reserve V for RealLinearSpace;
reserve p,q,r,u,v,w,y,u1,v1,w1 for Element of V;
reserve a,b,c,d,a1,b1,c1,a2,b2,c2,a3,b3,e,f for Real;
reserve x,y,z for object;
reserve V for non trivial RealLinearSpace;
reserve p,q,r,u,v,w for Element of V;

theorem
  u is not zero & v is not zero & w is not zero implies ([Dir(u),Dir(v),
  Dir(w)] in the Collinearity of ProjectiveSpace(V) iff u,v,w are_LinDep)
proof
  assume that
A1: u is not zero & v is not zero and
A2: w is not zero;
  now
    reconsider du = Dir(u), dv = Dir(v), dw = Dir(w) as set;
    assume [Dir(u),Dir(v),Dir(w)] in the Collinearity of ProjectiveSpace(V);
    then consider p,q,r such that
A3: du = Dir(p) & dv = Dir(q) and
A4: dw = Dir(r) and
A5: p is not zero & q is not zero and
A6: r is not zero and
A7: p,q,r are_LinDep by Def6;
A8: are_Prop r,w by A2,A4,A6,Th22;
    are_Prop p,u & are_Prop q,v by A1,A3,A5,Th22;
    hence u,v,w are_LinDep by A7,A8,Th4;
  end;
  hence thesis by A1,A2,Def6;
end;
