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 Th33:
  (ex u,v,u1,v1 st (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 non 2-dimensional
proof
  given u,v,u1,v1 such that
A1: 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;
  V is up-3-dimensional by A1,Lm42;
  then reconsider CS = ProjectiveSpace(V) as CollProjectiveSpace;
  take CS;
  thus CS = ProjectiveSpace(V);
A2: u1 is not zero & v1 is not zero by A1,Th2;
A3: u is not zero & v is not zero by A1,Th2;
  then reconsider p=Dir(u),p1=Dir(v),q=Dir(u1),q1=Dir(v1) as Element of CS by
A2,ANPROJ_1:26;
  take p,p1,q,q1;
  thus not ex r being Element of CS st (p,p1,r are_collinear & q,q1,r
  are_collinear)
  proof
    assume not thesis;
    then consider r being Element of CS such that
A4: p,p1,r are_collinear and
A5: q,q1,r are_collinear;
    consider y such that
A6: y is not zero and
A7: r=Dir(y) by ANPROJ_1:26;
    [q,q1,r] in the Collinearity of ProjectiveSpace(V) by A5;
    then
A8: u1,v1,y are_LinDep by A2,A6,A7,ANPROJ_1:25;
    [p,p1,r] in the Collinearity of ProjectiveSpace(V) by A4;
    then u,v,y are_LinDep by A3,A6,A7,ANPROJ_1:25;
    hence contradiction by A1,A6,A8,Th5;
  end;
end;
