theorem
  for V, W being non empty ModuleStr over INT.Ring, f,g being Form of V,W,
  v being Vector of V
  holds FunctionalFAF(f-g,v) = FunctionalFAF(f,v) - FunctionalFAF(g,v)
  proof
    let V, W be non empty ModuleStr over INT.Ring, f,g be Form of V,W,
      w be Vector of V;
    now
      let v be Vector of W;
      thus (FunctionalFAF(f-g,w)).v = (f-g).(w,v) by BLTh8
      .= f.(w,v) - g.(w,v) by BLDef7
      .= (FunctionalFAF(f,w)).v - g.(w,v) by BLTh8
      .= (FunctionalFAF(f,w)).v - (FunctionalFAF(g,w)).v by BLTh8
      .= (FunctionalFAF(f,w)).v + (-FunctionalFAF(g,w)).v by HAHNBAN1:def 4
      .= (FunctionalFAF(f,w) -FunctionalFAF(g,w)).v by HAHNBAN1:def 3;
    end;
    hence thesis by FUNCT_2:63;
  end;
