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 Th22:
  p is not zero & q is not zero implies (Dir(p) = Dir(q) iff are_Prop p,q)
proof
  assume that
A1: p is not zero and
A2: q is not zero;
A3: p in NonZero V by A1,STRUCT_0:1;
A4: now
    assume Dir(p) = Dir(q);
    then [p,q] in Proportionality_as_EqRel_of V by A3,EQREL_1:35;
    hence are_Prop p,q by Th20;
  end;
  now
    assume are_Prop p,q;
    then [p,q] in Proportionality_as_EqRel_of V by A1,A2,Th20;
    hence Dir(p) = Dir(q) by A3,EQREL_1:35;
  end;
  hence thesis by A4;
end;
