:: COLLSP semantic presentation
:: deftheorem Def1 defines Relation3 COLLSP:def 1 :
theorem Th1: :: COLLSP:1
canceled;
theorem Th2: :: COLLSP:2
for b
1 being
set holds
not ( not b
1 = {} & ( for b
2 being
set holds
( not
{b2} = b
1 & ( for b
3, b
4 being
set holds
not ( b
3 <> b
4 & b
3 in b
1 & b
4 in b
1 ) ) ) ) )
:: deftheorem Def2 defines is_collinear COLLSP:def 2 :
set c1 = {1};
Lemma4:
1 in {1}
by TARSKI:def 1;
Lemma5:
{[1,1,1]} c= [:{1},{1},{1}:]
reconsider c2 = {1} as non empty set by TARSKI:def 1;
reconsider c3 = {[1,1,1]} as Relation3 of c2 by Def1, Lemma5;
reconsider c4 = CollStr(# c2,c3 #) as non empty CollStr by STRUCT_0:def 1;
E6:
now
E7:
for b
1, b
2, b
3 being
Point of c
4 holds
[b1,b2,b3] in the
Collinearity of c
4
let c
5, c
6, c
7, c
8, c
9, c
10 be
Point of c
4;
thus
( not ( not c
5 = c
6 & not c
5 = c
7 & not c
6 = c
7 ) implies
[c5,c6,c7] in the
Collinearity of c
4 )
by E7;
thus
( c
5 <> c
6 &
[c5,c6,c8] in the
Collinearity of c
4 &
[c5,c6,c9] in the
Collinearity of c
4 &
[c5,c6,c10] in the
Collinearity of c
4 implies
[c8,c9,c10] in the
Collinearity of c
4 )
by E7;
end;
:: deftheorem Def3 defines reflexive COLLSP:def 3 :
definition
let c
5 be non
empty CollStr ;
attr a
1 is
transitive means :
Def4:
:: COLLSP:def 4
for b
1, b
2, b
3, b
4, b
5 being
Point of a
1 holds
( b
1 <> b
2 &
[b1,b2,b3] in the
Collinearity of a
1 &
[b1,b2,b4] in the
Collinearity of a
1 &
[b1,b2,b5] in the
Collinearity of a
1 implies
[b3,b4,b5] in the
Collinearity of a
1 );
end;
:: deftheorem Def4 defines transitive COLLSP:def 4 :
for b
1 being non
empty CollStr holds
( b
1 is
transitive iff for b
2, b
3, b
4, b
5, b
6 being
Point of b
1 holds
( b
2 <> b
3 &
[b2,b3,b4] in the
Collinearity of b
1 &
[b2,b3,b5] in the
Collinearity of b
1 &
[b2,b3,b6] in the
Collinearity of b
1 implies
[b4,b5,b6] in the
Collinearity of b
1 ) );
theorem Th3: :: COLLSP:3
canceled;
theorem Th4: :: COLLSP:4
canceled;
theorem Th5: :: COLLSP:5
canceled;
theorem Th6: :: COLLSP:6
canceled;
theorem Th7: :: COLLSP:7
for b
1 being
CollSpfor b
2, b
3, b
4 being
Point of b
1 holds
( not ( not b
2 = b
3 & not b
2 = b
4 & not b
3 = b
4 ) implies b
2,b
3,b
4 is_collinear )
theorem Th8: :: COLLSP:8
for b
1 being
CollSpfor b
2, b
3, b
4, b
5, b
6 being
Point of b
1 holds
( b
2 <> b
3 & b
2,b
3,b
4 is_collinear & b
2,b
3,b
5 is_collinear & b
2,b
3,b
6 is_collinear implies b
4,b
5,b
6 is_collinear )
theorem Th9: :: COLLSP:9
theorem Th10: :: COLLSP:10
theorem Th11: :: COLLSP:11
theorem Th12: :: COLLSP:12
theorem Th13: :: COLLSP:13
theorem Th14: :: COLLSP:14
for b
1 being
CollSpfor b
2, b
3, b
4, b
5, b
6 being
Point of b
1 holds
( b
2 <> b
3 & b
4,b
5,b
2 is_collinear & b
4,b
5,b
3 is_collinear & b
2,b
3,b
6 is_collinear implies b
4,b
5,b
6 is_collinear )
:: deftheorem Def5 defines Line COLLSP:def 5 :
theorem Th15: :: COLLSP:15
canceled;
theorem Th16: :: COLLSP:16
theorem Th17: :: COLLSP:17
set c5 = {1,2,3};
set c6 = { [b1,b2,b3] where B is Nat, B is Nat, B is Nat : ( not ( not b1 = b2 & not b2 = b3 & not b3 = b1 ) & b1 in {1,2,3} & b2 in {1,2,3} & b3 in {1,2,3} ) } ;
Lemma17:
{ [b1,b2,b3] where B is Nat, B is Nat, B is Nat : ( not ( not b1 = b2 & not b2 = b3 & not b3 = b1 ) & b1 in {1,2,3} & b2 in {1,2,3} & b3 in {1,2,3} ) } c= [:{1,2,3},{1,2,3},{1,2,3}:]
reconsider c7 = {1,2,3} as non empty set by ENUMSET1:def 1;
reconsider c8 = { [b1,b2,b3] where B is Nat, B is Nat, B is Nat : ( not ( not b1 = b2 & not b2 = b3 & not b3 = b1 ) & b1 in {1,2,3} & b2 in {1,2,3} & b3 in {1,2,3} ) } as Relation3 of c7 by Def1, Lemma17;
reconsider c9 = CollStr(# c7,c8 #) as non empty CollStr by STRUCT_0:def 1;
Lemma18:
for b1, b2, b3 being Point of c9 holds
( [b1,b2,b3] in c8 iff ( not ( not b1 = b2 & not b2 = b3 & not b3 = b1 ) & b1 in c7 & b2 in c7 & b3 in c7 ) )
Lemma19:
for b1, b2, b3, b4, b5, b6 being Point of c9 holds
( b1 <> b2 & [b1,b2,b4] in the Collinearity of c9 & [b1,b2,b5] in the Collinearity of c9 & [b1,b2,b6] in the Collinearity of c9 implies [b4,b5,b6] in the Collinearity of c9 )
Lemma20:
not for b1, b2, b3 being Point of c9 holds b1,b2,b3 is_collinear
Lemma21:
c9 is CollSp
:: deftheorem Def6 defines proper COLLSP:def 6 :
theorem Th18: :: COLLSP:18
canceled;
theorem Th19: :: COLLSP:19
:: deftheorem Def7 defines LINE COLLSP:def 7 :
theorem Th20: :: COLLSP:20
canceled;
theorem Th21: :: COLLSP:21
canceled;
theorem Th22: :: COLLSP:22
theorem Th23: :: COLLSP:23
theorem Th24: :: COLLSP:24
for b
1 being
proper CollSpfor b
2, b
3 being
Point of b
1 holds
not ( b
2 <> b
3 & ( for b
4 being
LINE of b
1 holds
not ( b
2 in b
4 & b
3 in b
4 ) ) )
theorem Th25: :: COLLSP:25
Lemma26:
for b1 being proper CollSp
for b2 being LINE of b1
for b3 being set holds
( b3 in b2 implies b3 is Point of b1 )
theorem Th26: :: COLLSP:26
theorem Th27: :: COLLSP:27
theorem Th28: :: COLLSP:28
theorem Th29: :: COLLSP:29
for b
1 being
proper CollSpfor b
2, b
3 being
Point of b
1for b
4, b
5 being
LINE of b
1 holds
( b
2 <> b
3 & b
2 in b
4 & b
3 in b
4 & b
2 in b
5 & b
3 in b
5 implies b
4 = b
5 )
theorem Th30: :: COLLSP:30
theorem Th31: :: COLLSP:31