reserve V for RealLinearSpace;
reserve u,u1,u2,v,v1,v2,w,w1,y for VECTOR of V;
reserve a,a1,a2,b,b1,b2,c1,c2 for Real;
reserve x,z for set;

theorem Th3:
  Gen w,y implies for p,q,p1,q1 being Element of (the carrier of
  Lambda(OASpace(V))) st p=u & q=v & p1=u1 & q1=v1 holds (p,q // p1,q1 iff u,v
  '||' u1,v1)
proof
  assume Gen w,y;
  then reconsider S = OASpace(V) as OAffinSpace by Th1;
  let p,q,p1,q1 be Element of (the carrier of Lambda(OASpace(V))) such that
A1: p=u & q=v & p1=u1 & q1=v1;
  Lambda(OASpace(V)) = AffinStruct (# the carrier of OASpace(V), lambda(
    the CONGR of OASpace(V)) #) by DIRAF:def 2;
  then reconsider p9=p,q9=q,p19=p1,q19=q1 as Element of S;
A2: now
    assume u,v '||' u1,v1;
    then u,v // u1,v1 or u,v // v1,u1;
    then p9,q9 // p19,q19 or p9,q9 // q19,p19 by A1,Th2;
    then p9,q9 '||' p19, q19 by DIRAF:def 4;
    hence p,q // p1,q1 by DIRAF:38;
  end;
  now
    assume p,q // p1,q1;
    then p9,q9 '||' p19,q19 by DIRAF:38;
    then p9,q9 // p19,q19 or p9,q9 // q19,p19 by DIRAF:def 4;
    then u,v // u1,v1 or u,v // v1,u1 by A1,Th2;
    hence u,v '||' u1,v1;
  end;
  hence thesis by A2;
end;
