let V be non trivial RealLinearSpace; :: thesis: ( ex u, v, u1, v1 being Element of 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 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 ) ; :: thesis: 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 ; :: thesis: ( CS = ProjectiveSpace V & not CS is 2-dimensional )
thus CS = ProjectiveSpace V ; :: thesis: 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 by A2, ANPROJ_1:42;
take p ; :: according to ANPROJ_2:def 14 :: thesis: ex p1, q, q1 being Element of st
for r being Element of holds
( not p,p1,r is_collinear or not q,q1,r is_collinear )
take p1 ; :: thesis: ex q, q1 being Element of st
for r being Element of holds
( not p,p1,r is_collinear or not q,q1,r is_collinear )
take q ; :: thesis: ex q1 being Element of st
for r being Element of holds
( not p,p1,r is_collinear or not q,q1,r is_collinear )
take q1 ; :: thesis: for r being Element of holds
( not p,p1,r is_collinear or not q,q1,r is_collinear )
thus for r being Element of holds
( not p,p1,r is_collinear or not q,q1,r is_collinear ) :: thesis: verum

proof

assume ex r being Element of st
( p,p1,r is_collinear & q,q1,r is_collinear ) ; :: thesis: contradiction
then consider r being Element of such that
A4: p,p1,r is_collinear and
A5: q,q1,r is_collinear ;
consider y being Element of such that
A6: not y is zero and
A7: r = Dir y by ANPROJ_1:42;
[q,q1,r] in the Collinearity of (ProjectiveSpace V) by A5, COLLSP:def 2;
then A8: u1,v1,y are_LinDep by A2, A6, A7, ANPROJ_1:41;
[p,p1,r] in the Collinearity of (ProjectiveSpace V) by A4, COLLSP:def 2;
then u,v,y are_LinDep by A3, A6, A7, ANPROJ_1:41;
hence contradiction by A1, A6, A8, Th5; :: thesis: verum

end;