:: AFVECT01 semantic presentation
Lemma1:
for b1 being WeakAffVect
for b2, b3, b4 being Element of b1 holds
( b2,b3 '||' b3,b4 & b2 <> b4 implies b2,b3 // b3,b4 )
Lemma2:
for b1 being WeakAffVect
for b2, b3 being Element of b1 holds
( b2,b3 // b3,b2 iff ex b4, b5 being Element of b1 st
( b2,b3 '||' b4,b5 & b2,b4 '||' b4,b3 & b2,b5 '||' b5,b3 ) )
Lemma3:
for b1 being WeakAffVect
for b2, b3, b4, b5 being Element of b1 holds
( b2,b3 '||' b4,b5 implies b3,b2 '||' b4,b5 )
Lemma4:
for b1 being WeakAffVect
for b2, b3 being Element of b1 holds b2,b3 '||' b3,b2
Lemma5:
for b1 being WeakAffVect
for b2, b3, b4 being Element of b1 holds
( b2,b3 '||' b4,b4 implies b2 = b3 )
Lemma6:
for b1 being WeakAffVect
for b2, b3, b4, b5, b6, b7 being Element of b1 holds
( b2,b3 '||' b6,b7 & b4,b5 '||' b6,b7 implies b2,b3 '||' b4,b5 )
Lemma7:
for b1 being WeakAffVect
for b2, b3 being Element of b1 holds
ex b4 being Element of b1 st b2,b4 '||' b4,b3
Lemma8:
for b1 being WeakAffVect
for b2, b3, b4, b5, b6 being Element of b1 holds
( b2 <> b3 & b4 <> b5 & b6,b2 '||' b6,b3 & b6,b4 '||' b6,b5 implies b2,b4 '||' b3,b5 )
Lemma9:
for b1 being WeakAffVect
for b2, b3 being Element of b1 holds
not ( not b2 = b3 & ( for b4 being Element of b1 holds
( not ( b2 <> b4 & b2,b3 '||' b3,b4 ) & ( for b5, b6 being Element of b1 holds
not ( b5 <> b6 & b2,b3 '||' b5,b6 & b2,b5 '||' b5,b3 & b2,b6 '||' b6,b3 ) ) ) ) )
Lemma10:
for b1 being WeakAffVect
for b2, b3, b4, b5, b6, b7 being Element of b1 holds
( b2,b3 '||' b3,b7 & b3,b4 '||' b5,b6 & b3,b5 '||' b5,b4 & b3,b6 '||' b6,b4 implies b2,b4 '||' b4,b7 )
Lemma11:
for b1 being WeakAffVect
for b2, b3, b4, b5 being Element of b1 holds
not ( b2 <> b5 & b3 <> b4 & b2,b3 '||' b3,b5 & b2,b4 '||' b4,b5 & ( for b6, b7 being Element of b1 holds
not ( b6 <> b7 & b3,b4 '||' b6,b7 & b3,b6 '||' b6,b4 & b3,b7 '||' b7,b4 ) ) )
Lemma12:
for b1 being WeakAffVect
for b2, b3, b4, b5, b6, b7, b8 being Element of b1 holds
not ( b2,b3 '||' b5,b6 & b2,b4 '||' b7,b8 & b2,b5 '||' b5,b3 & b2,b7 '||' b7,b4 & b2,b6 '||' b6,b3 & b2,b8 '||' b8,b4 & ( for b9, b10 being Element of b1 holds
not ( b3,b4 '||' b9,b10 & b3,b9 '||' b9,b4 & b3,b10 '||' b10,b4 ) ) )
consider c1 being WeakAffVect;
set c2 = the carrier of c1;
set c3 = [:the carrier of c1,the carrier of c1:];
defpred S1[ set , set ] means ex b1, b2, b3, b4 being Element of the carrier of c1 st
( a1 = [b1,b2] & a2 = [b3,b4] & b1,b2 '||' b3,b4 );
consider c4 being Relation of [:the carrier of c1,the carrier of c1:],[:the carrier of c1,the carrier of c1:] such that
Lemma13:
for b1, b2 being set holds
( [b1,b2] in c4 iff ( b1 in [:the carrier of c1,the carrier of c1:] & b2 in [:the carrier of c1,the carrier of c1:] & S1[b1,b2] ) )
from RELSET_1:sch 1();
Lemma14:
for b1, b2, b3, b4 being Element of the carrier of c1 holds
( [[b1,b2],[b3,b4]] in c4 iff b1,b2 '||' b3,b4 )
set c5 = AffinStruct(# the carrier of c1,c4 #);
Lemma15:
for b1, b2, b3, b4 being Element of c1
for b5, b6, b7, b8 being Element of the carrier of AffinStruct(# the carrier of c1,c4 #) holds
( b1 = b5 & b2 = b6 & b3 = b7 & b4 = b8 implies ( b1,b2 '||' b3,b4 iff b5,b6 // b7,b8 ) )
E16:
now
thus
not for b
1, b
2 being
Element of
AffinStruct(# the
carrier of c
1,c
4 #) holds not b
1 <> b
2
by REALSET2:def 7;
thus
for b
1, b
2 being
Element of
AffinStruct(# the
carrier of c
1,c
4 #) holds b
1,b
2 // b
2,b
1
thus
for b
1, b
2 being
Element of
AffinStruct(# the
carrier of c
1,c
4 #) holds
( b
1,b
2 // b
1,b
1 implies b
1 = b
2 )
thus
for b
1, b
2, b
3, b
4, b
5, b
6 being
Element of
AffinStruct(# the
carrier of c
1,c
4 #) holds
( b
1,b
2 // b
5,b
6 & b
3,b
4 // b
5,b
6 implies b
1,b
2 // b
3,b
4 )
proof
let c
6, c
7, c
8, c
9, c
10, c
11 be
Element of
AffinStruct(# the
carrier of c
1,c
4 #);
assume E17:
( c
6,c
7 // c
10,c
11 & c
8,c
9 // c
10,c
11 )
;
reconsider c
12 = c
6, c
13 = c
7, c
14 = c
8, c
15 = c
9, c
16 = c
10, c
17 = c
11 as
Element of c
1 ;
( c
12,c
13 '||' c
16,c
17 & c
14,c
15 '||' c
16,c
17 )
by E17, Lemma15;
then
c
12,c
13 '||' c
14,c
15
by Lemma6;
hence
c
6,c
7 // c
8,c
9
by Lemma15;
end;
thus
for b
1, b
2 being
Element of
AffinStruct(# the
carrier of c
1,c
4 #) holds
ex b
3 being
Element of the
carrier of
AffinStruct(# the
carrier of c
1,c
4 #) st b
1,b
3 // b
3,b
2
proof
let c
6, c
7 be
Element of
AffinStruct(# the
carrier of c
1,c
4 #);
reconsider c
8 = c
6, c
9 = c
7 as
Element of c
1 ;
consider c
10 being
Element of c
1 such that E17:
c
8,c
10 '||' c
10,c
9
by Lemma7;
reconsider c
11 = c
10 as
Element of
AffinStruct(# the
carrier of c
1,c
4 #) ;
c
6,c
11 // c
11,c
7
by E17, Lemma15;
hence
ex b
1 being
Element of the
carrier of
AffinStruct(# the
carrier of c
1,c
4 #) st c
6,b
1 // b
1,c
7
;
end;
thus
for b
1, b
2, b
3, b
4, b
5 being
Element of
AffinStruct(# the
carrier of c
1,c
4 #) holds
( b
1 <> b
2 & b
3 <> b
4 & b
5,b
1 // b
5,b
2 & b
5,b
3 // b
5,b
4 implies b
1,b
3 // b
2,b
4 )
proof
let c
6, c
7, c
8, c
9, c
10 be
Element of
AffinStruct(# the
carrier of c
1,c
4 #);
assume that E17:
( c
6 <> c
7 & c
8 <> c
9 )
and E18:
( c
10,c
6 // c
10,c
7 & c
10,c
8 // c
10,c
9 )
;
reconsider c
11 = c
6, c
12 = c
7, c
13 = c
8, c
14 = c
9, c
15 = c
10 as
Element of c
1 ;
( c
15,c
11 '||' c
15,c
12 & c
15,c
13 '||' c
15,c
14 )
by E18, Lemma15;
then
c
11,c
13 '||' c
12,c
14
by E17, Lemma8;
hence
c
6,c
8 // c
7,c
9
by Lemma15;
end;
thus
for b
1, b
2 being
Element of
AffinStruct(# the
carrier of c
1,c
4 #) holds
not ( not b
1 = b
2 & ( for b
3 being
Element of
AffinStruct(# the
carrier of c
1,c
4 #) holds
( not ( b
1 <> b
3 & b
1,b
2 // b
2,b
3 ) & ( for b
4, b
5 being
Element of
AffinStruct(# the
carrier of c
1,c
4 #) holds
not ( b
4 <> b
5 & b
1,b
2 // b
4,b
5 & b
1,b
4 // b
4,b
2 & b
1,b
5 // b
5,b
2 ) ) ) ) )
proof
let c
6, c
7 be
Element of
AffinStruct(# the
carrier of c
1,c
4 #);
assume E17:
not c
6 = c
7
;
reconsider c
8 = c
6, c
9 = c
7 as
Element of c
1 ;
now
given c
10, c
11 being
Element of c
1 such that E19:
( c
10 <> c
11 & c
8,c
9 '||' c
10,c
11 & c
8,c
10 '||' c
10,c
9 & c
8,c
11 '||' c
11,c
9 )
;
reconsider c
12 = c
10, c
13 = c
11 as
Element of
AffinStruct(# the
carrier of c
1,c
4 #) ;
( c
12 <> c
13 & c
6,c
7 // c
12,c
13 & c
6,c
12 // c
12,c
7 & c
6,c
13 // c
13,c
7 )
by E19, Lemma15;
hence
ex b
1, b
2 being
Element of
AffinStruct(# the
carrier of c
1,c
4 #) st
( b
1 <> b
2 & c
6,c
7 // b
1,b
2 & c
6,b
1 // b
1,c
7 & c
6,b
2 // b
2,c
7 )
;
end;
hence
not for b
1 being
Element of
AffinStruct(# the
carrier of c
1,c
4 #) holds
( not ( c
6 <> b
1 & c
6,c
7 // c
7,b
1 ) & ( for b
2, b
3 being
Element of
AffinStruct(# the
carrier of c
1,c
4 #) holds
not ( b
2 <> b
3 & c
6,c
7 // b
2,b
3 & c
6,b
2 // b
2,c
7 & c
6,b
3 // b
3,c
7 ) ) )
by E17, E18, Lemma9;
end;
thus
for b
1, b
2, b
3, b
4, b
5, b
6 being
Element of
AffinStruct(# the
carrier of c
1,c
4 #) holds
( b
1,b
2 // b
2,b
6 & b
2,b
3 // b
4,b
5 & b
2,b
4 // b
4,b
3 & b
2,b
5 // b
5,b
3 implies b
1,b
3 // b
3,b
6 )
proof
let c
6, c
7, c
8, c
9, c
10, c
11 be
Element of
AffinStruct(# the
carrier of c
1,c
4 #);
assume E17:
( c
6,c
7 // c
7,c
11 & c
7,c
8 // c
9,c
10 & c
7,c
9 // c
9,c
8 & c
7,c
10 // c
10,c
8 )
;
reconsider c
12 = c
6, c
13 = c
7, c
14 = c
8, c
15 = c
9, c
16 = c
10, c
17 = c
11 as
Element of c
1 ;
( c
12,c
13 '||' c
13,c
17 & c
13,c
14 '||' c
15,c
16 & c
13,c
15 '||' c
15,c
14 & c
13,c
16 '||' c
16,c
14 )
by E17, Lemma15;
then
c
12,c
14 '||' c
14,c
17
by Lemma10;
hence
c
6,c
8 // c
8,c
11
by Lemma15;
end;
thus
for b
1, b
2, b
3, b
4 being
Element of
AffinStruct(# the
carrier of c
1,c
4 #) holds
not ( b
1 <> b
4 & b
2 <> b
3 & b
1,b
2 // b
2,b
4 & b
1,b
3 // b
3,b
4 & ( for b
5, b
6 being
Element of
AffinStruct(# the
carrier of c
1,c
4 #) holds
not ( b
5 <> b
6 & b
2,b
3 // b
5,b
6 & b
2,b
5 // b
5,b
3 & b
2,b
6 // b
6,b
3 ) ) )
proof
let c
6, c
7, c
8, c
9 be
Element of
AffinStruct(# the
carrier of c
1,c
4 #);
assume that E17:
( c
6 <> c
9 & c
7 <> c
8 )
and E18:
( c
6,c
7 // c
7,c
9 & c
6,c
8 // c
8,c
9 )
;
reconsider c
10 = c
6, c
11 = c
7, c
12 = c
8, c
13 = c
9 as
Element of the
carrier of c
1 ;
( c
10,c
11 '||' c
11,c
13 & c
10,c
12 '||' c
12,c
13 )
by E18, Lemma15;
then consider c
14, c
15 being
Element of c
1 such that E19:
( c
14 <> c
15 & c
11,c
12 '||' c
14,c
15 & c
11,c
14 '||' c
14,c
12 & c
11,c
15 '||' c
15,c
12 )
by E17, Lemma11;
reconsider c
16 = c
14, c
17 = c
15 as
Element of
AffinStruct(# the
carrier of c
1,c
4 #) ;
( c
16 <> c
17 & c
7,c
8 // c
16,c
17 & c
7,c
16 // c
16,c
8 & c
7,c
17 // c
17,c
8 )
by E19, Lemma15;
hence
ex b
1, b
2 being
Element of
AffinStruct(# the
carrier of c
1,c
4 #) st
( b
1 <> b
2 & c
7,c
8 // b
1,b
2 & c
7,b
1 // b
1,c
8 & c
7,b
2 // b
2,c
8 )
;
end;
thus
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
Element of
AffinStruct(# the
carrier of c
1,c
4 #) holds
not ( b
1,b
2 // b
4,b
5 & b
1,b
3 // b
6,b
7 & b
1,b
4 // b
4,b
2 & b
1,b
6 // b
6,b
3 & b
1,b
5 // b
5,b
2 & b
1,b
7 // b
7,b
3 & ( for b
8, b
9 being
Element of
AffinStruct(# the
carrier of c
1,c
4 #) holds
not ( b
2,b
3 // b
8,b
9 & b
2,b
8 // b
8,b
3 & b
2,b
9 // b
9,b
3 ) ) )
proof
let c
6, c
7, c
8, c
9, c
10, c
11, c
12 be
Element of
AffinStruct(# the
carrier of c
1,c
4 #);
assume E17:
( c
6,c
7 // c
9,c
10 & c
6,c
8 // c
11,c
12 & c
6,c
9 // c
9,c
7 & c
6,c
11 // c
11,c
8 & c
6,c
10 // c
10,c
7 & c
6,c
12 // c
12,c
8 )
;
reconsider c
13 = c
6, c
14 = c
7, c
15 = c
8, c
16 = c
9, c
17 = c
10, c
18 = c
11, c
19 = c
12 as
Element of the
carrier of c
1 ;
( c
13,c
14 '||' c
16,c
17 & c
13,c
15 '||' c
18,c
19 & c
13,c
16 '||' c
16,c
14 & c
13,c
18 '||' c
18,c
15 & c
13,c
17 '||' c
17,c
14 & c
13,c
19 '||' c
19,c
15 )
by E17, Lemma15;
then consider c
20, c
21 being
Element of c
1 such that E18:
( c
14,c
15 '||' c
20,c
21 & c
14,c
20 '||' c
20,c
15 & c
14,c
21 '||' c
21,c
15 )
by Lemma12;
reconsider c
22 = c
20, c
23 = c
21 as
Element of
AffinStruct(# the
carrier of c
1,c
4 #) ;
( c
7,c
8 // c
22,c
23 & c
7,c
22 // c
22,c
8 & c
7,c
23 // c
23,c
8 )
by E18, Lemma15;
hence
ex b
1, b
2 being
Element of
AffinStruct(# the
carrier of c
1,c
4 #) st
( c
7,c
8 // b
1,b
2 & c
7,b
1 // b
1,c
8 & c
7,b
2 // b
2,c
8 )
;
end;
end;
definition
let c
6 be non
empty AffinStruct ;
canceled;attr a
1 is
WeakAffSegm-like means :
Def2:
:: AFVECT01:def 2
( ( for b
1, b
2 being
Element of a
1 holds b
1,b
2 // b
2,b
1 ) & ( for b
1, b
2 being
Element of a
1 holds
( b
1,b
2 // b
1,b
1 implies b
1 = b
2 ) ) & ( for b
1, b
2, b
3, b
4, b
5, b
6 being
Element of a
1 holds
( b
1,b
2 // b
5,b
6 & b
3,b
4 // b
5,b
6 implies b
1,b
2 // b
3,b
4 ) ) & ( for b
1, b
2 being
Element of a
1 holds
ex b
3 being
Element of a
1 st b
1,b
3 // b
3,b
2 ) & ( for b
1, b
2, b
3, b
4, b
5 being
Element of a
1 holds
( b
1 <> b
2 & b
3 <> b
4 & b
5,b
1 // b
5,b
2 & b
5,b
3 // b
5,b
4 implies b
1,b
3 // b
2,b
4 ) ) & ( for b
1, b
2 being
Element of a
1 holds
not ( not b
1 = b
2 & ( for b
3 being
Element of a
1 holds
( not ( b
1 <> b
3 & b
1,b
2 // b
2,b
3 ) & ( for b
4, b
5 being
Element of a
1 holds
not ( b
4 <> b
5 & b
1,b
2 // b
4,b
5 & b
1,b
4 // b
4,b
2 & b
1,b
5 // b
5,b
2 ) ) ) ) ) ) & ( for b
1, b
2, b
3, b
4, b
5, b
6 being
Element of a
1 holds
( b
1,b
2 // b
2,b
6 & b
2,b
3 // b
4,b
5 & b
2,b
4 // b
4,b
3 & b
2,b
5 // b
5,b
3 implies b
1,b
3 // b
3,b
6 ) ) & ( for b
1, b
2, b
3, b
4 being
Element of a
1 holds
not ( b
1 <> b
4 & b
2 <> b
3 & b
1,b
2 // b
2,b
4 & b
1,b
3 // b
3,b
4 & ( for b
5, b
6 being
Element of a
1 holds
not ( b
5 <> b
6 & b
2,b
3 // b
5,b
6 & b
2,b
5 // b
5,b
3 & b
2,b
6 // b
6,b
3 ) ) ) ) & ( for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
Element of a
1 holds
not ( b
1,b
2 // b
4,b
5 & b
1,b
3 // b
6,b
7 & b
1,b
4 // b
4,b
2 & b
1,b
6 // b
6,b
3 & b
1,b
5 // b
5,b
2 & b
1,b
7 // b
7,b
3 & ( for b
8, b
9 being
Element of a
1 holds
not ( b
2,b
3 // b
8,b
9 & b
2,b
8 // b
8,b
3 & b
2,b
9 // b
9,b
3 ) ) ) ) );
end;
:: deftheorem Def1 AFVECT01:def 1 :
canceled;
:: deftheorem Def2 defines WeakAffSegm-like AFVECT01:def 2 :
for b
1 being non
empty AffinStruct holds
( b
1 is
WeakAffSegm-like iff ( ( for b
2, b
3 being
Element of b
1 holds b
2,b
3 // b
3,b
2 ) & ( for b
2, b
3 being
Element of b
1 holds
( b
2,b
3 // b
2,b
2 implies b
2 = b
3 ) ) & ( for b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( b
2,b
3 // b
6,b
7 & b
4,b
5 // b
6,b
7 implies b
2,b
3 // b
4,b
5 ) ) & ( for b
2, b
3 being
Element of b
1 holds
ex b
4 being
Element of b
1 st b
2,b
4 // b
4,b
3 ) & ( for b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
( b
2 <> b
3 & b
4 <> b
5 & b
6,b
2 // b
6,b
3 & b
6,b
4 // b
6,b
5 implies b
2,b
4 // b
3,b
5 ) ) & ( for b
2, b
3 being
Element of b
1 holds
not ( not b
2 = b
3 & ( for b
4 being
Element of b
1 holds
( not ( b
2 <> b
4 & b
2,b
3 // b
3,b
4 ) & ( for b
5, b
6 being
Element of b
1 holds
not ( b
5 <> b
6 & b
2,b
3 // b
5,b
6 & b
2,b
5 // b
5,b
3 & b
2,b
6 // b
6,b
3 ) ) ) ) ) ) & ( for b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( b
2,b
3 // b
3,b
7 & b
3,b
4 // b
5,b
6 & b
3,b
5 // b
5,b
4 & b
3,b
6 // b
6,b
4 implies b
2,b
4 // b
4,b
7 ) ) & ( for b
2, b
3, b
4, b
5 being
Element of b
1 holds
not ( b
2 <> b
5 & b
3 <> b
4 & b
2,b
3 // b
3,b
5 & b
2,b
4 // b
4,b
5 & ( for b
6, b
7 being
Element of b
1 holds
not ( b
6 <> b
7 & b
3,b
4 // b
6,b
7 & b
3,b
6 // b
6,b
4 & b
3,b
7 // b
7,b
4 ) ) ) ) & ( for b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of b
1 holds
not ( b
2,b
3 // b
5,b
6 & b
2,b
4 // b
7,b
8 & b
2,b
5 // b
5,b
3 & b
2,b
7 // b
7,b
4 & b
2,b
6 // b
6,b
3 & b
2,b
8 // b
8,b
4 & ( for b
9, b
10 being
Element of b
1 holds
not ( b
3,b
4 // b
9,b
10 & b
3,b
9 // b
9,b
4 & b
3,b
10 // b
10,b
4 ) ) ) ) ) );
theorem Th1: :: AFVECT01:1
theorem Th2: :: AFVECT01:2
for b
1 being
WeakAffSegmfor b
2, b
3, b
4, b
5 being
Element of b
1 holds
( b
2,b
3 // b
4,b
5 implies b
4,b
5 // b
2,b
3 )
theorem Th3: :: AFVECT01:3
for b
1 being
WeakAffSegmfor b
2, b
3, b
4, b
5 being
Element of b
1 holds
( b
2,b
3 // b
4,b
5 implies b
2,b
3 // b
5,b
4 )
theorem Th4: :: AFVECT01:4
for b
1 being
WeakAffSegmfor b
2, b
3, b
4, b
5 being
Element of b
1 holds
( b
2,b
3 // b
4,b
5 implies b
3,b
2 // b
4,b
5 )
theorem Th5: :: AFVECT01:5
theorem Th6: :: AFVECT01:6
theorem Th7: :: AFVECT01:7
for b
1 being
WeakAffSegmfor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
( b
2,b
3 // b
4,b
5 & b
2,b
3 // b
3,b
6 & b
2,b
4 // b
4,b
3 & b
2,b
5 // b
5,b
3 implies b
2 = b
6 )
theorem Th8: :: AFVECT01:8
for b
1 being
WeakAffSegmfor b
2, b
3, b
4, b
5 being
Element of b
1 holds
not ( b
2,b
3 // b
2,b
4 & b
2,b
5 // b
2,b
4 & not b
5 = b
3 & not b
5 = b
4 & not b
3 = b
4 )
:: deftheorem Def3 AFVECT01:def 3 :
canceled;
:: deftheorem Def4 defines MDist AFVECT01:def 4 :
:: deftheorem Def5 defines Mid AFVECT01:def 5 :
for b
1 being
WeakAffSegmfor b
2, b
3, b
4 being
Element of b
1 holds
(
Mid b
2,b
3,b
4 iff not ( not ( b
2 = b
3 & b
3 = b
4 & b
2 = b
4 ) & not ( b
2 = b
4 &
MDist b
2,b
3 ) & not ( b
2 <> b
4 & b
2,b
3 // b
3,b
4 ) ) );
theorem Th9: :: AFVECT01:9
canceled;
theorem Th10: :: AFVECT01:10
canceled;
theorem Th11: :: AFVECT01:11
theorem Th12: :: AFVECT01:12
theorem Th13: :: AFVECT01:13