:: Some Properties of Rectangles on the Plane
:: by Artur Korni{\l}owicz and Yasunari Shidama
::
:: Received October 18, 2004
:: Copyright (c) 2004 Association of Mizar Users
set R = the carrier of R^1 ;
Lm1:
the carrier of [:R^1 ,R^1 :] = [:the carrier of R^1 ,the carrier of R^1 :]
by BORSUK_1:def 5;
theorem :: TOPREALA:1
theorem Th2: :: TOPREALA:2
theorem Th3: :: TOPREALA:3
theorem :: TOPREALA:4
theorem Th5: :: TOPREALA:5
theorem Th6: :: TOPREALA:6
theorem :: TOPREALA:7
registration
let r be
real number ;
let s be
real positive number ;
cluster K155(
r,
(r + s))
-> non
empty ;
coherence
not ].r,(r + s).[ is empty
cluster K153(
r,
(r + s))
-> non
empty ;
coherence
not [.r,(r + s).[ is empty
cluster K154(
r,
(r + s))
-> non
empty ;
coherence
not ].r,(r + s).] is empty
cluster K152(
r,
(r + s))
-> non
empty ;
coherence
not [.r,(r + s).] is empty
cluster K155(
(r - s),
r)
-> non
empty ;
coherence
not ].(r - s),r.[ is empty
cluster K153(
(r - s),
r)
-> non
empty ;
coherence
not [.(r - s),r.[ is empty
cluster K154(
(r - s),
r)
-> non
empty ;
coherence
not ].(r - s),r.] is empty
cluster K152(
(r - s),
r)
-> non
empty ;
coherence
not [.(r - s),r.] is empty
end;
theorem :: TOPREALA:8
canceled;
theorem :: TOPREALA:9
canceled;
theorem :: TOPREALA:10
canceled;
theorem :: TOPREALA:11
canceled;
theorem :: TOPREALA:12
canceled;
theorem :: TOPREALA:13
canceled;
theorem :: TOPREALA:14
canceled;
theorem :: TOPREALA:15
canceled;
theorem :: TOPREALA:16
canceled;
theorem :: TOPREALA:17
canceled;
theorem :: TOPREALA:18
canceled;
theorem :: TOPREALA:19
canceled;
theorem :: TOPREALA:20
canceled;
theorem :: TOPREALA:21
canceled;
theorem :: TOPREALA:22
theorem :: TOPREALA:23
theorem Th24: :: TOPREALA:24
theorem Th25: :: TOPREALA:25
theorem Th26: :: TOPREALA:26
theorem Th27: :: TOPREALA:27
theorem :: TOPREALA:28
theorem :: TOPREALA:29
theorem :: TOPREALA:30
theorem :: TOPREALA:31
theorem :: TOPREALA:32
theorem :: TOPREALA:33
theorem Th34: :: TOPREALA:34
theorem :: TOPREALA:35
theorem :: TOPREALA:36
theorem :: TOPREALA:37
theorem :: TOPREALA:38
theorem :: TOPREALA:39
theorem Th40: :: TOPREALA:40
theorem Th41: :: TOPREALA:41
theorem Th42: :: TOPREALA:42
theorem :: TOPREALA:43
theorem :: TOPREALA:44
theorem :: TOPREALA:45
theorem :: TOPREALA:46
theorem :: TOPREALA:47
theorem Th48: :: TOPREALA:48
theorem :: TOPREALA:49
theorem :: TOPREALA:50
theorem Th51: :: TOPREALA:51
for
a,
b,
r,
s being
real number holds
closed_inside_of_rectangle a,
b,
r,
s = product (1,2 --> [.a,b.],[.r,s.])
theorem Th52: :: TOPREALA:52
definition
let a,
b,
c,
d be
real number ;
func Trectangle a,
b,
c,
d -> SubSpace of
TOP-REAL 2
equals :: TOPREALA:def 1
(TOP-REAL 2) | (closed_inside_of_rectangle a,b,c,d);
coherence
(TOP-REAL 2) | (closed_inside_of_rectangle a,b,c,d) is SubSpace of TOP-REAL 2
;
end;
:: deftheorem defines Trectangle TOPREALA:def 1 :
theorem :: TOPREALA:53
canceled;
theorem Th54: :: TOPREALA:54
definition
func R2Homeomorphism -> Function of
[:R^1 ,R^1 :],
(TOP-REAL 2) means :
Def2:
:: TOPREALA:def 2
for
x,
y being
real number holds
it . [x,y] = <*x,y*>;
existence
ex b1 being Function of [:R^1 ,R^1 :],(TOP-REAL 2) st
for x, y being real number holds b1 . [x,y] = <*x,y*>
uniqueness
for b1, b2 being Function of [:R^1 ,R^1 :],(TOP-REAL 2) st ( for x, y being real number holds b1 . [x,y] = <*x,y*> ) & ( for x, y being real number holds b2 . [x,y] = <*x,y*> ) holds
b1 = b2
end;
:: deftheorem Def2 defines R2Homeomorphism TOPREALA:def 2 :
theorem Th55: :: TOPREALA:55
theorem Th56: :: TOPREALA:56
theorem Th57: :: TOPREALA:57
for
a,
b,
r,
s being
real number st
a <= b &
r <= s holds
R2Homeomorphism | the
carrier of
[:(Closed-Interval-TSpace a,b),(Closed-Interval-TSpace r,s):] is
Function of
[:(Closed-Interval-TSpace a,b),(Closed-Interval-TSpace r,s):],
(Trectangle a,b,r,s)
theorem Th58: :: TOPREALA:58
for
a,
b,
r,
s being
real number st
a <= b &
r <= s holds
for
h being
Function of
[:(Closed-Interval-TSpace a,b),(Closed-Interval-TSpace r,s):],
(Trectangle a,b,r,s) st
h = R2Homeomorphism | the
carrier of
[:(Closed-Interval-TSpace a,b),(Closed-Interval-TSpace r,s):] holds
h is
being_homeomorphism
theorem :: TOPREALA:59
for
a,
b,
r,
s being
real number st
a <= b &
r <= s holds
[:(Closed-Interval-TSpace a,b),(Closed-Interval-TSpace r,s):],
Trectangle a,
b,
r,
s are_homeomorphic
theorem Th60: :: TOPREALA:60
for
a,
b,
r,
s being
real number st
a <= b &
r <= s holds
for
A being
Subset of
(Closed-Interval-TSpace a,b) for
B being
Subset of
(Closed-Interval-TSpace r,s) holds
product (1,2 --> A,B) is
Subset of
(Trectangle a,b,r,s)
theorem :: TOPREALA:61
for
a,
b,
r,
s being
real number st
a <= b &
r <= s holds
for
A being
open Subset of
(Closed-Interval-TSpace a,b) for
B being
open Subset of
(Closed-Interval-TSpace r,s) holds
product (1,2 --> A,B) is
open Subset of
(Trectangle a,b,r,s)
theorem :: TOPREALA:62
for
a,
b,
r,
s being
real number st
a <= b &
r <= s holds
for
A being
closed Subset of
(Closed-Interval-TSpace a,b) for
B being
closed Subset of
(Closed-Interval-TSpace r,s) holds
product (1,2 --> A,B) is
closed Subset of
(Trectangle a,b,r,s)