:: MIDSP_3 semantic presentation
theorem Th1: :: MIDSP_3:1
theorem Th2: :: MIDSP_3:2
for b
1, b
2 being
Nat holds
not ( b
1 in Seg b
2 & ( for b
3, b
4 being
Nat holds
not ( b
2 = (b3 + 1) + b
4 & b
1 = b
3 + 1 ) ) )
theorem Th3: :: MIDSP_3:3
for b
1 being
Natfor b
2 being non
empty set for b
3 being
Element of b
2for b
4, b
5, b
6 being
FinSequence of b
2 holds
( b
4 = (b5 ^ <*b3*>) ^ b
6 & b
1 = (len b5) + 1 implies ( ( for b
7 being
Nat holds
( 1
<= b
7 & b
7 <= len b
5 implies b
4 . b
7 = b
5 . b
7 ) ) & b
4 . b
1 = b
3 & ( for b
7 being
Nat holds
( b
1 + 1
<= b
7 & b
7 <= len b
4 implies b
4 . b
7 = b
6 . (b7 - b1) ) ) ) )
theorem Th4: :: MIDSP_3:4
for b
1, b
2 being
Nat holds
not ( not b
1 <= b
2 & not b
1 = b
2 + 1 & not b
2 + 2
<= b
1 )
theorem Th5: :: MIDSP_3:5
for b
1, b
2, b
3, b
4 being
Nat holds
not ( b
1 in (Seg b2) \ {b3} & b
3 = b
4 + 1 & not ( 1
<= b
1 & b
1 <= b
4 ) & not ( b
3 + 1
<= b
1 & b
1 <= b
2 ) )
Lemma2:
for b1, b2 being Nat
for b3 being non empty set
for b4 being Element of b3
for b5 being Element of b1 -tuples_on b3 holds
( b2 in Seg b1 implies (b5 +* b2,b4) . b2 = b4 )
Lemma3:
for b1, b2 being Nat
for b3 being non empty set
for b4 being Element of b3
for b5 being Element of b1 -tuples_on b3
for b6 being Nat holds
( b6 in (dom b5) \ {b2} implies (b5 +* b2,b4) . b6 = b5 . b6 )
E4:
now
let c
1 be
Nat;
let c
2 be
MidSp;
let c
3 be
Function of
(c1 + 2) -tuples_on the
carrier of c
2,the
carrier of c
2;
set c
4 =
ReperAlgebraStr(# the
carrier of c
2,the
MIDPOINT of c
2,c
3 #);
thus
ReperAlgebraStr(# the
carrier of c
2,the
MIDPOINT of c
2,c
3 #) is
MidSp-like
proof
let c
5, c
6, c
7, c
8 be
Element of
ReperAlgebraStr(# the
carrier of c
2,the
MIDPOINT of c
2,c
3 #);
:: according to MIDSP_1:def 4
reconsider c
9 = c
5, c
10 = c
6, c
11 = c
7, c
12 = c
8 as
Element of c
2 ;
E5:
for b
1, b
2 being
Element of
ReperAlgebraStr(# the
carrier of c
2,the
MIDPOINT of c
2,c
3 #)
for b
3, b
4 being
Element of c
2 holds
( b
1 = b
3 & b
2 = b
4 implies b
1 @ b
2 = b
3 @ b
4 )
;
thus c
5 @ c
5 =
c
9 @ c
9
.=
c
5
by MIDSP_1:def 4
;
thus c
5 @ c
6 =
c
10 @ c
9
by E5
.=
c
6 @ c
5
;
thus (c5 @ c6) @ (c7 @ c8) =
(c9 @ c10) @ (c11 @ c12)
.=
(c9 @ c11) @ (c10 @ c12)
by MIDSP_1:def 4
.=
(c5 @ c7) @ (c6 @ c8)
;
consider c
13 being
Element of c
2 such that E6:
c
13 @ c
9 = c
10
by MIDSP_1:def 4;
reconsider c
14 = c
13 as
Element of
ReperAlgebraStr(# the
carrier of c
2,the
MIDPOINT of c
2,c
3 #) ;
take
c
14
;
thus
c
14 @ c
5 = c
6
by E6;
end;
end;
:: deftheorem Def1 MIDSP_3:def 1 :
canceled;
:: deftheorem Def2 defines *' MIDSP_3:def 2 :
theorem Th6: :: MIDSP_3:6
canceled;
theorem Th7: :: MIDSP_3:7
:: deftheorem Def3 MIDSP_3:def 3 :
canceled;
:: deftheorem Def4 defines Nat MIDSP_3:def 4 :
for b
1, b
2 being
Nat holds
( b
2 is
Nat of b
1 iff ( 1
<= b
2 & b
2 <= b
1 + 1 ) );
theorem Th8: :: MIDSP_3:8
for b
1, b
2 being
Nat holds
( b
1 is
Nat of b
2 iff b
1 in Seg (b2 + 1) )
theorem Th9: :: MIDSP_3:9
canceled;
theorem Th10: :: MIDSP_3:10
for b
1, b
2 being
Nat holds
( b
1 <= b
2 implies b
1 + 1 is
Nat of b
2 )
theorem Th11: :: MIDSP_3:11
theorem Th12: :: MIDSP_3:12
:: deftheorem Def5 defines being_invariance MIDSP_3:def 5 :
:: deftheorem Def6 defines has_property_of_zero_in MIDSP_3:def 6 :
:: deftheorem Def7 defines is_semi_additive_in MIDSP_3:def 7 :
theorem Th13: :: MIDSP_3:13
:: deftheorem Def8 defines is_additive_in MIDSP_3:def 8 :
:: deftheorem Def9 defines is_alternative_in MIDSP_3:def 9 :
theorem Th14: :: MIDSP_3:14
theorem Th15: :: MIDSP_3:15
theorem Th16: :: MIDSP_3:16
definition
let c
1 be
Nat;
let c
2 be non
empty MidSp-like ReperAlgebraStr of c
1 + 2;
let c
3 be
ATLAS of c
2;
let c
4 be
Point of c
2;
let c
5 be
Tuple of
(c1 + 1),c
3;
canceled;func c
4,c
5 . c
3 -> Tuple of
(a1 + 1),a
2 means :
Def11:
:: MIDSP_3:def 11
for b
1 being
Nat of a
1 holds a
6 . b
1 = a
4,
(a5 . b1) . a
3;
existence
ex b1 being Tuple of (c1 + 1),c2 st
for b2 being Nat of c1 holds b1 . b2 = c4,(c5 . b2) . c3
uniqueness
for b1, b2 being Tuple of (c1 + 1),c2 holds
( ( for b3 being Nat of c1 holds b1 . b3 = c4,(c5 . b3) . c3 ) & ( for b3 being Nat of c1 holds b2 . b3 = c4,(c5 . b3) . c3 ) implies b1 = b2 )
end;
:: deftheorem Def10 MIDSP_3:def 10 :
canceled;
:: deftheorem Def11 defines . MIDSP_3:def 11 :
definition
let c
1 be
Nat;
let c
2 be non
empty MidSp-like ReperAlgebraStr of c
1 + 2;
let c
3 be
ATLAS of c
2;
let c
4 be
Point of c
2;
let c
5 be
Tuple of
(c1 + 1),c
2;
func c
3 . c
4,c
5 -> Tuple of
(a1 + 1),a
3 means :
Def12:
:: MIDSP_3:def 12
for b
1 being
Nat of a
1 holds a
6 . b
1 = a
3 . a
4,
(a5 . b1);
existence
ex b1 being Tuple of (c1 + 1),c3 st
for b2 being Nat of c1 holds b1 . b2 = c3 . c4,(c5 . b2)
uniqueness
for b1, b2 being Tuple of (c1 + 1),c3 holds
( ( for b3 being Nat of c1 holds b1 . b3 = c3 . c4,(c5 . b3) ) & ( for b3 being Nat of c1 holds b2 . b3 = c3 . c4,(c5 . b3) ) implies b1 = b2 )
end;
:: deftheorem Def12 defines . MIDSP_3:def 12 :
theorem Th17: :: MIDSP_3:17
theorem Th18: :: MIDSP_3:18
theorem Th19: :: MIDSP_3:19
:: deftheorem Def13 defines Phi MIDSP_3:def 13 :
theorem Th20: :: MIDSP_3:20
theorem Th21: :: MIDSP_3:21
theorem Th22: :: MIDSP_3:22
theorem Th23: :: MIDSP_3:23
canceled;
theorem Th24: :: MIDSP_3:24
for b
1 being
Nat holds
1 is
Nat of b
1
:: deftheorem Def14 defines ReperAlgebra MIDSP_3:def 14 :
theorem Th25: :: MIDSP_3:25
:: deftheorem Def15 defines Phi MIDSP_3:def 15 :
Lemma28:
for b1 being Nat
for b2 being ReperAlgebra of b1
for b3 being Point of b2
for b4 being Tuple of (b1 + 1),b2
for b5 being ATLAS of b2
for b6 being Tuple of (b1 + 1),b5 holds
( b5 . b3,b4 = b6 implies Phi b6 = b5 . b3,(*' b3,b4) )
Lemma29:
for b1 being Nat
for b2 being ReperAlgebra of b1
for b3 being Point of b2
for b4 being Tuple of (b1 + 1),b2
for b5 being ATLAS of b2
for b6 being Tuple of (b1 + 1),b5 holds
( b3,b6 . b5 = b4 implies Phi b6 = b5 . b3,(*' b3,b4) )
theorem Th26: :: MIDSP_3:26
for b
1 being
Natfor b
2 being
ReperAlgebra of b
1for b
3, b
4 being
Point of b
2for b
5 being
Tuple of
(b1 + 1),b
2for b
6 being
ATLAS of b
2for b
7 being
Vector of b
6for b
8 being
Tuple of
(b1 + 1),b
6 holds
( b
6 . b
3,b
5 = b
8 & b
6 . b
3,b
4 = b
7 &
Phi b
8 = b
7 implies
*' b
3,b
5 = b
4 )
theorem Th27: :: MIDSP_3:27
for b
1 being
Natfor b
2 being
ReperAlgebra of b
1for b
3, b
4 being
Point of b
2for b
5 being
Tuple of
(b1 + 1),b
2for b
6 being
ATLAS of b
2for b
7 being
Vector of b
6for b
8 being
Tuple of
(b1 + 1),b
6 holds
( b
3,b
8 . b
6 = b
5 & b
3,b
7 . b
6 = b
4 &
*' b
3,b
5 = b
4 implies
Phi b
8 = b
7 )
theorem Th28: :: MIDSP_3:28
for b
1 being
Natfor b
2 being
Nat of b
1for b
3 being
ReperAlgebra of b
1for b
4, b
5 being
Point of b
3for b
6 being
Tuple of
(b1 + 1),b
3for b
7 being
ATLAS of b
3for b
8 being
Vector of b
7for b
9 being
Tuple of
(b1 + 1),b
7 holds
( b
7 . b
4,b
6 = b
9 & b
7 . b
4,b
5 = b
8 implies b
7 . b
4,
(b6 +* b2,b5) = b
9 +* b
2,b
8 )
theorem Th29: :: MIDSP_3:29
for b
1 being
Natfor b
2 being
Nat of b
1for b
3 being
ReperAlgebra of b
1for b
4, b
5 being
Point of b
3for b
6 being
Tuple of
(b1 + 1),b
3for b
7 being
ATLAS of b
3for b
8 being
Vector of b
7for b
9 being
Tuple of
(b1 + 1),b
7 holds
( b
4,b
9 . b
7 = b
6 & b
4,b
8 . b
7 = b
5 implies b
4,
(b9 +* b2,b8) . b
7 = b
6 +* b
2,b
5 )
theorem Th30: :: MIDSP_3:30
theorem Th31: :: MIDSP_3:31
theorem Th32: :: MIDSP_3:32
Lemma36:
for b1 being Nat
for b2 being Nat of b1
for b3 being ReperAlgebra of b1
for b4 being ATLAS of b3 holds
( b3 is_semi_additive_in b2 implies for b5, b6, b7 being Point of b3
for b8 being Tuple of (b1 + 1),b3 holds
( b5 @ b7 = (b8 . b2) @ b6 implies ( *' b5,(b8 +* b2,((b8 . b2) @ b6)) = (*' b5,b8) @ (*' b5,(b8 +* b2,b6)) iff b4 . b5,(*' b5,(b8 +* b2,b7)) = (b4 . b5,(*' b5,b8)) + (b4 . b5,(*' b5,(b8 +* b2,b6))) ) ) )
Lemma37:
for b1 being Nat
for b2 being Nat of b1
for b3 being ReperAlgebra of b1
for b4 being ATLAS of b3 holds
( ( for b5 being Tuple of (b1 + 1),b4
for b6 being Vector of b4 holds Phi (b5 +* b2,((b5 . b2) + b6)) = (Phi b5) + (Phi (b5 +* b2,b6)) ) implies b3 is_semi_additive_in b2 )
theorem Th33: :: MIDSP_3:33
theorem Th34: :: MIDSP_3:34
for b
1 being
Natfor b
2 being
Nat of b
1for b
3 being
ReperAlgebra of b
1for b
4 being
Point of b
3for b
5 being
Tuple of
(b1 + 1),b
3for b
6 being
ATLAS of b
3for b
7 being
Tuple of
(b1 + 1),b
6 holds
( b
6 . b
4,b
5 = b
7 & b
2 <= b
1 implies b
6 . b
4,
(b5 +* (b2 + 1),(b5 . b2)) = b
7 +* (b2 + 1),
(b7 . b2) )
theorem Th35: :: MIDSP_3:35
for b
1 being
Natfor b
2 being
Nat of b
1for b
3 being
ReperAlgebra of b
1for b
4 being
Point of b
3for b
5 being
Tuple of
(b1 + 1),b
3for b
6 being
ATLAS of b
3for b
7 being
Tuple of
(b1 + 1),b
6 holds
( b
4,b
7 . b
6 = b
5 & b
2 <= b
1 implies b
4,
(b7 +* (b2 + 1),(b7 . b2)) . b
6 = b
5 +* (b2 + 1),
(b5 . b2) )
theorem Th36: :: MIDSP_3:36