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 Th15:
  [[u,v],[u1,v1]] in lambda(DirPar(V)) iff ex a,b st a*(v-u) = b*(
  v1-u1) & (a<>0 or b<>0)
proof
  [[u,v],[u1,v1]] in lambda(DirPar(V)) iff [[u,v],[u1,v1]] in DirPar(V) or
  [[u,v],[v1,u1]] in DirPar(V) by DIRAF:def 1;
  then
  [[u,v],[u1,v1]] in lambda(DirPar(V)) iff (u,v // u1,v1 or u,v // v1,u1)
  by ANALOAF:22;
  hence thesis by Th14;
end;
