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);

theorem Th29:
  (ex u,v,w st (for a,b,c st a*u + b*v + c*w = 0.V holds a=0 & b=0
  & c = 0) & (for y ex a,b,c st y = a*u + b*v + c*w)) implies ex CS being
  CollProjectiveSpace st CS = ProjectiveSpace(V) & CS is 2-dimensional
proof
  given u,v,w such that
A1: for a,b,c st a*u + b*v + c*w = 0.V holds a=0 & b=0 & c = 0 and
A2: for y ex a,b,c st y = a*u + b*v + c*w;
  reconsider V9 = V as up-3-dimensional RealLinearSpace by A1,Def6;
  take ProjectiveSpace(V9);
  ex x1,x2 being Element of ProjectiveSpace(V) st (x1<>x2 & for r1,r2 ex q
  st x1,x2,q are_collinear & r1,r2,q are_collinear) by A1,A2,Th27;
  then for p,p1,q,q1 ex r st p,p1,r are_collinear & q,q1,r are_collinear
   by Th28;
  hence thesis;
end;
