theorem BLTh13:
  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 BLDef2
      .= (FunctionalFAF(f,w)).v + g.(w,v) by BLTh8
      .= (FunctionalFAF(f,w)).v + (FunctionalFAF(g,w)).v by BLTh8
      .= (FunctionalFAF(f,w) + FunctionalFAF(g,w)).v by HAHNBAN1:def 3;
  end;
  hence thesis by FUNCT_2:63;
end;
