
theorem
  for K be non empty addLoopStr for V be non empty ModuleStr over K for
  f,g be Functional of V, v be Vector of V holds (f-g).v = f.v - g.v
proof
  let K be non empty addLoopStr;
  let V be non empty ModuleStr over K;
  let f,g be Functional of V, v be Vector of V;
  thus (f-g).v = f.v + (-g).v by HAHNBAN1:def 3
    .= f.v - g.v by HAHNBAN1:def 4;
end;
