:: JORDAN5C semantic presentation
Lemma1:
for b1 being Real holds
( 0 <= b1 & b1 <= 1 implies ( 0 <= 1 - b1 & 1 - b1 <= 1 ) )
theorem Th1: :: JORDAN5C:1
definition
let c
1, c
2 be
Subset of
(TOP-REAL 2);
let c
3, c
4 be
Point of
(TOP-REAL 2);
assume E3:
( c
1 meets c
2 & c
1 /\ c
2 is
closed & c
1 is_an_arc_of c
3,c
4 )
;
func First_Point c
1,c
3,c
4,c
2 -> Point of
(TOP-REAL 2) means :
Def1:
:: JORDAN5C:def 1
( a
5 in a
1 /\ a
2 & ( for b
1 being
Function of
I[01] ,
((TOP-REAL 2) | a1)for b
2 being
Real holds
( b
1 is
being_homeomorphism & b
1 . 0
= a
3 & b
1 . 1
= a
4 & b
1 . b
2 = a
5 & 0
<= b
2 & b
2 <= 1 implies for b
3 being
Real holds
not ( 0
<= b
3 & b
3 < b
2 & b
1 . b
3 in a
2 ) ) ) );
existence
ex b1 being Point of (TOP-REAL 2) st
( b1 in c1 /\ c2 & ( for b2 being Function of I[01] ,((TOP-REAL 2) | c1)
for b3 being Real holds
( b2 is being_homeomorphism & b2 . 0 = c3 & b2 . 1 = c4 & b2 . b3 = b1 & 0 <= b3 & b3 <= 1 implies for b4 being Real holds
not ( 0 <= b4 & b4 < b3 & b2 . b4 in c2 ) ) ) )
uniqueness
for b1, b2 being Point of (TOP-REAL 2) holds
( b1 in c1 /\ c2 & ( for b3 being Function of I[01] ,((TOP-REAL 2) | c1)
for b4 being Real holds
( b3 is being_homeomorphism & b3 . 0 = c3 & b3 . 1 = c4 & b3 . b4 = b1 & 0 <= b4 & b4 <= 1 implies for b5 being Real holds
not ( 0 <= b5 & b5 < b4 & b3 . b5 in c2 ) ) ) & b2 in c1 /\ c2 & ( for b3 being Function of I[01] ,((TOP-REAL 2) | c1)
for b4 being Real holds
( b3 is being_homeomorphism & b3 . 0 = c3 & b3 . 1 = c4 & b3 . b4 = b2 & 0 <= b4 & b4 <= 1 implies for b5 being Real holds
not ( 0 <= b5 & b5 < b4 & b3 . b5 in c2 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def1 defines First_Point JORDAN5C:def 1 :
theorem Th2: :: JORDAN5C:2
theorem Th3: :: JORDAN5C:3
theorem Th4: :: JORDAN5C:4
definition
let c
1, c
2 be
Subset of
(TOP-REAL 2);
let c
3, c
4 be
Point of
(TOP-REAL 2);
assume E6:
( c
1 meets c
2 & c
1 /\ c
2 is
closed & c
1 is_an_arc_of c
3,c
4 )
;
func Last_Point c
1,c
3,c
4,c
2 -> Point of
(TOP-REAL 2) means :
Def2:
:: JORDAN5C:def 2
( a
5 in a
1 /\ a
2 & ( for b
1 being
Function of
I[01] ,
((TOP-REAL 2) | a1)for b
2 being
Real holds
( b
1 is
being_homeomorphism & b
1 . 0
= a
3 & b
1 . 1
= a
4 & b
1 . b
2 = a
5 & 0
<= b
2 & b
2 <= 1 implies for b
3 being
Real holds
not ( 1
>= b
3 & b
3 > b
2 & b
1 . b
3 in a
2 ) ) ) );
existence
ex b1 being Point of (TOP-REAL 2) st
( b1 in c1 /\ c2 & ( for b2 being Function of I[01] ,((TOP-REAL 2) | c1)
for b3 being Real holds
( b2 is being_homeomorphism & b2 . 0 = c3 & b2 . 1 = c4 & b2 . b3 = b1 & 0 <= b3 & b3 <= 1 implies for b4 being Real holds
not ( 1 >= b4 & b4 > b3 & b2 . b4 in c2 ) ) ) )
uniqueness
for b1, b2 being Point of (TOP-REAL 2) holds
( b1 in c1 /\ c2 & ( for b3 being Function of I[01] ,((TOP-REAL 2) | c1)
for b4 being Real holds
( b3 is being_homeomorphism & b3 . 0 = c3 & b3 . 1 = c4 & b3 . b4 = b1 & 0 <= b4 & b4 <= 1 implies for b5 being Real holds
not ( 1 >= b5 & b5 > b4 & b3 . b5 in c2 ) ) ) & b2 in c1 /\ c2 & ( for b3 being Function of I[01] ,((TOP-REAL 2) | c1)
for b4 being Real holds
( b3 is being_homeomorphism & b3 . 0 = c3 & b3 . 1 = c4 & b3 . b4 = b2 & 0 <= b4 & b4 <= 1 implies for b5 being Real holds
not ( 1 >= b5 & b5 > b4 & b3 . b5 in c2 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def2 defines Last_Point JORDAN5C:def 2 :
theorem Th5: :: JORDAN5C:5
theorem Th6: :: JORDAN5C:6
theorem Th7: :: JORDAN5C:7
:: deftheorem Def3 defines LE JORDAN5C:def 3 :
theorem Th8: :: JORDAN5C:8
for b
1 being
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) | b1)for b
7, b
8 being
Real holds
( b
1 is_an_arc_of b
2,b
3 & b
6 is
being_homeomorphism & 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 Th9: :: JORDAN5C:9
theorem Th10: :: JORDAN5C:10
for b
1 being
Subset of
(TOP-REAL 2)for b
2, b
3, b
4 being
Point of
(TOP-REAL 2) holds
( b
1 is_an_arc_of b
2,b
3 & b
4 in b
1 implies (
LE b
2,b
4,b
1,b
2,b
3 &
LE b
4,b
3,b
1,b
2,b
3 ) )
theorem Th11: :: JORDAN5C:11
theorem Th12: :: JORDAN5C:12
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
1 is_an_arc_of b
2,b
3 &
LE b
4,b
5,b
1,b
2,b
3 &
LE b
5,b
4,b
1,b
2,b
3 implies b
4 = b
5 )
theorem Th13: :: JORDAN5C:13
for b
1 being
Subset of
(TOP-REAL 2)for b
2, b
3, b
4, b
5, b
6 being
Point of
(TOP-REAL 2) holds
(
LE b
4,b
5,b
1,b
2,b
3 &
LE b
5,b
6,b
1,b
2,b
3 implies
LE b
4,b
6,b
1,b
2,b
3 )
theorem Th14: :: JORDAN5C:14
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
not ( b
1 is_an_arc_of b
2,b
3 & b
4 in b
1 & b
5 in b
1 & b
4 <> b
5 & not (
LE b
4,b
5,b
1,b
2,b
3 & not
LE b
5,b
4,b
1,b
2,b
3 ) & not (
LE b
5,b
4,b
1,b
2,b
3 & not
LE b
4,b
5,b
1,b
2,b
3 ) )
theorem Th15: :: JORDAN5C:15
theorem Th16: :: JORDAN5C:16
theorem Th17: :: JORDAN5C:17
for b
1, b
2, b
3, b
4 being
Point of
(TOP-REAL 2) holds
( b
3 <> b
4 &
LE b
1,b
2,
LSeg b
3,b
4,b
3,b
4 implies
LE b
1,b
2,b
3,b
4 )
theorem Th18: :: JORDAN5C:18
for b
1, b
2 being
Subset of
(TOP-REAL 2)for b
3, b
4 being
Point of
(TOP-REAL 2) holds
( b
1 is_an_arc_of b
3,b
4 & b
1 meets b
2 & b
1 /\ b
2 is
closed implies (
First_Point b
1,b
3,b
4,b
2 = Last_Point b
1,b
4,b
3,b
2 &
Last_Point b
1,b
3,b
4,b
2 = First_Point b
1,b
4,b
3,b
2 ) )
theorem Th19: :: JORDAN5C:19
for b
1 being
FinSequence of
(TOP-REAL 2)for b
2 being
Subset of
(TOP-REAL 2)for b
3 being
Nat holds
(
L~ b
1 meets b
2 & b
2 is
closed & b
1 is_S-Seq & 1
<= b
3 & b
3 + 1
<= len b
1 &
First_Point (L~ b1),
(b1 /. 1),
(b1 /. (len b1)),b
2 in LSeg b
1,b
3 implies
First_Point (L~ b1),
(b1 /. 1),
(b1 /. (len b1)),b
2 = First_Point (LSeg b1,b3),
(b1 /. b3),
(b1 /. (b3 + 1)),b
2 )
theorem Th20: :: JORDAN5C:20
for b
1 being
FinSequence of
(TOP-REAL 2)for b
2 being
Subset of
(TOP-REAL 2)for b
3 being
Nat holds
(
L~ b
1 meets b
2 & b
2 is
closed & b
1 is_S-Seq & 1
<= b
3 & b
3 + 1
<= len b
1 &
Last_Point (L~ b1),
(b1 /. 1),
(b1 /. (len b1)),b
2 in LSeg b
1,b
3 implies
Last_Point (L~ b1),
(b1 /. 1),
(b1 /. (len b1)),b
2 = Last_Point (LSeg b1,b3),
(b1 /. b3),
(b1 /. (b3 + 1)),b
2 )
theorem Th21: :: JORDAN5C:21
theorem Th22: :: JORDAN5C:22
theorem Th23: :: JORDAN5C:23
theorem Th24: :: JORDAN5C:24
Lemma22:
for b1 being FinSequence of (TOP-REAL 2)
for b2 being Subset of (TOP-REAL 2)
for b3 being Point of (TOP-REAL 2)
for b4 being Nat holds
( LSeg b1,b4 meets b2 & b1 is_S-Seq & b2 is closed & 1 <= b4 & b4 + 1 <= len b1 & b3 in LSeg b1,b4 & b3 in b2 implies LE First_Point (LSeg b1,b4),(b1 /. b4),(b1 /. (b4 + 1)),b2,b3,b1 /. b4,b1 /. (b4 + 1) )
Lemma23:
for b1 being FinSequence of (TOP-REAL 2)
for b2 being Subset of (TOP-REAL 2)
for b3 being Point of (TOP-REAL 2)
for b4 being Nat holds
( L~ b1 meets b2 & b1 is_S-Seq & b2 is closed & First_Point (L~ b1),(b1 /. 1),(b1 /. (len b1)),b2 in LSeg b1,b4 & 1 <= b4 & b4 + 1 <= len b1 & b3 in LSeg b1,b4 & b3 in b2 implies LE First_Point (L~ b1),(b1 /. 1),(b1 /. (len b1)),b2,b3,b1 /. b4,b1 /. (b4 + 1) )
Lemma24:
for b1 being FinSequence of (TOP-REAL 2)
for b2 being Subset of (TOP-REAL 2)
for b3 being Point of (TOP-REAL 2)
for b4 being Nat holds
( LSeg b1,b4 meets b2 & b1 is_S-Seq & b2 is closed & 1 <= b4 & b4 + 1 <= len b1 & b3 in LSeg b1,b4 & b3 in b2 implies LE b3, Last_Point (LSeg b1,b4),(b1 /. b4),(b1 /. (b4 + 1)),b2,b1 /. b4,b1 /. (b4 + 1) )
Lemma25:
for b1 being FinSequence of (TOP-REAL 2)
for b2 being Subset of (TOP-REAL 2)
for b3 being Point of (TOP-REAL 2)
for b4 being Nat holds
( L~ b1 meets b2 & b1 is_S-Seq & b2 is closed & Last_Point (L~ b1),(b1 /. 1),(b1 /. (len b1)),b2 in LSeg b1,b4 & 1 <= b4 & b4 + 1 <= len b1 & b3 in LSeg b1,b4 & b3 in b2 implies LE b3, Last_Point (L~ b1),(b1 /. 1),(b1 /. (len b1)),b2,b1 /. b4,b1 /. (b4 + 1) )
theorem Th25: :: JORDAN5C:25
theorem Th26: :: JORDAN5C:26
theorem Th27: :: JORDAN5C:27
for b
1 being
FinSequence of
(TOP-REAL 2)for b
2 being
Subset of
(TOP-REAL 2)for b
3 being
Point of
(TOP-REAL 2)for b
4, b
5 being
Nat holds
(
L~ b
1 meets b
2 & b
1 is_S-Seq & b
2 is
closed &
First_Point (L~ b1),
(b1 /. 1),
(b1 /. (len b1)),b
2 in LSeg b
1,b
4 & 1
<= b
4 & b
4 + 1
<= len b
1 & b
3 in LSeg b
1,b
5 & 1
<= b
5 & b
5 + 1
<= len b
1 & b
3 in b
2 &
First_Point (L~ b1),
(b1 /. 1),
(b1 /. (len b1)),b
2 <> b
3 implies ( b
4 <= b
5 & ( b
4 = b
5 implies
LE First_Point (L~ b1),
(b1 /. 1),
(b1 /. (len b1)),b
2,b
3,b
1 /. b
4,b
1 /. (b4 + 1) ) ) )
theorem Th28: :: JORDAN5C:28
for b
1 being
FinSequence of
(TOP-REAL 2)for b
2 being
Subset of
(TOP-REAL 2)for b
3 being
Point of
(TOP-REAL 2)for b
4, b
5 being
Nat holds
(
L~ b
1 meets b
2 & b
1 is_S-Seq & b
2 is
closed &
Last_Point (L~ b1),
(b1 /. 1),
(b1 /. (len b1)),b
2 in LSeg b
1,b
4 & 1
<= b
4 & b
4 + 1
<= len b
1 & b
3 in LSeg b
1,b
5 & 1
<= b
5 & b
5 + 1
<= len b
1 & b
3 in b
2 &
Last_Point (L~ b1),
(b1 /. 1),
(b1 /. (len b1)),b
2 <> b
3 implies ( b
4 >= b
5 & ( b
4 = b
5 implies
LE b
3,
Last_Point (L~ b1),
(b1 /. 1),
(b1 /. (len b1)),b
2,b
1 /. b
4,b
1 /. (b4 + 1) ) ) )
theorem Th29: :: JORDAN5C:29
for b
1 being
FinSequence of
(TOP-REAL 2)for b
2, b
3 being
Point of
(TOP-REAL 2)for b
4 being
Nat holds
( b
2 in LSeg b
1,b
4 & b
3 in LSeg b
1,b
4 & b
1 is_S-Seq & 1
<= b
4 & b
4 + 1
<= len b
1 &
LE b
2,b
3,
L~ b
1,b
1 /. 1,b
1 /. (len b1) implies
LE b
2,b
3,
LSeg b
1,b
4,b
1 /. b
4,b
1 /. (b4 + 1) )
theorem Th30: :: JORDAN5C:30
for b
1 being
FinSequence of
(TOP-REAL 2)for b
2, b
3 being
Point of
(TOP-REAL 2) holds
( b
2 in L~ b
1 & b
3 in L~ b
1 & b
1 is_S-Seq & b
2 <> b
3 implies (
LE b
2,b
3,
L~ b
1,b
1 /. 1,b
1 /. (len b1) iff for b
4, b
5 being
Nat holds
( b
2 in LSeg b
1,b
4 & b
3 in LSeg b
1,b
5 & 1
<= b
4 & b
4 + 1
<= len b
1 & 1
<= b
5 & b
5 + 1
<= len b
1 implies ( b
4 <= b
5 & ( b
4 = b
5 implies
LE b
2,b
3,b
1 /. b
4,b
1 /. (b4 + 1) ) ) ) ) )