
theorem Th14:
  for K be non empty doubleLoopStr for V,W be non empty ModuleStr
  over K for f be Form of V,W, a be Element of K, w be Vector of W holds
  FunctionalSAF(a*f,w) = a*FunctionalSAF(f,w)
proof
  let K be non empty doubleLoopStr, V,W be non empty ModuleStr over K, f be
  Form of V,W, a be Element of K, w be Vector of W;
  now
    let v be Vector of V;
    thus (FunctionalSAF(a*f,w)).v = (a*f).(v,w) by Th9
      .= a*f.(v,w) by Def3
      .= a*(FunctionalSAF(f,w)).v by Th9
      .= (a*(FunctionalSAF(f,w))).v by HAHNBAN1:def 6;
  end;
  hence thesis by FUNCT_2:63;
end;
