Article COLLSP, MML version 4.99.1005
:: COLLSP:modenot 1 => COLLSP:mode 1
definition
let a1 be set;
mode Relation3 of A1 means
it c= [:a1,a1,a1:];
end;
:: COLLSP:dfs 1
definiens
let a1, a2 be set;
To prove
a2 is Relation3 of a1
it is sufficient to prove
thus a2 c= [:a1,a1,a1:];
:: COLLSP:def 1
theorem
for b1, b2 being set holds
b2 is Relation3 of b1
iff
b2 c= [:b1,b1,b1:];
:: COLLSP:th 2
theorem
for b1 being set
st b1 <> {}
holds ex b2 being set st
({b2} <> b1 implies ex b3, b4 being set st
b3 <> b4 & b3 in b1 & b4 in b1);
:: COLLSP:structnot 1 => COLLSP:struct 1
definition
struct(1-sorted) CollStr(#
carrier -> set,
Collinearity -> Relation3 of the carrier of it
#);
end;
:: COLLSP:attrnot 1 => COLLSP:attr 1
definition
let a1 be CollStr;
attr a1 is strict;
end;
:: COLLSP:exreg 1
registration
cluster strict CollStr;
end;
:: COLLSP:aggrnot 1 => COLLSP:aggr 1
definition
let a1 be set;
let a2 be Relation3 of a1;
aggr CollStr(#a1,a2#) -> strict CollStr;
end;
:: COLLSP:selnot 1 => COLLSP:sel 1
definition
let a1 be CollStr;
sel the Collinearity of a1 -> Relation3 of the carrier of a1;
end;
:: COLLSP:exreg 2
registration
cluster non empty strict CollStr;
end;
:: COLLSP:prednot 1 => COLLSP:pred 1
definition
let a1 be non empty CollStr;
let a2, a3, a4 be Element of the carrier of a1;
pred A2,A3,A4 is_collinear means
[a2,a3,a4] in the Collinearity of a1;
end;
:: COLLSP:dfs 2
definiens
let a1 be non empty CollStr;
let a2, a3, a4 be Element of the carrier of a1;
To prove
a2,a3,a4 is_collinear
it is sufficient to prove
thus [a2,a3,a4] in the Collinearity of a1;
:: COLLSP:def 2
theorem
for b1 being non empty CollStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2,b3,b4 is_collinear
iff
[b2,b3,b4] in the Collinearity of b1;
:: COLLSP:attrnot 2 => COLLSP:attr 2
definition
let a1 be non empty CollStr;
attr a1 is reflexive means
for b1, b2, b3 being Element of the carrier of a1
st (b1 <> b2 & b1 <> b3 implies b2 = b3)
holds [b1,b2,b3] in the Collinearity of a1;
end;
:: COLLSP:dfs 3
definiens
let a1 be non empty CollStr;
To prove
a1 is reflexive
it is sufficient to prove
thus for b1, b2, b3 being Element of the carrier of a1
st (b1 <> b2 & b1 <> b3 implies b2 = b3)
holds [b1,b2,b3] in the Collinearity of a1;
:: COLLSP:def 3
theorem
for b1 being non empty CollStr holds
b1 is reflexive
iff
for b2, b3, b4 being Element of the carrier of b1
st (b2 <> b3 & b2 <> b4 implies b3 = b4)
holds [b2,b3,b4] in the Collinearity of b1;
:: COLLSP:attrnot 3 => COLLSP:attr 3
definition
let a1 be non empty CollStr;
attr a1 is transitive means
for b1, b2, b3, b4, b5 being Element of the carrier of a1
st b1 <> b2 & [b1,b2,b3] in the Collinearity of a1 & [b1,b2,b4] in the Collinearity of a1 & [b1,b2,b5] in the Collinearity of a1
holds [b3,b4,b5] in the Collinearity of a1;
end;
:: COLLSP:dfs 4
definiens
let a1 be non empty CollStr;
To prove
a1 is transitive
it is sufficient to prove
thus for b1, b2, b3, b4, b5 being Element of the carrier of a1
st b1 <> b2 & [b1,b2,b3] in the Collinearity of a1 & [b1,b2,b4] in the Collinearity of a1 & [b1,b2,b5] in the Collinearity of a1
holds [b3,b4,b5] in the Collinearity of a1;
:: COLLSP:def 4
theorem
for b1 being non empty CollStr holds
b1 is transitive
iff
for b2, b3, b4, b5, b6 being Element of the carrier of b1
st b2 <> b3 & [b2,b3,b4] in the Collinearity of b1 & [b2,b3,b5] in the Collinearity of b1 & [b2,b3,b6] in the Collinearity of b1
holds [b4,b5,b6] in the Collinearity of b1;
:: COLLSP:exreg 3
registration
cluster non empty strict reflexive transitive CollStr;
end;
:: COLLSP:modenot 2
definition
mode CollSp is non empty reflexive transitive CollStr;
end;
:: COLLSP:th 7
theorem
for b1 being non empty reflexive transitive CollStr
for b2, b3, b4 being Element of the carrier of b1
st (b2 <> b3 & b2 <> b4 implies b3 = b4)
holds b2,b3,b4 is_collinear;
:: COLLSP:th 8
theorem
for b1 being non empty reflexive transitive CollStr
for b2, b3, b4, b5, b6 being Element of the carrier of b1
st b2 <> b3 & b2,b3,b4 is_collinear & b2,b3,b5 is_collinear & b2,b3,b6 is_collinear
holds b4,b5,b6 is_collinear;
:: COLLSP:th 9
theorem
for b1 being non empty reflexive transitive CollStr
for b2, b3, b4 being Element of the carrier of b1
st b2,b3,b4 is_collinear
holds b3,b2,b4 is_collinear & b2,b4,b3 is_collinear;
:: COLLSP:th 10
theorem
for b1 being non empty reflexive transitive CollStr
for b2, b3 being Element of the carrier of b1 holds
b2,b3,b2 is_collinear;
:: COLLSP:th 11
theorem
for b1 being non empty reflexive transitive CollStr
for b2, b3, b4, b5 being Element of the carrier of b1
st b2 <> b3 & b2,b3,b4 is_collinear & b2,b3,b5 is_collinear
holds b2,b4,b5 is_collinear;
:: COLLSP:th 12
theorem
for b1 being non empty reflexive transitive CollStr
for b2, b3, b4 being Element of the carrier of b1
st b2,b3,b4 is_collinear
holds b3,b2,b4 is_collinear;
:: COLLSP:th 13
theorem
for b1 being non empty reflexive transitive CollStr
for b2, b3, b4 being Element of the carrier of b1
st b2,b3,b4 is_collinear
holds b3,b4,b2 is_collinear;
:: COLLSP:th 14
theorem
for b1 being non empty reflexive transitive CollStr
for b2, b3, b4, b5, b6 being Element of the carrier of b1
st b2 <> b3 & b4,b5,b2 is_collinear & b4,b5,b3 is_collinear & b2,b3,b6 is_collinear
holds b4,b5,b6 is_collinear;
:: COLLSP:funcnot 1 => COLLSP:func 1
definition
let a1 be non empty reflexive transitive CollStr;
let a2, a3 be Element of the carrier of a1;
func Line(A2,A3) -> set equals
{b1 where b1 is Element of the carrier of a1: a2,a3,b1 is_collinear};
end;
:: COLLSP:def 5
theorem
for b1 being non empty reflexive transitive CollStr
for b2, b3 being Element of the carrier of b1 holds
Line(b2,b3) = {b4 where b4 is Element of the carrier of b1: b2,b3,b4 is_collinear};
:: COLLSP:th 16
theorem
for b1 being non empty reflexive transitive CollStr
for b2, b3 being Element of the carrier of b1 holds
b2 in Line(b2,b3) & b3 in Line(b2,b3);
:: COLLSP:th 17
theorem
for b1 being non empty reflexive transitive CollStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2,b3,b4 is_collinear
iff
b4 in Line(b2,b3);
:: COLLSP:attrnot 4 => COLLSP:attr 4
definition
let a1 be non empty CollStr;
attr a1 is proper means
ex b1, b2, b3 being Element of the carrier of a1 st
not b1,b2,b3 is_collinear;
end;
:: COLLSP:dfs 6
definiens
let a1 be non empty CollStr;
To prove
a1 is proper
it is sufficient to prove
thus ex b1, b2, b3 being Element of the carrier of a1 st
not b1,b2,b3 is_collinear;
:: COLLSP:def 6
theorem
for b1 being non empty CollStr holds
b1 is proper
iff
ex b2, b3, b4 being Element of the carrier of b1 st
not b2,b3,b4 is_collinear;
:: COLLSP:exreg 4
registration
cluster non empty strict reflexive transitive proper CollStr;
end;
:: COLLSP:th 19
theorem
for b1 being non empty reflexive transitive proper CollStr
for b2, b3 being Element of the carrier of b1
st b2 <> b3
holds ex b4 being Element of the carrier of b1 st
not b2,b3,b4 is_collinear;
:: COLLSP:modenot 3 => COLLSP:mode 2
definition
let a1 be non empty reflexive transitive proper CollStr;
mode LINE of A1 means
ex b1, b2 being Element of the carrier of a1 st
b1 <> b2 & it = Line(b1,b2);
end;
:: COLLSP:dfs 7
definiens
let a1 be non empty reflexive transitive proper CollStr;
let a2 be set;
To prove
a2 is LINE of a1
it is sufficient to prove
thus ex b1, b2 being Element of the carrier of a1 st
b1 <> b2 & a2 = Line(b1,b2);
:: COLLSP:def 7
theorem
for b1 being non empty reflexive transitive proper CollStr
for b2 being set holds
b2 is LINE of b1
iff
ex b3, b4 being Element of the carrier of b1 st
b3 <> b4 & b2 = Line(b3,b4);
:: COLLSP:th 22
theorem
for b1 being non empty reflexive transitive proper CollStr
for b2, b3 being Element of the carrier of b1
st b2 = b3
holds Line(b2,b3) = the carrier of b1;
:: COLLSP:th 23
theorem
for b1 being non empty reflexive transitive proper CollStr
for b2 being LINE of b1 holds
ex b3, b4 being Element of the carrier of b1 st
b3 <> b4 & b3 in b2 & b4 in b2;
:: COLLSP:th 24
theorem
for b1 being non empty reflexive transitive proper CollStr
for b2, b3 being Element of the carrier of b1
st b2 <> b3
holds ex b4 being LINE of b1 st
b2 in b4 & b3 in b4;
:: COLLSP:th 25
theorem
for b1 being non empty reflexive transitive proper CollStr
for b2, b3, b4 being Element of the carrier of b1
for b5 being LINE of b1
st b2 in b5 & b3 in b5 & b4 in b5
holds b2,b3,b4 is_collinear;
:: COLLSP:th 26
theorem
for b1 being non empty reflexive transitive proper CollStr
for b2, b3 being LINE of b1
st b2 c= b3
holds b2 = b3;
:: COLLSP:th 27
theorem
for b1 being non empty reflexive transitive proper CollStr
for b2, b3 being Element of the carrier of b1
for b4 being LINE of b1
st b2 <> b3 & b2 in b4 & b3 in b4
holds Line(b2,b3) c= b4;
:: COLLSP:th 28
theorem
for b1 being non empty reflexive transitive proper CollStr
for b2, b3 being Element of the carrier of b1
for b4 being LINE of b1
st b2 <> b3 & b2 in b4 & b3 in b4
holds Line(b2,b3) = b4;
:: COLLSP:th 29
theorem
for b1 being non empty reflexive transitive proper CollStr
for b2, b3 being Element of the carrier of b1
for b4, b5 being LINE of b1
st b2 <> b3 & b2 in b4 & b3 in b4 & b2 in b5 & b3 in b5
holds b4 = b5;
:: COLLSP:th 30
theorem
for b1 being non empty reflexive transitive proper CollStr
for b2, b3 being LINE of b1
st b2 <> b3 & b2 meets b3
holds ex b4 being Element of the carrier of b1 st
b2 /\ b3 = {b4};
:: COLLSP:th 31
theorem
for b1 being non empty reflexive transitive proper CollStr
for b2, b3 being Element of the carrier of b1
st b2 <> b3
holds Line(b2,b3) <> the carrier of b1;