theorem Th32:
  (ex y,u,v,w st (for w1 ex a,b,c,c1 st w1 = a*y + b*u + c*v + c1*
  w) & (for a,b,a1,b1 st a*y + b*u + a1*v + b1*w = 0.V holds a=0 & b=0 & a1=0 &
b1=0)) implies ex CS being CollProjectiveSpace st CS = ProjectiveSpace(V) & CS
  is at_most-3-dimensional
proof
  given y,u,v,w such that
A1: ( for w1 ex a,b,c,c1 st w1 = a*y + b*u + c*v + c1*w)& for a,b,a1,b1
  st a*y + b *u + a1*v + b1*w = 0.V holds a=0 & b=0 & a1 =0 & b1=0;
  ProjectiveSpace(V) is proper at_least_3rank & ex p,q1,q2 st not p,q1,q2
  are_collinear & for r1,r2 ex q3,r3 st r1,r2,r3 are_collinear & q1,q2,q3
  are_collinear & p,r3,q3 are_collinear by A1,Lm43,Th30;
  hence thesis by Th31;
end;
