let V be non trivial RealLinearSpace; ( ex u, v, u1, v1 being Element of V st
for a, b, a1, b1 being Real st (((a * u) + (b * v)) + (a1 * u1)) + (b1 * v1) = 0. V holds
( a = 0 & b = 0 & a1 = 0 & b1 = 0 ) implies ex CS being CollProjectiveSpace st
( CS = ProjectiveSpace V & not CS is 2-dimensional ) )
given u, v, u1, v1 being Element of V such that A1:
for a, b, a1, b1 being Real st (((a * u) + (b * v)) + (a1 * u1)) + (b1 * v1) = 0. V holds
( a = 0 & b = 0 & a1 = 0 & b1 = 0 )
; ex CS being CollProjectiveSpace st
( CS = ProjectiveSpace V & not CS is 2-dimensional )
V is up-3-dimensional
by A1, Lm42;
then reconsider CS = ProjectiveSpace V as CollProjectiveSpace ;
take
CS
; ( CS = ProjectiveSpace V & not CS is 2-dimensional )
thus
CS = ProjectiveSpace V
; not CS is 2-dimensional
A2:
( not u1 is zero & not v1 is zero )
by A1, Th2;
A3:
( not u is zero & not v is zero )
by A1, Th2;
then reconsider p = Dir u, p1 = Dir v, q = Dir u1, q1 = Dir v1 as Element of CS by A2, ANPROJ_1:26;
take
p
; ANPROJ_2:def 14 ex p1, q, q1 being Element of CS st
for r being Element of CS holds
( not p,p1,r are_collinear or not q,q1,r are_collinear )
take
p1
; ex q, q1 being Element of CS st
for r being Element of CS holds
( not p,p1,r are_collinear or not q,q1,r are_collinear )
take
q
; ex q1 being Element of CS st
for r being Element of CS holds
( not p,p1,r are_collinear or not q,q1,r are_collinear )
take
q1
; for r being Element of CS holds
( not p,p1,r are_collinear or not q,q1,r are_collinear )
thus
for r being Element of CS holds
( not p,p1,r are_collinear or not q,q1,r are_collinear )
verumproof
assume
ex
r being
Element of
CS st
(
p,
p1,
r are_collinear &
q,
q1,
r are_collinear )
;
contradiction
then consider r being
Element of
CS such that A4:
p,
p1,
r are_collinear
and A5:
q,
q1,
r are_collinear
;
consider y being
Element of
V such that A6:
not
y is
zero
and A7:
r = Dir y
by ANPROJ_1:26;
[q,q1,r] in the
Collinearity of
(ProjectiveSpace V)
by A5;
then A8:
u1,
v1,
y are_LinDep
by A2, A6, A7, ANPROJ_1:25;
[p,p1,r] in the
Collinearity of
(ProjectiveSpace V)
by A4;
then
u,
v,
y are_LinDep
by A3, A6, A7, ANPROJ_1:25;
hence
contradiction
by A1, A6, A8, Th5;
verum
end;