:: NAGATA_2 semantic presentation
theorem Th1: :: NAGATA_2:1
for b
1 being
Nat holds
not ( b
1 > 0 & ( for b
2, b
3 being
Nat holds
not b
1 = (2 |^ b2) * ((2 * b3) + 1) ) )
definition
func PairFunc -> Function of
[:NAT ,NAT :],
NAT means :
Def1:
:: NAGATA_2:def 1
for b
1, b
2 being
Nat holds a
1 . [b1,b2] = ((2 |^ b1) * ((2 * b2) + 1)) - 1;
existence
ex b1 being Function of [:NAT ,NAT :], NAT st
for b2, b3 being Nat holds b1 . [b2,b3] = ((2 |^ b2) * ((2 * b3) + 1)) - 1
uniqueness
for b1, b2 being Function of [:NAT ,NAT :], NAT holds
( ( for b3, b4 being Nat holds b1 . [b3,b4] = ((2 |^ b3) * ((2 * b4) + 1)) - 1 ) & ( for b3, b4 being Nat holds b2 . [b3,b4] = ((2 |^ b3) * ((2 * b4) + 1)) - 1 ) implies b1 = b2 )
end;
:: deftheorem Def1 defines PairFunc NAGATA_2:def 1 :
theorem Th2: :: NAGATA_2:2
definition
let c
1 be
set ;
let c
2 be
Function of
[:c1,c1:],
REAL ;
let c
3 be
Element of c
1;
func dist c
2,c
3 -> Function of a
1,
REAL means :
Def2:
:: NAGATA_2:def 2
for b
1 being
Element of a
1 holds a
4 . b
1 = a
2 . a
3,b
1;
existence
ex b1 being Function of c1, REAL st
for b2 being Element of c1 holds b1 . b2 = c2 . c3,b2
uniqueness
for b1, b2 being Function of c1, REAL holds
( ( for b3 being Element of c1 holds b1 . b3 = c2 . c3,b3 ) & ( for b3 being Element of c1 holds b2 . b3 = c2 . c3,b3 ) implies b1 = b2 )
end;
:: deftheorem Def2 defines dist NAGATA_2:def 2 :
theorem Th3: :: NAGATA_2:3
theorem Th4: :: NAGATA_2:4
definition
let c
1 be non
empty set ;
let c
2 be
Function of
[:c1,c1:],
REAL ;
let c
3 be
Subset of c
1;
func inf c
2,c
3 -> Function of a
1,
REAL means :
Def3:
:: NAGATA_2:def 3
for b
1 being
Element of a
1 holds a
4 . b
1 = inf ((dist a2,b1) .: a3);
existence
ex b1 being Function of c1, REAL st
for b2 being Element of c1 holds b1 . b2 = inf ((dist c2,b2) .: c3)
uniqueness
for b1, b2 being Function of c1, REAL holds
( ( for b3 being Element of c1 holds b1 . b3 = inf ((dist c2,b3) .: c3) ) & ( for b3 being Element of c1 holds b2 . b3 = inf ((dist c2,b3) .: c3) ) implies b1 = b2 )
end;
:: deftheorem Def3 defines inf NAGATA_2:def 3 :
Lemma8:
for b1 being non empty set
for b2 being Function of [:b1,b1:], REAL holds
( b2 is_a_pseudometric_of b1 implies ( b2 is bounded_below & ( for b3 being non empty Subset of b1
for b4 being Element of b1 holds
( not (dist b2,b4) .: b3 is empty & (dist b2,b4) .: b3 is bounded_below ) ) ) )
theorem Th5: :: NAGATA_2:5
theorem Th6: :: NAGATA_2:6
theorem Th7: :: NAGATA_2:7
theorem Th8: :: NAGATA_2:8
theorem Th9: :: NAGATA_2:9
theorem Th10: :: NAGATA_2:10
theorem Th11: :: NAGATA_2:11
theorem Th12: :: NAGATA_2:12
theorem Th13: :: NAGATA_2:13
theorem Th14: :: NAGATA_2:14
theorem Th15: :: NAGATA_2:15
for b
1 being non
empty TopSpacefor b
2 being
Realfor b
3 being
Functional_Sequence of
[:the carrier of b1,the carrier of b1:],
REAL holds
( ( for b
4 being
Nat holds
ex b
5 being
Function of
[:the carrier of b1,the carrier of b1:],
REAL st
( b
3 . b
4 = b
5 & b
5 is_a_pseudometric_of the
carrier of b
1 & ( for b
6 being
Element of
[:the carrier of b1,the carrier of b1:] holds b
5 . b
6 <= b
2 ) & ( for b
6 being
RealMap of
[:b1,b1:] holds
( b
5 = b
6 implies b
6 is
continuous ) ) ) ) implies for b
4 being
Function of
[:the carrier of b1,the carrier of b1:],
REAL holds
( ( for b
5 being
Element of
[:the carrier of b1,the carrier of b1:] holds b
4 . b
5 = Sum (((1 / 2) GeoSeq ) (#) (b3 # b5)) ) implies ( b
4 is_a_pseudometric_of the
carrier of b
1 & ( for b
5 being
RealMap of
[:b1,b1:] holds
( b
4 = b
5 implies b
5 is
continuous ) ) ) ) )
theorem Th16: :: NAGATA_2:16
theorem Th17: :: NAGATA_2:17
for b
1 being non
empty TopSpace holds
( b
1 is_T1 implies for b
2 being
Realfor b
3 being
Functional_Sequence of
[:the carrier of b1,the carrier of b1:],
REAL holds
( ( for b
4 being
Nat holds
ex b
5 being
Function of
[:the carrier of b1,the carrier of b1:],
REAL st
( b
3 . b
4 = b
5 & b
5 is_a_pseudometric_of the
carrier of b
1 & ( for b
6 being
Element of
[:the carrier of b1,the carrier of b1:] holds b
5 . b
6 <= b
2 ) & ( for b
6 being
RealMap of
[:b1,b1:] holds
( b
5 = b
6 implies b
6 is
continuous ) ) ) ) & ( for b
4 being
Point of b
1for b
5 being non
empty Subset of b
1 holds
not ( not b
4 in b
5 & b
5 is
closed & ( for b
6 being
Nat holds
ex b
7 being
Function of
[:the carrier of b1,the carrier of b1:],
REAL st
( b
3 . b
6 = b
7 & not
(inf b7,b5) . b
4 > 0 ) ) ) ) implies ( ex b
4 being
Function of
[:the carrier of b1,the carrier of b1:],
REAL st
( b
4 is_metric_of the
carrier of b
1 & ( for b
5 being
Element of
[:the carrier of b1,the carrier of b1:] holds b
4 . b
5 = Sum (((1 / 2) GeoSeq ) (#) (b3 # b5)) ) ) & b
1 is
metrizable ) ) )
theorem Th18: :: NAGATA_2:18
theorem Th19: :: NAGATA_2:19
theorem Th20: :: NAGATA_2:20
theorem Th21: :: NAGATA_2:21
theorem Th22: :: NAGATA_2:22