:: The Euclidean Space
:: by Agata Darmochwa{\l}
::
:: Received November 21, 1991
:: Copyright (c) 1991 Association of Mizar Users
:: deftheorem defines REAL EUCLID:def 1 :
:: deftheorem Def2 defines absreal EUCLID:def 2 :
:: deftheorem defines abs EUCLID:def 3 :
:: deftheorem defines 0* EUCLID:def 4 :
:: deftheorem defines |. EUCLID:def 5 :
theorem :: EUCLID:1
canceled;
theorem :: EUCLID:2
canceled;
theorem Th3: :: EUCLID:3
theorem :: EUCLID:4
theorem :: EUCLID:5
canceled;
theorem Th6: :: EUCLID:6
Lm1:
for f being real-valued FinSequence holds f is FinSequence of REAL
Lm2:
for f being real-valued FinSequence holds f is Element of REAL (len f)
theorem :: EUCLID:7
theorem Th8: :: EUCLID:8
theorem Th9: :: EUCLID:9
theorem Th10: :: EUCLID:10
Lm3:
for f being real-valued FinSequence holds sqr (abs f) = sqr f
theorem Th11: :: EUCLID:11
theorem Th12: :: EUCLID:12
theorem Th13: :: EUCLID:13
theorem :: EUCLID:14
theorem Th15: :: EUCLID:15
theorem Th16: :: EUCLID:16
theorem :: EUCLID:17
theorem :: EUCLID:18
theorem Th19: :: EUCLID:19
theorem :: EUCLID:20
theorem Th21: :: EUCLID:21
theorem Th22: :: EUCLID:22
definition
let n be
Nat;
func Pitag_dist n -> Function of
[:(REAL n),(REAL n):],
REAL means :
Def6:
:: EUCLID:def 6
for
x,
y being
Element of
REAL n holds
it . x,
y = |.(x - y).|;
existence
ex b1 being Function of [:(REAL n),(REAL n):],REAL st
for x, y being Element of REAL n holds b1 . x,y = |.(x - y).|
uniqueness
for b1, b2 being Function of [:(REAL n),(REAL n):],REAL st ( for x, y being Element of REAL n holds b1 . x,y = |.(x - y).| ) & ( for x, y being Element of REAL n holds b2 . x,y = |.(x - y).| ) holds
b1 = b2
end;
:: deftheorem Def6 defines Pitag_dist EUCLID:def 6 :
theorem :: EUCLID:23
theorem Th24: :: EUCLID:24
:: deftheorem defines Euclid EUCLID:def 7 :
definition
let n be
Nat;
func TOP-REAL n -> strict RLTopStruct means :
Def8:
:: EUCLID:def 8
(
TopStruct(# the
carrier of
it,the
topology of
it #)
= TopSpaceMetr (Euclid n) &
RLSStruct(# the
carrier of
it,the
ZeroF of
it,the
addF of
it,the
Mult of
it #)
= RealVectSpace (Seg n) );
existence
ex b1 being strict RLTopStruct st
( TopStruct(# the carrier of b1,the topology of b1 #) = TopSpaceMetr (Euclid n) & RLSStruct(# the carrier of b1,the ZeroF of b1,the addF of b1,the Mult of b1 #) = RealVectSpace (Seg n) )
uniqueness
for b1, b2 being strict RLTopStruct st TopStruct(# the carrier of b1,the topology of b1 #) = TopSpaceMetr (Euclid n) & RLSStruct(# the carrier of b1,the ZeroF of b1,the addF of b1,the Mult of b1 #) = RealVectSpace (Seg n) & TopStruct(# the carrier of b2,the topology of b2 #) = TopSpaceMetr (Euclid n) & RLSStruct(# the carrier of b2,the ZeroF of b2,the addF of b2,the Mult of b2 #) = RealVectSpace (Seg n) holds
b1 = b2
;
end;
:: deftheorem Def8 defines TOP-REAL EUCLID:def 8 :
theorem Th25: :: EUCLID:25
theorem Th26: :: EUCLID:26
theorem Th27: :: EUCLID:27
Lm4:
for n being Nat
for p1, p2 being Point of (TOP-REAL n)
for r1, r2 being real-valued Function st p1 = r1 & p2 = r2 holds
p1 + p2 = r1 + r2
Lm5:
for n being Nat
for p being Point of (TOP-REAL n)
for x being real number
for r being real-valued Function st p = r holds
x * p = x (#) r
theorem :: EUCLID:28
canceled;
theorem :: EUCLID:29
theorem :: EUCLID:30
theorem :: EUCLID:31
theorem :: EUCLID:32
theorem :: EUCLID:33
theorem :: EUCLID:34
theorem :: EUCLID:35
theorem :: EUCLID:36
theorem :: EUCLID:37
theorem :: EUCLID:38
theorem :: EUCLID:39
theorem :: EUCLID:40
theorem :: EUCLID:41
theorem :: EUCLID:42
theorem :: EUCLID:43
theorem :: EUCLID:44
theorem :: EUCLID:45
theorem :: EUCLID:46
theorem :: EUCLID:47
theorem :: EUCLID:48
theorem :: EUCLID:49
theorem :: EUCLID:50
theorem :: EUCLID:51
theorem :: EUCLID:52
theorem :: EUCLID:53
theorem :: EUCLID:54
theorem :: EUCLID:55
:: deftheorem EUCLID:def 9 :
canceled;
:: deftheorem EUCLID:def 10 :
canceled;
:: deftheorem EUCLID:def 11 :
canceled;
:: deftheorem EUCLID:def 12 :
canceled;
:: deftheorem EUCLID:def 13 :
canceled;
:: deftheorem defines `1 EUCLID:def 14 :
:: deftheorem defines `2 EUCLID:def 15 :
theorem :: EUCLID:56
theorem Th57: :: EUCLID:57
theorem :: EUCLID:58
theorem Th59: :: EUCLID:59
theorem :: EUCLID:60
theorem Th61: :: EUCLID:61
theorem :: EUCLID:62
theorem Th63: :: EUCLID:63
theorem :: EUCLID:64
theorem Th65: :: EUCLID:65
theorem :: EUCLID:66
theorem :: EUCLID:67
theorem :: EUCLID:68
theorem :: EUCLID:69
theorem Th70: :: EUCLID:70
theorem :: EUCLID:71
theorem :: EUCLID:72
theorem :: EUCLID:73
theorem :: EUCLID:74
theorem :: EUCLID:75