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 Th35:
  (a*p + b*q + c*r = 0.(TOP-REAL 3) & (a <> 0 or b <> 0 or c <> 0))
  implies |{p,q,r}| = 0
  proof
    assume a*p + b*q + c*r = 0.(TOP-REAL 3) & (a<>0 or b<>0 or c <>0);
    then consider a,b,c such that
A1: a*p + b*q + c*r = 0.(TOP-REAL 3) and
A2: a <> 0 or b <> 0 or c <> 0;
    per cases by A2;
    suppose a <> 0;
      hence thesis by A1,Th12;
    end;
    suppose b <> 0;
      then
A3:   |{q,p,r}| = 0 by A1,Th12;
      |{p,q,r}| = - |{q,p,r}| by Th25;
      hence thesis by A3;
    end;
    suppose
A4:   c <> 0;
      c*r + a*p + b*q = 0.(TOP-REAL 3) by A1,RLVECT_1:def 3;
      then |{r,p,q}| = 0 by A4,Th12;
      hence thesis by EUCLID_5:33;
    end;
  end;
