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 Th75:
  for pf being FinSequence of R
  for N being Matrix of 3,R st
  len pf = 3 holds
  Line(N * (<*pf*>@),1) = <* (N * (<*pf*>@))*(1,1) *> &
  Line(N * (<*pf*>@),2) = <* (N * (<*pf*>@))*(2,1) *> &
  Line(N * (<*pf*>@),3) = <* (N * (<*pf*>@))*(3,1) *>
  proof
    let pf be FinSequence of R;
    let N be Matrix of 3,R;
    assume len pf = 3;
    then N * (<*pf*>@) is Matrix of 3,1,R by Th74;
    hence thesis by Th73;
  end;
