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 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;
