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
  x is Element of ProjectiveSpace(V) iff ex u st u is not zero & x = Dir (u)
proof
  now
    assume
A1: x is Element of ProjectiveSpace(V);
A2: ex Y being Subset-Family of NonZero V st Y = Class
    Proportionality_as_EqRel_of V & ProjectivePoints(V) = Y by Def5;
    then reconsider x9 = x as Subset of NonZero V by A1,TARSKI:def 3;
    consider y being object such that
A3: y in NonZero V and
A4: x9 = Class(Proportionality_as_EqRel_of V,y) by A1,A2,EQREL_1:def 3;
A5: y<>0.V by A3,ZFMISC_1:56;
    reconsider y as Element of V by A3;
    take y;
    thus y is not zero by A5;
    thus x = Dir(y) by A4;
  end;
  hence thesis by Th21;
end;
