theorem Th27: :: ANPROJ_2:27

for V being non trivial RealLinearSpace st ex u, v, w being Element of V st
( ( for a, b, c being Real st ((a * u) + (b * v)) + (c * w) = 0. V holds
( a = 0 & b = 0 & c = 0 ) ) & ( for y being Element of V ex a, b, c being Real st y = ((a * u) + (b * v)) + (c * w) ) ) holds
ex x1, x2 being Element of (ProjectiveSpace V) st
( x1 <> x2 & ( for r1, r2 being Element of (ProjectiveSpace V) ex q being Element of (ProjectiveSpace V) st
( x1,x2,q are_collinear & r1,r2,q are_collinear ) ) )