Article PROJPL_1, MML version 4.99.1005
:: PROJPL_1:prednot 1 => not INCSP_1:pred 1
notation
let a1 be IncProjStr;
let a2 be Element of the Points of a1;
let a3 be Element of the Lines of a1;
antonym a2 |' a3 for a2 on a3;
end;
:: PROJPL_1:prednot 2 => PROJPL_1:pred 1
definition
let a1 be IncProjStr;
let a2, a3 be Element of the Points of a1;
let a4 be Element of the Lines of a1;
pred A2,A3 |' A4 means
not a2 on a4 & not a3 on a4;
end;
:: PROJPL_1:dfs 1
definiens
let a1 be IncProjStr;
let a2, a3 be Element of the Points of a1;
let a4 be Element of the Lines of a1;
To prove
a2,a3 |' a4
it is sufficient to prove
thus not a2 on a4 & not a3 on a4;
:: PROJPL_1:def 1
theorem
for b1 being IncProjStr
for b2, b3 being Element of the Points of b1
for b4 being Element of the Lines of b1 holds
b2,b3 |' b4
iff
not b2 on b4 & not b3 on b4;
:: PROJPL_1:prednot 3 => PROJPL_1:pred 2
definition
let a1 be IncProjStr;
let a2 be Element of the Points of a1;
let a3, a4 be Element of the Lines of a1;
pred A2 on A3,A4 means
a2 on a3 & a2 on a4;
end;
:: PROJPL_1:dfs 2
definiens
let a1 be IncProjStr;
let a2 be Element of the Points of a1;
let a3, a4 be Element of the Lines of a1;
To prove
a2 on a3,a4
it is sufficient to prove
thus a2 on a3 & a2 on a4;
:: PROJPL_1:def 2
theorem
for b1 being IncProjStr
for b2 being Element of the Points of b1
for b3, b4 being Element of the Lines of b1 holds
b2 on b3,b4
iff
b2 on b3 & b2 on b4;
:: PROJPL_1:prednot 4 => PROJPL_1:pred 3
definition
let a1 be IncProjStr;
let a2 be Element of the Points of a1;
let a3, a4, a5 be Element of the Lines of a1;
pred A2 on A3,A4,A5 means
a2 on a3 & a2 on a4 & a2 on a5;
end;
:: PROJPL_1:dfs 3
definiens
let a1 be IncProjStr;
let a2 be Element of the Points of a1;
let a3, a4, a5 be Element of the Lines of a1;
To prove
a2 on a3,a4,a5
it is sufficient to prove
thus a2 on a3 & a2 on a4 & a2 on a5;
:: PROJPL_1:def 3
theorem
for b1 being IncProjStr
for b2 being Element of the Points of b1
for b3, b4, b5 being Element of the Lines of b1 holds
b2 on b3,b4,b5
iff
b2 on b3 & b2 on b4 & b2 on b5;
:: PROJPL_1:th 1
theorem
for b1 being IncProjStr
for b2, b3, b4 being Element of the Points of b1
for b5, b6, b7 being Element of the Lines of b1 holds
({b2,b3} on b5 implies {b3,b2} on b5) &
({b2,b3,b4} on b5 implies {b2,b4,b3} on b5 & {b3,b2,b4} on b5 & {b3,b4,b2} on b5 & {b4,b2,b3} on b5 & {b4,b3,b2} on b5) &
(b2 on b5,b6 implies b2 on b6,b5) &
(b2 on b5,b6,b7 implies b2 on b5,b7,b6 & b2 on b6,b5,b7 & b2 on b6,b7,b5 & b2 on b7,b5,b6 & b2 on b7,b6,b5);
:: PROJPL_1:attrnot 1 => PROJPL_1:attr 1
definition
let a1 be IncProjStr;
attr a1 is configuration means
for b1, b2 being Element of the Points of a1
for b3, b4 being Element of the Lines of a1
st b1 on b3 & b2 on b3 & b1 on b4 & b2 on b4 & b1 <> b2
holds b3 = b4;
end;
:: PROJPL_1:dfs 4
definiens
let a1 be IncProjStr;
To prove
a1 is configuration
it is sufficient to prove
thus for b1, b2 being Element of the Points of a1
for b3, b4 being Element of the Lines of a1
st b1 on b3 & b2 on b3 & b1 on b4 & b2 on b4 & b1 <> b2
holds b3 = b4;
:: PROJPL_1:def 4
theorem
for b1 being IncProjStr holds
b1 is configuration
iff
for b2, b3 being Element of the Points of b1
for b4, b5 being Element of the Lines of b1
st b2 on b4 & b3 on b4 & b2 on b5 & b3 on b5 & b2 <> b3
holds b4 = b5;
:: PROJPL_1:th 2
theorem
for b1 being IncProjStr holds
b1 is configuration
iff
for b2, b3 being Element of the Points of b1
for b4, b5 being Element of the Lines of b1
st {b2,b3} on b4 & {b2,b3} on b5 & b2 <> b3
holds b4 = b5;
:: PROJPL_1:th 3
theorem
for b1 being IncProjStr holds
b1 is configuration
iff
for b2, b3 being Element of the Points of b1
for b4, b5 being Element of the Lines of b1
st b2 on b4,b5 & b3 on b4,b5 & b2 <> b3
holds b4 = b5;
:: PROJPL_1:th 4
theorem
for b1 being IncProjStr holds
b1 is linear partial up-2-dimensional up-3-rank Vebleian IncProjStr
iff
b1 is configuration &
(for b2, b3 being Element of the Points of b1 holds
ex b4 being Element of the Lines of b1 st
{b2,b3} on b4) &
(ex b2 being Element of the Points of b1 st
ex b3 being Element of the Lines of b1 st
not b2 on b3) &
(for b2 being Element of the Lines of b1 holds
ex b3, b4, b5 being Element of the Points of b1 st
b3,b4,b5 are_mutually_different & {b3,b4,b5} on b2) &
(for b2, b3, b4, b5, b6 being Element of the Points of b1
for b7, b8, b9, b10 being Element of the Lines of b1
st {b2,b3,b6} on b7 & {b4,b5,b6} on b8 & {b2,b4} on b9 & {b3,b5} on b10 & not b6 on b9 & not b6 on b10 & b7 <> b8
holds ex b11 being Element of the Points of b1 st
b11 on b9,b10);
:: PROJPL_1:modenot 1
definition
mode IncProjectivePlane is linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional IncProjStr;
end;
:: PROJPL_1:prednot 5 => PROJPL_1:pred 4
definition
let a1 be IncProjStr;
let a2, a3, a4 be Element of the Points of a1;
pred A2,A3,A4 is_collinear means
ex b1 being Element of the Lines of a1 st
{a2,a3,a4} on b1;
end;
:: PROJPL_1:dfs 5
definiens
let a1 be IncProjStr;
let a2, a3, a4 be Element of the Points of a1;
To prove
a2,a3,a4 is_collinear
it is sufficient to prove
thus ex b1 being Element of the Lines of a1 st
{a2,a3,a4} on b1;
:: PROJPL_1:def 5
theorem
for b1 being IncProjStr
for b2, b3, b4 being Element of the Points of b1 holds
b2,b3,b4 is_collinear
iff
ex b5 being Element of the Lines of b1 st
{b2,b3,b4} on b5;
:: PROJPL_1:prednot 6 => not PROJPL_1:pred 4
notation
let a1 be IncProjStr;
let a2, a3, a4 be Element of the Points of a1;
antonym a2,a3,a4 is_a_triangle for a2,a3,a4 is_collinear;
end;
:: PROJPL_1:th 5
theorem
for b1 being IncProjStr
for b2, b3, b4 being Element of the Points of b1 holds
b2,b3,b4 is_collinear
iff
ex b5 being Element of the Lines of b1 st
b2 on b5 & b3 on b5 & b4 on b5;
:: PROJPL_1:th 6
theorem
for b1 being IncProjStr
for b2, b3, b4 being Element of the Points of b1 holds
b2,b3,b4 is_a_triangle
iff
for b5 being Element of the Lines of b1
st b2 on b5 & b3 on b5
holds not b4 on b5;
:: PROJPL_1:prednot 7 => PROJPL_1:pred 5
definition
let a1 be IncProjStr;
let a2, a3, a4, a5 be Element of the Points of a1;
pred A2,A3,A4,A5 is_a_quadrangle means
a2,a3,a4 is_a_triangle & a3,a4,a5 is_a_triangle & a4,a5,a2 is_a_triangle & a5,a2,a3 is_a_triangle;
end;
:: PROJPL_1:dfs 6
definiens
let a1 be IncProjStr;
let a2, a3, a4, a5 be Element of the Points of a1;
To prove
a2,a3,a4,a5 is_a_quadrangle
it is sufficient to prove
thus a2,a3,a4 is_a_triangle & a3,a4,a5 is_a_triangle & a4,a5,a2 is_a_triangle & a5,a2,a3 is_a_triangle;
:: PROJPL_1:def 6
theorem
for b1 being IncProjStr
for b2, b3, b4, b5 being Element of the Points of b1 holds
b2,b3,b4,b5 is_a_quadrangle
iff
b2,b3,b4 is_a_triangle & b3,b4,b5 is_a_triangle & b4,b5,b2 is_a_triangle & b5,b2,b3 is_a_triangle;
:: PROJPL_1:th 7
theorem
for b1 being IncProjStr
st b1 is linear partial up-2-dimensional up-3-rank Vebleian IncProjStr
holds ex b2, b3 being Element of the Lines of b1 st
b2 <> b3;
:: PROJPL_1:th 8
theorem
for b1 being IncProjStr
for b2 being Element of the Points of b1
for b3 being Element of the Lines of b1
st b1 is linear partial up-2-dimensional up-3-rank Vebleian IncProjStr &
b2 on b3
holds ex b4, b5 being Element of the Points of b1 st
{b4,b5} on b3 & b2,b4,b5 are_mutually_different;
:: PROJPL_1:th 9
theorem
for b1 being IncProjStr
for b2 being Element of the Points of b1
for b3, b4 being Element of the Lines of b1
st b1 is linear partial up-2-dimensional up-3-rank Vebleian IncProjStr &
b2 on b3 &
b3 <> b4
holds ex b5 being Element of the Points of b1 st
b5 on b3 & not b5 on b4 & b2 <> b5;
:: PROJPL_1:th 10
theorem
for b1 being IncProjStr
for b2, b3, b4 being Element of the Points of b1
for b5 being Element of the Lines of b1
st b1 is configuration & {b2,b3} on b5 & b2 <> b3 & not b4 on b5
holds b2,b3,b4 is_a_triangle;
:: PROJPL_1:th 11
theorem
for b1 being IncProjStr
for b2, b3, b4 being Element of the Points of b1
st b2,b3,b4 is_collinear
holds b2,b4,b3 is_collinear & b3,b2,b4 is_collinear & b3,b4,b2 is_collinear & b4,b2,b3 is_collinear & b4,b3,b2 is_collinear;
:: PROJPL_1:th 12
theorem
for b1 being IncProjStr
for b2, b3, b4 being Element of the Points of b1
st b2,b3,b4 is_a_triangle
holds b2,b4,b3 is_a_triangle & b3,b2,b4 is_a_triangle & b3,b4,b2 is_a_triangle & b4,b2,b3 is_a_triangle & b4,b3,b2 is_a_triangle;
:: PROJPL_1:th 13
theorem
for b1 being IncProjStr
for b2, b3, b4, b5 being Element of the Points of b1
st b2,b3,b4,b5 is_a_quadrangle
holds b2,b4,b3,b5 is_a_quadrangle & b3,b2,b4,b5 is_a_quadrangle & b3,b4,b2,b5 is_a_quadrangle & b4,b2,b3,b5 is_a_quadrangle & b4,b3,b2,b5 is_a_quadrangle & b2,b3,b5,b4 is_a_quadrangle & b2,b4,b5,b3 is_a_quadrangle & b3,b2,b5,b4 is_a_quadrangle & b3,b4,b5,b2 is_a_quadrangle & b4,b2,b5,b3 is_a_quadrangle & b4,b3,b5,b2 is_a_quadrangle & b2,b5,b3,b4 is_a_quadrangle & b2,b5,b4,b3 is_a_quadrangle & b3,b5,b2,b4 is_a_quadrangle & b3,b5,b4,b2 is_a_quadrangle & b4,b5,b2,b3 is_a_quadrangle & b4,b5,b3,b2 is_a_quadrangle & b5,b2,b3,b4 is_a_quadrangle & b5,b2,b4,b3 is_a_quadrangle & b5,b3,b2,b4 is_a_quadrangle & b5,b3,b4,b2 is_a_quadrangle & b5,b4,b2,b3 is_a_quadrangle & b5,b4,b3,b2 is_a_quadrangle;
:: PROJPL_1:th 14
theorem
for b1 being IncProjStr
for b2, b3, b4, b5 being Element of the Points of b1
for b6, b7 being Element of the Lines of b1
st b1 is configuration & {b2,b3} on b6 & {b4,b5} on b7 & b2,b3 |' b7 & b4,b5 |' b6 & b2 <> b3 & b4 <> b5
holds b2,b3,b4,b5 is_a_quadrangle;
:: PROJPL_1:th 15
theorem
for b1 being IncProjStr
st b1 is linear partial up-2-dimensional up-3-rank Vebleian IncProjStr
holds ex b2, b3, b4, b5 being Element of the Points of b1 st
b2,b3,b4,b5 is_a_quadrangle;
:: PROJPL_1:modenot 2 => PROJPL_1:mode 1
definition
let a1 be linear partial up-2-dimensional up-3-rank Vebleian IncProjStr;
mode Quadrangle of A1 -> Element of [:the Points of a1,the Points of a1,the Points of a1,the Points of a1:] means
it `1,it `2,it `3,it `4 is_a_quadrangle;
end;
:: PROJPL_1:dfs 7
definiens
let a1 be linear partial up-2-dimensional up-3-rank Vebleian IncProjStr;
let a2 be Element of [:the Points of a1,the Points of a1,the Points of a1,the Points of a1:];
To prove
a2 is Quadrangle of a1
it is sufficient to prove
thus a2 `1,a2 `2,a2 `3,a2 `4 is_a_quadrangle;
:: PROJPL_1:def 7
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian IncProjStr
for b2 being Element of [:the Points of b1,the Points of b1,the Points of b1,the Points of b1:] holds
b2 is Quadrangle of b1
iff
b2 `1,b2 `2,b2 `3,b2 `4 is_a_quadrangle;
:: PROJPL_1:funcnot 1 => PROJPL_1:func 1
definition
let a1 be linear partial up-2-dimensional up-3-rank Vebleian IncProjStr;
let a2, a3 be Element of the Points of a1;
assume a2 <> a3;
func A2 * A3 -> Element of the Lines of a1 means
{a2,a3} on it;
end;
:: PROJPL_1:def 8
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian IncProjStr
for b2, b3 being Element of the Points of b1
st b2 <> b3
for b4 being Element of the Lines of b1 holds
b4 = b2 * b3
iff
{b2,b3} on b4;
:: PROJPL_1:th 16
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian IncProjStr
for b2, b3 being Element of the Points of b1
for b4 being Element of the Lines of b1
st b2 <> b3
holds b2 on b2 * b3 &
b3 on b2 * b3 &
b2 * b3 = b3 * b2 &
(b2 on b4 & b3 on b4 implies b4 = b2 * b3);
:: PROJPL_1:th 17
theorem
for b1 being IncProjStr
st (ex b2, b3, b4 being Element of the Points of b1 st
b2,b3,b4 is_a_triangle) &
(for b2, b3 being Element of the Points of b1 holds
ex b4 being Element of the Lines of b1 st
{b2,b3} on b4)
holds ex b2 being Element of the Points of b1 st
ex b3 being Element of the Lines of b1 st
not b2 on b3;
:: PROJPL_1:th 18
theorem
for b1 being IncProjStr
st ex b2, b3, b4, b5 being Element of the Points of b1 st
b2,b3,b4,b5 is_a_quadrangle
holds ex b2, b3, b4 being Element of the Points of b1 st
b2,b3,b4 is_a_triangle;
:: PROJPL_1:th 19
theorem
for b1 being IncProjStr
for b2, b3, b4 being Element of the Points of b1
for b5, b6 being Element of the Lines of b1
st b2,b3,b4 is_a_triangle & {b2,b3} on b5 & {b2,b4} on b6
holds b5 <> b6;
:: PROJPL_1:th 20
theorem
for b1 being IncProjStr
for b2, b3, b4, b5 being Element of the Points of b1
for b6, b7, b8 being Element of the Lines of b1
st b2,b3,b4,b5 is_a_quadrangle & {b2,b3} on b6 & {b2,b4} on b7 & {b2,b5} on b8
holds b6,b7,b8 are_mutually_different;
:: PROJPL_1:th 21
theorem
for b1 being IncProjStr
for b2, b3, b4, b5 being Element of the Points of b1
for b6, b7, b8, b9 being Element of the Lines of b1
st b1 is configuration & b2 on b6,b7,b8 & b6,b7,b8 are_mutually_different & not b2 on b9 & b3 on b9,b6 & b4 on b9,b7 & b5 on b9,b8
holds b3,b4,b5 are_mutually_different;
:: PROJPL_1:th 22
theorem
for b1 being IncProjStr
st b1 is configuration &
(for b2, b3 being Element of the Points of b1 holds
ex b4 being Element of the Lines of b1 st
{b2,b3} on b4) &
(for b2, b3 being Element of the Lines of b1 holds
ex b4 being Element of the Points of b1 st
b4 on b2,b3) &
(ex b2, b3, b4, b5 being Element of the Points of b1 st
b2,b3,b4,b5 is_a_quadrangle)
for b2 being Element of the Lines of b1 holds
ex b3, b4, b5 being Element of the Points of b1 st
b3,b4,b5 are_mutually_different & {b3,b4,b5} on b2;
:: PROJPL_1:th 23
theorem
for b1 being IncProjStr holds
b1 is linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional IncProjStr
iff
b1 is configuration &
(for b2, b3 being Element of the Points of b1 holds
ex b4 being Element of the Lines of b1 st
{b2,b3} on b4) &
(for b2, b3 being Element of the Lines of b1 holds
ex b4 being Element of the Points of b1 st
b4 on b2,b3) &
(ex b2, b3, b4, b5 being Element of the Points of b1 st
b2,b3,b4,b5 is_a_quadrangle);
:: PROJPL_1:funcnot 2 => PROJPL_1:func 2
definition
let a1 be linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional IncProjStr;
let a2, a3 be Element of the Lines of a1;
assume a2 <> a3;
func A2 * A3 -> Element of the Points of a1 means
it on a2,a3;
end;
:: PROJPL_1:def 9
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional IncProjStr
for b2, b3 being Element of the Lines of b1
st b2 <> b3
for b4 being Element of the Points of b1 holds
b4 = b2 * b3
iff
b4 on b2,b3;
:: PROJPL_1:th 24
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional IncProjStr
for b2 being Element of the Points of b1
for b3, b4 being Element of the Lines of b1
st b3 <> b4
holds b3 * b4 on b3 &
b3 * b4 on b4 &
b3 * b4 = b4 * b3 &
(b2 on b3 & b2 on b4 implies b2 = b3 * b4);
:: PROJPL_1:th 25
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional IncProjStr
for b2, b3 being Element of the Points of b1
for b4, b5 being Element of the Lines of b1
st b4 <> b5 & b2 on b4 & not b3 on b4 & b2 <> b4 * b5
holds (b3 * b2) * b5 on b5 & not (b3 * b2) * b5 on b4;
:: PROJPL_1:th 26
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian IncProjStr
for b2, b3, b4 being Element of the Points of b1
st b2,b3,b4 is_a_triangle
holds b2,b3,b4 are_mutually_different;
:: PROJPL_1:th 27
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian IncProjStr
for b2, b3, b4, b5 being Element of the Points of b1
st b2,b3,b4,b5 is_a_quadrangle
holds b2,b3,b4,b5 are_mutually_different;
:: PROJPL_1:th 28
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional IncProjStr
for b2, b3, b4, b5 being Element of the Points of b1
st b2 * b4 = b3 * b5 & b2 <> b4 & b3 <> b5 & b4 <> b5
holds b2 * b4 = b4 * b5;
:: PROJPL_1:th 29
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional IncProjStr
for b2, b3, b4, b5 being Element of the Points of b1
st b2 * b4 = b3 * b5 & b2 <> b4 & b3 <> b5 & b4 <> b5
holds b2 on b4 * b5;
:: PROJPL_1:th 30
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional IncProjStr
for b2, b3, b4 being Element of the Points of b1
for b5 being Element of the Lines of b1
st b3 on b5 & b4 on b5 & b3 <> b4 & not b2 on b5
holds b3 * b2 <> b4 * b2 & b2 * b3 <> b2 * b4;
:: PROJPL_1:th 31
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional IncProjStr
for b2 being Element of the Points of b1
for b3, b4, b5 being Element of the Lines of b1
st b2 on b4 & b2 on b5 & b4 <> b5 & not b2 on b3
holds b3 * b4 <> b3 * b5 & b4 * b3 <> b5 * b3;
:: PROJPL_1:th 32
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian IncProjStr
for b2, b3, b4, b5 being Element of the Points of b1
st b4 on b2 * b3 & b4 on b2 * b5 & b4 <> b2 & b5 <> b2 & b2 <> b3
holds b5 on b2 * b3;
:: PROJPL_1:th 33
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian IncProjStr
for b2, b3, b4 being Element of the Points of b1
st b4 on b2 * b3 & b2 <> b4
holds b3 on b2 * b4;
:: PROJPL_1:th 34
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional IncProjStr
for b2, b3, b4, b5 being Element of the Points of b1
for b6 being Element of the Lines of b1
st b4 <> b5 & b4 on b6 & b5 on b6 & not b2 on b6 & not b3 on b6
holds b2 * b4 <> b3 * b5;
:: PROJPL_1:th 35
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional IncProjStr
for b2, b3, b4 being Element of the Points of b1
st b2 on b3 * b4 & b2 <> b4 & b3 <> b4
holds b3 on b4 * b2;
:: PROJPL_1:th 36
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional IncProjStr
for b2, b3, b4 being Element of the Points of b1
st b2 on b3 * b4 & b3 <> b2 & b3 <> b4
holds b4 on b3 * b2;
:: PROJPL_1:th 37
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional IncProjStr
for b2, b3, b4, b5, b6 being Element of the Points of b1
for b7, b8 being Element of the Lines of b1
st b3 on b8 & b4 on b8 & b2 on b7 & not b2 on b8 & b3 <> b4 & b5 <> b2 & b6 <> b2 & b5 on b2 * b3 & b6 on b2 * b4
holds ex b9 being Element of the Points of b1 st
b9 on b5 * b6 & b9 on b7 & b9 <> b2;