:: JORDAN semantic presentation
set c1 = TOP-REAL 2;
Lemma1:
for b1, b2, b3, b4 being set holds
( b1 c= b4 & b2 c= b4 & b3 c= b4 implies (b1 \/ b2) \/ b3 c= b4 )
Lemma2:
for b1, b2, b3, b4, b5 being set holds
( b1 c= b5 & b2 c= b5 & b3 c= b5 & b4 c= b5 implies ((b1 \/ b2) \/ b3) \/ b4 c= b5 )
Lemma3:
for b1, b2, b3, b4, b5 being set holds
( b1 misses b5 & b2 misses b5 & b3 misses b5 & b4 misses b5 implies ((b1 \/ b2) \/ b3) \/ b4 misses b5 )
theorem Th1: :: JORDAN:1
theorem Th2: :: JORDAN:2
theorem Th3: :: JORDAN:3
theorem Th4: :: JORDAN:4
theorem Th5: :: JORDAN:5
theorem Th6: :: JORDAN:6
theorem Th7: :: JORDAN:7
theorem Th8: :: JORDAN:8
theorem Th9: :: JORDAN:9
theorem Th10: :: JORDAN:10
theorem Th11: :: JORDAN:11
Lemma12:
for b1 being non empty convex Subset of (TOP-REAL 2) holds b1 is connected
;
theorem Th12: :: JORDAN:12
theorem Th13: :: JORDAN:13
theorem Th14: :: JORDAN:14
theorem Th15: :: JORDAN:15
Lemma16:
for b1 being Point of (TOP-REAL 2)
for b2 being Simple_closed_curve
for b3 being Subset of ((TOP-REAL 2) | (b2 ` )) holds
( b1 in b2 implies {b1} misses b3 )
set c2 = Closed-Interval-TSpace 0,1;
set c3 = Closed-Interval-TSpace (- 1),1;
set c4 = (#) (- 1),1;
set c5 = (- 1),1 (#) ;
set c6 = L[01] ((#) (- 1),1),((- 1),1 (#) );
Lemma17:
the carrier of [:(TOP-REAL 2),(TOP-REAL 2):] = [:the carrier of (TOP-REAL 2),the carrier of (TOP-REAL 2):]
by BORSUK_1:def 5;
theorem Th16: :: JORDAN:16
theorem Th17: :: JORDAN:17
theorem Th18: :: JORDAN:18
theorem Th19: :: JORDAN:19
theorem Th20: :: JORDAN:20
theorem Th21: :: JORDAN:21
theorem Th22: :: JORDAN:22
theorem Th23: :: JORDAN:23
theorem Th24: :: JORDAN:24
theorem Th25: :: JORDAN:25
theorem Th26: :: JORDAN:26
theorem Th27: :: JORDAN:27
theorem Th28: :: JORDAN:28
theorem Th29: :: JORDAN:29
theorem Th30: :: JORDAN:30
Lemma33:
for b1 being non empty TopSpace
for b2, b3 being Point of b1
for b4 being Path of b2,b3 holds
( b2,b3 are_connected implies rng b4 c= rng (- b4) )
theorem Th31: :: JORDAN:31
theorem Th32: :: JORDAN:32
theorem Th33: :: JORDAN:33
theorem Th34: :: JORDAN:34
theorem Th35: :: JORDAN:35
theorem Th36: :: JORDAN:36
theorem Th37: :: JORDAN:37
theorem Th38: :: JORDAN:38
theorem Th39: :: JORDAN:39
theorem Th40: :: JORDAN:40
Lemma41:
for b1 being non empty TopSpace
for b2, b3, b4, b5, b6 being Point of b1
for b7 being Path of b2,b3
for b8 being Path of b3,b4
for b9 being Path of b4,b5
for b10 being Path of b5,b6 holds
( b2,b3 are_connected & b3,b4 are_connected & b4,b5 are_connected & b5,b6 are_connected implies rng (((b7 + b8) + b9) + b10) = (((rng b7) \/ (rng b8)) \/ (rng b9)) \/ (rng b10) )
Lemma42:
for b1 being non empty arcwise_connected TopSpace
for b2, b3, b4, b5, b6 being Point of b1
for b7 being Path of b2,b3
for b8 being Path of b3,b4
for b9 being Path of b4,b5
for b10 being Path of b5,b6 holds rng (((b7 + b8) + b9) + b10) = (((rng b7) \/ (rng b8)) \/ (rng b9)) \/ (rng b10)
Lemma43:
for b1 being non empty TopSpace
for b2, b3, b4, b5, b6, b7 being Point of b1
for b8 being Path of b2,b3
for b9 being Path of b3,b4
for b10 being Path of b4,b5
for b11 being Path of b5,b6
for b12 being Path of b6,b7 holds
( b2,b3 are_connected & b3,b4 are_connected & b4,b5 are_connected & b5,b6 are_connected & b6,b7 are_connected implies rng ((((b8 + b9) + b10) + b11) + b12) = ((((rng b8) \/ (rng b9)) \/ (rng b10)) \/ (rng b11)) \/ (rng b12) )
Lemma44:
for b1 being non empty arcwise_connected TopSpace
for b2, b3, b4, b5, b6, b7 being Point of b1
for b8 being Path of b2,b3
for b9 being Path of b3,b4
for b10 being Path of b4,b5
for b11 being Path of b5,b6
for b12 being Path of b6,b7 holds rng ((((b8 + b9) + b10) + b11) + b12) = ((((rng b8) \/ (rng b9)) \/ (rng b10)) \/ (rng b11)) \/ (rng b12)
theorem Th41: :: JORDAN:41
theorem Th42: :: JORDAN:42
theorem Th43: :: JORDAN:43
theorem Th44: :: JORDAN:44
theorem Th45: :: JORDAN:45
theorem Th46: :: JORDAN:46
theorem Th47: :: JORDAN:47
theorem Th48: :: JORDAN:48
theorem Th49: :: JORDAN:49
for b
1, b
2, b
3, b
4 being
real number holds
(closed_inside_of_rectangle b1,b2,b3,b4) /\ (inside_of_rectangle b1,b2,b3,b4) = inside_of_rectangle b
1,b
2,b
3,b
4
theorem Th50: :: JORDAN:50
theorem Th51: :: JORDAN:51
for b
1, b
2, b
3, b
4 being
real number holds
( b
1 <= b
2 & b
3 <= b
4 implies
(closed_inside_of_rectangle b1,b2,b3,b4) \ (inside_of_rectangle b1,b2,b3,b4) = rectangle b
1,b
2,b
3,b
4 )
theorem Th52: :: JORDAN:52
theorem Th53: :: JORDAN:53
theorem Th54: :: JORDAN:54
theorem Th55: :: JORDAN:55
theorem Th56: :: JORDAN:56
theorem Th57: :: JORDAN:57
for b
1, b
2, b
3, b
4 being
real number for b
5, b
6 being
Point of
(TOP-REAL 2)for b
7 being
Subset of
(TOP-REAL 2) holds
( b
1 < b
2 & b
3 < b
4 & b
5 in closed_inside_of_rectangle b
1,b
2,b
3,b
4 & not b
6 in closed_inside_of_rectangle b
1,b
2,b
3,b
4 & b
7 is_an_arc_of b
5,b
6 implies
Segment b
7,b
5,b
6,b
5,
(First_Point b7,b5,b6,(rectangle b1,b2,b3,b4)) c= closed_inside_of_rectangle b
1,b
2,b
3,b
4 )
definition
let c
7 be
Point of
(TOP-REAL 2);
func diffX2_1 c
1 -> RealMap of
[:(TOP-REAL 2),(TOP-REAL 2):] means :
Def1:
:: JORDAN:def 1
for b
1 being
Point of
[:(TOP-REAL 2),(TOP-REAL 2):] holds a
2 . b
1 = ((b1 `2 ) `1 ) - (a1 `1 );
existence
ex b1 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] st
for b2 being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . b2 = ((b2 `2 ) `1 ) - (c7 `1 )
uniqueness
for b1, b2 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] holds
( ( for b3 being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . b3 = ((b3 `2 ) `1 ) - (c7 `1 ) ) & ( for b3 being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b2 . b3 = ((b3 `2 ) `1 ) - (c7 `1 ) ) implies b1 = b2 )
func diffX2_2 c
1 -> RealMap of
[:(TOP-REAL 2),(TOP-REAL 2):] means :
Def2:
:: JORDAN:def 2
for b
1 being
Point of
[:(TOP-REAL 2),(TOP-REAL 2):] holds a
2 . b
1 = ((b1 `2 ) `2 ) - (a1 `2 );
existence
ex b1 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] st
for b2 being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . b2 = ((b2 `2 ) `2 ) - (c7 `2 )
uniqueness
for b1, b2 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] holds
( ( for b3 being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . b3 = ((b3 `2 ) `2 ) - (c7 `2 ) ) & ( for b3 being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b2 . b3 = ((b3 `2 ) `2 ) - (c7 `2 ) ) implies b1 = b2 )
end;
:: deftheorem Def1 defines diffX2_1 JORDAN:def 1 :
:: deftheorem Def2 defines diffX2_2 JORDAN:def 2 :
definition
func diffX1_X2_1 -> RealMap of
[:(TOP-REAL 2),(TOP-REAL 2):] means :
Def3:
:: JORDAN:def 3
for b
1 being
Point of
[:(TOP-REAL 2),(TOP-REAL 2):] holds a
1 . b
1 = ((b1 `1 ) `1 ) - ((b1 `2 ) `1 );
existence
ex b1 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] st
for b2 being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . b2 = ((b2 `1 ) `1 ) - ((b2 `2 ) `1 )
uniqueness
for b1, b2 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] holds
( ( for b3 being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . b3 = ((b3 `1 ) `1 ) - ((b3 `2 ) `1 ) ) & ( for b3 being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b2 . b3 = ((b3 `1 ) `1 ) - ((b3 `2 ) `1 ) ) implies b1 = b2 )
func diffX1_X2_2 -> RealMap of
[:(TOP-REAL 2),(TOP-REAL 2):] means :
Def4:
:: JORDAN:def 4
for b
1 being
Point of
[:(TOP-REAL 2),(TOP-REAL 2):] holds a
1 . b
1 = ((b1 `1 ) `2 ) - ((b1 `2 ) `2 );
existence
ex b1 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] st
for b2 being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . b2 = ((b2 `1 ) `2 ) - ((b2 `2 ) `2 )
uniqueness
for b1, b2 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] holds
( ( for b3 being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . b3 = ((b3 `1 ) `2 ) - ((b3 `2 ) `2 ) ) & ( for b3 being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b2 . b3 = ((b3 `1 ) `2 ) - ((b3 `2 ) `2 ) ) implies b1 = b2 )
func Proj2_1 -> RealMap of
[:(TOP-REAL 2),(TOP-REAL 2):] means :
Def5:
:: JORDAN:def 5
for b
1 being
Point of
[:(TOP-REAL 2),(TOP-REAL 2):] holds a
1 . b
1 = (b1 `2 ) `1 ;
existence
ex b1 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] st
for b2 being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . b2 = (b2 `2 ) `1
uniqueness
for b1, b2 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] holds
( ( for b3 being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . b3 = (b3 `2 ) `1 ) & ( for b3 being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b2 . b3 = (b3 `2 ) `1 ) implies b1 = b2 )
func Proj2_2 -> RealMap of
[:(TOP-REAL 2),(TOP-REAL 2):] means :
Def6:
:: JORDAN:def 6
for b
1 being
Point of
[:(TOP-REAL 2),(TOP-REAL 2):] holds a
1 . b
1 = (b1 `2 ) `2 ;
existence
ex b1 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] st
for b2 being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . b2 = (b2 `2 ) `2
uniqueness
for b1, b2 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] holds
( ( for b3 being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . b3 = (b3 `2 ) `2 ) & ( for b3 being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b2 . b3 = (b3 `2 ) `2 ) implies b1 = b2 )
end;
:: deftheorem Def3 defines diffX1_X2_1 JORDAN:def 3 :
:: deftheorem Def4 defines diffX1_X2_2 JORDAN:def 4 :
:: deftheorem Def5 defines Proj2_1 JORDAN:def 5 :
:: deftheorem Def6 defines Proj2_2 JORDAN:def 6 :
theorem Th58: :: JORDAN:58
theorem Th59: :: JORDAN:59
theorem Th60: :: JORDAN:60
theorem Th61: :: JORDAN:61
theorem Th62: :: JORDAN:62
theorem Th63: :: JORDAN:63
definition
let c
7 be non
empty Nat;
let c
8, c
9 be
Point of
(TOP-REAL c7);
let c
10 be
positive real number ;
assume E70:
c
9 is
Point of
(Tdisk c8,c10)
;
set c
11 =
(TOP-REAL c7) | ((cl_Ball c8,c10) \ {c9});
func DiskProj c
2,c
4,c
3 -> Function of
((TOP-REAL a1) | ((cl_Ball a2,a4) \ {a3})),
(Tcircle a2,a4) means :
Def7:
:: JORDAN:def 7
for b
1 being
Point of
((TOP-REAL a1) | ((cl_Ball a2,a4) \ {a3})) holds
ex b
2 being
Point of
(TOP-REAL a1) st
( b
1 = b
2 & a
5 . b
1 = HC a
3,b
2,a
2,a
4 );
existence
ex b1 being Function of ((TOP-REAL c7) | ((cl_Ball c8,c10) \ {c9})),(Tcircle c8,c10) st
for b2 being Point of ((TOP-REAL c7) | ((cl_Ball c8,c10) \ {c9})) holds
ex b3 being Point of (TOP-REAL c7) st
( b2 = b3 & b1 . b2 = HC c9,b3,c8,c10 )
uniqueness
for b1, b2 being Function of ((TOP-REAL c7) | ((cl_Ball c8,c10) \ {c9})),(Tcircle c8,c10) holds
( ( for b3 being Point of ((TOP-REAL c7) | ((cl_Ball c8,c10) \ {c9})) holds
ex b4 being Point of (TOP-REAL c7) st
( b3 = b4 & b1 . b3 = HC c9,b4,c8,c10 ) ) & ( for b3 being Point of ((TOP-REAL c7) | ((cl_Ball c8,c10) \ {c9})) holds
ex b4 being Point of (TOP-REAL c7) st
( b3 = b4 & b2 . b3 = HC c9,b4,c8,c10 ) ) implies b1 = b2 )
end;
:: deftheorem Def7 defines DiskProj JORDAN:def 7 :
for b
1 being non
empty Natfor b
2, b
3 being
Point of
(TOP-REAL b1)for b
4 being
positive real number holds
( b
3 is
Point of
(Tdisk b2,b4) implies for b
5 being
Function of
((TOP-REAL b1) | ((cl_Ball b2,b4) \ {b3})),
(Tcircle b2,b4) holds
( b
5 = DiskProj b
2,b
4,b
3 iff for b
6 being
Point of
((TOP-REAL b1) | ((cl_Ball b2,b4) \ {b3})) holds
ex b
7 being
Point of
(TOP-REAL b1) st
( b
6 = b
7 & b
5 . b
6 = HC b
3,b
7,b
2,b
4 ) ) );
theorem Th64: :: JORDAN:64
theorem Th65: :: JORDAN:65
definition
let c
7 be non
empty Nat;
let c
8, c
9 be
Point of
(TOP-REAL c7);
let c
10 be
positive real number ;
assume E73:
c
9 in Ball c
8,c
10
;
set c
11 =
Tcircle c
8,c
10;
func RotateCircle c
2,c
4,c
3 -> Function of
(Tcircle a2,a4),
(Tcircle a2,a4) means :
Def8:
:: JORDAN:def 8
for b
1 being
Point of
(Tcircle a2,a4) holds
ex b
2 being
Point of
(TOP-REAL a1) st
( b
1 = b
2 & a
5 . b
1 = HC b
2,a
3,a
2,a
4 );
existence
ex b1 being Function of (Tcircle c8,c10),(Tcircle c8,c10) st
for b2 being Point of (Tcircle c8,c10) holds
ex b3 being Point of (TOP-REAL c7) st
( b2 = b3 & b1 . b2 = HC b3,c9,c8,c10 )
uniqueness
for b1, b2 being Function of (Tcircle c8,c10),(Tcircle c8,c10) holds
( ( for b3 being Point of (Tcircle c8,c10) holds
ex b4 being Point of (TOP-REAL c7) st
( b3 = b4 & b1 . b3 = HC b4,c9,c8,c10 ) ) & ( for b3 being Point of (Tcircle c8,c10) holds
ex b4 being Point of (TOP-REAL c7) st
( b3 = b4 & b2 . b3 = HC b4,c9,c8,c10 ) ) implies b1 = b2 )
end;
:: deftheorem Def8 defines RotateCircle JORDAN:def 8 :
for b
1 being non
empty Natfor b
2, b
3 being
Point of
(TOP-REAL b1)for b
4 being
positive real number holds
( b
3 in Ball b
2,b
4 implies for b
5 being
Function of
(Tcircle b2,b4),
(Tcircle b2,b4) holds
( b
5 = RotateCircle b
2,b
4,b
3 iff for b
6 being
Point of
(Tcircle b2,b4) holds
ex b
7 being
Point of
(TOP-REAL b1) st
( b
6 = b
7 & b
5 . b
6 = HC b
7,b
3,b
2,b
4 ) ) );
theorem Th66: :: JORDAN:66
theorem Th67: :: JORDAN:67
theorem Th68: :: JORDAN:68
theorem Th69: :: JORDAN:69
Lemma78:
for b1, b2, b3 being Point of (TOP-REAL 2)
for b4 being Subset of (TOP-REAL 2)
for b5 being non negative real number holds
not ( b4 is_an_arc_of b1,b2 & b4 is Subset of (Tdisk b3,b5) & ( for b6 being Function of (Tdisk b3,b5),((TOP-REAL 2) | b4) holds
not ( b6 is continuous & b6 | b4 = id b4 ) ) )
Lemma79:
for b1, b2, b3 being Point of (TOP-REAL 2)
for b4 being Simple_closed_curve
for b5, b6, b7 being Subset of (TOP-REAL 2)
for b8 being Subset of ((TOP-REAL 2) | (b4 ` ))
for b9 being positive real number holds
( b5 is_an_arc_of b1,b2 & b5 c= b4 & b4 c= Ball b3,b9 & b3 in b8 & (Cl b6) /\ (b6 ` ) c= b5 & b6 c= Ball b3,b9 implies for b10 being Function of (Tdisk b3,b9),((TOP-REAL 2) | b5) holds
not ( b10 is continuous & b10 | b5 = id b5 & b8 = b6 & b8 is_a_component_of (TOP-REAL 2) | (b4 ` ) & b7 = (cl_Ball b3,b9) \ {b3} & ( for b11 being Function of (Tdisk b3,b9),((TOP-REAL 2) | b7) holds
not ( b11 is continuous & ( for b12 being Point of (Tdisk b3,b9) holds
( ( b12 in Cl b6 implies b11 . b12 = b10 . b12 ) & ( b12 in b6 ` implies b11 . b12 = b12 ) ) ) ) ) ) )
Lemma80:
for b1 being Point of (TOP-REAL 2)
for b2 being Simple_closed_curve
for b3, b4 being Subset of (TOP-REAL 2)
for b5, b6 being Subset of ((TOP-REAL 2) | (b2 ` ))
for b7 being non empty Subset of (TOP-REAL 2) holds
( b5 <> b6 implies for b8 being positive real number holds
( b7 c= b2 & b2 c= Ball b1,b8 & b1 in b6 & (Cl b3) /\ (b3 ` ) c= b7 & Ball b1,b8 meets b3 implies for b9 being Function of (Tdisk b1,b8),((TOP-REAL 2) | b7) holds
not ( b9 is continuous & b9 | b7 = id b7 & b5 = b3 & b5 is_a_component_of (TOP-REAL 2) | (b2 ` ) & b6 is_a_component_of (TOP-REAL 2) | (b2 ` ) & b4 = (cl_Ball b1,b8) \ {b1} & ( for b10 being Function of (Tdisk b1,b8),((TOP-REAL 2) | b4) holds
not ( b10 is continuous & ( for b11 being Point of (Tdisk b1,b8) holds
( ( b11 in Cl b3 implies b10 . b11 = b11 ) & ( b11 in b3 ` implies b10 . b11 = b9 . b11 ) ) ) ) ) ) ) )
Lemma81:
for b1 being Simple_closed_curve
for b2 being Subset of (TOP-REAL 2)
for b3 being Subset of ((TOP-REAL 2) | (b1 ` )) holds
( not BDD b1 is empty & b3 = b2 & b3 is_a_component_of (TOP-REAL 2) | (b1 ` ) implies b1 = Fr b2 )
set c7 = 1;
set c8 = - 1;
set c9 = 3;
set c10 = - 3;
set c11 = |[(- 1),0]|;
set c12 = |[1,0]|;
set c13 = |[0,3]|;
set c14 = |[0,(- 3)]|;
set c15 = |[(- 1),3]|;
set c16 = |[1,3]|;
set c17 = |[(- 1),(- 3)]|;
set c18 = |[1,(- 3)]|;
set c19 = closed_inside_of_rectangle (- 1),1,(- 3),3;
set c20 = rectangle (- 1),1,(- 3),3;
set c21 = Trectangle (- 1),1,(- 3),3;
Lemma82:
|[(- 1),0]| `1 = - 1
by EUCLID:56;
Lemma83:
|[1,0]| `1 = 1
by EUCLID:56;
Lemma84:
|[(- 1),0]| `2 = 0
by EUCLID:56;
Lemma85:
|[1,0]| `2 = 0
by EUCLID:56;
Lemma86:
|[0,3]| `1 = 0
by EUCLID:56;
Lemma87:
|[0,3]| `2 = 3
by EUCLID:56;
Lemma88:
|[0,(- 3)]| `1 = 0
by EUCLID:56;
Lemma89:
|[0,(- 3)]| `2 = - 3
by EUCLID:56;
Lemma90:
|[(- 1),3]| `1 = - 1
by EUCLID:56;
Lemma91:
|[(- 1),3]| `2 = 3
by EUCLID:56;
Lemma92:
|[(- 1),(- 3)]| `1 = - 1
by EUCLID:56;
Lemma93:
|[(- 1),(- 3)]| `2 = - 3
by EUCLID:56;
Lemma94:
|[1,3]| `1 = 1
by EUCLID:56;
Lemma95:
|[1,3]| `2 = 3
by EUCLID:56;
Lemma96:
|[1,(- 3)]| `1 = 1
by EUCLID:56;
Lemma97:
|[1,(- 3)]| `2 = - 3
by EUCLID:56;
Lemma98:
|[(- 1),(- 3)]| = |[(|[(- 1),(- 3)]| `1 ),(|[(- 1),(- 3)]| `2 )]|
by EUCLID:57;
Lemma99:
|[(- 1),3]| = |[(|[(- 1),3]| `1 ),(|[(- 1),3]| `2 )]|
by EUCLID:57;
Lemma100:
|[1,(- 3)]| = |[(|[1,(- 3)]| `1 ),(|[1,(- 3)]| `2 )]|
by EUCLID:57;
Lemma101:
|[1,3]| = |[(|[1,3]| `1 ),(|[1,3]| `2 )]|
by EUCLID:57;
Lemma102:
rectangle (- 1),1,(- 3),3 = ((LSeg |[(- 1),(- 3)]|,|[(- 1),3]|) \/ (LSeg |[(- 1),3]|,|[1,3]|)) \/ ((LSeg |[1,3]|,|[1,(- 3)]|) \/ (LSeg |[1,(- 3)]|,|[(- 1),(- 3)]|))
by SPPOL_2:def 3;
( LSeg |[(- 1),(- 3)]|,|[(- 1),3]| c= (LSeg |[(- 1),(- 3)]|,|[(- 1),3]|) \/ (LSeg |[(- 1),3]|,|[1,3]|) & (LSeg |[(- 1),(- 3)]|,|[(- 1),3]|) \/ (LSeg |[(- 1),3]|,|[1,3]|) c= rectangle (- 1),1,(- 3),3 )
by Lemma102, XBOOLE_1:7;
then Lemma103:
LSeg |[(- 1),(- 3)]|,|[(- 1),3]| c= rectangle (- 1),1,(- 3),3
by XBOOLE_1:1;
( LSeg |[(- 1),3]|,|[1,3]| c= (LSeg |[(- 1),(- 3)]|,|[(- 1),3]|) \/ (LSeg |[(- 1),3]|,|[1,3]|) & (LSeg |[(- 1),(- 3)]|,|[(- 1),3]|) \/ (LSeg |[(- 1),3]|,|[1,3]|) c= rectangle (- 1),1,(- 3),3 )
by Lemma102, XBOOLE_1:7;
then Lemma104:
LSeg |[(- 1),3]|,|[1,3]| c= rectangle (- 1),1,(- 3),3
by XBOOLE_1:1;
( LSeg |[1,3]|,|[1,(- 3)]| c= (LSeg |[1,3]|,|[1,(- 3)]|) \/ (LSeg |[1,(- 3)]|,|[(- 1),(- 3)]|) & (LSeg |[1,3]|,|[1,(- 3)]|) \/ (LSeg |[1,(- 3)]|,|[(- 1),(- 3)]|) c= rectangle (- 1),1,(- 3),3 )
by Lemma102, XBOOLE_1:7;
then Lemma105:
LSeg |[1,3]|,|[1,(- 3)]| c= rectangle (- 1),1,(- 3),3
by XBOOLE_1:1;
( LSeg |[1,(- 3)]|,|[(- 1),(- 3)]| c= (LSeg |[1,3]|,|[1,(- 3)]|) \/ (LSeg |[1,(- 3)]|,|[(- 1),(- 3)]|) & (LSeg |[1,3]|,|[1,(- 3)]|) \/ (LSeg |[1,(- 3)]|,|[(- 1),(- 3)]|) c= rectangle (- 1),1,(- 3),3 )
by Lemma102, XBOOLE_1:7;
then Lemma106:
LSeg |[1,(- 3)]|,|[(- 1),(- 3)]| c= rectangle (- 1),1,(- 3),3
by XBOOLE_1:1;
Lemma107:
LSeg |[(- 1),(- 3)]|,|[(- 1),3]| is vertical
by Lemma90, Lemma92, SPPOL_1:37;
Lemma108:
LSeg |[1,(- 3)]|,|[1,3]| is vertical
by Lemma94, Lemma96, SPPOL_1:37;
Lemma109:
LSeg |[(- 1),0]|,|[(- 1),3]| is vertical
by Lemma82, Lemma90, SPPOL_1:37;
Lemma110:
LSeg |[(- 1),0]|,|[(- 1),(- 3)]| is vertical
by Lemma82, Lemma92, SPPOL_1:37;
Lemma111:
LSeg |[1,0]|,|[1,3]| is vertical
by Lemma83, Lemma94, SPPOL_1:37;
Lemma112:
LSeg |[1,0]|,|[1,(- 3)]| is vertical
by Lemma83, Lemma96, SPPOL_1:37;
Lemma113:
LSeg |[(- 1),(- 3)]|,|[0,(- 3)]| is horizontal
by Lemma93, Lemma89, SPPOL_1:36;
Lemma114:
LSeg |[1,(- 3)]|,|[0,(- 3)]| is horizontal
by Lemma97, Lemma89, SPPOL_1:36;
Lemma115:
LSeg |[(- 1),3]|,|[0,3]| is horizontal
by Lemma91, Lemma87, SPPOL_1:36;
Lemma116:
LSeg |[1,3]|,|[0,3]| is horizontal
by Lemma95, Lemma87, SPPOL_1:36;
Lemma117:
LSeg |[(- 1),3]|,|[1,3]| is horizontal
by Lemma91, Lemma95, SPPOL_1:36;
Lemma118:
LSeg |[(- 1),(- 3)]|,|[1,(- 3)]| is horizontal
by Lemma93, Lemma97, SPPOL_1:36;
Lemma119:
LSeg |[(- 1),0]|,|[(- 1),3]| c= LSeg |[(- 1),(- 3)]|,|[(- 1),3]|
by Lemma107, Lemma109, Lemma82, Lemma92, Lemma84, Lemma91, Lemma93, GOBOARD7:65;
Lemma120:
LSeg |[(- 1),0]|,|[(- 1),(- 3)]| c= LSeg |[(- 1),(- 3)]|,|[(- 1),3]|
by Lemma107, Lemma110, Lemma92, Lemma84, Lemma91, Lemma93, GOBOARD7:65;
Lemma121:
LSeg |[1,0]|,|[1,3]| c= LSeg |[1,(- 3)]|,|[1,3]|
by Lemma111, Lemma108, Lemma83, Lemma96, Lemma95, Lemma97, Lemma85, GOBOARD7:65;
Lemma122:
LSeg |[1,0]|,|[1,(- 3)]| c= LSeg |[1,(- 3)]|,|[1,3]|
by Lemma112, Lemma108, Lemma96, Lemma85, Lemma95, Lemma97, GOBOARD7:65;
Lemma123:
rectangle (- 1),1,(- 3),3 = { b1 where B is Point of (TOP-REAL 2) : not ( not ( b1 `1 = - 1 & b1 `2 <= 3 & b1 `2 >= - 3 ) & not ( b1 `1 <= 1 & b1 `1 >= - 1 & b1 `2 = 3 ) & not ( b1 `1 <= 1 & b1 `1 >= - 1 & b1 `2 = - 3 ) & not ( b1 `1 = 1 & b1 `2 <= 3 & b1 `2 >= - 3 ) ) }
by SPPOL_2:58;
then Lemma124:
|[0,3]| in rectangle (- 1),1,(- 3),3
by Lemma86, Lemma87;
Lemma125:
|[0,(- 3)]| in rectangle (- 1),1,(- 3),3
by Lemma88, Lemma89, Lemma123;
Lemma126:
(2 + 1) ^2 = (4 + 4) + 1
;
then Lemma127:
sqrt 9 = 3
by SQUARE_1:def 4;
E128: dist |[(- 1),0]|,|[1,0]| =
sqrt ((((|[(- 1),0]| `1 ) - (|[1,0]| `1 )) ^2 ) + (((|[(- 1),0]| `2 ) - (|[1,0]| `2 )) ^2 ))
by TOPREAL6:101
.=
- (- 2)
by Lemma82, Lemma83, Lemma84, Lemma85, SQUARE_1:90
;
theorem Th70: :: JORDAN:70
theorem Th71: :: JORDAN:71
Lemma131:
rectangle (- 1),1,(- 3),3 c= closed_inside_of_rectangle (- 1),1,(- 3),3
by Th45;
( |[(- 1),3]| `2 = |[(- 1),3]| `2 & |[(- 1),3]| `1 <= |[0,3]| `1 & |[0,3]| `1 <= |[1,3]| `1 )
by Lemma90, Lemma94, EUCLID:56;
then
LSeg |[(- 1),3]|,|[0,3]| c= LSeg |[(- 1),3]|,|[1,3]|
by Lemma115, Lemma117, GOBOARD7:66;
then Lemma132:
LSeg |[(- 1),3]|,|[0,3]| c= rectangle (- 1),1,(- 3),3
by Lemma104, XBOOLE_1:1;
LSeg |[1,3]|,|[0,3]| c= LSeg |[(- 1),3]|,|[1,3]|
by Lemma116, Lemma117, Lemma86, Lemma87, Lemma90, Lemma91, Lemma94, GOBOARD7:66;
then Lemma133:
LSeg |[1,3]|,|[0,3]| c= rectangle (- 1),1,(- 3),3
by Lemma104, XBOOLE_1:1;
( |[(- 1),(- 3)]| `2 = |[(- 1),(- 3)]| `2 & |[(- 1),(- 3)]| `1 <= |[0,(- 3)]| `1 & |[0,(- 3)]| `1 <= |[1,(- 3)]| `1 )
by Lemma92, Lemma96, EUCLID:56;
then
LSeg |[(- 1),(- 3)]|,|[0,(- 3)]| c= LSeg |[(- 1),(- 3)]|,|[1,(- 3)]|
by Lemma113, Lemma118, GOBOARD7:66;
then Lemma134:
LSeg |[(- 1),(- 3)]|,|[0,(- 3)]| c= rectangle (- 1),1,(- 3),3
by Lemma106, XBOOLE_1:1;
LSeg |[1,(- 3)]|,|[0,(- 3)]| c= LSeg |[(- 1),(- 3)]|,|[1,(- 3)]|
by Lemma114, Lemma118, Lemma88, Lemma89, Lemma92, Lemma93, Lemma96, GOBOARD7:66;
then Lemma135:
LSeg |[1,(- 3)]|,|[0,(- 3)]| c= rectangle (- 1),1,(- 3),3
by Lemma106, XBOOLE_1:1;
Lemma136:
for b1 being Point of (TOP-REAL 2) holds
not ( 0 <= b1 `2 & b1 in rectangle (- 1),1,(- 3),3 & not b1 in LSeg |[(- 1),0]|,|[(- 1),3]| & not b1 in LSeg |[(- 1),3]|,|[0,3]| & not b1 in LSeg |[0,3]|,|[1,3]| & not b1 in LSeg |[1,3]|,|[1,0]| )
Lemma137:
for b1 being Point of (TOP-REAL 2) holds
not ( b1 `2 <= 0 & b1 in rectangle (- 1),1,(- 3),3 & not b1 in LSeg |[(- 1),0]|,|[(- 1),(- 3)]| & not b1 in LSeg |[(- 1),(- 3)]|,|[0,(- 3)]| & not b1 in LSeg |[0,(- 3)]|,|[1,(- 3)]| & not b1 in LSeg |[1,(- 3)]|,|[1,0]| )
theorem Th72: :: JORDAN:72
theorem Th73: :: JORDAN:73
theorem Th74: :: JORDAN:74
Lemma141:
for b1 being Simple_closed_curve holds
( |[(- 1),0]|,|[1,0]| realize-max-dist-in b1 implies LSeg |[(- 1),3]|,|[0,3]| misses b1 )
Lemma142:
for b1 being Simple_closed_curve holds
( |[(- 1),0]|,|[1,0]| realize-max-dist-in b1 implies LSeg |[1,3]|,|[0,3]| misses b1 )
Lemma143:
for b1 being Simple_closed_curve holds
( |[(- 1),0]|,|[1,0]| realize-max-dist-in b1 implies LSeg |[(- 1),(- 3)]|,|[0,(- 3)]| misses b1 )
Lemma144:
for b1 being Simple_closed_curve holds
( |[(- 1),0]|,|[1,0]| realize-max-dist-in b1 implies LSeg |[1,(- 3)]|,|[0,(- 3)]| misses b1 )
Lemma145:
for b1 being Point of (TOP-REAL 2)
for b2 being Simple_closed_curve holds
( |[(- 1),0]|,|[1,0]| realize-max-dist-in b2 & b1 in b2 ` & b1 in LSeg |[(- 1),0]|,|[(- 1),3]| implies LSeg b1,|[(- 1),3]| misses b2 )
Lemma146:
for b1 being Point of (TOP-REAL 2)
for b2 being Simple_closed_curve holds
( |[(- 1),0]|,|[1,0]| realize-max-dist-in b2 & b1 in b2 ` & b1 in LSeg |[1,0]|,|[1,3]| implies LSeg b1,|[1,3]| misses b2 )
Lemma147:
for b1 being Point of (TOP-REAL 2)
for b2 being Simple_closed_curve holds
( |[(- 1),0]|,|[1,0]| realize-max-dist-in b2 & b1 in b2 ` & b1 in LSeg |[(- 1),0]|,|[(- 1),(- 3)]| implies LSeg b1,|[(- 1),(- 3)]| misses b2 )
Lemma148:
for b1 being Point of (TOP-REAL 2)
for b2 being Simple_closed_curve holds
( |[(- 1),0]|,|[1,0]| realize-max-dist-in b2 & b1 in b2 ` & b1 in LSeg |[1,0]|,|[1,(- 3)]| implies LSeg b1,|[1,(- 3)]| misses b2 )
Lemma149:
for b1 being real number holds
not ( |[0,b1]| in rectangle (- 1),1,(- 3),3 & not b1 = - 3 & not b1 = 3 )
theorem Th75: :: JORDAN:75
theorem Th76: :: JORDAN:76
theorem Th77: :: JORDAN:77
theorem Th78: :: JORDAN:78
theorem Th79: :: JORDAN:79
theorem Th80: :: JORDAN:80
Lemma156:
for b1 being Subset of (TOP-REAL 2) holds
( |[(- 1),0]|,|[1,0]| realize-max-dist-in b1 implies |[0,3]| `1 = ((W-bound b1) + (E-bound b1)) / 2 )
Lemma157:
for b1 being Subset of (TOP-REAL 2) holds
( |[(- 1),0]|,|[1,0]| realize-max-dist-in b1 implies |[0,(- 3)]| `1 = ((W-bound b1) + (E-bound b1)) / 2 )
theorem Th81: :: JORDAN:81
theorem Th82: :: JORDAN:82
theorem Th83: :: JORDAN:83
theorem Th84: :: JORDAN:84
theorem Th85: :: JORDAN:85
theorem Th86: :: JORDAN:86
theorem Th87: :: JORDAN:87
theorem Th88: :: JORDAN:88
theorem Th89: :: JORDAN:89
theorem Th90: :: JORDAN:90
theorem Th91: :: JORDAN:91
theorem Th92: :: JORDAN:92
theorem Th93: :: JORDAN:93
Lemma170:
for b1 being Point of (TOP-REAL 2) holds
( b1 in closed_inside_of_rectangle (- 1),1,(- 3),3 implies closed_inside_of_rectangle (- 1),1,(- 3),3 c= Ball b1,10 )
theorem Th94: :: JORDAN:94
Lemma171:
for b1 being Simple_closed_curve holds
not ( |[(- 1),0]|,|[1,0]| realize-max-dist-in b1 & ( for b2, b3 being compact with_the_max_arc Subset of (TOP-REAL 2) holds
not ( b2 is_an_arc_of |[(- 1),0]|,|[1,0]| & b3 is_an_arc_of |[(- 1),0]|,|[1,0]| & b1 = b2 \/ b3 & b2 /\ b3 = {|[(- 1),0]|,|[1,0]|} & UMP b1 in b2 & LMP b1 in b3 & W-bound b1 = W-bound b2 & E-bound b1 = E-bound b2 ) ) )
theorem Th95: :: JORDAN:95
for b
1 being
Simple_closed_curve holds
(
|[(- 1),0]|,
|[1,0]| realize-max-dist-in b
1 implies for b
2, b
3 being
compact with_the_max_arc Subset of
(TOP-REAL 2) holds
( b
2 is_an_arc_of |[(- 1),0]|,
|[1,0]| & b
3 is_an_arc_of |[(- 1),0]|,
|[1,0]| & b
1 = b
2 \/ b
3 & b
2 /\ b
3 = {|[(- 1),0]|,|[1,0]|} &
UMP b
1 in b
2 &
LMP b
1 in b
3 &
W-bound b
1 = W-bound b
2 &
E-bound b
1 = E-bound b
2 implies for b
4 being
Subset of
(TOP-REAL 2) holds
( b
4 = skl (Down ((1 / 2) * ((UMP ((LSeg (LMP b2),|[0,(- 3)]|) /\ b3)) + (LMP b2))),(b1 ` )) implies ( b
4 is_inside_component_of b
1 & ( for b
5 being
Subset of
(TOP-REAL 2) holds
( b
5 is_inside_component_of b
1 implies b
5 = b
4 ) ) ) ) ) )
theorem Th96: :: JORDAN:96
for b
1 being
Simple_closed_curve holds
(
|[(- 1),0]|,
|[1,0]| realize-max-dist-in b
1 implies for b
2, b
3 being
compact with_the_max_arc Subset of
(TOP-REAL 2) holds
( b
2 is_an_arc_of |[(- 1),0]|,
|[1,0]| & b
3 is_an_arc_of |[(- 1),0]|,
|[1,0]| & b
1 = b
2 \/ b
3 & b
2 /\ b
3 = {|[(- 1),0]|,|[1,0]|} &
UMP b
1 in b
2 &
LMP b
1 in b
3 &
W-bound b
1 = W-bound b
2 &
E-bound b
1 = E-bound b
2 implies
BDD b
1 = skl (Down ((1 / 2) * ((UMP ((LSeg (LMP b2),|[0,(- 3)]|) /\ b3)) + (LMP b2))),(b1 ` )) ) )
Lemma174:
for b1 being Simple_closed_curve holds
( |[(- 1),0]|,|[1,0]| realize-max-dist-in b1 implies b1 is Jordan )
Lemma175:
for b1 being Simple_closed_curve holds b1 is Jordan
theorem Th97: :: JORDAN:97
theorem Th98: :: JORDAN:98
theorem Th99: :: JORDAN:99