theorem BLTh25:
  for V, W being non empty ModuleStr over INT.Ring, f being Functional of V,
  g being Functional of W, w being Vector of W
  holds FunctionalSAF(FormFunctional(f,g),w) = g.w * f
  proof
    let V, W be non empty ModuleStr over INT.Ring;
    let f be Functional of V, h be Functional of W, y be Vector of W;
    set F = FormFunctional(f,h), FF = FunctionalSAF(F,y);
    now
      let v be Vector of V;
      thus FF.v = F.(v,y) by BLTh9
      .= f.v * h.y by BLDef10
      .= (h.y * f).v by HAHNBAN1:def 6;
    end;
    hence thesis by FUNCT_2:63;
  end;
