reserve D for non empty set,
  i,j,k,l for Nat,
  n for Nat,
  x for set,
  a,b,c,r,r1,r2 for Real,
  p,q for FinSequence of REAL,
  MR,MR1 for Matrix of REAL;

theorem Th12:
  len p = len q & p has_onlyone_value_in k implies mlt(p,q)
  has_onlyone_value_in k & mlt(p,q).k = p.k * q.k
proof
  assume that
A1: len p = len q and
A2: p has_onlyone_value_in k;
  len mlt(p,q) = len p by A1,MATRPROB:30;
  then
A3: dom mlt(p,q) = dom p by FINSEQ_3:29;
A4: now
    let i such that
A5: i in dom mlt(p,q) and
A6: i <> k;
    thus (mlt(p,q)).i = p.i * q.i by RVSUM_1:59
      .= 0*q.i by A2,A3,A5,A6
      .= 0;
  end;
  k in dom mlt(p,q) by A2,A3;
  hence thesis by A4,RVSUM_1:59;
end;
