reserve a,b,c,d,e,f for Real,
        k,m for Nat,
        D for non empty set,
        V for non trivial RealLinearSpace,
        u,v,w for Element of V,
        p,q,r for Element of ProjectiveSpace(V);
reserve o,p,q,r,s,t for Point of TOP-REAL 3,
        M for Matrix of 3,F_Real;

theorem Th42:
  0.TOP-REAL 3 <X> p = 0.TOP-REAL 3 & p <X> 0.TOP-REAL 3 = 0.TOP-REAL 3
  proof
A1: 0.TOP-REAL 3 <X> p = |[0,0,0]| <X> |[p`1,p`2,p`3]| by EUCLID_5:3,EUCLID_5:4
                      .= 0.TOP-REAL 3 by EUCLID_5:19;
    p <X> 0.TOP-REAL 3 = - 0.TOP-REAL 3 by A1,EUCLID_5:17
                      .= 0.TOP-REAL 3 by RLVECT_1:12;
    hence thesis by A1;
  end;
