:: 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
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 Th14: :: 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 :
:: deftheorem defines TOP-REAL EUCLID:def 8 :
theorem :: EUCLID:25
theorem Th26: :: EUCLID:26
theorem Th27: :: EUCLID:27
theorem Th28: :: EUCLID:28
:: deftheorem EUCLID:def 9 :
canceled;
:: deftheorem EUCLID:def 10 :
canceled;
theorem :: EUCLID:29
theorem :: EUCLID:30
theorem :: EUCLID:31
:: deftheorem EUCLID:def 11 :
canceled;
theorem Th32: :: EUCLID:32
theorem :: EUCLID:33
theorem :: EUCLID:34
theorem :: EUCLID:35
theorem :: EUCLID:36
theorem :: EUCLID:37
theorem :: EUCLID:38
:: deftheorem EUCLID:def 12 :
canceled;
theorem :: EUCLID:39
theorem :: EUCLID:40
theorem :: EUCLID:41
theorem :: EUCLID:42
theorem :: EUCLID:43
theorem Th44: :: EUCLID:44
:: deftheorem EUCLID:def 13 :
canceled;
theorem :: EUCLID:45
theorem :: EUCLID:46
theorem :: EUCLID:47
theorem :: EUCLID:48
theorem Th49: :: EUCLID:49
theorem Th50: :: EUCLID:50
theorem :: EUCLID:51
theorem :: EUCLID:52
theorem :: EUCLID:53
theorem :: EUCLID:54
theorem :: EUCLID:55
:: 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