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;
reserve pf for FinSequence of D;
reserve PQR for Matrix of 3,F_Real;
reserve R for Ring;

theorem
  for pt,qt,rt being FinSequence of 1-tuples_on REAL
  st M = <* M2F pt,M2F qt,M2F rt*> & Det M = 0 & M2F pt = p &
  M2F qt = q & M2F rt = r holds |{ p,q,r }| = 0
  proof
    let pt,qt,rt be FinSequence of 1-tuples_on REAL;
    assume that
A1: M = <* M2F pt,M2F qt,M2F rt*> and
A2: Det M = 0 and
A3: M2F pt = p and
A4: M2F qt = q and
A5: M2F rt = r;
    p = <*p`1,p`2,p`3*> & q = <*q`1,q`2,q`3*> &
      r = <*r`1,r`2,r`3*> by EUCLID_5:3;
    hence thesis by A2,A1,A3,A4,A5,Th29;
  end;
