Article PROJRED1, MML version 4.99.1005
:: PROJRED1:th 1
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian IncProjStr
for b2 being Element of the Lines of b1 holds
ex b3 being Element of the Points of b1 st
not b3 on b2;
:: PROJRED1:th 2
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian IncProjStr
for b2 being Element of the Points of b1 holds
ex b3 being Element of the Lines of b1 st
not b2 on b3;
:: PROJRED1:th 3
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian IncProjStr
for b2, b3 being Element of the Lines of b1
st b2 <> b3
holds ex b4, b5 being Element of the Points of b1 st
b4 on b2 & not b4 on b3 & b5 on b3 & not b5 on b2;
:: PROJRED1:th 4
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
holds ex b4, b5 being Element of the Lines of b1 st
b2 on b4 & not b2 on b5 & b3 on b5 & not b3 on b4;
:: PROJRED1:th 5
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian IncProjStr
for b2 being Element of the Points of b1 holds
ex b3, b4, b5 being Element of the Lines of b1 st
b2 on b3 & b2 on b4 & b2 on b5 & b3 <> b4 & b4 <> b5 & b5 <> b3;
:: PROJRED1:th 6
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian IncProjStr
for b2, b3 being Element of the Lines of b1 holds
ex b4 being Element of the Points of b1 st
not b4 on b2 & not b4 on b3;
:: PROJRED1:th 7
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian IncProjStr
for b2 being Element of the Lines of b1 holds
ex b3 being Element of the Points of b1 st
b3 on b2;
:: PROJRED1:th 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
for b4 being Element of the Lines of b1 holds
ex b5 being Element of the Points of b1 st
b5 on b4 & b5 <> b2 & b5 <> b3;
:: PROJRED1:th 9
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian IncProjStr
for b2, b3 being Element of the Points of b1 holds
ex b4 being Element of the Lines of b1 st
not b2 on b4 & not b3 on b4;
:: PROJRED1:th 12
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
for b6, b7, b8, b9 being Element of the Lines of b1
st b2 on b6 & b2 on b7 & b6 <> b7 & b3 on b6 & b2 <> b3 & b4 on b7 & b5 on b7 & b4 <> b5 & b3 on b8 & b4 on b8 & b3 on b9 & b5 on b9
holds b8 <> b9;
:: PROJRED1:th 13
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
for b5 being Element of the Lines of b1
st {b2,b3,b4} on b5
holds {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;
:: PROJRED1:th 14
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian Desarguesian IncProjStr
for b2, b3, b4, b5, b6, b7, b8, b9, b10, b11 being Element of the Points of b1
for b12, b13, b14, b15, b16, b17, b18, b19, b20 being Element of the Lines of b1
st {b2,b3,b4} on b12 & {b2,b6,b5} on b13 & {b2,b8,b7} on b14 & {b8,b6,b11} on b15 & {b8,b9,b4} on b16 & {b6,b10,b4} on b17 & {b11,b5,b7} on b18 & {b3,b9,b7} on b19 & {b3,b10,b5} on b20 & b12,b13,b14 are_mutually_different & b2 <> b8 & b2 <> b3 & b2 <> b5 & b6 <> b5
holds ex b21 being Element of the Lines of b1 st
{b9,b10,b11} on b21;
:: PROJRED1:th 15
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian IncProjStr
st ex b2 being Element of the Lines of b1 st
ex b3, b4, b5, b6 being Element of the Points of b1 st
b3 on b2 & b4 on b2 & b5 on b2 & b6 on b2 & b3,b4,b5,b6 are_mutually_different
for b2 being Element of the Lines of b1 holds
ex b3, b4, b5, b6 being Element of the Points of b1 st
b3 on b2 & b4 on b2 & b5 on b2 & b6 on b2 & b3,b4,b5,b6 are_mutually_different;
:: PROJRED1:th 16
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian Fanoian IncProjStr holds
ex b2, b3, b4, b5, b6, b7, b8 being Element of the Points of b1 st
ex b9, b10, b11, b12, b13, b14, b15, b16 being Element of the Lines of b1 st
not b3 on b13 & not b4 on b13 & not b2 on b12 & not b5 on b12 & not b2 on b14 & not b4 on b14 & not b3 on b15 & not b5 on b15 & {b6,b2,b5} on b13 & {b6,b3,b4} on b12 & {b7,b3,b5} on b14 & {b7,b2,b4} on b15 & {b8,b2,b3} on b9 & {b8,b4,b5} on b10 & {b6,b7} on b11 & not b8 on b11;
:: PROJRED1:th 17
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian Fanoian IncProjStr holds
ex b2 being Element of the Points of b1 st
ex b3, b4, b5, b6 being Element of the Lines of b1 st
b2 on b3 & b2 on b4 & b2 on b5 & b2 on b6 & b3,b4,b5,b6 are_mutually_different;
:: PROJRED1:th 18
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian Fanoian IncProjStr holds
ex b2, b3, b4, b5 being Element of the Points of b1 st
ex b6 being Element of the Lines of b1 st
b2 on b6 & b3 on b6 & b4 on b6 & b5 on b6 & b2,b3,b4,b5 are_mutually_different;
:: PROJRED1:th 19
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian Fanoian IncProjStr
for b2 being Element of the Lines of b1 holds
ex b3, b4, b5, b6 being Element of the Points of b1 st
b3 on b2 & b4 on b2 & b5 on b2 & b6 on b2 & b3,b4,b5,b6 are_mutually_different;
:: PROJRED1:funcnot 1 => PROJRED1:func 1
definition
let a1 be linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional Desarguesian IncProjStr;
let a2, a3 be Element of the Lines of a1;
let a4 be Element of the Points of a1;
assume not a4 on a2 & not a4 on a3;
func IncProj(A2,A4,A3) -> Function-like Relation of the Points of a1,the Points of a1 means
dom it c= the Points of a1 &
(for b1 being Element of the Points of a1 holds
b1 in dom it
iff
b1 on a2) &
(for b1, b2 being Element of the Points of a1
st b1 on a2 & b2 on a3
holds it . b1 = b2
iff
ex b3 being Element of the Lines of a1 st
a4 on b3 & b1 on b3 & b2 on b3);
end;
:: PROJRED1:def 1
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional Desarguesian IncProjStr
for b2, b3 being Element of the Lines of b1
for b4 being Element of the Points of b1
st not b4 on b2 & not b4 on b3
for b5 being Function-like Relation of the Points of b1,the Points of b1 holds
b5 = IncProj(b2,b4,b3)
iff
dom b5 c= the Points of b1 &
(for b6 being Element of the Points of b1 holds
b6 in dom b5
iff
b6 on b2) &
(for b6, b7 being Element of the Points of b1
st b6 on b2 & b7 on b3
holds b5 . b6 = b7
iff
ex b8 being Element of the Lines of b1 st
b4 on b8 & b6 on b8 & b7 on b8);
:: PROJRED1:th 21
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional Desarguesian IncProjStr
for b2 being Element of the Points of b1
for b3 being Element of the Lines of b1
st not b2 on b3
for b4 being Element of the Points of b1
st b4 on b3
holds (IncProj(b3,b2,b3)) . b4 = b4;
:: PROJRED1:th 22
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional Desarguesian IncProjStr
for b2, b3 being Element of the Points of b1
for b4, b5 being Element of the Lines of b1
st not b2 on b4 & not b2 on b5 & b3 on b4
holds (IncProj(b4,b2,b5)) . b3 is Element of the Points of b1;
:: PROJRED1:th 23
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional Desarguesian IncProjStr
for b2, b3, b4 being Element of the Points of b1
for b5, b6 being Element of the Lines of b1
st not b2 on b5 & not b2 on b6 & b3 on b5 & b4 = (IncProj(b5,b2,b6)) . b3
holds b4 on b6;
:: PROJRED1:th 24
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional Desarguesian IncProjStr
for b2, b3 being Element of the Points of b1
for b4, b5 being Element of the Lines of b1
st not b2 on b4 & not b2 on b5 & b3 in proj2 IncProj(b4,b2,b5)
holds b3 on b5;
:: PROJRED1:th 25
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional Desarguesian IncProjStr
for b2, b3 being Element of the Points of b1
for b4, b5, b6 being Element of the Lines of b1
st not b2 on b4 & not b2 on b5 & not b3 on b5 & not b3 on b6
holds proj1 ((IncProj(b4,b2,b5)) * IncProj(b5,b3,b6)) = dom IncProj(b4,b2,b5) &
proj2 ((IncProj(b4,b2,b5)) * IncProj(b5,b3,b6)) = proj2 IncProj(b5,b3,b6);
:: PROJRED1:th 26
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional Desarguesian IncProjStr
for b2 being Element of the Points of b1
for b3, b4 being Element of the Lines of b1
for b5, b6, b7, b8 being Element of the Points of b1
st not b2 on b3 & not b2 on b4 & b5 on b3 & b6 on b3 & (IncProj(b3,b2,b4)) . b5 = b7 & (IncProj(b3,b2,b4)) . b6 = b8 & b7 = b8
holds b5 = b6;
:: PROJRED1:th 27
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional Desarguesian IncProjStr
for b2, b3 being Element of the Points of b1
for b4, b5 being Element of the Lines of b1
st not b2 on b4 & not b2 on b5 & b3 on b4 & b3 on b5
holds (IncProj(b4,b2,b5)) . b3 = b3;
:: PROJRED1:th 28
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional Desarguesian IncProjStr
for b2, b3, b4 being Element of the Points of b1
for b5, b6, b7 being Element of the Lines of b1
st not b2 on b5 & not b2 on b6 & not b3 on b6 & not b3 on b7 & b4 on b5 & b4 on b6 & b4 on b7 & b5 <> b7
holds ex b8 being Element of the Points of b1 st
not b8 on b5 &
not b8 on b7 &
(IncProj(b5,b2,b6)) * IncProj(b6,b3,b7) = IncProj(b5,b8,b7);