
theorem Th13:
  for K be non empty addLoopStr for V,W be non empty ModuleStr
  over K for f,g be Form of V,W, v be Vector of V holds FunctionalFAF(f+g,v) =
  FunctionalFAF(f,v) + FunctionalFAF(g,v)
proof
  let K be non empty addLoopStr, V,W be non empty ModuleStr over K, 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 Th8
      .= f.(w,v) + g.(w,v) by Def2
      .= (FunctionalFAF(f,w)).v + g.(w,v) by Th8
      .= (FunctionalFAF(f,w)).v + (FunctionalFAF(g,w)).v by Th8
      .= (FunctionalFAF(f,w) + FunctionalFAF(g,w)).v by HAHNBAN1:def 3;
  end;
  hence thesis by FUNCT_2:63;
end;
