reserve X for set,
        n,m,k for Nat,
        K for Field,
        f for n-element real-valued FinSequence,
        M for Matrix of n,m,F_Real;

theorem Th34:
  for S, T be RealNormSpace,
      I be LinearOperator of S, T,
      s1 be sequence of S
    st I is one-to-one onto isometric-like
     & s1 is convergent
  holds
    I * s1 is convergent
      &
    lim (I*s1) = I.lim s1
  proof
    let S, T be RealNormSpace;
    let I be LinearOperator of S, T, s1 be sequence of S;
    assume
    A1: I is one-to-one onto isometric-like
      & s1 is convergent;
    dom I = the carrier of S by FUNCT_2:def 1; then
    A3: rng s1 c= dom I;
    I is_continuous_in lim s1 by A1,Th32;
    then I/*s1 is convergent & I/.lim s1 = lim (I/*s1) by A1,A3;
    hence thesis by A3,FUNCT_2:def 11;
  end;
