:: TOPREALA semantic presentation
set c1 = the carrier of R^1 ;
Lemma1:
the carrier of [:R^1 ,R^1 :] = [:the carrier of R^1 ,the carrier of R^1 :]
by BORSUK_1:def 5;
reconsider c2 = 1 as positive real number ;
theorem Th1: :: TOPREALA:1
theorem Th2: :: TOPREALA:2
theorem Th3: :: TOPREALA:3
theorem Th4: :: TOPREALA:4
theorem Th5: :: TOPREALA:5
theorem Th6: :: TOPREALA:6
theorem Th7: :: TOPREALA:7
registration
let c
3 be
real number ;
let c
4 be
positive real number ;
cluster ].a1,(a1 + a2).[ -> non
empty ;
coherence
not ].c3,(c3 + c4).[ is empty
cluster [.a1,(a1 + a2).[ -> non
empty ;
coherence
not [.c3,(c3 + c4).[ is empty
cluster ].a1,(a1 + a2).] -> non
empty ;
coherence
not ].c3,(c3 + c4).] is empty
cluster [.a1,(a1 + a2).] -> non
empty ;
coherence
not [.c3,(c3 + c4).] is empty
cluster ].(a1 - a2),a1.[ -> non
empty ;
coherence
not ].(c3 - c4),c3.[ is empty
cluster [.(a1 - a2),a1.[ -> non
empty ;
coherence
not [.(c3 - c4),c3.[ is empty
cluster ].(a1 - a2),a1.] -> non
empty ;
coherence
not ].(c3 - c4),c3.] is empty
cluster [.(a1 - a2),a1.] -> non
empty ;
coherence
not [.(c3 - c4),c3.] is empty
end;
theorem Th8: :: TOPREALA:8
theorem Th9: :: TOPREALA:9
theorem Th10: :: TOPREALA:10
theorem Th11: :: TOPREALA:11
theorem Th12: :: TOPREALA:12
theorem Th13: :: TOPREALA:13
theorem Th14: :: TOPREALA:14
theorem Th15: :: TOPREALA:15
theorem Th16: :: TOPREALA:16
theorem Th17: :: TOPREALA:17
theorem Th18: :: TOPREALA:18
theorem Th19: :: TOPREALA:19
theorem Th20: :: TOPREALA:20
theorem Th21: :: TOPREALA:21
theorem Th22: :: TOPREALA:22
theorem Th23: :: TOPREALA:23
theorem Th24: :: TOPREALA:24
theorem Th25: :: TOPREALA:25
theorem Th26: :: TOPREALA:26
theorem Th27: :: TOPREALA:27
theorem Th28: :: TOPREALA:28
theorem Th29: :: TOPREALA:29
theorem Th30: :: TOPREALA:30
theorem Th31: :: TOPREALA:31
theorem Th32: :: TOPREALA:32
theorem Th33: :: TOPREALA:33
theorem Th34: :: TOPREALA:34
theorem Th35: :: TOPREALA:35
theorem Th36: :: TOPREALA:36
theorem Th37: :: TOPREALA:37
theorem Th38: :: TOPREALA:38
theorem Th39: :: TOPREALA:39
theorem Th40: :: TOPREALA:40
theorem Th41: :: TOPREALA:41
theorem Th42: :: TOPREALA:42
theorem Th43: :: TOPREALA:43
theorem Th44: :: TOPREALA:44
theorem Th45: :: TOPREALA:45
theorem Th46: :: TOPREALA:46
theorem Th47: :: TOPREALA:47
theorem Th48: :: TOPREALA:48
theorem Th49: :: TOPREALA:49
theorem Th50: :: TOPREALA:50
theorem Th51: :: TOPREALA:51
for b
1, b
2, b
3, b
4 being
real number holds
closed_inside_of_rectangle b
1,b
2,b
3,b
4 = product (1,2 --> [.b1,b2.],[.b3,b4.])
theorem Th52: :: TOPREALA:52
definition
let c
3, c
4, c
5, c
6 be
real number ;
func Trectangle c
1,c
2,c
3,c
4 -> SubSpace of
TOP-REAL 2
equals :: TOPREALA:def 1
(TOP-REAL 2) | (closed_inside_of_rectangle a1,a2,a3,a4);
coherence
(TOP-REAL 2) | (closed_inside_of_rectangle c3,c4,c5,c6) is SubSpace of TOP-REAL 2
;
end;
:: deftheorem Def1 defines Trectangle TOPREALA:def 1 :
theorem Th53: :: TOPREALA:53
theorem Th54: :: TOPREALA:54
definition
func R2Homeomorphism -> Function of
[:R^1 ,R^1 :],
(TOP-REAL 2) means :
Def2:
:: TOPREALA:def 2
for b
1, b
2 being
real number holds a
1 . [b1,b2] = <*b1,b2*>;
existence
ex b1 being Function of [:R^1 ,R^1 :],(TOP-REAL 2) st
for b2, b3 being real number holds b1 . [b2,b3] = <*b2,b3*>
uniqueness
for b1, b2 being Function of [:R^1 ,R^1 :],(TOP-REAL 2) holds
( ( for b3, b4 being real number holds b1 . [b3,b4] = <*b3,b4*> ) & ( for b3, b4 being real number holds b2 . [b3,b4] = <*b3,b4*> ) implies b1 = b2 )
end;
:: deftheorem Def2 defines R2Homeomorphism TOPREALA:def 2 :
theorem Th55: :: TOPREALA:55
theorem Th56: :: TOPREALA:56
theorem Th57: :: TOPREALA:57
for b
1, b
2, b
3, b
4 being
real number holds
( b
1 <= b
2 & b
3 <= b
4 implies
R2Homeomorphism | the
carrier of
[:(Closed-Interval-TSpace b1,b2),(Closed-Interval-TSpace b3,b4):] is
Function of
[:(Closed-Interval-TSpace b1,b2),(Closed-Interval-TSpace b3,b4):],
(Trectangle b1,b2,b3,b4) )
theorem Th58: :: TOPREALA:58
for b
1, b
2, b
3, b
4 being
real number holds
( b
1 <= b
2 & b
3 <= b
4 implies for b
5 being
Function of
[:(Closed-Interval-TSpace b1,b2),(Closed-Interval-TSpace b3,b4):],
(Trectangle b1,b2,b3,b4) holds
( b
5 = R2Homeomorphism | the
carrier of
[:(Closed-Interval-TSpace b1,b2),(Closed-Interval-TSpace b3,b4):] implies b
5 is_homeomorphism ) )
theorem Th59: :: TOPREALA:59
for b
1, b
2, b
3, b
4 being
real number holds
( b
1 <= b
2 & b
3 <= b
4 implies
[:(Closed-Interval-TSpace b1,b2),(Closed-Interval-TSpace b3,b4):],
Trectangle b
1,b
2,b
3,b
4 are_homeomorphic )
theorem Th60: :: TOPREALA:60
for b
1, b
2, b
3, b
4 being
real number holds
( b
1 <= b
2 & b
3 <= b
4 implies for b
5 being
Subset of
(Closed-Interval-TSpace b1,b2)for b
6 being
Subset of
(Closed-Interval-TSpace b3,b4) holds
product (1,2 --> b5,b6) is
Subset of
(Trectangle b1,b2,b3,b4) )
theorem Th61: :: TOPREALA:61
for b
1, b
2, b
3, b
4 being
real number holds
( b
1 <= b
2 & b
3 <= b
4 implies for b
5 being
open Subset of
(Closed-Interval-TSpace b1,b2)for b
6 being
open Subset of
(Closed-Interval-TSpace b3,b4) holds
product (1,2 --> b5,b6) is
open Subset of
(Trectangle b1,b2,b3,b4) )
theorem Th62: :: TOPREALA:62
for b
1, b
2, b
3, b
4 being
real number holds
( b
1 <= b
2 & b
3 <= b
4 implies for b
5 being
closed Subset of
(Closed-Interval-TSpace b1,b2)for b
6 being
closed Subset of
(Closed-Interval-TSpace b3,b4) holds
product (1,2 --> b5,b6) is
closed Subset of
(Trectangle b1,b2,b3,b4) )