reserve V for RealLinearSpace;
reserve p,q,u,v,w,y for VECTOR of V;
reserve a,b,c,d for Real;
reserve AS for non empty AffinStruct;
reserve a,b,c,d for Element of AS;
reserve x,z for object;

theorem
  (ex u,v st (for a,b being Real st a*u + b*v = 0.V holds a=0 & b=0) &
  (for w ex a,b being Real st w = a*u + b*v)) implies
   OASpace(V) is OAffinPlane
proof
  set S=OASpace(V);
  assume
A1: ex u,v st (for a,b being Real st a*u + b*v = 0.V holds a=0 & b=0) &
  for w ex a,b being Real st w = a*u + b*v;
  then
  for a,b,c,d being Element of S st not a,b // c,d & not a,b // d,c holds
ex p being Element of S st (a,b // a,p or a,b // p,a) & (c,d // c,p or c,d // p
  ,c) by Th24;
  hence thesis by A1,Def6,Th26;
end;
