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
  for CS being non empty CollStr holds CS is 2-dimensional
  CollProjectiveSpace iff (CS is at_least_3rank proper CollSp & for p,p1,q,q1
  being Element of CS ex r being Element of CS st p,p1,r are_collinear & q,q1,r
  are_collinear)
proof
  let CS be non empty CollStr;
  thus CS is 2-dimensional CollProjectiveSpace implies CS is at_least_3rank
proper CollSp & for p,p1,q,q1 being Element of CS ex r being Element of CS st p
  ,p1,r are_collinear & q,q1,r are_collinear by Def14;
  assume that
A1: CS is at_least_3rank proper CollSp and
A2: for p,p1,q,q1 being Element of CS ex r being Element of CS st p,p1,r
  are_collinear & q,q1,r are_collinear;
  CS is Vebleian
  by A2;
  hence thesis by A1,A2,Def14;
end;
