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 Th47:
  p is non zero & |(p,q)| = 0 & |(p,r)| = 0 & |(p,s)| = 0 implies |{q,r,s}| = 0
  proof
    assume that
A1: p is non zero and
A2: |(p,q)| = 0 and
A3: |(p,r)| = 0 and
A4: |(p,s)| = 0;
A5: p`1 * q`1 + p`2 * q`2 + p`3 * q`3 = 0 by A2,EUCLID_5:29;
A6: p`1 * r`1 + p`2 * r`2 + p`3 * r`3 = 0 by A3,EUCLID_5:29;
A7: p`1 * s`1 + p`2 * s`2 + p`3 * s`3 = 0 by A4,EUCLID_5:29;
    per cases by A1,EUCLID_5:3,EUCLID_5:4;
    suppose
A8:   p`1 <> 0;
      set l2 = p`2 / p`1, l3 = p`3 / p`1;
A9:   q`1 = - l2 * q`2 - l3 * q`3 by Th11,A8,A5;
A10:  r`1 = - l2 * r`2 - l3 * r`3 by Th11,A8,A6;
A11:  s`1 = - l2 * s`2 - l3 * s`3 by Th11,A8,A7;
      |{q,r,s}| = (- l2 * q`2 - l3 * q`3) * r`2 * s`3 -
             q`3 * r`2 * (- l2 * s`2 - l3 * s`3) -
             (- l2 * q`2 - l3 * q`3) * r`3 * s`2 +
             q`2*r`3*(- l2 * s`2 - l3 * s`3) -
             q`2*(- l2 * r`2 - l3 * r`3) * s`3 +
             q`3*(- l2 * r`2 - l3 * r`3)*s`2 by A9,A10,A11,Th23
               .= 0;
      hence thesis;
    end;
    suppose
A12:  p`2 <> 0;
      set l1 = p`1 / p`2, l3 = p`3 / p`2;
A13:  q`2 = - l1 * q`1 - l3 * q`3 by Th11,A12,A5;
A14:  r`2 = - l1 * r`1 - l3 * r`3 by Th11,A12,A6;
A15:  s`2 = - l1 * s`1 - l3 * s`3 by Th11,A12,A7;
      |{q,r,s}| = q`1 * r`2 * s`3 - q`3*r`2*s`1 -
                  q`1*r`3*s`2 + q`2*r`3*s`1 -
                  q`2*r`1*s`3 + q`3*r`1*s`2 by Th23
               .= 0 by A13,A14,A15;
      hence thesis;
    end;
    suppose
A16:  p`3 <> 0;
      set l1 = p`1 / p`3, l2 = p`2 / p`3;
      p`3 * q`3 + p`1 * q`1 + p`2 * q`2 = 0 by A5; then
A17:  q`3 = - l1 * q`1 - l2 * q`2 by Th11,A16;
      p`3 * r`3 + p`1 * r`1 + p`2 * r`2 = 0 by A6; then
A18:  r`3 = - l1 * r`1 - l2 * r`2 by Th11,A16;
      p`1 * s`1 + p`2 * s`2 + p`3 * s`3 = 0 by A4,EUCLID_5:29; then
      p`3 * s`3 + p`1 * s`1 + p`2 * s`2 = 0; then
A19:  s`3 = - l1 * s`1 - l2 * s`2 by Th11,A16;
      |{q,r,s}| = q`1 * r`2 * s`3 - q`3*r`2*s`1 -
                  q`1*r`3*s`2 + q`2*r`3*s`1 -
                  q`2*r`1*s`3 + q`3*r`1*s`2 by Th23
               .= 0 by A17,A18,A19;
      hence thesis;
    end;
  end;
