
theorem Th25:
  for K be commutative non empty multMagma for V,W be non empty
ModuleStr over K for f be Functional of V, g be Functional of W, w be Vector of
  W holds FunctionalSAF(FormFunctional(f,g),w) = g.w * f
proof
  let K be commutative non empty multMagma, V,W be non empty ModuleStr over
  K;
  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 Th9
      .= f.v * h.y by Def10
      .= (h.y * f).v by HAHNBAN1:def 6;
  end;
  hence thesis by FUNCT_2:63;
end;
