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;
reserve p,p1,q,q1 for Element of Lambda(OASpace(V));

theorem
  p=u & q=v & p1=u1 & q1=v1 implies (p,q // p1,q1 iff ex a,b st a*(v-u)
  = b*(v1-u1) & (a<>0 or b<>0) )
proof
  assume
A1: p=u & q=v & p1=u1 & q1=v1;
  hereby
    assume p,q // p1,q1;
    then [[p,q],[p1,q1]] in the CONGR of Lambda(OASpace(V)) by ANALOAF:def 2;
    then [[u,v],[u1,v1]] in lambda(DirPar(V)) by A1,Th17;
    hence ex a,b st a*(v-u) = b*(v1-u1) & (a<>0 or b<>0) by Th15;
  end;
  given a,b such that
A2: a*(v-u) = b*(v1-u1) &( a<>0 or b<>0);
  [[u,v],[u1,v1]] in lambda(DirPar(V)) by A2,Th15;
  then [[p,q],[p1,q1]] in the CONGR of Lambda(OASpace(V)) by A1,Th17;
  hence thesis by ANALOAF:def 2;
end;
