reserve V for RealLinearSpace,
  o,p,q,r,s,u,v,w,y,y1,u1,v1,w1,u2,v2,w2 for Element of V,
  a,b,c,d,a1,b1,c1,d1,a2,b2,c2,d2,a3,b3,c3,d3 for Real,
  z for set;
reserve A for non empty set;
reserve f,g,h,f1 for Element of Funcs(A,REAL);
reserve x1,x2,x3,x4 for Element of A;
reserve V for non trivial RealLinearSpace;
reserve u,v,w,y,u1,v1,w1,u2,w2 for Element of V;
reserve p,p1,p2,p3,q,q1,q2,q3,r,r1,r2,r3 for Element of ProjectiveSpace(V);
reserve x,z,x1,y1,z1,x2,x3,y2,z2,p4,q4 for Element of ProjectiveSpace(V);

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;
