
theorem NISOM02:
  for X,Y be RealNormSpace,
      L be Lipschitzian LinearOperator of X,Y,
      seq be sequence of X holds
  seq is convergent implies L * seq is convergent & lim(L * seq) = L.(lim seq)
  proof
    let X,Y be RealNormSpace,
        L be Lipschitzian LinearOperator of X,Y,
        seq be sequence of X;
    assume
    A1: seq is convergent;
    A2: L is_continuous_on the carrier of X by LOPBAN_7:6;
    A3: the carrier of X = dom L by FUNCT_2:def 1;
    A4: rng seq c= the carrier of X;
    lim seq in the carrier of X; then
    A5: L /* seq is convergent & L /. (lim seq) = lim(L /* seq)
      by A1,A2,A4,NFCONT_1:18;
    rng seq c= dom L by A3;
    hence thesis by A5,FUNCT_2:def 11;
  end;
