:: METRIC_2 semantic presentation
:: deftheorem Def1 defines tolerates METRIC_2:def 1 :
:: deftheorem Def2 defines -neighbour METRIC_2:def 2 :
:: deftheorem Def3 defines equivalence_class METRIC_2:def 3 :
Lemma3:
for b1 being non empty Reflexive MetrStruct
for b2 being Element of b1 holds b2 tolerates b2
;
theorem Th1: :: METRIC_2:1
canceled;
theorem Th2: :: METRIC_2:2
canceled;
theorem Th3: :: METRIC_2:3
canceled;
theorem Th4: :: METRIC_2:4
canceled;
theorem Th5: :: METRIC_2:5
canceled;
theorem Th6: :: METRIC_2:6
theorem Th7: :: METRIC_2:7
theorem Th8: :: METRIC_2:8
theorem Th9: :: METRIC_2:9
theorem Th10: :: METRIC_2:10
theorem Th11: :: METRIC_2:11
theorem Th12: :: METRIC_2:12
theorem Th13: :: METRIC_2:13
theorem Th14: :: METRIC_2:14
theorem Th15: :: METRIC_2:15
canceled;
theorem Th16: :: METRIC_2:16
theorem Th17: :: METRIC_2:17
theorem Th18: :: METRIC_2:18
theorem Th19: :: METRIC_2:19
theorem Th20: :: METRIC_2:20
theorem Th21: :: METRIC_2:21
:: deftheorem Def4 defines -neighbour METRIC_2:def 4 :
theorem Th22: :: METRIC_2:22
canceled;
theorem Th23: :: METRIC_2:23
theorem Th24: :: METRIC_2:24
theorem Th25: :: METRIC_2:25
canceled;
theorem Th26: :: METRIC_2:26
theorem Th27: :: METRIC_2:27
theorem Th28: :: METRIC_2:28
theorem Th29: :: METRIC_2:29
:: deftheorem Def5 defines is_dst METRIC_2:def 5 :
theorem Th30: :: METRIC_2:30
canceled;
theorem Th31: :: METRIC_2:31
theorem Th32: :: METRIC_2:32
:: deftheorem Def6 defines ev_eq_1 METRIC_2:def 6 :
theorem Th33: :: METRIC_2:33
canceled;
theorem Th34: :: METRIC_2:34
definition
let c1 be non
empty MetrStruct ;
let c2 be
Element of
REAL ;
func ev_eq_2 c2,
c1 -> Subset of
[:(a1 -neighbour ),(a1 -neighbour ):] equals :: METRIC_2:def 7
{ b1 where B is Element of [:(a1 -neighbour ),(a1 -neighbour ):] : ex b1, b2 being Element of a1 -neighbour st
( b1 = [b2,b3] & b2,b3 is_dst a2 ) } ;
coherence
{ b1 where B is Element of [:(c1 -neighbour ),(c1 -neighbour ):] : ex b1, b2 being Element of c1 -neighbour st
( b1 = [b2,b3] & b2,b3 is_dst c2 ) } is Subset of [:(c1 -neighbour ),(c1 -neighbour ):]
end;
:: deftheorem Def7 defines ev_eq_2 METRIC_2:def 7 :
theorem Th35: :: METRIC_2:35
canceled;
theorem Th36: :: METRIC_2:36
:: deftheorem Def8 defines real_in_rel METRIC_2:def 8 :
theorem Th37: :: METRIC_2:37
canceled;
theorem Th38: :: METRIC_2:38
:: deftheorem Def9 defines elem_in_rel_1 METRIC_2:def 9 :
theorem Th39: :: METRIC_2:39
canceled;
theorem Th40: :: METRIC_2:40
:: deftheorem Def10 defines elem_in_rel_2 METRIC_2:def 10 :
theorem Th41: :: METRIC_2:41
canceled;
theorem Th42: :: METRIC_2:42
definition
let c1 be non
empty MetrStruct ;
func elem_in_rel c1 -> Subset of
[:(a1 -neighbour ),(a1 -neighbour ):] equals :: METRIC_2:def 11
{ b1 where B is Element of [:(a1 -neighbour ),(a1 -neighbour ):] : ex b1, b2 being Element of a1 -neighbour ex b3 being Element of REAL st
( b1 = [b2,b3] & b2,b3 is_dst b4 ) } ;
coherence
{ b1 where B is Element of [:(c1 -neighbour ),(c1 -neighbour ):] : ex b1, b2 being Element of c1 -neighbour ex b3 being Element of REAL st
( b1 = [b2,b3] & b2,b3 is_dst b4 ) } is Subset of [:(c1 -neighbour ),(c1 -neighbour ):]
end;
:: deftheorem Def11 defines elem_in_rel METRIC_2:def 11 :
theorem Th43: :: METRIC_2:43
canceled;
theorem Th44: :: METRIC_2:44
definition
let c1 be non
empty MetrStruct ;
func set_in_rel c1 -> Subset of
[:(a1 -neighbour ),(a1 -neighbour ),REAL :] equals :: METRIC_2:def 12
{ b1 where B is Element of [:(a1 -neighbour ),(a1 -neighbour ),REAL :] : ex b1, b2 being Element of a1 -neighbour ex b3 being Element of REAL st
( b1 = [b2,b3,b4] & b2,b3 is_dst b4 ) } ;
coherence
{ b1 where B is Element of [:(c1 -neighbour ),(c1 -neighbour ),REAL :] : ex b1, b2 being Element of c1 -neighbour ex b3 being Element of REAL st
( b1 = [b2,b3,b4] & b2,b3 is_dst b4 ) } is Subset of [:(c1 -neighbour ),(c1 -neighbour ),REAL :]
end;
:: deftheorem Def12 defines set_in_rel METRIC_2:def 12 :
theorem Th45: :: METRIC_2:45
canceled;
theorem Th46: :: METRIC_2:46
theorem Th47: :: METRIC_2:47
theorem Th48: :: METRIC_2:48
theorem Th49: :: METRIC_2:49
canceled;
theorem Th50: :: METRIC_2:50
theorem Th51: :: METRIC_2:51
canceled;
theorem Th52: :: METRIC_2:52
definition
let c1 be
PseudoMetricSpace;
func nbourdist c1 -> Function of
[:(a1 -neighbour ),(a1 -neighbour ):],
REAL means :
Def13:
:: METRIC_2:def 13
for
b1,
b2 being
Element of
a1 -neighbour for
b3,
b4 being
Element of
a1 st
b3 in b1 &
b4 in b2 holds
a2 . b1,
b2 = dist b3,
b4;
existence
ex b1 being Function of [:(c1 -neighbour ),(c1 -neighbour ):], REAL st
for b2, b3 being Element of c1 -neighbour
for b4, b5 being Element of c1 st b4 in b2 & b5 in b3 holds
b1 . b2,b3 = dist b4,b5
uniqueness
for b1, b2 being Function of [:(c1 -neighbour ),(c1 -neighbour ):], REAL st ( for b3, b4 being Element of c1 -neighbour
for b5, b6 being Element of c1 st b5 in b3 & b6 in b4 holds
b1 . b3,b4 = dist b5,b6 ) & ( for b3, b4 being Element of c1 -neighbour
for b5, b6 being Element of c1 st b5 in b3 & b6 in b4 holds
b2 . b3,b4 = dist b5,b6 ) holds
b1 = b2
end;
:: deftheorem Def13 defines nbourdist METRIC_2:def 13 :
theorem Th53: :: METRIC_2:53
canceled;
theorem Th54: :: METRIC_2:54
theorem Th55: :: METRIC_2:55
theorem Th56: :: METRIC_2:56
:: deftheorem Def14 defines Eq_classMetricSpace METRIC_2:def 14 :