:: JGRAPH_5 semantic presentation
Lemma1:
for b1, b2 being real number holds
not ( b1 >= 0 & (b2 - b1) * (b2 + b1) >= 0 & not - b1 >= b2 & not b2 >= b1 )
by XREAL_1:97;
theorem Th1: :: JGRAPH_5:1
canceled;
theorem Th2: :: JGRAPH_5:2
theorem Th3: :: JGRAPH_5:3
theorem Th4: :: JGRAPH_5:4
theorem Th5: :: JGRAPH_5:5
theorem Th6: :: JGRAPH_5:6
theorem Th7: :: JGRAPH_5:7
theorem Th8: :: JGRAPH_5:8
theorem Th9: :: JGRAPH_5:9
theorem Th10: :: JGRAPH_5:10
theorem Th11: :: JGRAPH_5:11
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Realfor b
9 being
Function of
(Closed-Interval-TSpace b1,b2),
(Closed-Interval-TSpace b3,b4) holds
( b
9 is_homeomorphism & b
9 . b
5 = b
7 & b
9 . b
6 = b
8 & b
9 . b
1 = b
3 & b
9 . b
2 = b
4 & b
3 <= b
4 & b
7 <= b
8 & b
5 in [.b1,b2.] & b
6 in [.b1,b2.] implies b
5 <= b
6 )
theorem Th12: :: JGRAPH_5:12
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Realfor b
9 being
Function of
(Closed-Interval-TSpace b1,b2),
(Closed-Interval-TSpace b3,b4) holds
( b
9 is_homeomorphism & b
9 . b
5 = b
7 & b
9 . b
6 = b
8 & b
9 . b
1 = b
4 & b
9 . b
2 = b
3 & b
4 >= b
3 & b
7 >= b
8 & b
5 in [.b1,b2.] & b
6 in [.b1,b2.] implies b
5 <= b
6 )
theorem Th13: :: JGRAPH_5:13
theorem Th14: :: JGRAPH_5:14
Lemma12:
( 0 in [.0,1.] & 1 in [.0,1.] )
by RCOMP_1:15;
theorem Th15: :: JGRAPH_5:15
theorem Th16: :: JGRAPH_5:16
theorem Th17: :: JGRAPH_5:17
theorem Th18: :: JGRAPH_5:18
theorem Th19: :: JGRAPH_5:19
theorem Th20: :: JGRAPH_5:20
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
( b
5 = { b6 where B is Point of (TOP-REAL 2) : |.b6.| >= 1 } &
|.b1.| = 1 &
|.b2.| = 1 &
|.b3.| = 1 &
|.b4.| = 1 & ex b
6 being
Function of
(TOP-REAL 2),
(TOP-REAL 2) st
( b
6 is_homeomorphism & b
6 .: b
5 c= b
5 & b
6 . b
1 = |[(- 1),0]| & b
6 . b
2 = |[0,1]| & b
6 . b
3 = |[1,0]| & b
6 . b
4 = |[0,(- 1)]| ) implies for b
6, b
7 being
Function of
I[01] ,
(TOP-REAL 2) holds
not ( b
6 is
continuous & b
6 is
one-to-one & b
7 is
continuous & b
7 is
one-to-one & b
6 . 0
= b
1 & b
6 . 1
= b
3 & b
7 . 0
= b
4 & b
7 . 1
= b
2 &
rng b
6 c= b
5 &
rng b
7 c= b
5 & not
rng b
6 meets rng b
7 ) )
theorem Th21: :: JGRAPH_5:21
theorem Th22: :: JGRAPH_5:22
theorem Th23: :: JGRAPH_5:23
theorem Th24: :: JGRAPH_5:24
theorem Th25: :: JGRAPH_5:25
theorem Th26: :: JGRAPH_5:26
theorem Th27: :: JGRAPH_5:27
theorem Th28: :: JGRAPH_5:28
theorem Th29: :: JGRAPH_5:29
theorem Th30: :: JGRAPH_5:30
E26:
now
let c
1 be non
empty compact Subset of
(TOP-REAL 2);
assume E27:
c
1 = { b1 where B is Point of (TOP-REAL 2) : |.b1.| = 1 }
;
E28:
proj1 .: c
1 c= [.(- 1),1.]
[.(- 1),1.] c= proj1 .: c
1
proof
let c
2 be
set ;
:: according to TARSKI:def 3
assume
c
2 in [.(- 1),1.]
;
then
c
2 in { b1 where B is Real : ( - 1 <= b1 & b1 <= 1 ) }
by RCOMP_1:def 1;
then consider c
3 being
Real such that E29:
( c
2 = c
3 &
- 1
<= c
3 & c
3 <= 1 )
;
set c
4 =
|[c3,(sqrt (1 - (c3 ^2 )))]|;
E30:
dom proj1 = the
carrier of
(TOP-REAL 2)
by FUNCT_2:def 1;
E31:
(
|[c3,(sqrt (1 - (c3 ^2 )))]| `1 = c
3 &
|[c3,(sqrt (1 - (c3 ^2 )))]| `2 = sqrt (1 - (c3 ^2 )) )
by EUCLID:56;
1
^2 >= c
3 ^2
by E29, JGRAPH_2:7;
then E32:
1
- (c3 ^2 ) >= 0
by XREAL_1:50;
|.|[c3,(sqrt (1 - (c3 ^2 )))]|.| =
sqrt ((c3 ^2 ) + ((sqrt (1 - (c3 ^2 ))) ^2 ))
by E31, JGRAPH_3:10
.=
sqrt ((c3 ^2 ) + (1 - (c3 ^2 )))
by E32, SQUARE_1:def 4
.=
1
by SQUARE_1:83
;
then E33:
|[c3,(sqrt (1 - (c3 ^2 )))]| in c
1
by E27;
proj1 . |[c3,(sqrt (1 - (c3 ^2 )))]| =
|[c3,(sqrt (1 - (c3 ^2 )))]| `1
by PSCOMP_1:def 28
.=
c
3
by EUCLID:56
;
hence
c
2 in proj1 .: c
1
by E29, E30, E33, FUNCT_1:def 12;
end;
hence
proj1 .: c
1 = [.(- 1),1.]
by E28, XBOOLE_0:def 10;
E29:
proj2 .: c
1 c= [.(- 1),1.]
[.(- 1),1.] c= proj2 .: c
1
proof
let c
2 be
set ;
:: according to TARSKI:def 3
assume
c
2 in [.(- 1),1.]
;
then
c
2 in { b1 where B is Real : ( - 1 <= b1 & b1 <= 1 ) }
by RCOMP_1:def 1;
then consider c
3 being
Real such that E30:
( c
2 = c
3 &
- 1
<= c
3 & c
3 <= 1 )
;
set c
4 =
|[(sqrt (1 - (c3 ^2 ))),c3]|;
E31:
dom proj2 = the
carrier of
(TOP-REAL 2)
by FUNCT_2:def 1;
E32:
(
|[(sqrt (1 - (c3 ^2 ))),c3]| `2 = c
3 &
|[(sqrt (1 - (c3 ^2 ))),c3]| `1 = sqrt (1 - (c3 ^2 )) )
by EUCLID:56;
1
^2 >= c
3 ^2
by E30, JGRAPH_2:7;
then E33:
1
- (c3 ^2 ) >= 0
by XREAL_1:50;
|.|[(sqrt (1 - (c3 ^2 ))),c3]|.| =
sqrt (((sqrt (1 - (c3 ^2 ))) ^2 ) + (c3 ^2 ))
by E32, JGRAPH_3:10
.=
sqrt ((1 - (c3 ^2 )) + (c3 ^2 ))
by E33, SQUARE_1:def 4
.=
1
by SQUARE_1:83
;
then E34:
|[(sqrt (1 - (c3 ^2 ))),c3]| in c
1
by E27;
proj2 . |[(sqrt (1 - (c3 ^2 ))),c3]| =
|[(sqrt (1 - (c3 ^2 ))),c3]| `2
by PSCOMP_1:def 29
.=
c
3
by EUCLID:56
;
hence
c
2 in proj2 .: c
1
by E30, E31, E34, FUNCT_1:def 12;
end;
hence
proj2 .: c
1 = [.(- 1),1.]
by E29, XBOOLE_0:def 10;
end;
Lemma27:
for b1 being non empty compact Subset of (TOP-REAL 2) holds
( b1 = { b2 where B is Point of (TOP-REAL 2) : |.b2.| = 1 } implies W-bound b1 = - 1 )
theorem Th31: :: JGRAPH_5:31
theorem Th32: :: JGRAPH_5:32
theorem Th33: :: JGRAPH_5:33
theorem Th34: :: JGRAPH_5:34
theorem Th35: :: JGRAPH_5:35
theorem Th36: :: JGRAPH_5:36
theorem Th37: :: JGRAPH_5:37
theorem Th38: :: JGRAPH_5:38
theorem Th39: :: JGRAPH_5:39
theorem Th40: :: JGRAPH_5:40
theorem Th41: :: JGRAPH_5:41
theorem Th42: :: JGRAPH_5:42
E39:
now
let c
1 be non
empty compact Subset of
(TOP-REAL 2);
assume E40:
c
1 = { b1 where B is Point of (TOP-REAL 2) : |.b1.| = 1 }
;
reconsider c
2 =
proj1 as
Function of
(TOP-REAL 2),
R^1 by TOPMETR:24;
set c
3 =
Lower_Arc c
1;
reconsider c
4 = c
2 | (Lower_Arc c1) as
Function of
((TOP-REAL 2) | (Lower_Arc c1)),
R^1 by PRE_TOPC:30;
E41:
for b
1 being
Point of
((TOP-REAL 2) | (Lower_Arc c1)) holds c
4 . b
1 = proj1 . b
1
then E42:
c
4 is
continuous
by JGRAPH_2:39;
E43:
dom c
4 = the
carrier of
((TOP-REAL 2) | (Lower_Arc c1))
by FUNCT_2:def 1;
then E44:
dom c
4 = Lower_Arc c
1
by PRE_TOPC:29;
E45:
Lower_Arc c
1 c= c
1
by E40, Th36;
E46:
rng c
4 c= the
carrier of
(Closed-Interval-TSpace (- 1),1)
then reconsider c
5 = c
4 as
Function of
((TOP-REAL 2) | (Lower_Arc c1)),
(Closed-Interval-TSpace (- 1),1) by E43, FUNCT_2:4;
dom c
5 = the
carrier of
((TOP-REAL 2) | (Lower_Arc c1))
by FUNCT_2:def 1;
then
dom c
5 = [#] ((TOP-REAL 2) | (Lower_Arc c1))
by PRE_TOPC:12;
then E47:
dom c
5 = Lower_Arc c
1
by PRE_TOPC:def 10;
E48:
rng c
4 c= [#] (Closed-Interval-TSpace (- 1),1)
by E46, PRE_TOPC:12;
E49:
[#] (Closed-Interval-TSpace (- 1),1) c= rng c
5
proof
let c
6 be
set ;
:: according to TARSKI:def 3
assume
c
6 in [#] (Closed-Interval-TSpace (- 1),1)
;
then
c
6 in the
carrier of
(Closed-Interval-TSpace (- 1),1)
;
then E50:
c
6 in [.(- 1),1.]
by TOPMETR:25;
then reconsider c
7 = c
6 as
Real ;
set c
8 =
|[c7,(- (sqrt (1 - (c7 ^2 ))))]|;
E51:
|.|[c7,(- (sqrt (1 - (c7 ^2 ))))]|.| =
sqrt (((|[c7,(- (sqrt (1 - (c7 ^2 ))))]| `1 ) ^2 ) + ((|[c7,(- (sqrt (1 - (c7 ^2 ))))]| `2 ) ^2 ))
by JGRAPH_3:10
.=
sqrt ((c7 ^2 ) + ((|[c7,(- (sqrt (1 - (c7 ^2 ))))]| `2 ) ^2 ))
by EUCLID:56
.=
sqrt ((c7 ^2 ) + ((- (sqrt (1 - (c7 ^2 )))) ^2 ))
by EUCLID:56
.=
sqrt ((c7 ^2 ) + ((sqrt (1 - (c7 ^2 ))) ^2 ))
;
(
- 1
<= c
7 & c
7 <= 1 )
by E50, RCOMP_1:48;
then
1
^2 >= c
7 ^2
by JGRAPH_2:7;
then E52:
1
- (c7 ^2 ) >= 0
by XREAL_1:50;
then
0
<= sqrt (1 - (c7 ^2 ))
by SQUARE_1:def 4;
then E53:
- (sqrt (1 - (c7 ^2 ))) <= 0
by XREAL_1:60;
|.|[c7,(- (sqrt (1 - (c7 ^2 ))))]|.| =
sqrt ((c7 ^2 ) + (1 - (c7 ^2 )))
by E51, E52, SQUARE_1:def 4
.=
1
by SQUARE_1:83
;
then E54:
|[c7,(- (sqrt (1 - (c7 ^2 ))))]| in c
1
by E40;
|[c7,(- (sqrt (1 - (c7 ^2 ))))]| `2 = - (sqrt (1 - (c7 ^2 )))
by EUCLID:56;
then
|[c7,(- (sqrt (1 - (c7 ^2 ))))]| in { b1 where B is Point of (TOP-REAL 2) : ( b1 in c1 & b1 `2 <= 0 ) }
by E53, E54;
then E55:
|[c7,(- (sqrt (1 - (c7 ^2 ))))]| in dom c
5
by E40, E47, Th38;
then c
5 . |[c7,(- (sqrt (1 - (c7 ^2 ))))]| =
proj1 . |[c7,(- (sqrt (1 - (c7 ^2 ))))]|
by E41
.=
|[c7,(- (sqrt (1 - (c7 ^2 ))))]| `1
by PSCOMP_1:def 28
.=
c
7
by EUCLID:56
;
hence
c
6 in rng c
5
by E55, FUNCT_1:def 5;
end;
reconsider c
6 =
[.(- 1),1.] as non
empty Subset of
R^1 by TOPMETR:24, RCOMP_1:48;
E50:
Closed-Interval-TSpace (- 1),1
= R^1 | c
6
by TOPMETR:26;
for b
1, b
2 being
set holds
( b
1 in dom c
5 & b
2 in dom c
5 & c
5 . b
1 = c
5 . b
2 implies b
1 = b
2 )
proof
let c
7, c
8 be
set ;
assume E51:
( c
7 in dom c
5 & c
8 in dom c
5 & c
5 . c
7 = c
5 . c
8 )
;
then reconsider c
9 = c
7 as
Point of
(TOP-REAL 2) by E47;
reconsider c
10 = c
8 as
Point of
(TOP-REAL 2) by E47, E51;
E52: c
5 . c
7 =
proj1 . c
9
by E41, E51
.=
c
9 `1
by PSCOMP_1:def 28
;
E53: c
5 . c
8 =
proj1 . c
10
by E41, E51
.=
c
10 `1
by PSCOMP_1:def 28
;
E54:
c
9 in c
1
by E45, E47, E51;
c
10 in c
1
by E45, E47, E51;
then consider c
11 being
Point of
(TOP-REAL 2) such that E55:
( c
10 = c
11 &
|.c11.| = 1 )
by E40;
1
^2 = ((c10 `1 ) ^2 ) + ((c10 `2 ) ^2 )
by E55, JGRAPH_3:10;
then E56:
(1 ^2 ) - ((c10 `1 ) ^2 ) = (c10 `2 ) ^2
;
consider c
12 being
Point of
(TOP-REAL 2) such that E57:
( c
9 = c
12 &
|.c12.| = 1 )
by E40, E54;
1
^2 = ((c9 `1 ) ^2 ) + ((c9 `2 ) ^2 )
by E57, JGRAPH_3:10;
then
(1 ^2 ) - ((c9 `1 ) ^2 ) = (c9 `2 ) ^2
;
then
(- (c9 `2 )) ^2 = (c10 `2 ) ^2
by E51, E52, E53, E56;
then E58:
(- (c9 `2 )) ^2 = (- (c10 `2 )) ^2
;
c
9 in { b1 where B is Point of (TOP-REAL 2) : ( b1 in c1 & b1 `2 <= 0 ) }
by E40, E47, E51, Th38;
then consider c
13 being
Point of
(TOP-REAL 2) such that E59:
( c
9 = c
13 & c
13 in c
1 & c
13 `2 <= 0 )
;
- (- (c9 `2 )) <= 0
by E59;
then
- (c9 `2 ) >= 0
by XREAL_1:60;
then E60:
- (c9 `2 ) = sqrt ((- (c10 `2 )) ^2 )
by E58, SQUARE_1:89;
c
10 in { b1 where B is Point of (TOP-REAL 2) : ( b1 in c1 & b1 `2 <= 0 ) }
by E40, E47, E51, Th38;
then consider c
14 being
Point of
(TOP-REAL 2) such that E61:
( c
10 = c
14 & c
14 in c
1 & c
14 `2 <= 0 )
;
- (- (c10 `2 )) <= 0
by E61;
then
- (c10 `2 ) >= 0
by XREAL_1:60;
then
- (c9 `2 ) = - (c10 `2 )
by E60, SQUARE_1:89;
then
- (- (c9 `2 )) = c
10 `2
;
then c
9 =
|[(c10 `1 ),(c10 `2 )]|
by E51, E52, E53, EUCLID:57
.=
c
10
by EUCLID:57
;
hence
c
7 = c
8
;
end;
hence
(
proj1 | (Lower_Arc c1) is
continuous Function of
((TOP-REAL 2) | (Lower_Arc c1)),
(Closed-Interval-TSpace (- 1),1) &
proj1 | (Lower_Arc c1) is
one-to-one &
rng (proj1 | (Lower_Arc c1)) = [#] (Closed-Interval-TSpace (- 1),1) )
by E42, E48, E49, E50, FUNCT_1:def 8, JGRAPH_1:63, XBOOLE_0:def 10;
end;
E40:
now
let c
1 be non
empty compact Subset of
(TOP-REAL 2);
assume E41:
c
1 = { b1 where B is Point of (TOP-REAL 2) : |.b1.| = 1 }
;
reconsider c
2 =
proj1 as
Function of
(TOP-REAL 2),
R^1 by TOPMETR:24;
set c
3 =
Upper_Arc c
1;
reconsider c
4 = c
2 | (Upper_Arc c1) as
Function of
((TOP-REAL 2) | (Upper_Arc c1)),
R^1 by PRE_TOPC:30;
E42:
for b
1 being
Point of
((TOP-REAL 2) | (Upper_Arc c1)) holds c
4 . b
1 = proj1 . b
1
then E43:
c
4 is
continuous
by JGRAPH_2:39;
E44:
dom c
4 = the
carrier of
((TOP-REAL 2) | (Upper_Arc c1))
by FUNCT_2:def 1;
then E45:
dom c
4 = Upper_Arc c
1
by PRE_TOPC:29;
E46:
Upper_Arc c
1 c= c
1
by E41, Th36;
E47:
rng c
4 c= the
carrier of
(Closed-Interval-TSpace (- 1),1)
then reconsider c
5 = c
4 as
Function of
((TOP-REAL 2) | (Upper_Arc c1)),
(Closed-Interval-TSpace (- 1),1) by E44, FUNCT_2:4;
dom c
5 = the
carrier of
((TOP-REAL 2) | (Upper_Arc c1))
by FUNCT_2:def 1;
then
dom c
5 = [#] ((TOP-REAL 2) | (Upper_Arc c1))
by PRE_TOPC:12;
then E48:
dom c
5 = Upper_Arc c
1
by PRE_TOPC:def 10;
E49:
rng c
4 c= [#] (Closed-Interval-TSpace (- 1),1)
by E47, PRE_TOPC:12;
E50:
[#] (Closed-Interval-TSpace (- 1),1) c= rng c
5
proof
let c
6 be
set ;
:: according to TARSKI:def 3
assume
c
6 in [#] (Closed-Interval-TSpace (- 1),1)
;
then
c
6 in the
carrier of
(Closed-Interval-TSpace (- 1),1)
;
then E51:
c
6 in [.(- 1),1.]
by TOPMETR:25;
then reconsider c
7 = c
6 as
Real ;
set c
8 =
|[c7,(sqrt (1 - (c7 ^2 )))]|;
E52:
|.|[c7,(sqrt (1 - (c7 ^2 )))]|.| =
sqrt (((|[c7,(sqrt (1 - (c7 ^2 )))]| `1 ) ^2 ) + ((|[c7,(sqrt (1 - (c7 ^2 )))]| `2 ) ^2 ))
by JGRAPH_3:10
.=
sqrt ((c7 ^2 ) + ((|[c7,(sqrt (1 - (c7 ^2 )))]| `2 ) ^2 ))
by EUCLID:56
.=
sqrt ((c7 ^2 ) + ((sqrt (1 - (c7 ^2 ))) ^2 ))
by EUCLID:56
;
(
- 1
<= c
7 & c
7 <= 1 )
by E51, RCOMP_1:48;
then
1
^2 >= c
7 ^2
by JGRAPH_2:7;
then E53:
1
- (c7 ^2 ) >= 0
by XREAL_1:50;
then E54:
0
<= sqrt (1 - (c7 ^2 ))
by SQUARE_1:def 4;
|.|[c7,(sqrt (1 - (c7 ^2 )))]|.| =
sqrt ((c7 ^2 ) + (1 - (c7 ^2 )))
by E52, E53, SQUARE_1:def 4
.=
1
by SQUARE_1:83
;
then E55:
|[c7,(sqrt (1 - (c7 ^2 )))]| in c
1
by E41;
|[c7,(sqrt (1 - (c7 ^2 )))]| `2 = sqrt (1 - (c7 ^2 ))
by EUCLID:56;
then
|[c7,(sqrt (1 - (c7 ^2 )))]| in { b1 where B is Point of (TOP-REAL 2) : ( b1 in c1 & b1 `2 >= 0 ) }
by E54, E55;
then E56:
|[c7,(sqrt (1 - (c7 ^2 )))]| in dom c
5
by E41, E48, Th37;
then c
5 . |[c7,(sqrt (1 - (c7 ^2 )))]| =
proj1 . |[c7,(sqrt (1 - (c7 ^2 )))]|
by E42
.=
|[c7,(sqrt (1 - (c7 ^2 )))]| `1
by PSCOMP_1:def 28
.=
c
7
by EUCLID:56
;
hence
c
6 in rng c
5
by E56, FUNCT_1:def 5;
end;
reconsider c
6 =
[.(- 1),1.] as non
empty Subset of
R^1 by TOPMETR:24, RCOMP_1:48;
E51:
Closed-Interval-TSpace (- 1),1
= R^1 | c
6
by TOPMETR:26;
for b
1, b
2 being
set holds
( b
1 in dom c
5 & b
2 in dom c
5 & c
5 . b
1 = c
5 . b
2 implies b
1 = b
2 )
proof
let c
7, c
8 be
set ;
assume E52:
( c
7 in dom c
5 & c
8 in dom c
5 & c
5 . c
7 = c
5 . c
8 )
;
then reconsider c
9 = c
7 as
Point of
(TOP-REAL 2) by E48;
reconsider c
10 = c
8 as
Point of
(TOP-REAL 2) by E48, E52;
E53: c
5 . c
7 =
proj1 . c
9
by E42, E52
.=
c
9 `1
by PSCOMP_1:def 28
;
E54: c
5 . c
8 =
proj1 . c
10
by E42, E52
.=
c
10 `1
by PSCOMP_1:def 28
;
E55:
c
9 in c
1
by E46, E48, E52;
c
10 in c
1
by E46, E48, E52;
then consider c
11 being
Point of
(TOP-REAL 2) such that E56:
( c
10 = c
11 &
|.c11.| = 1 )
by E41;
1
^2 = ((c10 `1 ) ^2 ) + ((c10 `2 ) ^2 )
by E56, JGRAPH_3:10;
then E57:
(1 ^2 ) - ((c10 `1 ) ^2 ) = (c10 `2 ) ^2
;
consider c
12 being
Point of
(TOP-REAL 2) such that E58:
( c
9 = c
12 &
|.c12.| = 1 )
by E41, E55;
1
^2 = ((c9 `1 ) ^2 ) + ((c9 `2 ) ^2 )
by E58, JGRAPH_3:10;
then E59:
(c9 `2 ) ^2 = (c10 `2 ) ^2
by E52, E53, E54, E57;
c
9 in { b1 where B is Point of (TOP-REAL 2) : ( b1 in c1 & b1 `2 >= 0 ) }
by E41, E48, E52, Th37;
then consider c
13 being
Point of
(TOP-REAL 2) such that E60:
( c
9 = c
13 & c
13 in c
1 & c
13 `2 >= 0 )
;
E61:
c
9 `2 = sqrt ((c10 `2 ) ^2 )
by E59, E60, SQUARE_1:89;
c
10 in { b1 where B is Point of (TOP-REAL 2) : ( b1 in c1 & b1 `2 >= 0 ) }
by E41, E48, E52, Th37;
then consider c
14 being
Point of
(TOP-REAL 2) such that E62:
( c
10 = c
14 & c
14 in c
1 & c
14 `2 >= 0 )
;
c
9 `2 = c
10 `2
by E61, E62, SQUARE_1:89;
then c
9 =
|[(c10 `1 ),(c10 `2 )]|
by E52, E53, E54, EUCLID:57
.=
c
10
by EUCLID:57
;
hence
c
7 = c
8
;
end;
hence
(
proj1 | (Upper_Arc c1) is
continuous Function of
((TOP-REAL 2) | (Upper_Arc c1)),
(Closed-Interval-TSpace (- 1),1) &
proj1 | (Upper_Arc c1) is
one-to-one &
rng (proj1 | (Upper_Arc c1)) = [#] (Closed-Interval-TSpace (- 1),1) )
by E43, E49, E50, E51, FUNCT_1:def 8, JGRAPH_1:63, XBOOLE_0:def 10;
end;
theorem Th43: :: JGRAPH_5:43
theorem Th44: :: JGRAPH_5:44
theorem Th45: :: JGRAPH_5:45
theorem Th46: :: JGRAPH_5:46
theorem Th47: :: JGRAPH_5:47
theorem Th48: :: JGRAPH_5:48
theorem Th49: :: JGRAPH_5:49
theorem Th50: :: JGRAPH_5:50
theorem Th51: :: JGRAPH_5:51
theorem Th52: :: JGRAPH_5:52
theorem Th53: :: JGRAPH_5:53
theorem Th54: :: JGRAPH_5:54
theorem Th55: :: JGRAPH_5:55
theorem Th56: :: JGRAPH_5:56
theorem Th57: :: JGRAPH_5:57
theorem Th58: :: JGRAPH_5:58
theorem Th59: :: JGRAPH_5:59
theorem Th60: :: JGRAPH_5:60
theorem Th61: :: JGRAPH_5:61
theorem Th62: :: JGRAPH_5:62
for b
1, b
2, b
3, b
4 being
Point of
(TOP-REAL 2)for b
5 being non
empty compact Subset of
(TOP-REAL 2) holds
not ( b
5 = { b6 where B is Point of (TOP-REAL 2) : |.b6.| = 1 } &
LE b
1,b
2,b
5 &
LE b
2,b
3,b
5 &
LE b
3,b
4,b
5 & b
1 `1 < 0 & b
1 `2 >= 0 & b
2 `1 < 0 & b
2 `2 >= 0 & b
3 `1 < 0 & b
3 `2 >= 0 & b
4 `1 < 0 & b
4 `2 >= 0 & ( for b
6 being
Function of
(TOP-REAL 2),
(TOP-REAL 2)for b
7, b
8, b
9, b
10 being
Point of
(TOP-REAL 2) holds
not ( b
6 is_homeomorphism & ( for b
11 being
Point of
(TOP-REAL 2) holds
|.(b6 . b11).| = |.b11.| ) & b
7 = b
6 . b
1 & b
8 = b
6 . b
2 & b
9 = b
6 . b
3 & b
10 = b
6 . b
4 & b
7 `1 < 0 & b
7 `2 < 0 & b
8 `1 < 0 & b
8 `2 < 0 & b
9 `1 < 0 & b
9 `2 < 0 & b
10 `1 < 0 & b
10 `2 < 0 &
LE b
7,b
8,b
5 &
LE b
8,b
9,b
5 &
LE b
9,b
10,b
5 ) ) )
theorem Th63: :: JGRAPH_5:63
for b
1, b
2, b
3, b
4 being
Point of
(TOP-REAL 2)for b
5 being non
empty compact Subset of
(TOP-REAL 2) holds
not ( b
5 = { b6 where B is Point of (TOP-REAL 2) : |.b6.| = 1 } &
LE b
1,b
2,b
5 &
LE b
2,b
3,b
5 &
LE b
3,b
4,b
5 & b
1 `2 >= 0 & b
2 `2 >= 0 & b
3 `2 >= 0 & b
4 `2 > 0 & ( for b
6 being
Function of
(TOP-REAL 2),
(TOP-REAL 2)for b
7, b
8, b
9, b
10 being
Point of
(TOP-REAL 2) holds
not ( b
6 is_homeomorphism & ( for b
11 being
Point of
(TOP-REAL 2) holds
|.(b6 . b11).| = |.b11.| ) & b
7 = b
6 . b
1 & b
8 = b
6 . b
2 & b
9 = b
6 . b
3 & b
10 = b
6 . b
4 & b
7 `1 < 0 & b
7 `2 >= 0 & b
8 `1 < 0 & b
8 `2 >= 0 & b
9 `1 < 0 & b
9 `2 >= 0 & b
10 `1 < 0 & b
10 `2 >= 0 &
LE b
7,b
8,b
5 &
LE b
8,b
9,b
5 &
LE b
9,b
10,b
5 ) ) )
theorem Th64: :: JGRAPH_5:64
for b
1, b
2, b
3, b
4 being
Point of
(TOP-REAL 2)for b
5 being non
empty compact Subset of
(TOP-REAL 2) holds
not ( b
5 = { b6 where B is Point of (TOP-REAL 2) : |.b6.| = 1 } &
LE b
1,b
2,b
5 &
LE b
2,b
3,b
5 &
LE b
3,b
4,b
5 & b
1 `2 >= 0 & b
2 `2 >= 0 & b
3 `2 >= 0 & b
4 `2 > 0 & ( for b
6 being
Function of
(TOP-REAL 2),
(TOP-REAL 2)for b
7, b
8, b
9, b
10 being
Point of
(TOP-REAL 2) holds
not ( b
6 is_homeomorphism & ( for b
11 being
Point of
(TOP-REAL 2) holds
|.(b6 . b11).| = |.b11.| ) & b
7 = b
6 . b
1 & b
8 = b
6 . b
2 & b
9 = b
6 . b
3 & b
10 = b
6 . b
4 & b
7 `1 < 0 & b
7 `2 < 0 & b
8 `1 < 0 & b
8 `2 < 0 & b
9 `1 < 0 & b
9 `2 < 0 & b
10 `1 < 0 & b
10 `2 < 0 &
LE b
7,b
8,b
5 &
LE b
8,b
9,b
5 &
LE b
9,b
10,b
5 ) ) )
theorem Th65: :: JGRAPH_5:65
for b
1, b
2, b
3, b
4 being
Point of
(TOP-REAL 2)for b
5 being non
empty compact Subset of
(TOP-REAL 2) holds
not ( b
5 = { b6 where B is Point of (TOP-REAL 2) : |.b6.| = 1 } &
LE b
1,b
2,b
5 &
LE b
2,b
3,b
5 &
LE b
3,b
4,b
5 & ( b
1 `2 >= 0 or b
1 `1 >= 0 ) & ( b
2 `2 >= 0 or b
2 `1 >= 0 ) & ( b
3 `2 >= 0 or b
3 `1 >= 0 ) & not ( not b
4 `2 > 0 & not b
4 `1 > 0 ) & ( for b
6 being
Function of
(TOP-REAL 2),
(TOP-REAL 2)for b
7, b
8, b
9, b
10 being
Point of
(TOP-REAL 2) holds
not ( b
6 is_homeomorphism & ( for b
11 being
Point of
(TOP-REAL 2) holds
|.(b6 . b11).| = |.b11.| ) & b
7 = b
6 . b
1 & b
8 = b
6 . b
2 & b
9 = b
6 . b
3 & b
10 = b
6 . b
4 & b
7 `2 >= 0 & b
8 `2 >= 0 & b
9 `2 >= 0 & b
10 `2 > 0 &
LE b
7,b
8,b
5 &
LE b
8,b
9,b
5 &
LE b
9,b
10,b
5 ) ) )
theorem Th66: :: JGRAPH_5:66
for b
1, b
2, b
3, b
4 being
Point of
(TOP-REAL 2)for b
5 being non
empty compact Subset of
(TOP-REAL 2) holds
not ( b
5 = { b6 where B is Point of (TOP-REAL 2) : |.b6.| = 1 } &
LE b
1,b
2,b
5 &
LE b
2,b
3,b
5 &
LE b
3,b
4,b
5 & ( b
1 `2 >= 0 or b
1 `1 >= 0 ) & ( b
2 `2 >= 0 or b
2 `1 >= 0 ) & ( b
3 `2 >= 0 or b
3 `1 >= 0 ) & not ( not b
4 `2 > 0 & not b
4 `1 > 0 ) & ( for b
6 being
Function of
(TOP-REAL 2),
(TOP-REAL 2)for b
7, b
8, b
9, b
10 being
Point of
(TOP-REAL 2) holds
not ( b
6 is_homeomorphism & ( for b
11 being
Point of
(TOP-REAL 2) holds
|.(b6 . b11).| = |.b11.| ) & b
7 = b
6 . b
1 & b
8 = b
6 . b
2 & b
9 = b
6 . b
3 & b
10 = b
6 . b
4 & b
7 `1 < 0 & b
7 `2 < 0 & b
8 `1 < 0 & b
8 `2 < 0 & b
9 `1 < 0 & b
9 `2 < 0 & b
10 `1 < 0 & b
10 `2 < 0 &
LE b
7,b
8,b
5 &
LE b
8,b
9,b
5 &
LE b
9,b
10,b
5 ) ) )
theorem Th67: :: JGRAPH_5:67
for b
1, b
2, b
3, b
4 being
Point of
(TOP-REAL 2)for b
5 being non
empty compact Subset of
(TOP-REAL 2) holds
not ( b
5 = { b6 where B is Point of (TOP-REAL 2) : |.b6.| = 1 } & b
4 = W-min b
5 &
LE b
1,b
2,b
5 &
LE b
2,b
3,b
5 &
LE b
3,b
4,b
5 & ( for b
6 being
Function of
(TOP-REAL 2),
(TOP-REAL 2)for b
7, b
8, b
9, b
10 being
Point of
(TOP-REAL 2) holds
not ( b
6 is_homeomorphism & ( for b
11 being
Point of
(TOP-REAL 2) holds
|.(b6 . b11).| = |.b11.| ) & b
7 = b
6 . b
1 & b
8 = b
6 . b
2 & b
9 = b
6 . b
3 & b
10 = b
6 . b
4 & b
7 `1 < 0 & b
7 `2 < 0 & b
8 `1 < 0 & b
8 `2 < 0 & b
9 `1 < 0 & b
9 `2 < 0 & b
10 `1 < 0 & b
10 `2 < 0 &
LE b
7,b
8,b
5 &
LE b
8,b
9,b
5 &
LE b
9,b
10,b
5 ) ) )
theorem Th68: :: JGRAPH_5:68
for b
1, b
2, b
3, b
4 being
Point of
(TOP-REAL 2)for b
5 being non
empty compact Subset of
(TOP-REAL 2) holds
not ( b
5 = { b6 where B is Point of (TOP-REAL 2) : |.b6.| = 1 } &
LE b
1,b
2,b
5 &
LE b
2,b
3,b
5 &
LE b
3,b
4,b
5 & ( for b
6 being
Function of
(TOP-REAL 2),
(TOP-REAL 2)for b
7, b
8, b
9, b
10 being
Point of
(TOP-REAL 2) holds
not ( b
6 is_homeomorphism & ( for b
11 being
Point of
(TOP-REAL 2) holds
|.(b6 . b11).| = |.b11.| ) & b
7 = b
6 . b
1 & b
8 = b
6 . b
2 & b
9 = b
6 . b
3 & b
10 = b
6 . b
4 & b
7 `1 < 0 & b
7 `2 < 0 & b
8 `1 < 0 & b
8 `2 < 0 & b
9 `1 < 0 & b
9 `2 < 0 & b
10 `1 < 0 & b
10 `2 < 0 &
LE b
7,b
8,b
5 &
LE b
8,b
9,b
5 &
LE b
9,b
10,b
5 ) ) )
theorem Th69: :: JGRAPH_5:69
for b
1, b
2, b
3, b
4 being
Point of
(TOP-REAL 2)for b
5 being non
empty compact Subset of
(TOP-REAL 2) holds
not ( b
5 = { b6 where B is Point of (TOP-REAL 2) : |.b6.| = 1 } &
LE b
1,b
2,b
5 &
LE b
2,b
3,b
5 &
LE b
3,b
4,b
5 & b
1 <> b
2 & b
2 <> b
3 & b
3 <> b
4 & b
1 `1 < 0 & b
2 `1 < 0 & b
3 `1 < 0 & b
4 `1 < 0 & b
1 `2 < 0 & b
2 `2 < 0 & b
3 `2 < 0 & b
4 `2 < 0 & ( for b
6 being
Function of
(TOP-REAL 2),
(TOP-REAL 2) holds
not ( b
6 is_homeomorphism & ( for b
7 being
Point of
(TOP-REAL 2) holds
|.(b6 . b7).| = |.b7.| ) &
|[(- 1),0]| = b
6 . b
1 &
|[0,1]| = b
6 . b
2 &
|[1,0]| = b
6 . b
3 &
|[0,(- 1)]| = b
6 . b
4 ) ) )
theorem Th70: :: JGRAPH_5:70
for b
1, b
2, b
3, b
4 being
Point of
(TOP-REAL 2)for b
5 being non
empty compact Subset of
(TOP-REAL 2) holds
not ( b
5 = { b6 where B is Point of (TOP-REAL 2) : |.b6.| = 1 } &
LE b
1,b
2,b
5 &
LE b
2,b
3,b
5 &
LE b
3,b
4,b
5 & b
1 <> b
2 & b
2 <> b
3 & b
3 <> b
4 & ( for b
6 being
Function of
(TOP-REAL 2),
(TOP-REAL 2) holds
not ( b
6 is_homeomorphism & ( for b
7 being
Point of
(TOP-REAL 2) holds
|.(b6 . b7).| = |.b7.| ) &
|[(- 1),0]| = b
6 . b
1 &
|[0,1]| = b
6 . b
2 &
|[1,0]| = b
6 . b
3 &
|[0,(- 1)]| = b
6 . b
4 ) ) )
Lemma67:
( |[(- 1),0]| `1 = - 1 & |[(- 1),0]| `2 = 0 & |[1,0]| `1 = 1 & |[1,0]| `2 = 0 & |[0,(- 1)]| `1 = 0 & |[0,(- 1)]| `2 = - 1 & |[0,1]| `1 = 0 & |[0,1]| `2 = 1 )
by EUCLID:56;
E68:
now
thus |.|[(- 1),0]|.| =
sqrt (((- 1) ^2 ) + (0 ^2 ))
by Lemma67, JGRAPH_3:10
.=
1
by SQUARE_1:83
;
thus |.|[1,0]|.| =
sqrt (1 + 0)
by Lemma67, JGRAPH_3:10, SQUARE_1:59, SQUARE_1:60
.=
1
by SQUARE_1:83
;
thus |.|[0,(- 1)]|.| =
sqrt ((0 ^2 ) + ((- 1) ^2 ))
by Lemma67, JGRAPH_3:10
.=
1
by SQUARE_1:83
;
thus |.|[0,1]|.| =
sqrt ((0 ^2 ) + (1 ^2 ))
by Lemma67, JGRAPH_3:10
.=
1
by SQUARE_1:83
;
end;
Lemma69:
0 in [.0,1.]
by RCOMP_1:48;
Lemma70:
1 in [.0,1.]
by RCOMP_1:48;
theorem Th71: :: JGRAPH_5:71
for b
1, b
2, b
3, b
4 being
Point of
(TOP-REAL 2)for b
5 being non
empty compact Subset of
(TOP-REAL 2)for b
6 being
Subset of
(TOP-REAL 2) holds
( b
5 = { b7 where B is Point of (TOP-REAL 2) : |.b7.| = 1 } &
LE b
1,b
2,b
5 &
LE b
2,b
3,b
5 &
LE b
3,b
4,b
5 implies for b
7, b
8 being
Function of
I[01] ,
(TOP-REAL 2) holds
not ( b
7 is
continuous & b
7 is
one-to-one & b
8 is
continuous & b
8 is
one-to-one & b
6 = { b9 where B is Point of (TOP-REAL 2) : |.b9.| <= 1 } & b
7 . 0
= b
1 & b
7 . 1
= b
3 & b
8 . 0
= b
2 & b
8 . 1
= b
4 &
rng b
7 c= b
6 &
rng b
8 c= b
6 & not
rng b
7 meets rng b
8 ) )
theorem Th72: :: JGRAPH_5:72
for b
1, b
2, b
3, b
4 being
Point of
(TOP-REAL 2)for b
5 being non
empty compact Subset of
(TOP-REAL 2)for b
6 being
Subset of
(TOP-REAL 2) holds
( b
5 = { b7 where B is Point of (TOP-REAL 2) : |.b7.| = 1 } &
LE b
1,b
2,b
5 &
LE b
2,b
3,b
5 &
LE b
3,b
4,b
5 implies for b
7, b
8 being
Function of
I[01] ,
(TOP-REAL 2) holds
not ( b
7 is
continuous & b
7 is
one-to-one & b
8 is
continuous & b
8 is
one-to-one & b
6 = { b9 where B is Point of (TOP-REAL 2) : |.b9.| <= 1 } & b
7 . 0
= b
1 & b
7 . 1
= b
3 & b
8 . 0
= b
4 & b
8 . 1
= b
2 &
rng b
7 c= b
6 &
rng b
8 c= b
6 & not
rng b
7 meets rng b
8 ) )
theorem Th73: :: JGRAPH_5:73
for b
1, b
2, b
3, b
4 being
Point of
(TOP-REAL 2)for b
5 being non
empty compact Subset of
(TOP-REAL 2)for b
6 being
Subset of
(TOP-REAL 2) holds
( b
5 = { b7 where B is Point of (TOP-REAL 2) : |.b7.| = 1 } &
LE b
1,b
2,b
5 &
LE b
2,b
3,b
5 &
LE b
3,b
4,b
5 implies for b
7, b
8 being
Function of
I[01] ,
(TOP-REAL 2) holds
not ( b
7 is
continuous & b
7 is
one-to-one & b
8 is
continuous & b
8 is
one-to-one & b
6 = { b9 where B is Point of (TOP-REAL 2) : |.b9.| >= 1 } & b
7 . 0
= b
1 & b
7 . 1
= b
3 & b
8 . 0
= b
4 & b
8 . 1
= b
2 &
rng b
7 c= b
6 &
rng b
8 c= b
6 & not
rng b
7 meets rng b
8 ) )
theorem Th74: :: JGRAPH_5:74
for b
1, b
2, b
3, b
4 being
Point of
(TOP-REAL 2)for b
5 being non
empty compact Subset of
(TOP-REAL 2)for b
6 being
Subset of
(TOP-REAL 2) holds
( b
5 = { b7 where B is Point of (TOP-REAL 2) : |.b7.| = 1 } &
LE b
1,b
2,b
5 &
LE b
2,b
3,b
5 &
LE b
3,b
4,b
5 implies for b
7, b
8 being
Function of
I[01] ,
(TOP-REAL 2) holds
not ( b
7 is
continuous & b
7 is
one-to-one & b
8 is
continuous & b
8 is
one-to-one & b
6 = { b9 where B is Point of (TOP-REAL 2) : |.b9.| >= 1 } & b
7 . 0
= b
1 & b
7 . 1
= b
3 & b
8 . 0
= b
2 & b
8 . 1
= b
4 &
rng b
7 c= b
6 &
rng b
8 c= b
6 & not
rng b
7 meets rng b
8 ) )