Article PROJRED2, MML version 4.99.1005
:: PROJRED2:prednot 1 => PROJRED2:pred 1
definition
let a1 be linear partial up-2-dimensional up-3-rank Vebleian IncProjStr;
let a2, a3, a4 be Element of the Lines of a1;
pred A2,A3,A4 are_concurrent means
ex b1 being Element of the Points of a1 st
b1 on a2 & b1 on a3 & b1 on a4;
end;
:: PROJRED2:dfs 1
definiens
let a1 be linear partial up-2-dimensional up-3-rank Vebleian IncProjStr;
let a2, a3, a4 be Element of the Lines of a1;
To prove
a2,a3,a4 are_concurrent
it is sufficient to prove
thus ex b1 being Element of the Points of a1 st
b1 on a2 & b1 on a3 & b1 on a4;
:: PROJRED2:def 1
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian IncProjStr
for b2, b3, b4 being Element of the Lines of b1 holds
b2,b3,b4 are_concurrent
iff
ex b5 being Element of the Points of b1 st
b5 on b2 & b5 on b3 & b5 on b4;
:: PROJRED2:funcnot 1 => PROJRED2:func 1
definition
let a1 be linear partial up-2-dimensional up-3-rank Vebleian IncProjStr;
let a2 be Element of the Lines of a1;
func CHAIN A2 -> Element of bool the Points of a1 equals
{b1 where b1 is Element of the Points of a1: b1 on a2};
end;
:: PROJRED2:def 2
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian IncProjStr
for b2 being Element of the Lines of b1 holds
CHAIN b2 = {b3 where b3 is Element of the Points of b1: b3 on b2};
:: PROJRED2:modenot 1 => PROJRED2:mode 1
definition
let a1 be linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional Desarguesian IncProjStr;
mode Projection of A1 -> Function-like Relation of the Points of a1,the Points of a1 means
ex b1 being Element of the Points of a1 st
ex b2, b3 being Element of the Lines of a1 st
not b1 on b2 & not b1 on b3 & it = IncProj(b2,b1,b3);
end;
:: PROJRED2:dfs 3
definiens
let a1 be linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional Desarguesian IncProjStr;
let a2 be Function-like Relation of the Points of a1,the Points of a1;
To prove
a2 is Projection of a1
it is sufficient to prove
thus ex b1 being Element of the Points of a1 st
ex b2, b3 being Element of the Lines of a1 st
not b1 on b2 & not b1 on b3 & a2 = IncProj(b2,b1,b3);
:: PROJRED2:def 3
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional Desarguesian IncProjStr
for b2 being Function-like Relation of the Points of b1,the Points of b1 holds
b2 is Projection of b1
iff
ex b3 being Element of the Points of b1 st
ex b4, b5 being Element of the Lines of b1 st
not b3 on b4 & not b3 on b5 & b2 = IncProj(b4,b3,b5);
:: PROJRED2:th 1
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 Lines of b1
st (b2 <> b3 & b3 <> b4 implies b4 = b2)
holds b2,b3,b4 are_concurrent;
:: PROJRED2:th 2
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 Lines of b1
st b2,b3,b4 are_concurrent
holds b2,b4,b3 are_concurrent & b3,b2,b4 are_concurrent & b3,b4,b2 are_concurrent & b4,b2,b3 are_concurrent & b4,b3,b2 are_concurrent;
:: PROJRED2:th 3
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 b5
holds ex b6 being Element of the Points of b1 st
b6 on b4 & (IncProj(b4,b2,b5)) . b6 = b3;
:: PROJRED2:th 5
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
st not b2 on b3 & not b2 on b4
holds dom IncProj(b3,b2,b4) = CHAIN b3;
:: PROJRED2:th 6
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
st not b2 on b3 & not b2 on b4
holds proj2 IncProj(b3,b2,b4) = CHAIN b4;
:: PROJRED2:th 7
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional Desarguesian IncProjStr
for b2 being Element of the Lines of b1
for b3 being set holds
b3 in CHAIN b2
iff
ex b4 being Element of the Points of b1 st
b3 = b4 & b4 on b2;
:: PROJRED2:th 8
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
st not b2 on b3 & not b2 on b4
holds IncProj(b3,b2,b4) is one-to-one;
:: PROJRED2:th 9
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
st not b2 on b3 & not b2 on b4
holds (IncProj(b3,b2,b4)) " = IncProj(b4,b2,b3);
:: PROJRED2:th 10
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional Desarguesian IncProjStr
for b2 being Projection of b1 holds
b2 " is Projection of b1;
:: PROJRED2:th 11
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
holds IncProj(b3,b2,b3) = id CHAIN b3;
:: PROJRED2:th 12
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional Desarguesian IncProjStr
for b2 being Element of the Lines of b1 holds
id CHAIN b2 is Projection of b1;
:: PROJRED2:th 13
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, b5 being Element of the Lines of b1
st not b2 on b3 & not b2 on b4 & not b2 on b5
holds (IncProj(b3,b2,b5)) * IncProj(b5,b2,b4) = IncProj(b3,b2,b4);
:: PROJRED2:th 14
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 & b4,b5,b6 are_concurrent & b4 <> b6
holds ex b7 being Element of the Points of b1 st
not b7 on b4 &
not b7 on b6 &
(IncProj(b4,b2,b5)) * IncProj(b5,b3,b6) = IncProj(b4,b7,b6);
:: PROJRED2:th 15
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional Desarguesian IncProjStr
for b2, b3, b4, b5, b6, b7, b8 being Element of the Points of b1
for b9, b10, b11, b12, b13, b14, b15, b16 being Element of the Lines of b1
st not b2 on b9 & not b3 on b10 & not b2 on b11 & not b3 on b11 & not b9,b10,b11 are_concurrent & b4 on b9 & b4 on b11 & b4 on b12 & not b3 on b12 & b9 <> b12 & b2 <> b3 & b3 <> b5 & b2 on b13 & b3 on b13 & not b10,b11,b13 are_concurrent & b6 on b11 & b6 on b10 & b2 on b14 & b6 on b14 & b7 on b9 & b7 on b14 & b5 on b13 & b5 on b15 & b7 on b15 & b8 on b15 & b6 on b16 & b3 on b16 & b8 on b16 & b8 on b12 & b12 <> b11 & b5 <> b2 & not b5 on b9 & not b5 on b12
holds (IncProj(b9,b2,b11)) * IncProj(b11,b3,b10) = (IncProj(b9,b5,b12)) * IncProj(b12,b3,b10);
:: PROJRED2:th 16
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional Desarguesian IncProjStr
for b2, b3, b4, b5, b6, b7, b8, b9, b10 being Element of the Points of b1
for b11, b12, b13, b14, b15, b16, b17, b18 being Element of the Lines of b1
st not b2 on b11 & not b2 on b12 & not b3 on b13 & not b3 on b12 & not b3 on b14 & not b11,b13,b12 are_concurrent & b2 <> b3 & b3 <> b4 & b11 <> b14 & {b5,b6} on b11 & {b6,b7,b8} on b13 & {b5,b8,b9} on b12 & {b2,b3,b8} on b15 & {b5,b10} on b14 & {b2,b6,b9} on b16 & {b3,b9,b10} on b17 & {b6,b10,b4} on b18 & b4 on b15
holds (IncProj(b11,b2,b12)) * IncProj(b12,b3,b13) = (IncProj(b11,b4,b14)) * IncProj(b14,b3,b13);
:: PROJRED2:th 17
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional Desarguesian IncProjStr
for b2, b3, b4, b5, b6, b7, b8 being Element of the Points of b1
for b9, b10, b11, b12, b13, b14, b15, b16 being Element of the Lines of b1
st not b2 on b9 & not b2 on b10 & not b3 on b11 & not b3 on b10 & not b3 on b12 & not b9,b11,b10 are_concurrent & not b11,b10,b13 are_concurrent & b9 <> b12 & b12 <> b10 & b2 <> b3 & {b4,b5} on b9 & b6 on b11 & {b4,b6} on b10 & {b2,b3,b7} on b13 & {b4,b8} on b12 & {b2,b6,b5} on b14 & {b7,b5,b8} on b15 & {b3,b6,b8} on b16
holds b7 <> b2 & b7 <> b3 & not b7 on b9 & not b7 on b12;
:: PROJRED2:th 18
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional Desarguesian IncProjStr
for b2, b3, b4, b5, b6, b7, b8, b9, b10 being Element of the Points of b1
for b11, b12, b13, b14, b15, b16, b17, b18 being Element of the Lines of b1
st not b2 on b11 & not b2 on b12 & not b3 on b13 & not b3 on b12 & not b3 on b14 & not b11,b13,b12 are_concurrent & b2 <> b3 & b11 <> b14 & {b4,b5} on b11 & {b5,b6,b7} on b13 & {b4,b7,b8} on b12 & {b2,b3,b7} on b15 & {b4,b9} on b14 & {b2,b5,b8} on b16 & {b3,b8,b9} on b17 & {b5,b9,b10} on b18 & b10 on b15
holds not b10 on b11 & not b10 on b14 & b3 <> b10;
:: PROJRED2:th 19
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional Desarguesian IncProjStr
for b2, b3, b4, b5, b6, b7, b8 being Element of the Points of b1
for b9, b10, b11, b12, b13, b14, b15, b16 being Element of the Lines of b1
st not b2 on b9 & not b2 on b10 & not b3 on b11 & not b3 on b10 & not b4 on b9 & not b9,b11,b10 are_concurrent & not b11,b10,b12 are_concurrent & b2 <> b3 & b3 <> b4 & b4 <> b2 & {b5,b6} on b9 & b7 on b11 & {b5,b7} on b10 & {b2,b3,b4} on b12 & {b5,b8} on b13 & {b2,b7,b6} on b14 & {b4,b6,b8} on b15 & {b3,b7,b8} on b16
holds b13 <> b9 & b13 <> b10 & not b4 on b13 & not b3 on b13;
:: PROJRED2:th 20
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional Desarguesian IncProjStr
for b2, b3, b4, b5, b6, b7, b8, b9, b10 being Element of the Points of b1
for b11, b12, b13, b14, b15, b16, b17, b18 being Element of the Lines of b1
st not b2 on b11 & not b2 on b12 & not b3 on b13 & not b3 on b12 & not b4 on b11 & not b11,b13,b12 are_concurrent & b2 <> b3 & b3 <> b4 & {b5,b6} on b11 & {b6,b7,b8} on b13 & {b5,b8,b9} on b12 & {b2,b3,b8} on b14 & {b5,b10} on b15 & {b2,b6,b9} on b16 & {b3,b9,b10} on b17 & {b6,b10,b4} on b18 & b4 on b14
holds not b3 on b15 & not b4 on b15 & b11 <> b15;
:: PROJRED2:th 21
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, b7, b8 being Element of the Lines of b1
st not b2 on b4 & not b3 on b5 & not b2 on b6 & not b3 on b6 & not b4,b5,b6 are_concurrent & b4,b6,b7 are_concurrent & not b3 on b7 & b4 <> b7 & b2 <> b3 & b2 on b8 & b3 on b8
holds ex b9 being Element of the Points of b1 st
b9 on b8 &
not b9 on b4 &
not b9 on b7 &
(IncProj(b4,b2,b6)) * IncProj(b6,b3,b5) = (IncProj(b4,b9,b7)) * IncProj(b7,b3,b5);
:: PROJRED2: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, b6, b7, b8 being Element of the Lines of b1
st not b2 on b4 & not b3 on b5 & not b2 on b6 & not b3 on b6 & not b4,b5,b6 are_concurrent & b5,b6,b7 are_concurrent & not b2 on b7 & b5 <> b7 & b2 <> b3 & b2 on b8 & b3 on b8
holds ex b9 being Element of the Points of b1 st
b9 on b8 &
not b9 on b5 &
not b9 on b7 &
(IncProj(b4,b2,b6)) * IncProj(b6,b3,b5) = (IncProj(b4,b2,b7)) * IncProj(b7,b9,b5);
:: PROJRED2:th 23
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional Desarguesian IncProjStr
for b2, b3, b4, b5, b6, b7 being Element of the Points of b1
for b8, b9, b10, b11, b12, b13 being Element of the Lines of b1
st not b2 on b8 & not b3 on b9 & not b2 on b10 & not b3 on b10 & not b2 on b9 & not b3 on b8 & b4 on b8 & b4 on b10 & b5 on b9 & b5 on b10 & b2 on b11 & b5 on b11 & b4 on b12 & b3 on b12 & b6 on b8 & b6 on b11 & b7 on b9 & b7 on b12 & b6 on b13 & b7 on b13 & not b8,b9,b10 are_concurrent
holds (IncProj(b8,b2,b10)) * IncProj(b10,b3,b9) = (IncProj(b8,b3,b13)) * IncProj(b13,b2,b9);
:: PROJRED2:th 24
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, b8 being Element of the Lines of b1
st not b2 on b5 & not b3 on b6 & not b2 on b7 & not b3 on b7 & b2 <> b3 & b2 on b8 & b3 on b8 & b4 on b8 & not b4 on b5 & b4 <> b3 & not b5,b6,b7 are_concurrent
holds ex b9 being Element of the Lines of b1 st
b5,b7,b9 are_concurrent &
not b3 on b9 &
not b4 on b9 &
(IncProj(b5,b2,b7)) * IncProj(b7,b3,b6) = (IncProj(b5,b4,b9)) * IncProj(b9,b3,b6);
:: PROJRED2:th 25
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, b8 being Element of the Lines of b1
st not b2 on b5 & not b3 on b6 & not b2 on b7 & not b3 on b7 & b2 <> b3 & b2 on b8 & b3 on b8 & b4 on b8 & not b4 on b6 & b4 <> b2 & not b5,b6,b7 are_concurrent
holds ex b9 being Element of the Lines of b1 st
b6,b7,b9 are_concurrent &
not b2 on b9 &
not b4 on b9 &
(IncProj(b5,b2,b7)) * IncProj(b7,b3,b6) = (IncProj(b5,b2,b9)) * IncProj(b9,b4,b6);