theorem BLTh12:
  for V, W being non empty ModuleStr over INT.Ring, f, g being Form of V,W,
  w being Vector of W holds
  FunctionalSAF(f+g,w) = FunctionalSAF(f,w) + FunctionalSAF(g,w)
  proof
    let V, W be non empty ModuleStr over INT.Ring, f,g be Form of V,W,
        w be Vector of W;
    now
      let v be Vector of V;
      thus (FunctionalSAF(f+g,w)).v = (f+g).(v,w) by BLTh9
      .= f.(v,w) + g.(v,w) by BLDef2
      .= (FunctionalSAF(f,w)).v + g.(v,w) by BLTh9
      .= (FunctionalSAF(f,w)).v + (FunctionalSAF(g,w)).v by BLTh9
      .= (FunctionalSAF(f,w) +FunctionalSAF(g,w)).v by HAHNBAN1:def 3;
    end;
    hence thesis by FUNCT_2:63;
  end;
