
theorem
  for K be Field, V be non trivial VectSp of K for v,w be Vector of V, a
  be Scalar of V for W be Linear_Compl of Lin{v} st v <> 0.V & w in W holds (
  coeffFunctional(v,W)).(a*v+w) = a
proof
  let K be Field, V be non trivial VectSp of K, v,w be Vector of V, a be
  Scalar of V, W be Linear_Compl of Lin{v};
  assume that
A1: v <> 0.V and
A2: w in W;
  set cf = coeffFunctional(v,W);
  thus cf.(a*v+w) = cf.(a*v)+cf.w by VECTSP_1:def 20
    .= cf.(a*v) + 0.K by A1,A2,Th30
    .= cf.(a*v) by RLVECT_1:def 4
    .= a by A1,Th29;
end;
