:: AFPROJ semantic presentation
theorem Th1: :: AFPROJ:1
theorem Th2: :: AFPROJ:2
theorem Th3: :: AFPROJ:3
theorem Th4: :: AFPROJ:4
theorem Th5: :: AFPROJ:5
theorem Th6: :: AFPROJ:6
:: deftheorem Def1 defines AfLines AFPROJ:def 1 :
:: deftheorem Def2 defines AfPlanes AFPROJ:def 2 :
theorem Th7: :: AFPROJ:7
theorem Th8: :: AFPROJ:8
:: deftheorem Def3 defines LinesParallelity AFPROJ:def 3 :
:: deftheorem Def4 defines PlanesParallelity AFPROJ:def 4 :
:: deftheorem Def5 defines LDir AFPROJ:def 5 :
:: deftheorem Def6 defines PDir AFPROJ:def 6 :
theorem Th9: :: AFPROJ:9
theorem Th10: :: AFPROJ:10
theorem Th11: :: AFPROJ:11
theorem Th12: :: AFPROJ:12
theorem Th13: :: AFPROJ:13
:: deftheorem Def7 defines Dir_of_Lines AFPROJ:def 7 :
:: deftheorem Def8 defines Dir_of_Planes AFPROJ:def 8 :
theorem Th14: :: AFPROJ:14
theorem Th15: :: AFPROJ:15
theorem Th16: :: AFPROJ:16
theorem Th17: :: AFPROJ:17
theorem Th18: :: AFPROJ:18
theorem Th19: :: AFPROJ:19
:: deftheorem Def9 defines ProjectivePoints AFPROJ:def 9 :
:: deftheorem Def10 defines ProjectiveLines AFPROJ:def 10 :
definition
let AS be
AffinSpace;
func Proj_Inc c1 -> Relation of
ProjectivePoints a1,
ProjectiveLines a1 means :
Def11:
:: AFPROJ:def 11
for
x,
y being
set holds
(
[x,y] in it iff ( ex
K being
Subset of
AS st
(
K is_line &
y = [K,1] & ( (
x in the
carrier of
AS &
x in K ) or
x = LDir K ) ) or ex
K,
X being
Subset of
AS st
(
K is_line &
X is_plane &
x = LDir K &
y = [(PDir X),2] &
K '||' X ) ) );
existence
ex b1 being Relation of ProjectivePoints AS, ProjectiveLines AS st
for x, y being set holds
( [x,y] in b1 iff ( ex K being Subset of AS st
( K is_line & y = [K,1] & ( ( x in the carrier of AS & x in K ) or x = LDir K ) ) or ex K, X being Subset of AS st
( K is_line & X is_plane & x = LDir K & y = [(PDir X),2] & K '||' X ) ) )
uniqueness
for b1, b2 being Relation of ProjectivePoints AS, ProjectiveLines AS st ( for x, y being set holds
( [x,y] in b1 iff ( ex K being Subset of AS st
( K is_line & y = [K,1] & ( ( x in the carrier of AS & x in K ) or x = LDir K ) ) or ex K, X being Subset of AS st
( K is_line & X is_plane & x = LDir K & y = [(PDir X),2] & K '||' X ) ) ) ) & ( for x, y being set holds
( [x,y] in b2 iff ( ex K being Subset of AS st
( K is_line & y = [K,1] & ( ( x in the carrier of AS & x in K ) or x = LDir K ) ) or ex K, X being Subset of AS st
( K is_line & X is_plane & x = LDir K & y = [(PDir X),2] & K '||' X ) ) ) ) holds
b1 = b2
end;
:: deftheorem Def11 defines Proj_Inc AFPROJ:def 11 :
definition
let AS be
AffinSpace;
func Inc_of_Dir c1 -> Relation of
Dir_of_Lines a1,
Dir_of_Planes a1 means :
Def12:
:: AFPROJ:def 12
for
x,
y being
set holds
(
[x,y] in it iff ex
A,
X being
Subset of
AS st
(
x = LDir A &
y = PDir X &
A is_line &
X is_plane &
A '||' X ) );
existence
ex b1 being Relation of Dir_of_Lines AS, Dir_of_Planes AS st
for x, y being set holds
( [x,y] in b1 iff ex A, X being Subset of AS st
( x = LDir A & y = PDir X & A is_line & X is_plane & A '||' X ) )
uniqueness
for b1, b2 being Relation of Dir_of_Lines AS, Dir_of_Planes AS st ( for x, y being set holds
( [x,y] in b1 iff ex A, X being Subset of AS st
( x = LDir A & y = PDir X & A is_line & X is_plane & A '||' X ) ) ) & ( for x, y being set holds
( [x,y] in b2 iff ex A, X being Subset of AS st
( x = LDir A & y = PDir X & A is_line & X is_plane & A '||' X ) ) ) holds
b1 = b2
end;
:: deftheorem Def12 defines Inc_of_Dir AFPROJ:def 12 :
:: deftheorem Def13 defines IncProjSp_of AFPROJ:def 13 :
:: deftheorem Def14 defines ProjHorizon AFPROJ:def 14 :
theorem Th20: :: AFPROJ:20
theorem Th21: :: AFPROJ:21
theorem Th22: :: AFPROJ:22
theorem Th23: :: AFPROJ:23
theorem Th24: :: AFPROJ:24
theorem Th25: :: AFPROJ:25
theorem Th26: :: AFPROJ:26
theorem Th27: :: AFPROJ:27
theorem Th28: :: AFPROJ:28
theorem Th29: :: AFPROJ:29
theorem Th30: :: AFPROJ:30
theorem Th31: :: AFPROJ:31
theorem Th32: :: AFPROJ:32
theorem Th33: :: AFPROJ:33
theorem Th34: :: AFPROJ:34
theorem Th35: :: AFPROJ:35
theorem Th36: :: AFPROJ:36
theorem Th37: :: AFPROJ:37
theorem Th38: :: AFPROJ:38
theorem Th39: :: AFPROJ:39
theorem Th40: :: AFPROJ:40
Lemma113:
for AS being AffinSpace st AS is not AffinPlane holds
ex a being Element of the Points of (ProjHorizon AS) ex A being Element of the Lines of (ProjHorizon AS) st not a on A
Lemma114:
for AS being AffinSpace
for a, b being POINT of (IncProjSp_of AS)
for A, K being LINE of (IncProjSp_of AS) st a on A & a on K & b on A & b on K & not a = b holds
A = K
Lemma115:
for AS being AffinSpace
for A being LINE of (IncProjSp_of AS) ex a, b, c being POINT of (IncProjSp_of AS) st
( a on A & b on A & c on A & a <> b & b <> c & c <> a )
Lemma116:
for AS being AffinSpace
for a, b being POINT of (IncProjSp_of AS) ex A being LINE of (IncProjSp_of AS) st
( a on A & b on A )
Lemma117:
for AS being AffinSpace
for A, K being LINE of (IncProjSp_of AS) st AS is AffinPlane holds
ex a being POINT of (IncProjSp_of AS) st
( a on A & a on K )
Lemma122:
for AS being AffinSpace holds
not for a being POINT of (IncProjSp_of AS)
for A being LINE of (IncProjSp_of AS) holds a on A
theorem Th41: :: AFPROJ:41
theorem Th42: :: AFPROJ:42
Lemma127:
for AS being AffinSpace
for x being Element of AS
for X, X', Y, Y' being Subset of AS
for p, b, c being POINT of (IncProjSp_of AS)
for A, K, M being LINE of (IncProjSp_of AS) st X is_line & X' is_line & Y is_plane & X c= Y & X' c= Y & x = p & A = [X,1] & K = [X',1] & A <> K & b on A & c on K & b on M & c on M & b <> c & M = [Y',1] & Y' is_line holds
Y' c= Y
Lemma130:
for AS being AffinSpace
for x being Element of AS
for X, X', Y, Y' being Subset of AS
for p, b, c being POINT of (IncProjSp_of AS)
for A, K, M being LINE of (IncProjSp_of AS) st X is_line & X' is_line & Y is_plane & X c= Y & X' c= Y & x = p & A = [X,1] & K = [X',1] & A <> K & b on A & c on K & b on M & c on M & b <> c & M = [(PDir Y'),2] & Y' is_plane holds
( Y' '||' Y & Y '||' Y' )
theorem Th43: :: AFPROJ:43
Lemma132:
for AS being AffinSpace
for a, b, c, d, p being POINT of (IncProjSp_of AS)
for M, N, P, Q being LINE of (IncProjSp_of AS) st a on M & b on M & c on N & d on N & p on M & p on N & a on P & c on P & b on Q & d on Q & not p on P & not p on Q & M <> N & p is Element of AS holds
ex q being POINT of (IncProjSp_of AS) st
( q on P & q on Q )
Lemma144:
for AS being AffinSpace
for a, b, c, d, p being POINT of (IncProjSp_of AS)
for M, N, P, Q being LINE of (IncProjSp_of AS) st a on M & b on M & c on N & d on N & p on M & p on N & a on P & c on P & b on Q & d on Q & not p on P & not p on Q & M <> N & p is not Element of AS & a is Element of AS holds
ex q being POINT of (IncProjSp_of AS) st
( q on P & q on Q )
Lemma154:
for AS being AffinSpace
for a, b, c, d, p being POINT of (IncProjSp_of AS)
for M, N, P, Q being LINE of (IncProjSp_of AS) st a on M & b on M & c on N & d on N & p on M & p on N & a on P & c on P & b on Q & d on Q & not p on P & not p on Q & M <> N & p is not Element of AS & ( a is Element of AS or b is Element of AS or c is Element of AS or d is Element of AS ) holds
ex q being POINT of (IncProjSp_of AS) st
( q on P & q on Q )
Lemma155:
for AS being AffinSpace
for a, b, c, d, p being POINT of (IncProjSp_of AS)
for M, N, P, Q being LINE of (IncProjSp_of AS) st a on M & b on M & c on N & d on N & p on M & p on N & a on P & c on P & b on Q & d on Q & not p on P & not p on Q & M <> N holds
ex q being POINT of (IncProjSp_of AS) st
( q on P & q on Q )
theorem Th44: :: AFPROJ:44
theorem Th45: :: AFPROJ:45
theorem Th46: :: AFPROJ:46
theorem Th47: :: AFPROJ:47
theorem Th48: :: AFPROJ:48
for
AS being
AffinSpace for
M,
N being
Subset of
AS for
o,
a,
b,
c,
a',
b',
c' being
Element of the
carrier of
AS st
M is_line &
N is_line &
M <> N &
o in M &
o in N &
o <> a &
o <> a' &
o <> b &
o <> b' &
o <> c &
o <> c' &
a in M &
b in M &
c in M &
a' in N &
b' in N &
c' in N &
a,
b' // b,
a' &
b,
c' // c,
b' & (
a = b or
b = c or
a = c ) holds
a,
c' // c,
a'
theorem Th49: :: AFPROJ:49
theorem Th50: :: AFPROJ:50
for
AS being
AffinSpace for
A,
P,
C being
Subset of
AS for
o,
a,
b,
c,
a',
b',
c' being
Element of the
carrier of
AS st
o in A &
o in P &
o in C &
o <> a &
o <> b &
o <> c &
a in A &
a' in A &
b in P &
b' in P &
c in C &
c' in C &
A is_line &
P is_line &
C is_line &
A <> P &
A <> C &
a,
b // a',
b' &
a,
c // a',
c' & (
o = a' or
a = a' ) holds
b,
c // b',
c'
theorem Th51: :: AFPROJ:51
theorem Th52: :: AFPROJ:52