theorem Th34:
  (ex u,v,u1,v1 st (for w ex a,b,a1,b1 st w = a*u + b*v + a1*u1 +
b1*v1) & (for a,b,a1,b1 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)
  & CS is up-3-dimensional at_most-3-dimensional
proof
  assume ex u,v,u1,v1 st (for w ex a,b,a1,b1 st w = a*u + b*v + a1*u1 + b1*v1
  ) & for a,b,a1,b1 st a*u + b*v + a1*u1 + b1*v1 = 0.V holds a=0 & b=0 & a1=0 &
  b1=0;
  then
  (ex CS1 being CollProjectiveSpace st CS1 = ProjectiveSpace (V) & CS1 is
up-3-dimensional )& ex CS2 being CollProjectiveSpace st CS2 = ProjectiveSpace (
  V) & CS2 is at_most-3-dimensional by Th32,Th33;
  hence thesis;
end;
