:: Sequences in Metric Spaces
:: by Stanis{\l}awa Kanas and Adam Lecko
::
:: Received December 12, 1991
:: Copyright (c) 1991 Association of Mizar Users
theorem Th1: :: METRIC_6:1
theorem Th2: :: METRIC_6:2
theorem Th3: :: METRIC_6:3
theorem :: METRIC_6:4
theorem :: METRIC_6:5
definition
let A be non
empty set ;
let G be
Function of
[:A,A:],
REAL ;
canceled;canceled;canceled;func bounded_metric A,
G -> Function of
[:A,A:],
REAL means :
Def4:
:: METRIC_6:def 4
for
a,
b being
Element of
A holds
it . a,
b = (G . a,b) / (1 + (G . a,b));
existence
ex b1 being Function of [:A,A:], REAL st
for a, b being Element of A holds b1 . a,b = (G . a,b) / (1 + (G . a,b))
uniqueness
for b1, b2 being Function of [:A,A:], REAL st ( for a, b being Element of A holds b1 . a,b = (G . a,b) / (1 + (G . a,b)) ) & ( for a, b being Element of A holds b2 . a,b = (G . a,b) / (1 + (G . a,b)) ) holds
b1 = b2
end;
:: deftheorem METRIC_6:def 1 :
canceled;
:: deftheorem METRIC_6:def 2 :
canceled;
:: deftheorem METRIC_6:def 3 :
canceled;
:: deftheorem Def4 defines bounded_metric METRIC_6:def 4 :
theorem :: METRIC_6:6
theorem :: METRIC_6:7
canceled;
theorem :: METRIC_6:8
canceled;
theorem :: METRIC_6:9
canceled;
theorem Th10: :: METRIC_6:10
:: deftheorem METRIC_6:def 5 :
canceled;
:: deftheorem METRIC_6:def 6 :
canceled;
:: deftheorem METRIC_6:def 7 :
canceled;
:: deftheorem Def8 defines is_convergent_in_metrspace_to METRIC_6:def 8 :
:: deftheorem METRIC_6:def 9 :
canceled;
:: deftheorem defines bounded METRIC_6:def 10 :
:: deftheorem Def11 defines bounded METRIC_6:def 11 :
:: deftheorem Def12 defines contains_almost_all_sequence METRIC_6:def 12 :
theorem :: METRIC_6:11
canceled;
theorem :: METRIC_6:12
canceled;
theorem :: METRIC_6:13
canceled;
theorem :: METRIC_6:14
canceled;
theorem :: METRIC_6:15
canceled;
theorem :: METRIC_6:16
canceled;
theorem :: METRIC_6:17
canceled;
theorem :: METRIC_6:18
canceled;
theorem :: METRIC_6:19
canceled;
theorem Th20: :: METRIC_6:20
theorem Th21: :: METRIC_6:21
theorem Th22: :: METRIC_6:22
:: deftheorem METRIC_6:def 13 :
canceled;
:: deftheorem Def14 defines dist_to_point METRIC_6:def 14 :
:: deftheorem Def15 defines sequence_of_dist METRIC_6:def 15 :
theorem :: METRIC_6:23
canceled;
theorem :: METRIC_6:24
canceled;
theorem :: METRIC_6:25
canceled;
theorem Th26: :: METRIC_6:26
theorem Th27: :: METRIC_6:27
theorem Th28: :: METRIC_6:28
theorem Th29: :: METRIC_6:29
theorem Th30: :: METRIC_6:30
theorem Th31: :: METRIC_6:31
theorem Th32: :: METRIC_6:32
theorem :: METRIC_6:33
theorem :: METRIC_6:34
theorem :: METRIC_6:35
theorem Th36: :: METRIC_6:36
theorem :: METRIC_6:37
theorem Th38: :: METRIC_6:38
theorem :: METRIC_6:39
theorem :: METRIC_6:40
theorem :: METRIC_6:41
canceled;
theorem :: METRIC_6:42
theorem :: METRIC_6:43
theorem Th44: :: METRIC_6:44