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;

theorem Th67:
  for p being FinSequence of 1-tuples_on REAL st
  len p = 3 holds M2F (a * p) = a * (M2F p)
  proof
    let p be FinSequence of 1-tuples_on REAL;
    assume
A1: len p = 3;
    then consider p1,p2,p3 be Real such that
A2: p1 = (p.1).1 & p2 = (p.2).1 & p3 = (p.3).1 and
A3: a * p = <* <* a * p1 *>, <* a * p2 *> , <* a * p3 *> *> by DEF3;
    (a * p).1 = <* a * p1 *> & (a * p).2 = <* a * p2 *> &
      (a * p).3 = <* a * p3 *> by A3; then
A4: ((a * p).1).1 = a * p1 & ((a * p).2).1 = a * p2 &
      ((a * p).3).1 = a * p3;
    len (a * p) = 3 by A3,FINSEQ_1:45; then
A5: M2F (a * p) = |[ a * p1, a * p2, a * p3 ]| by A4,DEF2;
    reconsider q = M2F p as FinSequence of F_Real;
    M2F p = |[ p1,p2,p3 ]| by A1,A2,DEF2;
    hence thesis by A5,EUCLID_8:59;
  end;
