:: METRIC_4 semantic presentation
definition
let c
1, c
2 be non
empty MetrSpace;
func dist_cart2S c
1,c
2 -> Function of
[:[:the carrier of a1,the carrier of a2:],[:the carrier of a1,the carrier of a2:]:],
REAL means :
Def1:
:: METRIC_4:def 1
for b
1, b
2 being
Element of a
1for b
3, b
4 being
Element of a
2for b
5, b
6 being
Element of
[:the carrier of a1,the carrier of a2:] holds
( b
5 = [b1,b3] & b
6 = [b2,b4] implies a
3 . b
5,b
6 = sqrt (((dist b1,b2) ^2 ) + ((dist b3,b4) ^2 )) );
existence
ex b1 being Function of [:[:the carrier of c1,the carrier of c2:],[:the carrier of c1,the carrier of c2:]:], REAL st
for b2, b3 being Element of c1
for b4, b5 being Element of c2
for b6, b7 being Element of [:the carrier of c1,the carrier of c2:] holds
( b6 = [b2,b4] & b7 = [b3,b5] implies b1 . b6,b7 = sqrt (((dist b2,b3) ^2 ) + ((dist b4,b5) ^2 )) )
uniqueness
for b1, b2 being Function of [:[:the carrier of c1,the carrier of c2:],[:the carrier of c1,the carrier of c2:]:], REAL holds
( ( for b3, b4 being Element of c1
for b5, b6 being Element of c2
for b7, b8 being Element of [:the carrier of c1,the carrier of c2:] holds
( b7 = [b3,b5] & b8 = [b4,b6] implies b1 . b7,b8 = sqrt (((dist b3,b4) ^2 ) + ((dist b5,b6) ^2 )) ) ) & ( for b3, b4 being Element of c1
for b5, b6 being Element of c2
for b7, b8 being Element of [:the carrier of c1,the carrier of c2:] holds
( b7 = [b3,b5] & b8 = [b4,b6] implies b2 . b7,b8 = sqrt (((dist b3,b4) ^2 ) + ((dist b5,b6) ^2 )) ) ) implies b1 = b2 )
end;
:: deftheorem Def1 defines dist_cart2S METRIC_4:def 1 :
for b
1, b
2 being non
empty MetrSpacefor b
3 being
Function of
[:[:the carrier of b1,the carrier of b2:],[:the carrier of b1,the carrier of b2:]:],
REAL holds
( b
3 = dist_cart2S b
1,b
2 iff for b
4, b
5 being
Element of b
1for b
6, b
7 being
Element of b
2for b
8, b
9 being
Element of
[:the carrier of b1,the carrier of b2:] holds
( b
8 = [b4,b6] & b
9 = [b5,b7] implies b
3 . b
8,b
9 = sqrt (((dist b4,b5) ^2 ) + ((dist b6,b7) ^2 )) ) );
theorem Th1: :: METRIC_4:1
canceled;
theorem Th2: :: METRIC_4:2
for b
1, b
2 being
real number holds
( 0
<= b
1 & 0
<= b
2 implies (
sqrt (b1 + b2) = 0 iff ( b
1 = 0 & b
2 = 0 ) ) )
theorem Th3: :: METRIC_4:3
theorem Th4: :: METRIC_4:4
theorem Th5: :: METRIC_4:5
theorem Th6: :: METRIC_4:6
for b
1, b
2 being non
empty MetrSpacefor b
3, b
4, b
5 being
Element of
[:the carrier of b1,the carrier of b2:] holds
(dist_cart2S b1,b2) . b
3,b
5 <= ((dist_cart2S b1,b2) . b3,b4) + ((dist_cart2S b1,b2) . b4,b5)
:: deftheorem Def2 defines dist2S METRIC_4:def 2 :
:: deftheorem Def3 defines MetrSpaceCart2S METRIC_4:def 3 :
definition
let c
1, c
2, c
3 be non
empty MetrSpace;
func dist_cart3S c
1,c
2,c
3 -> Function of
[:[:the carrier of a1,the carrier of a2,the carrier of a3:],[:the carrier of a1,the carrier of a2,the carrier of a3:]:],
REAL means :
Def4:
:: METRIC_4:def 4
for b
1, b
2 being
Element of a
1for b
3, b
4 being
Element of a
2for b
5, b
6 being
Element of a
3for b
7, b
8 being
Element of
[:the carrier of a1,the carrier of a2,the carrier of a3:] holds
( b
7 = [b1,b3,b5] & b
8 = [b2,b4,b6] implies a
4 . b
7,b
8 = sqrt ((((dist b1,b2) ^2 ) + ((dist b3,b4) ^2 )) + ((dist b5,b6) ^2 )) );
existence
ex b1 being Function of [:[:the carrier of c1,the carrier of c2,the carrier of c3:],[:the carrier of c1,the carrier of c2,the carrier of c3:]:], REAL st
for b2, b3 being Element of c1
for b4, b5 being Element of c2
for b6, b7 being Element of c3
for b8, b9 being Element of [:the carrier of c1,the carrier of c2,the carrier of c3:] holds
( b8 = [b2,b4,b6] & b9 = [b3,b5,b7] implies b1 . b8,b9 = sqrt ((((dist b2,b3) ^2 ) + ((dist b4,b5) ^2 )) + ((dist b6,b7) ^2 )) )
uniqueness
for b1, b2 being Function of [:[:the carrier of c1,the carrier of c2,the carrier of c3:],[:the carrier of c1,the carrier of c2,the carrier of c3:]:], REAL holds
( ( for b3, b4 being Element of c1
for b5, b6 being Element of c2
for b7, b8 being Element of c3
for b9, b10 being Element of [:the carrier of c1,the carrier of c2,the carrier of c3:] holds
( b9 = [b3,b5,b7] & b10 = [b4,b6,b8] implies b1 . b9,b10 = sqrt ((((dist b3,b4) ^2 ) + ((dist b5,b6) ^2 )) + ((dist b7,b8) ^2 )) ) ) & ( for b3, b4 being Element of c1
for b5, b6 being Element of c2
for b7, b8 being Element of c3
for b9, b10 being Element of [:the carrier of c1,the carrier of c2,the carrier of c3:] holds
( b9 = [b3,b5,b7] & b10 = [b4,b6,b8] implies b2 . b9,b10 = sqrt ((((dist b3,b4) ^2 ) + ((dist b5,b6) ^2 )) + ((dist b7,b8) ^2 )) ) ) implies b1 = b2 )
end;
:: deftheorem Def4 defines dist_cart3S METRIC_4:def 4 :
for b
1, b
2, b
3 being non
empty MetrSpacefor b
4 being
Function of
[:[:the carrier of b1,the carrier of b2,the carrier of b3:],[:the carrier of b1,the carrier of b2,the carrier of b3:]:],
REAL holds
( b
4 = dist_cart3S b
1,b
2,b
3 iff for b
5, b
6 being
Element of b
1for b
7, b
8 being
Element of b
2for b
9, b
10 being
Element of b
3for b
11, b
12 being
Element of
[:the carrier of b1,the carrier of b2,the carrier of b3:] holds
( b
11 = [b5,b7,b9] & b
12 = [b6,b8,b10] implies b
4 . b
11,b
12 = sqrt ((((dist b5,b6) ^2 ) + ((dist b7,b8) ^2 )) + ((dist b9,b10) ^2 )) ) );
theorem Th7: :: METRIC_4:7
canceled;
theorem Th8: :: METRIC_4:8
canceled;
theorem Th9: :: METRIC_4:9
canceled;
theorem Th10: :: METRIC_4:10
theorem Th11: :: METRIC_4:11
for b
1, b
2, b
3 being non
empty MetrSpacefor b
4, b
5 being
Element of
[:the carrier of b1,the carrier of b2,the carrier of b3:] holds
(dist_cart3S b1,b2,b3) . b
4,b
5 = (dist_cart3S b1,b2,b3) . b
5,b
4
theorem Th12: :: METRIC_4:12
theorem Th13: :: METRIC_4:13
for b
1, b
2, b
3, b
4, b
5, b
6 being
real number holds
(((2 * (b1 * b4)) * (b3 * b2)) + ((2 * (b1 * b6)) * (b5 * b3))) + ((2 * (b2 * b6)) * (b5 * b4)) <= ((((((b1 * b4) ^2 ) + ((b3 * b2) ^2 )) + ((b1 * b6) ^2 )) + ((b5 * b3) ^2 )) + ((b2 * b6) ^2 )) + ((b5 * b4) ^2 )
theorem Th14: :: METRIC_4:14
canceled;
theorem Th15: :: METRIC_4:15
Lemma12:
for b1, b2, b3, b4, b5, b6 being real number holds
( 0 <= b1 & 0 <= b2 & 0 <= b3 & 0 <= b4 & 0 <= b5 & 0 <= b6 implies sqrt ((((b1 + b3) ^2 ) + ((b2 + b4) ^2 )) + ((b5 + b6) ^2 )) <= (sqrt (((b1 ^2 ) + (b2 ^2 )) + (b5 ^2 ))) + (sqrt (((b3 ^2 ) + (b4 ^2 )) + (b6 ^2 ))) )
theorem Th16: :: METRIC_4:16
for b
1, b
2, b
3 being non
empty MetrSpacefor b
4, b
5, b
6 being
Element of
[:the carrier of b1,the carrier of b2,the carrier of b3:] holds
(dist_cart3S b1,b2,b3) . b
4,b
6 <= ((dist_cart3S b1,b2,b3) . b4,b5) + ((dist_cart3S b1,b2,b3) . b5,b6)
definition
let c
1, c
2, c
3 be non
empty MetrSpace;
let c
4, c
5 be
Element of
[:the carrier of c1,the carrier of c2,the carrier of c3:];
func dist3S c
4,c
5 -> Real equals :: METRIC_4:def 5
(dist_cart3S a1,a2,a3) . a
4,a
5;
coherence
(dist_cart3S c1,c2,c3) . c4,c5 is Real
;
end;
:: deftheorem Def5 defines dist3S METRIC_4:def 5 :
definition
let c
1, c
2, c
3 be non
empty MetrSpace;
func MetrSpaceCart3S c
1,c
2,c
3 -> non
empty strict MetrSpace equals :: METRIC_4:def 6
MetrStruct(#
[:the carrier of a1,the carrier of a2,the carrier of a3:],
(dist_cart3S a1,a2,a3) #);
coherence
MetrStruct(# [:the carrier of c1,the carrier of c2,the carrier of c3:],(dist_cart3S c1,c2,c3) #) is non empty strict MetrSpace
end;
:: deftheorem Def6 defines MetrSpaceCart3S METRIC_4:def 6 :
theorem Th17: :: METRIC_4:17
canceled;
theorem Th18: :: METRIC_4:18
canceled;
definition
func taxi_dist2 -> Function of
[:[:REAL ,REAL :],[:REAL ,REAL :]:],
REAL means :
Def7:
:: METRIC_4:def 7
for b
1, b
2, b
3, b
4 being
Element of
REAL for b
5, b
6 being
Element of
[:REAL ,REAL :] holds
( b
5 = [b1,b3] & b
6 = [b2,b4] implies a
1 . b
5,b
6 = (real_dist . b1,b2) + (real_dist . b3,b4) );
existence
ex b1 being Function of [:[:REAL ,REAL :],[:REAL ,REAL :]:], REAL st
for b2, b3, b4, b5 being Element of REAL
for b6, b7 being Element of [:REAL ,REAL :] holds
( b6 = [b2,b4] & b7 = [b3,b5] implies b1 . b6,b7 = (real_dist . b2,b3) + (real_dist . b4,b5) )
uniqueness
for b1, b2 being Function of [:[:REAL ,REAL :],[:REAL ,REAL :]:], REAL holds
( ( for b3, b4, b5, b6 being Element of REAL
for b7, b8 being Element of [:REAL ,REAL :] holds
( b7 = [b3,b5] & b8 = [b4,b6] implies b1 . b7,b8 = (real_dist . b3,b4) + (real_dist . b5,b6) ) ) & ( for b3, b4, b5, b6 being Element of REAL
for b7, b8 being Element of [:REAL ,REAL :] holds
( b7 = [b3,b5] & b8 = [b4,b6] implies b2 . b7,b8 = (real_dist . b3,b4) + (real_dist . b5,b6) ) ) implies b1 = b2 )
end;
:: deftheorem Def7 defines taxi_dist2 METRIC_4:def 7 :
for b
1 being
Function of
[:[:REAL ,REAL :],[:REAL ,REAL :]:],
REAL holds
( b
1 = taxi_dist2 iff for b
2, b
3, b
4, b
5 being
Element of
REAL for b
6, b
7 being
Element of
[:REAL ,REAL :] holds
( b
6 = [b2,b4] & b
7 = [b3,b5] implies b
1 . b
6,b
7 = (real_dist . b2,b3) + (real_dist . b4,b5) ) );
theorem Th19: :: METRIC_4:19
theorem Th20: :: METRIC_4:20
theorem Th21: :: METRIC_4:21
:: deftheorem Def8 defines RealSpaceCart2 METRIC_4:def 8 :
definition
func Eukl_dist2 -> Function of
[:[:REAL ,REAL :],[:REAL ,REAL :]:],
REAL means :
Def9:
:: METRIC_4:def 9
for b
1, b
2, b
3, b
4 being
Element of
REAL for b
5, b
6 being
Element of
[:REAL ,REAL :] holds
( b
5 = [b1,b3] & b
6 = [b2,b4] implies a
1 . b
5,b
6 = sqrt (((real_dist . b1,b2) ^2 ) + ((real_dist . b3,b4) ^2 )) );
existence
ex b1 being Function of [:[:REAL ,REAL :],[:REAL ,REAL :]:], REAL st
for b2, b3, b4, b5 being Element of REAL
for b6, b7 being Element of [:REAL ,REAL :] holds
( b6 = [b2,b4] & b7 = [b3,b5] implies b1 . b6,b7 = sqrt (((real_dist . b2,b3) ^2 ) + ((real_dist . b4,b5) ^2 )) )
uniqueness
for b1, b2 being Function of [:[:REAL ,REAL :],[:REAL ,REAL :]:], REAL holds
( ( for b3, b4, b5, b6 being Element of REAL
for b7, b8 being Element of [:REAL ,REAL :] holds
( b7 = [b3,b5] & b8 = [b4,b6] implies b1 . b7,b8 = sqrt (((real_dist . b3,b4) ^2 ) + ((real_dist . b5,b6) ^2 )) ) ) & ( for b3, b4, b5, b6 being Element of REAL
for b7, b8 being Element of [:REAL ,REAL :] holds
( b7 = [b3,b5] & b8 = [b4,b6] implies b2 . b7,b8 = sqrt (((real_dist . b3,b4) ^2 ) + ((real_dist . b5,b6) ^2 )) ) ) implies b1 = b2 )
end;
:: deftheorem Def9 defines Eukl_dist2 METRIC_4:def 9 :
for b
1 being
Function of
[:[:REAL ,REAL :],[:REAL ,REAL :]:],
REAL holds
( b
1 = Eukl_dist2 iff for b
2, b
3, b
4, b
5 being
Element of
REAL for b
6, b
7 being
Element of
[:REAL ,REAL :] holds
( b
6 = [b2,b4] & b
7 = [b3,b5] implies b
1 . b
6,b
7 = sqrt (((real_dist . b2,b3) ^2 ) + ((real_dist . b4,b5) ^2 )) ) );
theorem Th22: :: METRIC_4:22
theorem Th23: :: METRIC_4:23
theorem Th24: :: METRIC_4:24
:: deftheorem Def10 defines EuklSpace2 METRIC_4:def 10 :
definition
func taxi_dist3 -> Function of
[:[:REAL ,REAL ,REAL :],[:REAL ,REAL ,REAL :]:],
REAL means :
Def11:
:: METRIC_4:def 11
for b
1, b
2, b
3, b
4, b
5, b
6 being
Element of
REAL for b
7, b
8 being
Element of
[:REAL ,REAL ,REAL :] holds
( b
7 = [b1,b3,b5] & b
8 = [b2,b4,b6] implies a
1 . b
7,b
8 = ((real_dist . b1,b2) + (real_dist . b3,b4)) + (real_dist . b5,b6) );
existence
ex b1 being Function of [:[:REAL ,REAL ,REAL :],[:REAL ,REAL ,REAL :]:], REAL st
for b2, b3, b4, b5, b6, b7 being Element of REAL
for b8, b9 being Element of [:REAL ,REAL ,REAL :] holds
( b8 = [b2,b4,b6] & b9 = [b3,b5,b7] implies b1 . b8,b9 = ((real_dist . b2,b3) + (real_dist . b4,b5)) + (real_dist . b6,b7) )
uniqueness
for b1, b2 being Function of [:[:REAL ,REAL ,REAL :],[:REAL ,REAL ,REAL :]:], REAL holds
( ( for b3, b4, b5, b6, b7, b8 being Element of REAL
for b9, b10 being Element of [:REAL ,REAL ,REAL :] holds
( b9 = [b3,b5,b7] & b10 = [b4,b6,b8] implies b1 . b9,b10 = ((real_dist . b3,b4) + (real_dist . b5,b6)) + (real_dist . b7,b8) ) ) & ( for b3, b4, b5, b6, b7, b8 being Element of REAL
for b9, b10 being Element of [:REAL ,REAL ,REAL :] holds
( b9 = [b3,b5,b7] & b10 = [b4,b6,b8] implies b2 . b9,b10 = ((real_dist . b3,b4) + (real_dist . b5,b6)) + (real_dist . b7,b8) ) ) implies b1 = b2 )
end;
:: deftheorem Def11 defines taxi_dist3 METRIC_4:def 11 :
for b
1 being
Function of
[:[:REAL ,REAL ,REAL :],[:REAL ,REAL ,REAL :]:],
REAL holds
( b
1 = taxi_dist3 iff for b
2, b
3, b
4, b
5, b
6, b
7 being
Element of
REAL for b
8, b
9 being
Element of
[:REAL ,REAL ,REAL :] holds
( b
8 = [b2,b4,b6] & b
9 = [b3,b5,b7] implies b
1 . b
8,b
9 = ((real_dist . b2,b3) + (real_dist . b4,b5)) + (real_dist . b6,b7) ) );
theorem Th25: :: METRIC_4:25
theorem Th26: :: METRIC_4:26
theorem Th27: :: METRIC_4:27
:: deftheorem Def12 defines RealSpaceCart3 METRIC_4:def 12 :
definition
func Eukl_dist3 -> Function of
[:[:REAL ,REAL ,REAL :],[:REAL ,REAL ,REAL :]:],
REAL means :
Def13:
:: METRIC_4:def 13
for b
1, b
2, b
3, b
4, b
5, b
6 being
Element of
REAL for b
7, b
8 being
Element of
[:REAL ,REAL ,REAL :] holds
( b
7 = [b1,b3,b5] & b
8 = [b2,b4,b6] implies a
1 . b
7,b
8 = sqrt ((((real_dist . b1,b2) ^2 ) + ((real_dist . b3,b4) ^2 )) + ((real_dist . b5,b6) ^2 )) );
existence
ex b1 being Function of [:[:REAL ,REAL ,REAL :],[:REAL ,REAL ,REAL :]:], REAL st
for b2, b3, b4, b5, b6, b7 being Element of REAL
for b8, b9 being Element of [:REAL ,REAL ,REAL :] holds
( b8 = [b2,b4,b6] & b9 = [b3,b5,b7] implies b1 . b8,b9 = sqrt ((((real_dist . b2,b3) ^2 ) + ((real_dist . b4,b5) ^2 )) + ((real_dist . b6,b7) ^2 )) )
uniqueness
for b1, b2 being Function of [:[:REAL ,REAL ,REAL :],[:REAL ,REAL ,REAL :]:], REAL holds
( ( for b3, b4, b5, b6, b7, b8 being Element of REAL
for b9, b10 being Element of [:REAL ,REAL ,REAL :] holds
( b9 = [b3,b5,b7] & b10 = [b4,b6,b8] implies b1 . b9,b10 = sqrt ((((real_dist . b3,b4) ^2 ) + ((real_dist . b5,b6) ^2 )) + ((real_dist . b7,b8) ^2 )) ) ) & ( for b3, b4, b5, b6, b7, b8 being Element of REAL
for b9, b10 being Element of [:REAL ,REAL ,REAL :] holds
( b9 = [b3,b5,b7] & b10 = [b4,b6,b8] implies b2 . b9,b10 = sqrt ((((real_dist . b3,b4) ^2 ) + ((real_dist . b5,b6) ^2 )) + ((real_dist . b7,b8) ^2 )) ) ) implies b1 = b2 )
end;
:: deftheorem Def13 defines Eukl_dist3 METRIC_4:def 13 :
for b
1 being
Function of
[:[:REAL ,REAL ,REAL :],[:REAL ,REAL ,REAL :]:],
REAL holds
( b
1 = Eukl_dist3 iff for b
2, b
3, b
4, b
5, b
6, b
7 being
Element of
REAL for b
8, b
9 being
Element of
[:REAL ,REAL ,REAL :] holds
( b
8 = [b2,b4,b6] & b
9 = [b3,b5,b7] implies b
1 . b
8,b
9 = sqrt ((((real_dist . b2,b3) ^2 ) + ((real_dist . b4,b5) ^2 )) + ((real_dist . b6,b7) ^2 )) ) );
theorem Th28: :: METRIC_4:28
theorem Th29: :: METRIC_4:29
theorem Th30: :: METRIC_4:30
:: deftheorem Def14 defines EuklSpace3 METRIC_4:def 14 :