:: TBSP_1 semantic presentation
theorem Th1: :: TBSP_1:1
theorem Th2: :: TBSP_1:2
theorem Th3: :: TBSP_1:3
for b
1 being
Real holds
( 0
< b
1 & b
1 < 1 implies for b
2 being
Real holds
not ( 0
< b
2 & ( for b
3 being
Nat holds not b
1 to_power b
3 < b
2 ) ) )
:: deftheorem Def1 defines totally_bounded TBSP_1:def 1 :
Lemma5:
for b1 being non empty MetrStruct
for b2 being Function holds
( b2 is sequence of b1 iff ( dom b2 = NAT & ( for b3 being set holds
( b3 in NAT implies b2 . b3 is Element of b1 ) ) ) )
theorem Th4: :: TBSP_1:4
canceled;
theorem Th5: :: TBSP_1:5
:: deftheorem Def2 TBSP_1:def 2 :
canceled;
:: deftheorem Def3 defines convergent TBSP_1:def 3 :
:: deftheorem Def4 defines lim TBSP_1:def 4 :
:: deftheorem Def5 defines Cauchy TBSP_1:def 5 :
:: deftheorem Def6 defines complete TBSP_1:def 6 :
theorem Th6: :: TBSP_1:6
canceled;
theorem Th7: :: TBSP_1:7
theorem Th8: :: TBSP_1:8
theorem Th9: :: TBSP_1:9
theorem Th10: :: TBSP_1:10
theorem Th11: :: TBSP_1:11
canceled;
theorem Th12: :: TBSP_1:12
:: deftheorem Def7 TBSP_1:def 7 :
canceled;
:: deftheorem Def8 defines bounded TBSP_1:def 8 :
:: deftheorem Def9 defines bounded TBSP_1:def 9 :
theorem Th13: :: TBSP_1:13
canceled;
theorem Th14: :: TBSP_1:14
theorem Th15: :: TBSP_1:15
theorem Th16: :: TBSP_1:16
theorem Th17: :: TBSP_1:17
Lemma17:
for b1 being non empty Reflexive symmetric triangle MetrStruct
for b2 being Element of b1
for b3 being real number holds
( 0 < b3 implies Ball b2,b3 is bounded )
theorem Th18: :: TBSP_1:18
canceled;
theorem Th19: :: TBSP_1:19
theorem Th20: :: TBSP_1:20
theorem Th21: :: TBSP_1:21
theorem Th22: :: TBSP_1:22
theorem Th23: :: TBSP_1:23
theorem Th24: :: TBSP_1:24
theorem Th25: :: TBSP_1:25
theorem Th26: :: TBSP_1:26
definition
let c
1 be non
empty Reflexive MetrStruct ;
let c
2 be
Subset of c
1;
assume E25:
c
2 is
bounded
;
func diameter c
2 -> Real means :
Def10:
:: TBSP_1:def 10
( ( for b
1, b
2 being
Point of a
1 holds
( b
1 in a
2 & b
2 in a
2 implies
dist b
1,b
2 <= a
3 ) ) & ( for b
1 being
Real holds
( ( for b
2, b
3 being
Point of a
1 holds
( b
2 in a
2 & b
3 in a
2 implies
dist b
2,b
3 <= b
1 ) ) implies a
3 <= b
1 ) ) )
if a
2 <> {} otherwise a
3 = 0;
consistency
for b1 being Real holds
verum
;
existence
( not ( c2 <> {} & ( for b1 being Real holds
not ( ( for b2, b3 being Point of c1 holds
( b2 in c2 & b3 in c2 implies dist b2,b3 <= b1 ) ) & ( for b2 being Real holds
( ( for b3, b4 being Point of c1 holds
( b3 in c2 & b4 in c2 implies dist b3,b4 <= b2 ) ) implies b1 <= b2 ) ) ) ) ) & not ( not c2 <> {} & ( for b1 being Real holds
not b1 = 0 ) ) )
uniqueness
for b1, b2 being Real holds
( ( c2 <> {} & ( for b3, b4 being Point of c1 holds
( b3 in c2 & b4 in c2 implies dist b3,b4 <= b1 ) ) & ( for b3 being Real holds
( ( for b4, b5 being Point of c1 holds
( b4 in c2 & b5 in c2 implies dist b4,b5 <= b3 ) ) implies b1 <= b3 ) ) & ( for b3, b4 being Point of c1 holds
( b3 in c2 & b4 in c2 implies dist b3,b4 <= b2 ) ) & ( for b3 being Real holds
( ( for b4, b5 being Point of c1 holds
( b4 in c2 & b5 in c2 implies dist b4,b5 <= b3 ) ) implies b2 <= b3 ) ) implies b1 = b2 ) & ( not c2 <> {} & b1 = 0 & b2 = 0 implies b1 = b2 ) )
end;
:: deftheorem Def10 defines diameter TBSP_1:def 10 :
theorem Th27: :: TBSP_1:27
canceled;
theorem Th28: :: TBSP_1:28
theorem Th29: :: TBSP_1:29
theorem Th30: :: TBSP_1:30
theorem Th31: :: TBSP_1:31
theorem Th32: :: TBSP_1:32
theorem Th33: :: TBSP_1:33
theorem Th34: :: TBSP_1:34