
theorem
  for v,w being VECTOR of Linear_Space_of_RealSequences holds
  v + w = seq_id(v) + seq_id(w)
proof
  let v,w be VECTOR of Linear_Space_of_RealSequences;
  reconsider v1 = v, w1 = w as Element of Funcs(NAT,REAL);
  (RealFuncAdd NAT).(v1,w1) = seq_id(v)+seq_id(w)
  proof
    let n be Element of NAT;
    thus (RealFuncAdd NAT).(v1,w1).n = v1.n+w1.n by FUNCSDOM:1
    .= (seq_id(v)+seq_id(w)).n by VALUED_1:1;
  end;
  hence thesis;
end;
