:: METRIC_3 semantic presentation
scheme :: METRIC_3:sch 1
s1{ F
1()
-> non
empty set , F
2()
-> non
empty set , F
3()
-> non
empty set , F
4(
set ,
set ,
set ,
set )
-> Element of F
3() } :
ex b
1 being
Function of
[:[:F1(),F2():],[:F1(),F2():]:],F
3() st
for b
2, b
3 being
Element of F
1()
for b
4, b
5 being
Element of F
2()
for b
6, b
7 being
Element of
[:F1(),F2():] holds
( b
6 = [b2,b4] & b
7 = [b3,b5] implies b
1 . b
6,b
7 = F
4(b
2,b
3,b
4,b
5) )
definition
let c
1, c
2 be non
empty MetrSpace;
func dist_cart2 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_3: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 = (dist b1,b2) + (dist b3,b4) );
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 = (dist b2,b3) + (dist b4,b5) )
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 = (dist b3,b4) + (dist b5,b6) ) ) & ( 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 = (dist b3,b4) + (dist b5,b6) ) ) implies b1 = b2 )
end;
:: deftheorem Def1 defines dist_cart2 METRIC_3: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_cart2 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 = (dist b4,b5) + (dist b6,b7) ) );
theorem Th1: :: METRIC_3:1
canceled;
theorem Th2: :: METRIC_3:2
canceled;
theorem Th3: :: METRIC_3:3
canceled;
theorem Th4: :: METRIC_3:4
canceled;
theorem Th5: :: METRIC_3:5
theorem Th6: :: METRIC_3:6
theorem Th7: :: METRIC_3:7
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_cart2 b1,b2) . b
3,b
5 <= ((dist_cart2 b1,b2) . b3,b4) + ((dist_cart2 b1,b2) . b4,b5)
:: deftheorem Def2 defines dist2 METRIC_3:def 2 :
:: deftheorem Def3 defines MetrSpaceCart2 METRIC_3:def 3 :
scheme :: METRIC_3:sch 2
s2{ F
1()
-> non
empty set , F
2()
-> non
empty set , F
3()
-> non
empty set , F
4()
-> non
empty set , F
5(
set ,
set ,
set ,
set ,
set ,
set )
-> Element of F
4() } :
ex b
1 being
Function of
[:[:F1(),F2(),F3():],[:F1(),F2(),F3():]:],F
4() st
for b
2, b
3 being
Element of F
1()
for b
4, b
5 being
Element of F
2()
for b
6, b
7 being
Element of F
3()
for b
8, b
9 being
Element of
[:F1(),F2(),F3():] holds
( b
8 = [b2,b4,b6] & b
9 = [b3,b5,b7] implies b
1 . b
8,b
9 = F
5(b
2,b
3,b
4,b
5,b
6,b
7) )
definition
let c
1, c
2, c
3 be non
empty MetrSpace;
func dist_cart3 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_3: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 = ((dist b1,b2) + (dist b3,b4)) + (dist b5,b6) );
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 = ((dist b2,b3) + (dist b4,b5)) + (dist b6,b7) )
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 = ((dist b3,b4) + (dist b5,b6)) + (dist b7,b8) ) ) & ( 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 = ((dist b3,b4) + (dist b5,b6)) + (dist b7,b8) ) ) implies b1 = b2 )
end;
:: deftheorem Def4 defines dist_cart3 METRIC_3: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_cart3 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 = ((dist b5,b6) + (dist b7,b8)) + (dist b9,b10) ) );
theorem Th8: :: METRIC_3:8
canceled;
theorem Th9: :: METRIC_3:9
canceled;
theorem Th10: :: METRIC_3:10
canceled;
theorem Th11: :: METRIC_3:11
canceled;
theorem Th12: :: METRIC_3:12
theorem Th13: :: METRIC_3:13
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_cart3 b1,b2,b3) . b
4,b
5 = (dist_cart3 b1,b2,b3) . b
5,b
4
theorem Th14: :: METRIC_3:14
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_cart3 b1,b2,b3) . b
4,b
6 <= ((dist_cart3 b1,b2,b3) . b4,b5) + ((dist_cart3 b1,b2,b3) . b5,b6)
definition
let c
1, c
2, c
3 be non
empty MetrSpace;
func MetrSpaceCart3 c
1,c
2,c
3 -> non
empty strict MetrSpace equals :: METRIC_3:def 5
MetrStruct(#
[:the carrier of a1,the carrier of a2,the carrier of a3:],
(dist_cart3 a1,a2,a3) #);
coherence
MetrStruct(# [:the carrier of c1,the carrier of c2,the carrier of c3:],(dist_cart3 c1,c2,c3) #) is non empty strict MetrSpace
end;
:: deftheorem Def5 defines MetrSpaceCart3 METRIC_3:def 5 :
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 dist3 c
4,c
5 -> Real equals :: METRIC_3:def 6
(dist_cart3 a1,a2,a3) . a
4,a
5;
coherence
(dist_cart3 c1,c2,c3) . c4,c5 is Real
;
end;
:: deftheorem Def6 defines dist3 METRIC_3:def 6 :
scheme :: METRIC_3:sch 3
s3{ F
1()
-> non
empty set , F
2()
-> non
empty set , F
3()
-> non
empty set , F
4()
-> non
empty set , F
5()
-> non
empty set , F
6(
set ,
set ,
set ,
set ,
set ,
set ,
set ,
set )
-> Element of F
5() } :
ex b
1 being
Function of
[:[:F1(),F2(),F3(),F4():],[:F1(),F2(),F3(),F4():]:],F
5() st
for b
2, b
3 being
Element of F
1()
for b
4, b
5 being
Element of F
2()
for b
6, b
7 being
Element of F
3()
for b
8, b
9 being
Element of F
4()
for b
10, b
11 being
Element of
[:F1(),F2(),F3(),F4():] holds
( b
10 = [b2,b4,b6,b8] & b
11 = [b3,b5,b7,b9] implies b
1 . b
10,b
11 = F
6(b
2,b
3,b
4,b
5,b
6,b
7,b
8,b
9) )
definition
let c
1, c
2, c
3, c
4 be non
empty MetrSpace;
func dist_cart4 c
1,c
2,c
3,c
4 -> Function of
[:[:the carrier of a1,the carrier of a2,the carrier of a3,the carrier of a4:],[:the carrier of a1,the carrier of a2,the carrier of a3,the carrier of a4:]:],
REAL means :
Def7:
:: METRIC_3:def 7
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 a
4for b
9, b
10 being
Element of
[:the carrier of a1,the carrier of a2,the carrier of a3,the carrier of a4:] holds
( b
9 = [b1,b3,b5,b7] & b
10 = [b2,b4,b6,b8] implies a
5 . b
9,b
10 = ((dist b1,b2) + (dist b3,b4)) + ((dist b5,b6) + (dist b7,b8)) );
existence
ex b1 being Function of [:[:the carrier of c1,the carrier of c2,the carrier of c3,the carrier of c4:],[:the carrier of c1,the carrier of c2,the carrier of c3,the carrier of c4:]:], 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 c4
for b10, b11 being Element of [:the carrier of c1,the carrier of c2,the carrier of c3,the carrier of c4:] holds
( b10 = [b2,b4,b6,b8] & b11 = [b3,b5,b7,b9] implies b1 . b10,b11 = ((dist b2,b3) + (dist b4,b5)) + ((dist b6,b7) + (dist b8,b9)) )
uniqueness
for b1, b2 being Function of [:[:the carrier of c1,the carrier of c2,the carrier of c3,the carrier of c4:],[:the carrier of c1,the carrier of c2,the carrier of c3,the carrier of c4:]:], 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 c4
for b11, b12 being Element of [:the carrier of c1,the carrier of c2,the carrier of c3,the carrier of c4:] holds
( b11 = [b3,b5,b7,b9] & b12 = [b4,b6,b8,b10] implies b1 . b11,b12 = ((dist b3,b4) + (dist b5,b6)) + ((dist b7,b8) + (dist b9,b10)) ) ) & ( 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 c4
for b11, b12 being Element of [:the carrier of c1,the carrier of c2,the carrier of c3,the carrier of c4:] holds
( b11 = [b3,b5,b7,b9] & b12 = [b4,b6,b8,b10] implies b2 . b11,b12 = ((dist b3,b4) + (dist b5,b6)) + ((dist b7,b8) + (dist b9,b10)) ) ) implies b1 = b2 )
end;
:: deftheorem Def7 defines dist_cart4 METRIC_3:def 7 :
for b
1, b
2, b
3, b
4 being non
empty MetrSpacefor b
5 being
Function of
[:[:the carrier of b1,the carrier of b2,the carrier of b3,the carrier of b4:],[:the carrier of b1,the carrier of b2,the carrier of b3,the carrier of b4:]:],
REAL holds
( b
5 = dist_cart4 b
1,b
2,b
3,b
4 iff for b
6, b
7 being
Element of b
1for b
8, b
9 being
Element of b
2for b
10, b
11 being
Element of b
3for b
12, b
13 being
Element of b
4for b
14, b
15 being
Element of
[:the carrier of b1,the carrier of b2,the carrier of b3,the carrier of b4:] holds
( b
14 = [b6,b8,b10,b12] & b
15 = [b7,b9,b11,b13] implies b
5 . b
14,b
15 = ((dist b6,b7) + (dist b8,b9)) + ((dist b10,b11) + (dist b12,b13)) ) );
theorem Th15: :: METRIC_3:15
canceled;
theorem Th16: :: METRIC_3:16
canceled;
theorem Th17: :: METRIC_3:17
canceled;
theorem Th18: :: METRIC_3:18
canceled;
theorem Th19: :: METRIC_3:19
for b
1, b
2, b
3, b
4 being non
empty MetrSpacefor b
5, b
6 being
Element of
[:the carrier of b1,the carrier of b2,the carrier of b3,the carrier of b4:] holds
(
(dist_cart4 b1,b2,b3,b4) . b
5,b
6 = 0 iff b
5 = b
6 )
theorem Th20: :: METRIC_3:20
for b
1, b
2, b
3, b
4 being non
empty MetrSpacefor b
5, b
6 being
Element of
[:the carrier of b1,the carrier of b2,the carrier of b3,the carrier of b4:] holds
(dist_cart4 b1,b2,b3,b4) . b
5,b
6 = (dist_cart4 b1,b2,b3,b4) . b
6,b
5
theorem Th21: :: METRIC_3:21
for b
1, b
2, b
3, b
4 being non
empty MetrSpacefor b
5, b
6, b
7 being
Element of
[:the carrier of b1,the carrier of b2,the carrier of b3,the carrier of b4:] holds
(dist_cart4 b1,b2,b3,b4) . b
5,b
7 <= ((dist_cart4 b1,b2,b3,b4) . b5,b6) + ((dist_cart4 b1,b2,b3,b4) . b6,b7)
definition
let c
1, c
2, c
3, c
4 be non
empty MetrSpace;
func MetrSpaceCart4 c
1,c
2,c
3,c
4 -> non
empty strict MetrSpace equals :: METRIC_3:def 8
MetrStruct(#
[:the carrier of a1,the carrier of a2,the carrier of a3,the carrier of a4:],
(dist_cart4 a1,a2,a3,a4) #);
coherence
MetrStruct(# [:the carrier of c1,the carrier of c2,the carrier of c3,the carrier of c4:],(dist_cart4 c1,c2,c3,c4) #) is non empty strict MetrSpace
end;
:: deftheorem Def8 defines MetrSpaceCart4 METRIC_3:def 8 :
for b
1, b
2, b
3, b
4 being non
empty MetrSpace holds
MetrSpaceCart4 b
1,b
2,b
3,b
4 = MetrStruct(#
[:the carrier of b1,the carrier of b2,the carrier of b3,the carrier of b4:],
(dist_cart4 b1,b2,b3,b4) #);
definition
let c
1, c
2, c
3, c
4 be non
empty MetrSpace;
let c
5, c
6 be
Element of
[:the carrier of c1,the carrier of c2,the carrier of c3,the carrier of c4:];
func dist4 c
5,c
6 -> Real equals :: METRIC_3:def 9
(dist_cart4 a1,a2,a3,a4) . a
5,a
6;
coherence
(dist_cart4 c1,c2,c3,c4) . c5,c6 is Real
;
end;
:: deftheorem Def9 defines dist4 METRIC_3:def 9 :
for b
1, b
2, b
3, b
4 being non
empty MetrSpacefor b
5, b
6 being
Element of
[:the carrier of b1,the carrier of b2,the carrier of b3,the carrier of b4:] holds
dist4 b
5,b
6 = (dist_cart4 b1,b2,b3,b4) . b
5,b
6;