:: COLLSP semantic presentation
:: deftheorem Def1 defines Relation3 COLLSP:def 1 :
theorem Th1: :: COLLSP:1
canceled;
theorem Th2: :: COLLSP:2
for
X being
set holds
(
X = {} or ex
a being
set st
(
{a} = X or ex
a,
b being
set st
(
a <> b &
a in X &
b in X ) ) )
:: deftheorem Def2 defines is_collinear COLLSP:def 2 :
set Z = {1};
Lemma29:
1 in {1}
by TARSKI:def 1;
Lemma30:
{[1,1,1]} c= [:{1},{1},{1}:]
reconsider Z = {1} as non empty set by TARSKI:def 1;
reconsider RR = {[1,1,1]} as Relation3 of Z by , ;
reconsider CLS = CollStr(# Z,RR #) as non empty CollStr by STRUCT_0:def 1;
E34:
now
E19:
for
z1,
z2,
z3 being
Point of
CLS holds
[z1,z2,z3] in the
Collinearity of
CLS
let a be
Point of
CLS,
b be
Point of
CLS,
c be
Point of
CLS,
p be
Point of
CLS,
q be
Point of
CLS,
r be
Point of
CLS;
thus
( (
a = b or
a = c or
b = c ) implies
[a,b,c] in the
Collinearity of
CLS )
by ;
thus
(
a <> b &
[a,b,p] in the
Collinearity of
CLS &
[a,b,q] in the
Collinearity of
CLS &
[a,b,r] in the
Collinearity of
CLS implies
[p,q,r] in the
Collinearity of
CLS )
by ;
end;
:: deftheorem Def3 defines reflexive COLLSP:def 3 :
definition
let IT be non
empty CollStr ;
attr a1 is
transitive means :
Def4:
:: COLLSP:def 4
for
a,
b,
p,
q,
r being
Point of
Z st
a <> b &
[a,b,p] in the
Collinearity of
Z &
[a,b,q] in the
Collinearity of
Z &
[a,b,r] in the
Collinearity of
Z holds
[p,q,r] in the
Collinearity of
Z;
end;
:: deftheorem Def4 defines transitive COLLSP:def 4 :
for
IT being non
empty CollStr holds
(
IT is
transitive iff for
a,
b,
p,
q,
r being
Point of
IT st
a <> b &
[a,b,p] in the
Collinearity of
IT &
[a,b,q] in the
Collinearity of
IT &
[a,b,r] in the
Collinearity of
IT holds
[p,q,r] in the
Collinearity of
IT );
theorem Th3: :: COLLSP:3
canceled;
theorem Th4: :: COLLSP:4
canceled;
theorem Th5: :: COLLSP:5
canceled;
theorem Th6: :: COLLSP:6
canceled;
theorem Th7: :: COLLSP:7
theorem Th8: :: COLLSP:8
for
CLSP being
CollSp for
a,
b,
p,
q,
r being
Point of
CLSP st
a <> b &
a,
b,
p is_collinear &
a,
b,
q is_collinear &
a,
b,
r is_collinear holds
p,
q,
r 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
CLSP being
CollSp for
p,
q,
a,
b,
r being
Point of
CLSP st
p <> q &
a,
b,
p is_collinear &
a,
b,
q is_collinear &
p,
q,
r is_collinear holds
a,
b,
r is_collinear
:: deftheorem Def5 defines Line COLLSP:def 5 :
theorem Th15: :: COLLSP:15
canceled;
theorem Th16: :: COLLSP:16
theorem Th17: :: COLLSP:17
set Z = {1,2,3};
set RR = { [i,j,k] where i is Element of NAT , j is Element of NAT , k is Element of NAT : ( ( i = j or j = k or k = i ) & i in {1,2,3} & j in {1,2,3} & k in {1,2,3} ) } ;
Lemma56:
{ [i,j,k] where i is Element of NAT , j is Element of NAT , k is Element of NAT : ( ( i = j or j = k or k = i ) & i in {1,2,3} & j in {1,2,3} & k in {1,2,3} ) } c= [:{1,2,3},{1,2,3},{1,2,3}:]
reconsider Z = {1,2,3} as non empty set by ENUMSET1:def 1;
reconsider RR = { [i,j,k] where i is Element of NAT , j is Element of NAT , k is Element of NAT : ( ( i = j or j = k or k = i ) & i in {1,2,3} & j in {1,2,3} & k in {1,2,3} ) } as Relation3 of Z by , ;
reconsider CLS = CollStr(# Z,RR #) as non empty CollStr by STRUCT_0:def 1;
Lemma57:
for a, b, c being Point of CLS holds
( [a,b,c] in RR iff ( ( a = b or b = c or c = a ) & a in Z & b in Z & c in Z ) )
Lemma58:
for a, b, c, p, q, r being Point of CLS st a <> b & [a,b,p] in the Collinearity of CLS & [a,b,q] in the Collinearity of CLS & [a,b,r] in the Collinearity of CLS holds
[p,q,r] in the Collinearity of CLS
Lemma59:
not for a, b, c being Point of CLS holds a,b,c is_collinear
Lemma60:
CLS 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
theorem Th25: :: COLLSP:25
Lemma70:
for CLSP being proper CollSp
for P being LINE of CLSP
for x being set st x in P holds
x is Point of CLSP
theorem Th26: :: COLLSP:26
theorem Th27: :: COLLSP:27
theorem Th28: :: COLLSP:28
theorem Th29: :: COLLSP:29
theorem Th30: :: COLLSP:30
theorem Th31: :: COLLSP:31