOP-REAL 2) : ( LE b2,b4,b1 & LE b4,b3,b1 ) } ) & ( not b
3 <> W-min b
1 implies
Segment b
2,b
3,b
1 = { b4 where B is Point of (TOP-REAL 2) : ( LE b2,b4,b1 or ( b2 in b1 & b4 = W-min b1 ) ) } ) );
theorem Th4: :: JORDAN7:4
theorem Th5: :: JORDAN7:5
theorem Th6: :: JORDAN7:6
theorem Th7: :: JORDAN7:7
theorem Th8: :: JORDAN7:8
theorem Th9: :: JORDAN7:9
theorem Th10: :: JORDAN7:10
for b
1 being non
empty compact Subset of
(TOP-REAL 2)for b
2, b
3, b
4 being
Point of
(TOP-REAL 2) holds
( b
1 is_simple_closed_curve &
LE b
2,b
3,b
1 &
LE b
3,b
4,b
1 & not ( b
2 = b
3 & b
2 = W-min b
1 ) & b
2 <> b
4 & not ( b
3 = b
4 & b
3 = W-min b
1 ) implies
(Segment b2,b3,b1) /\ (Segment b3,b4,b1) = {b3} )
theorem Th11: :: JORDAN7:11
theorem Th12: :: JORDAN7:12
theorem Th13: :: JORDAN7:13
for b
1 being non
empty compact Subset of
(TOP-REAL 2)for b
2, b
3, b
4, b
5 being
Point of
(TOP-REAL 2) holds
( b
1 is_simple_closed_curve &
LE b
2,b
3,b
1 &
LE b
3,b
4,b
1 &
LE b
4,b
5,b
1 & b
2 <> b
3 & b
3 <> b
4 implies
Segment b
2,b
3,b
1 misses Segment b
4,b
5,b
1 )
theorem Th14: :: JORDAN7:14
for b
1 being non
empty compact Subset of
(TOP-REAL 2)for b
2, b
3, b
4 being
Point of
(TOP-REAL 2) holds
( b
1 is_simple_closed_curve &
LE b
2,b
3,b
1 &
LE b
3,b
4,b
1 & b
2 <> W-min b
1 & b
2 <> b
3 & b
3 <> b
4 implies
Segment b
2,b
3,b
1 misses Segment b
4,
(W-min b1),b
1 )
theorem Th15: :: JORDAN7:15
theorem Th16: :: JORDAN7:16
theorem Th17: :: JORDAN7:17
Lemma20:
( 0 in [.0,1.] & 1 in [.0,1.] )
by RCOMP_1:15;
theorem Th18: :: JORDAN7:18
theorem Th19: :: JORDAN7:19
for b
1 being non
empty Subset of
(TOP-REAL 2)for b
2, b
3, b
4, b
5 being
Point of
(TOP-REAL 2)for b
6 being
Function of
I[01] ,
(TOP-REAL 2)for b
7, b
8 being
Real holds
( b
1 is_an_arc_of b
2,b
3 & b
6 is
continuous & b
6 is
one-to-one &
rng b
6 = b
1 & b
6 . 0
= b
2 & b
6 . 1
= b
3 & b
6 . b
7 = b
4 & 0
<= b
7 & b
7 <= 1 & b
6 . b
8 = b
5 & 0
<= b
8 & b
8 <= 1 & b
7 <= b
8 implies
LE b
4,b
5,b
1,b
2,b
3 )
theorem Th20: :: JORDAN7:20
theorem Th21: :: JORDAN7:21
for b
1 being non
empty compact Subset of
(TOP-REAL 2)for b
2 being
Real holds
not ( b
1 is_simple_closed_curve & b
2 > 0 & ( for b
3 being
FinSequence of the
carrier of
(TOP-REAL 2) holds
not ( b
3 . 1
= W-min b
1 & b
3 is
one-to-one & 8
<= len b
3 &
rng b
3 c= b
1 & ( for b
4 being
Nat holds
( 1
<= b
4 & b
4 < len b
3 implies
LE b
3 /. b
4,b
3 /. (b4 + 1),b
1 ) ) & ( for b
4 being
Natfor b
5 being
Subset of
(Euclid 2) holds
not ( 1
<= b
4 & b
4 < len b
3 & b
5 = Segment (b3 /. b4),
(b3 /. (b4 + 1)),b
1 & not
diameter b
5 < b
2 ) ) & ( for b
4 being
Subset of
(Euclid 2) holds
not ( b
4 = Segment (b3 /. (len b3)),
(b3 /. 1),b
1 & not
diameter b
4 < b
2 ) ) & ( for b
4 being
Nat holds
( 1
<= b
4 & b
4 + 1
< len b
3 implies
(Segment (b3 /. b4),(b3 /. (b4 + 1)),b1) /\ (Segment (b3 /. (b4 + 1)),(b3 /. (b4 + 2)),b1) = {(b3 /. (b4 + 1))} ) ) &
(Segment (b3 /. (len b3)),(b3 /. 1),b1) /\ (Segment (b3 /. 1),(b3 /. 2),b1) = {(b3 /. 1)} &
(Segment (b3 /. ((len b3) -' 1)),(b3 /. (len b3)),b1) /\ (Segment (b3 /. (len b3)),(b3 /. 1),b1) = {(b3 /. (len b3))} &
Segment (b3 /. ((len b3) -' 1)),
(b3 /. (len b3)),b
1 misses Segment (b3 /. 1),
(b3 /. 2),b
1 & ( for b
4, b
5 being
Nat holds
( 1
<= b
4 & b
4 < b
5 & b
5 < len b
3 & not b
4,b
5 are_adjacent1 implies
Segment (b3 /. b4),
(b3 /. (b4 + 1)),b
1 misses Segment (b3 /. b5),
(b3 /. (b5 + 1)),b
1 ) ) & ( for b
4 being
Nat holds
( 1
< b
4 & b
4 + 1
< len b
3 implies
Segment (b3 /. (len b3)),
(b3 /. 1),b
1 misses Segment (b3 /. b4),
(b3 /. (b4 + 1)),b
1 ) ) ) ) )