Article COMBGRAS, MML version 4.99.1005
:: COMBGRAS:th 1
theorem
for b1 being Element of NAT
for b2, b3 being set
st b2 <> b3 & Card b2 = b1 & Card b3 = b1
holds Card (b2 /\ b3) in b1 & b1 + 1 c= Card (b2 \/ b3);
:: COMBGRAS:th 2
theorem
for b1, b2 being Element of NAT
for b3, b4 being set
st Card b3 = b1 + b2 & Card b4 = b1 + b2
holds Card (b3 /\ b4) = b1
iff
Card (b3 \/ b4) = b1 + (2 * b2);
:: COMBGRAS:th 3
theorem
for b1, b2 being set holds
Card b1 c= Card b2
iff
ex b3 being Relation-like Function-like set st
b3 is one-to-one & b1 c= proj1 b3 & b3 .: b1 c= b2;
:: COMBGRAS:th 4
theorem
for b1 being set
for b2 being Relation-like Function-like set
st b2 is one-to-one & b1 c= proj1 b2
holds Card (b2 .: b1) = Card b1;
:: COMBGRAS:th 5
theorem
for b1, b2, b3 being set
st b1 \ b2 = b1 \ b3 & b2 c= b1 & b3 c= b1
holds b2 = b3;
:: COMBGRAS:th 6
theorem
for b1 being set
for b2 being non empty set
for b3 being Function-like quasi_total Relation of b1,b2
st b3 is one-to-one
for b4, b5 being Element of bool b1
st b4 <> b5
holds b3 .: b4 <> b3 .: b5;
:: COMBGRAS:th 7
theorem
for b1 being Element of NAT
for b2, b3, b4 being set
st Card b2 = b1 - 1 & Card b3 = b1 - 1 & Card b4 = b1 - 1 & Card (b2 /\ b3) = b1 - 2 & Card (b2 /\ b4) = b1 - 2 & Card (b3 /\ b4) = b1 - 2 & 2 <= b1
holds (3 <= b1 &
(Card ((b2 /\ b3) /\ b4) = b1 - 2 implies Card ((b2 \/ b3) \/ b4) <> b1 + 1) implies Card ((b2 /\ b3) /\ b4) = b1 - 3 &
Card ((b2 \/ b3) \/ b4) = b1) &
(b1 = 2 implies Card ((b2 /\ b3) /\ b4) = b1 - 2 &
Card ((b2 \/ b3) \/ b4) = b1 + 1);
:: COMBGRAS:th 8
theorem
for b1, b2 being IncProjStr
st IncProjStr(#the Points of b1,the Lines of b1,the Inc of b1#) = IncProjStr(#the Points of b2,the Lines of b2,the Inc of b2#)
for b3 being Element of the Points of b1
for b4 being Element of the Points of b2
st b3 = b4
for b5 being Element of the Lines of b1
for b6 being Element of the Lines of b2
st b5 = b6 & b3 on b5
holds b4 on b6;
:: COMBGRAS:th 9
theorem
for b1, b2 being IncProjStr
st IncProjStr(#the Points of b1,the Lines of b1,the Inc of b1#) = IncProjStr(#the Points of b2,the Lines of b2,the Inc of b2#)
for b3 being Element of bool the Points of b1
for b4 being Element of bool the Points of b2
st b3 = b4
for b5 being Element of the Lines of b1
for b6 being Element of the Lines of b2
st b5 = b6 & b3 on b5
holds b4 on b6;
:: COMBGRAS:exreg 1
registration
cluster strict with_non-trivial_lines linear up-2-rank IncProjStr;
end;
:: COMBGRAS:modenot 1
definition
mode PartialLinearSpace is with_non-trivial_lines up-2-rank IncProjStr;
end;
:: COMBGRAS:funcnot 1 => COMBGRAS:func 1
definition
let a1 be Element of NAT;
let a2 be non empty set;
assume 0 < a1 & a1 + 1 c= Card a2;
func G_(A1,A2) -> strict with_non-trivial_lines up-2-rank IncProjStr means
the Points of it = {b1 where b1 is Element of bool a2: Card b1 = a1} &
the Lines of it = {b1 where b1 is Element of bool a2: Card b1 = a1 + 1} &
the Inc of it = (RelIncl bool a2) /\ [:the Points of it,the Lines of it:];
end;
:: COMBGRAS:def 1
theorem
for b1 being Element of NAT
for b2 being non empty set
st 0 < b1 & b1 + 1 c= Card b2
for b3 being strict with_non-trivial_lines up-2-rank IncProjStr holds
b3 = G_(b1,b2)
iff
the Points of b3 = {b4 where b4 is Element of bool b2: Card b4 = b1} &
the Lines of b3 = {b4 where b4 is Element of bool b2: Card b4 = b1 + 1} &
the Inc of b3 = (RelIncl bool b2) /\ [:the Points of b3,the Lines of b3:];
:: COMBGRAS:th 10
theorem
for b1 being Element of NAT
for b2 being non empty set
st 0 < b1 & b1 + 1 c= Card b2
for b3 being Element of the Points of G_(b1,b2)
for b4 being Element of the Lines of G_(b1,b2) holds
b3 on b4
iff
b3 c= b4;
:: COMBGRAS:th 11
theorem
for b1 being Element of NAT
for b2 being non empty set
st 0 < b1 & b1 + 1 c= Card b2
holds G_(b1,b2) is Vebleian;
:: COMBGRAS:th 12
theorem
for b1 being Element of NAT
for b2 being non empty set
st 0 < b1 & b1 + 1 c= Card b2
for b3, b4, b5, b6, b7, b8 being Element of the Points of G_(b1,b2)
for b9, b10, b11, b12 being Element of the Lines of G_(b1,b2)
st b3 on b9 & b4 on b9 & b5 on b10 & b6 on b10 & b7 on b9 & b7 on b10 & b3 on b11 & b5 on b11 & b4 on b12 & b6 on b12 & not b7 on b11 & not b7 on b12 & b9 <> b10 & b11 <> b12
holds ex b13 being Element of the Points of G_(b1,b2) st
b13 on b11 & b13 on b12 & b13 = (b3 /\ b4) \/ (b5 /\ b6);
:: COMBGRAS:th 13
theorem
for b1 being Element of NAT
for b2 being non empty set
st 0 < b1 & b1 + 1 c= Card b2
holds G_(b1,b2) is Desarguesian;
:: COMBGRAS:attrnot 1 => COMBGRAS:attr 1
definition
let a1 be IncProjStr;
let a2 be Element of bool the Points of a1;
attr a2 is clique means
for b1, b2 being Element of the Points of a1
st b1 in a2 & b2 in a2
holds ex b3 being Element of the Lines of a1 st
{b1,b2} on b3;
end;
:: COMBGRAS:dfs 2
definiens
let a1 be IncProjStr;
let a2 be Element of bool the Points of a1;
To prove
a2 is clique
it is sufficient to prove
thus for b1, b2 being Element of the Points of a1
st b1 in a2 & b2 in a2
holds ex b3 being Element of the Lines of a1 st
{b1,b2} on b3;
:: COMBGRAS:def 2
theorem
for b1 being IncProjStr
for b2 being Element of bool the Points of b1 holds
b2 is clique(b1)
iff
for b3, b4 being Element of the Points of b1
st b3 in b2 & b4 in b2
holds ex b5 being Element of the Lines of b1 st
{b3,b4} on b5;
:: COMBGRAS:attrnot 2 => COMBGRAS:attr 2
definition
let a1 be IncProjStr;
let a2 be Element of bool the Points of a1;
attr a2 is maximal_clique means
a2 is clique(a1) &
(for b1 being Element of bool the Points of a1
st b1 is clique(a1) & a2 c= b1
holds b1 = a2);
end;
:: COMBGRAS:dfs 3
definiens
let a1 be IncProjStr;
let a2 be Element of bool the Points of a1;
To prove
a2 is maximal_clique
it is sufficient to prove
thus a2 is clique(a1) &
(for b1 being Element of bool the Points of a1
st b1 is clique(a1) & a2 c= b1
holds b1 = a2);
:: COMBGRAS:def 3
theorem
for b1 being IncProjStr
for b2 being Element of bool the Points of b1 holds
b2 is maximal_clique(b1)
iff
b2 is clique(b1) &
(for b3 being Element of bool the Points of b1
st b3 is clique(b1) & b2 c= b3
holds b3 = b2);
:: COMBGRAS:attrnot 3 => COMBGRAS:attr 3
definition
let a1 be Element of NAT;
let a2 be non empty set;
let a3 be Element of bool the Points of G_(a1,a2);
attr a3 is STAR means
ex b1 being Element of bool a2 st
Card b1 = a1 - 1 &
a3 = {b2 where b2 is Element of bool a2: Card b2 = a1 & b1 c= b2};
end;
:: COMBGRAS:dfs 4
definiens
let a1 be Element of NAT;
let a2 be non empty set;
let a3 be Element of bool the Points of G_(a1,a2);
To prove
a3 is STAR
it is sufficient to prove
thus ex b1 being Element of bool a2 st
Card b1 = a1 - 1 &
a3 = {b2 where b2 is Element of bool a2: Card b2 = a1 & b1 c= b2};
:: COMBGRAS:def 4
theorem
for b1 being Element of NAT
for b2 being non empty set
for b3 being Element of bool the Points of G_(b1,b2) holds
b3 is STAR(b1, b2)
iff
ex b4 being Element of bool b2 st
Card b4 = b1 - 1 &
b3 = {b5 where b5 is Element of bool b2: Card b5 = b1 & b4 c= b5};
:: COMBGRAS:attrnot 4 => COMBGRAS:attr 4
definition
let a1 be Element of NAT;
let a2 be non empty set;
let a3 be Element of bool the Points of G_(a1,a2);
attr a3 is TOP means
ex b1 being Element of bool a2 st
Card b1 = a1 + 1 &
a3 = {b2 where b2 is Element of bool a2: Card b2 = a1 & b2 c= b1};
end;
:: COMBGRAS:dfs 5
definiens
let a1 be Element of NAT;
let a2 be non empty set;
let a3 be Element of bool the Points of G_(a1,a2);
To prove
a3 is TOP
it is sufficient to prove
thus ex b1 being Element of bool a2 st
Card b1 = a1 + 1 &
a3 = {b2 where b2 is Element of bool a2: Card b2 = a1 & b2 c= b1};
:: COMBGRAS:def 5
theorem
for b1 being Element of NAT
for b2 being non empty set
for b3 being Element of bool the Points of G_(b1,b2) holds
b3 is TOP(b1, b2)
iff
ex b4 being Element of bool b2 st
Card b4 = b1 + 1 &
b3 = {b5 where b5 is Element of bool b2: Card b5 = b1 & b5 c= b4};
:: COMBGRAS:th 14
theorem
for b1 being Element of NAT
for b2 being non empty set
st 2 <= b1 & b1 + 2 c= Card b2
for b3 being Element of bool the Points of G_(b1,b2)
st (b3 is STAR(b1, b2) or b3 is TOP(b1, b2))
holds b3 is maximal_clique(G_(b1,b2));
:: COMBGRAS:th 15
theorem
for b1 being Element of NAT
for b2 being non empty set
st 2 <= b1 & b1 + 2 c= Card b2
for b3 being Element of bool the Points of G_(b1,b2)
st b3 is maximal_clique(G_(b1,b2)) & b3 is not STAR(b1, b2)
holds b3 is TOP(b1, b2);
:: COMBGRAS:structnot 1 => COMBGRAS:struct 1
definition
let a1, a2 be IncProjStr;
struct() IncProjMap(#
point-map -> Function-like quasi_total Relation of the Points of A1,the Points of A2,
line-map -> Function-like quasi_total Relation of the Lines of A1,the Lines of A2
#);
end;
:: COMBGRAS:attrnot 5 => COMBGRAS:attr 5
definition
let a1, a2 be IncProjStr;
let a3 be IncProjMap over a1,a2;
attr a3 is strict;
end;
:: COMBGRAS:exreg 2
registration
let a1, a2 be IncProjStr;
cluster strict IncProjMap over a1,a2;
end;
:: COMBGRAS:aggrnot 1 => COMBGRAS:aggr 1
definition
let a1, a2 be IncProjStr;
let a3 be Function-like quasi_total Relation of the Points of a1,the Points of a2;
let a4 be Function-like quasi_total Relation of the Lines of a1,the Lines of a2;
aggr IncProjMap(#a3,a4#) -> strict IncProjMap over a1,a2;
end;
:: COMBGRAS:selnot 1 => COMBGRAS:sel 1
definition
let a1, a2 be IncProjStr;
let a3 be IncProjMap over a1,a2;
sel the point-map of a3 -> Function-like quasi_total Relation of the Points of a1,the Points of a2;
end;
:: COMBGRAS:selnot 2 => COMBGRAS:sel 2
definition
let a1, a2 be IncProjStr;
let a3 be IncProjMap over a1,a2;
sel the line-map of a3 -> Function-like quasi_total Relation of the Lines of a1,the Lines of a2;
end;
:: COMBGRAS:funcnot 2 => COMBGRAS:func 2
definition
let a1, a2 be IncProjStr;
let a3 be IncProjMap over a1,a2;
let a4 be Element of the Points of a1;
func A3 . A4 -> Element of the Points of a2 equals
(the point-map of a3) . a4;
end;
:: COMBGRAS:def 6
theorem
for b1, b2 being IncProjStr
for b3 being IncProjMap over b1,b2
for b4 being Element of the Points of b1 holds
b3 . b4 = (the point-map of b3) . b4;
:: COMBGRAS:funcnot 3 => COMBGRAS:func 3
definition
let a1, a2 be IncProjStr;
let a3 be IncProjMap over a1,a2;
let a4 be Element of the Lines of a1;
func A3 . A4 -> Element of the Lines of a2 equals
(the line-map of a3) . a4;
end;
:: COMBGRAS:def 7
theorem
for b1, b2 being IncProjStr
for b3 being IncProjMap over b1,b2
for b4 being Element of the Lines of b1 holds
b3 . b4 = (the line-map of b3) . b4;
:: COMBGRAS:th 16
theorem
for b1, b2 being IncProjStr
for b3, b4 being IncProjMap over b1,b2
st (for b5 being Element of the Points of b1 holds
b3 . b5 = b4 . b5) &
(for b5 being Element of the Lines of b1 holds
b3 . b5 = b4 . b5)
holds IncProjMap(#the point-map of b3,the line-map of b3#) = IncProjMap(#the point-map of b4,the line-map of b4#);
:: COMBGRAS:attrnot 6 => COMBGRAS:attr 6
definition
let a1, a2 be IncProjStr;
let a3 be IncProjMap over a1,a2;
attr a3 is incidence_preserving means
for b1 being Element of the Points of a1
for b2 being Element of the Lines of a1 holds
b1 on b2
iff
a3 . b1 on a3 . b2;
end;
:: COMBGRAS:dfs 8
definiens
let a1, a2 be IncProjStr;
let a3 be IncProjMap over a1,a2;
To prove
a3 is incidence_preserving
it is sufficient to prove
thus for b1 being Element of the Points of a1
for b2 being Element of the Lines of a1 holds
b1 on b2
iff
a3 . b1 on a3 . b2;
:: COMBGRAS:def 8
theorem
for b1, b2 being IncProjStr
for b3 being IncProjMap over b1,b2 holds
b3 is incidence_preserving(b1, b2)
iff
for b4 being Element of the Points of b1
for b5 being Element of the Lines of b1 holds
b4 on b5
iff
b3 . b4 on b3 . b5;
:: COMBGRAS:th 17
theorem
for b1, b2 being IncProjStr
for b3, b4 being IncProjMap over b1,b2
st IncProjMap(#the point-map of b3,the line-map of b3#) = IncProjMap(#the point-map of b4,the line-map of b4#) &
b3 is incidence_preserving(b1, b2)
holds b4 is incidence_preserving(b1, b2);
:: COMBGRAS:attrnot 7 => COMBGRAS:attr 7
definition
let a1 be IncProjStr;
let a2 be IncProjMap over a1,a1;
attr a2 is automorphism means
the line-map of a2 is bijective(the Lines of a1, the Lines of a1) & the point-map of a2 is bijective(the Points of a1, the Points of a1) & a2 is incidence_preserving(a1, a1);
end;
:: COMBGRAS:dfs 9
definiens
let a1 be IncProjStr;
let a2 be IncProjMap over a1,a1;
To prove
a2 is automorphism
it is sufficient to prove
thus the line-map of a2 is bijective(the Lines of a1, the Lines of a1) & the point-map of a2 is bijective(the Points of a1, the Points of a1) & a2 is incidence_preserving(a1, a1);
:: COMBGRAS:def 9
theorem
for b1 being IncProjStr
for b2 being IncProjMap over b1,b1 holds
b2 is automorphism(b1)
iff
the line-map of b2 is bijective(the Lines of b1, the Lines of b1) & the point-map of b2 is bijective(the Points of b1, the Points of b1) & b2 is incidence_preserving(b1, b1);
:: COMBGRAS:funcnot 4 => COMBGRAS:func 4
definition
let a1, a2 be IncProjStr;
let a3 be IncProjMap over a1,a2;
let a4 be Element of bool the Points of a1;
func A3 .: A4 -> Element of bool the Points of a2 equals
(the point-map of a3) .: a4;
end;
:: COMBGRAS:def 10
theorem
for b1, b2 being IncProjStr
for b3 being IncProjMap over b1,b2
for b4 being Element of bool the Points of b1 holds
b3 .: b4 = (the point-map of b3) .: b4;
:: COMBGRAS:funcnot 5 => COMBGRAS:func 5
definition
let a1, a2 be IncProjStr;
let a3 be IncProjMap over a1,a2;
let a4 be Element of bool the Points of a2;
func A3 " A4 -> Element of bool the Points of a1 equals
(the point-map of a3) " a4;
end;
:: COMBGRAS:def 11
theorem
for b1, b2 being IncProjStr
for b3 being IncProjMap over b1,b2
for b4 being Element of bool the Points of b2 holds
b3 " b4 = (the point-map of b3) " b4;
:: COMBGRAS:funcnot 6 => COMBGRAS:func 6
definition
let a1 be set;
let a2 be finite set;
func ^^(A2,A1) -> Element of bool bool a1 equals
{b1 where b1 is Element of bool a1: Card b1 = (card a2) + 1 & a2 c= b1};
end;
:: COMBGRAS:def 12
theorem
for b1 being set
for b2 being finite set holds
^^(b2,b1) = {b3 where b3 is Element of bool b1: Card b3 = (card b2) + 1 & b2 c= b3};
:: COMBGRAS:funcnot 7 => COMBGRAS:func 7
definition
let a1 be Element of NAT;
let a2 be non empty set;
let a3 be finite set;
assume 0 < a1 & a1 + 1 c= Card a2 & Card a3 = a1 - 1 & a3 c= a2;
func ^^(A3,A2,A1) -> Element of bool the Points of G_(a1,a2) equals
^^(a3,a2);
end;
:: COMBGRAS:def 13
theorem
for b1 being Element of NAT
for b2 being non empty set
st 0 < b1 & b1 + 1 c= Card b2
for b3 being finite set
st Card b3 = b1 - 1 & b3 c= b2
holds ^^(b3,b2,b1) = ^^(b3,b2);
:: COMBGRAS:th 18
theorem
for b1, b2 being IncProjStr
for b3 being IncProjMap over b1,b2
for b4 being Element of bool the Points of b1 holds
b3 .: b4 = {b5 where b5 is Element of the Points of b2: ex b6 being Element of the Points of b1 st
b6 in b4 & b3 . b6 = b5};
:: COMBGRAS:th 19
theorem
for b1, b2 being IncProjStr
for b3 being IncProjMap over b1,b2
for b4 being Element of bool the Points of b2 holds
b3 " b4 = {b5 where b5 is Element of the Points of b1: ex b6 being Element of the Points of b2 st
b6 in b4 & b3 . b5 = b6};
:: COMBGRAS:th 20
theorem
for b1 being IncProjStr
for b2 being IncProjMap over b1,b1
for b3 being Element of bool the Points of b1
st b2 is incidence_preserving(b1, b1) & b3 is clique(b1)
holds b2 .: b3 is clique(b1);
:: COMBGRAS:th 21
theorem
for b1 being IncProjStr
for b2 being IncProjMap over b1,b1
for b3 being Element of bool the Points of b1
st b2 is incidence_preserving(b1, b1) & the line-map of b2 is onto(the Lines of b1, the Lines of b1) & b3 is clique(b1)
holds b2 " b3 is clique(b1);
:: COMBGRAS:th 22
theorem
for b1 being IncProjStr
for b2 being IncProjMap over b1,b1
for b3 being Element of bool the Points of b1
st b2 is automorphism(b1) & b3 is maximal_clique(b1)
holds b2 .: b3 is maximal_clique(b1) & b2 " b3 is maximal_clique(b1);
:: COMBGRAS:th 23
theorem
for b1 being Element of NAT
for b2 being non empty set
st 2 <= b1 & b1 + 2 c= Card b2
for b3 being IncProjMap over G_(b1,b2),G_(b1,b2)
st b3 is automorphism(G_(b1,b2))
for b4 being Element of bool the Points of G_(b1,b2)
st b4 is STAR(b1, b2)
holds b3 .: b4 is STAR(b1, b2) & b3 " b4 is STAR(b1, b2);
:: COMBGRAS:funcnot 8 => COMBGRAS:func 8
definition
let a1 be Element of NAT;
let a2 be non empty set;
let a3 be Function-like quasi_total bijective Relation of a2,a2;
assume 0 < a1 & a1 + 1 c= Card a2;
func incprojmap(A1,A3) -> strict IncProjMap over G_(a1,a2),G_(a1,a2) means
(for b1 being Element of the Points of G_(a1,a2) holds
it . b1 = a3 .: b1) &
(for b1 being Element of the Lines of G_(a1,a2) holds
it . b1 = a3 .: b1);
end;
:: COMBGRAS:def 14
theorem
for b1 being Element of NAT
for b2 being non empty set
st 0 < b1 & b1 + 1 c= Card b2
for b3 being Function-like quasi_total bijective Relation of b2,b2
for b4 being strict IncProjMap over G_(b1,b2),G_(b1,b2) holds
b4 = incprojmap(b1,b3)
iff
(for b5 being Element of the Points of G_(b1,b2) holds
b4 . b5 = b3 .: b5) &
(for b5 being Element of the Lines of G_(b1,b2) holds
b4 . b5 = b3 .: b5);
:: COMBGRAS:th 24
theorem
for b1 being Element of NAT
for b2 being non empty set
st b1 = 1 & b1 + 1 c= Card b2
for b3 being IncProjMap over G_(b1,b2),G_(b1,b2)
st b3 is automorphism(G_(b1,b2))
holds ex b4 being Function-like quasi_total bijective Relation of b2,b2 st
IncProjMap(#the point-map of b3,the line-map of b3#) = incprojmap(b1,b4);
:: COMBGRAS:th 25
theorem
for b1 being Element of NAT
for b2 being non empty set
st 1 < b1 & Card b2 = b1 + 1
for b3 being IncProjMap over G_(b1,b2),G_(b1,b2)
st b3 is automorphism(G_(b1,b2))
holds ex b4 being Function-like quasi_total bijective Relation of b2,b2 st
IncProjMap(#the point-map of b3,the line-map of b3#) = incprojmap(b1,b4);
:: COMBGRAS:th 26
theorem
for b1 being Element of NAT
for b2 being non empty set
st 0 < b1 & b1 + 1 c= Card b2
for b3 being Element of bool the Points of G_(b1,b2)
for b4 being Element of bool b2
st Card b4 = b1 - 1 &
b3 = {b5 where b5 is Element of bool b2: Card b5 = b1 & b4 c= b5}
holds b4 = meet b3;
:: COMBGRAS:th 27
theorem
for b1 being Element of NAT
for b2 being non empty set
st 0 < b1 & b1 + 1 c= Card b2
for b3 being Element of bool the Points of G_(b1,b2)
st b3 is STAR(b1, b2)
for b4 being Element of bool b2
st b4 = meet b3
holds Card b4 = b1 - 1 &
b3 = {b5 where b5 is Element of bool b2: Card b5 = b1 & b4 c= b5};
:: COMBGRAS:th 28
theorem
for b1 being Element of NAT
for b2 being non empty set
st 0 < b1 & b1 + 1 c= Card b2
for b3, b4 being Element of bool the Points of G_(b1,b2)
st b3 is STAR(b1, b2) & b4 is STAR(b1, b2) & meet b3 = meet b4
holds b3 = b4;
:: COMBGRAS:th 29
theorem
for b1 being Element of NAT
for b2 being non empty set
st 0 < b1 & b1 + 1 c= Card b2
for b3 being finite Element of bool b2
st Card b3 = b1 - 1
holds ^^(b3,b2,b1) is STAR(b1, b2);
:: COMBGRAS:th 30
theorem
for b1 being Element of NAT
for b2 being non empty set
st 0 < b1 & b1 + 1 c= Card b2
for b3 being finite Element of bool b2
st Card b3 = b1 - 1
holds meet ^^(b3,b2,b1) = b3;
:: COMBGRAS:th 31
theorem
for b1 being Element of NAT
for b2 being non empty set
st 0 < b1 & b1 + 3 c= Card b2
for b3 being IncProjMap over G_(b1 + 1,b2),G_(b1 + 1,b2)
st b3 is automorphism(G_(b1 + 1,b2))
holds ex b4 being IncProjMap over G_(b1,b2),G_(b1,b2) st
b4 is automorphism(G_(b1,b2)) &
the line-map of b4 = the point-map of b3 &
(for b5 being Element of the Points of G_(b1,b2)
for b6 being finite set
st b6 = b5
holds b4 . b5 = meet (b3 .: ^^(b6,b2,b1 + 1)));
:: COMBGRAS:th 32
theorem
for b1 being Element of NAT
for b2 being non empty set
st 0 < b1 & b1 + 3 c= Card b2
for b3 being IncProjMap over G_(b1 + 1,b2),G_(b1 + 1,b2)
st b3 is automorphism(G_(b1 + 1,b2))
for b4 being IncProjMap over G_(b1,b2),G_(b1,b2)
st b4 is automorphism(G_(b1,b2)) &
the line-map of b4 = the point-map of b3 &
(for b5 being Element of the Points of G_(b1,b2)
for b6 being finite set
st b6 = b5
holds b4 . b5 = meet (b3 .: ^^(b6,b2,b1 + 1)))
for b5 being Function-like quasi_total bijective Relation of b2,b2
st IncProjMap(#the point-map of b4,the line-map of b4#) = incprojmap(b1,b5)
holds IncProjMap(#the point-map of b3,the line-map of b3#) = incprojmap(b1 + 1,b5);
:: COMBGRAS:th 33
theorem
for b1 being Element of NAT
for b2 being non empty set
st 2 <= b1 & b1 + 2 c= Card b2
for b3 being IncProjMap over G_(b1,b2),G_(b1,b2)
st b3 is automorphism(G_(b1,b2))
holds ex b4 being Function-like quasi_total bijective Relation of b2,b2 st
IncProjMap(#the point-map of b3,the line-map of b3#) = incprojmap(b1,b4);
:: COMBGRAS:th 34
theorem
for b1 being Element of NAT
for b2 being non empty set
st 0 < b1 & b1 + 1 c= Card b2
for b3 being Function-like quasi_total bijective Relation of b2,b2 holds
incprojmap(b1,b3) is automorphism(G_(b1,b2));
:: COMBGRAS:th 35
theorem
for b1 being Element of NAT
for b2 being non empty set
st 0 < b1 & b1 + 1 c= Card b2
for b3 being IncProjMap over G_(b1,b2),G_(b1,b2) holds
b3 is automorphism(G_(b1,b2))
iff
ex b4 being Function-like quasi_total bijective Relation of b2,b2 st
IncProjMap(#the point-map of b3,the line-map of b3#) = incprojmap(b1,b4);