
theorem Th11:
  for X be RealLinearSpace-Sequence
  holds
    for v,w be Element of product carr X,
        i be Element of dom carr X
    holds
      ([:addop X:].(v,w)).i = (the addF of X.i).(v.i, w.i)
    & for vi,wi be VECTOR of X.i
        st vi = v.i & wi = w.i
      holds ([:addop X:].(v,w)).i = vi + wi
  proof
    let X be RealLinearSpace-Sequence;
    let v,w be Element of product carr X;
    let i be Element of dom carr X;
    ([:addop X:].(v,w)).i = ((addop X).i).(v.i,w.i) by PRVECT_1:def 8;
    hence thesis by PRVECT_1:def 12;
  end;
