
theorem Th2:
  for v,w being VECTOR of Linear_Space_of_ComplexSequences holds
  v + w = seq_id(v) + seq_id(w)
proof
  let v,w be VECTOR of Linear_Space_of_ComplexSequences;
  reconsider v1 = v, w1 = w as Element of Funcs(NAT,COMPLEX);
  reconsider f = (ComplexFuncAdd NAT).(v1,w1) as Function of NAT,COMPLEX
  by FUNCT_2:66;
  f = seq_id(v)+seq_id(w)
  proof
    let n be Element of NAT;
    thus f.n = v1.n+w1.n by CFUNCDOM:1
    .= (seq_id(v)+seq_id(w)).n by VALUED_1:1;
  end;
  hence thesis;
end;
