:: EUCLID semantic presentation
:: deftheorem Def1 defines REAL EUCLID:def 1 :
:: deftheorem Def2 defines absreal EUCLID:def 2 :
:: deftheorem Def3 defines abs EUCLID:def 3 :
:: deftheorem Def4 defines 0* EUCLID:def 4 :
:: deftheorem Def5 defines |. EUCLID:def 5 :
theorem Th1: :: EUCLID:1
canceled;
theorem Th2: :: EUCLID:2
theorem Th3: :: EUCLID:3
theorem Th4: :: EUCLID:4
theorem Th5: :: EUCLID:5
Lemma50:
for n being Element of NAT
for x being Element of REAL n holds dom (abs x) = dom x
theorem Th6: :: EUCLID:6
Lemma54:
for n being Element of NAT
for x being Element of REAL n holds sqr (abs x) = sqr x
theorem Th7: :: EUCLID:7
theorem Th8: :: EUCLID:8
theorem Th9: :: EUCLID:9
theorem Th10: :: EUCLID:10
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 Th17: :: EUCLID:17
theorem Th18: :: EUCLID:18
theorem Th19: :: EUCLID:19
theorem Th20: :: EUCLID:20
theorem Th21: :: EUCLID:21
theorem Th22: :: EUCLID:22
definition
let n be
Element of
NAT ;
func Pitag_dist c1 -> Function of
[:(REAL a1),(REAL a1):],
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 Th23: :: EUCLID:23
theorem Th24: :: EUCLID:24
:: deftheorem Def7 defines Euclid EUCLID:def 7 :
:: deftheorem Def8 defines TOP-REAL EUCLID:def 8 :
theorem Th25: :: EUCLID:25
theorem Th26: :: EUCLID:26
theorem Th27: :: EUCLID:27
theorem Th28: :: EUCLID:28
:: deftheorem Def9 defines 0.REAL EUCLID:def 9 :
:: deftheorem Def10 defines + EUCLID:def 10 :
theorem Th29: :: EUCLID:29
theorem Th30: :: EUCLID:30
theorem Th31: :: EUCLID:31
:: deftheorem Def11 defines * EUCLID:def 11 :
theorem Th32: :: EUCLID:32
theorem Th33: :: EUCLID:33
theorem Th34: :: EUCLID:34
theorem Th35: :: EUCLID:35
theorem Th36: :: EUCLID:36
theorem Th37: :: EUCLID:37
theorem Th38: :: EUCLID:38
:: deftheorem Def12 defines - EUCLID:def 12 :
theorem Th39: :: EUCLID:39
theorem Th40: :: EUCLID:40
theorem Th41: :: EUCLID:41
theorem Th42: :: EUCLID:42
theorem Th43: :: EUCLID:43
theorem Th44: :: EUCLID:44
:: deftheorem Def13 defines - EUCLID:def 13 :
theorem Th45: :: EUCLID:45
theorem Th46: :: EUCLID:46
theorem Th47: :: EUCLID:47
theorem Th48: :: EUCLID:48
theorem Th49: :: EUCLID:49
theorem Th50: :: EUCLID:50
theorem Th51: :: EUCLID:51
theorem Th52: :: EUCLID:52
theorem Th53: :: EUCLID:53
theorem Th54: :: EUCLID:54
theorem Th55: :: EUCLID:55
:: deftheorem Def14 defines `1 EUCLID:def 14 :
:: deftheorem Def15 defines `2 EUCLID:def 15 :
:: deftheorem Def16 defines |[ EUCLID:def 16 :
theorem Th56: :: EUCLID:56
theorem Th57: :: EUCLID:57
theorem Th58: :: EUCLID:58
theorem Th59: :: EUCLID:59
theorem Th60: :: EUCLID:60
theorem Th61: :: EUCLID:61
theorem Th62: :: EUCLID:62
theorem Th63: :: EUCLID:63
theorem Th64: :: EUCLID:64
theorem Th65: :: EUCLID:65
theorem Th66: :: EUCLID:66