
theorem NISOM04:
  for X,Y be RealNormSpace,
      L be Lipschitzian LinearOperator of X,Y,
      seq be sequence of X
  st L is isomorphism
  holds
  seq is Cauchy_sequence_by_Norm implies L * seq is Cauchy_sequence_by_Norm
  proof
    let X,Y be RealNormSpace,
        L be Lipschitzian LinearOperator of X,Y,
        seq be sequence of X;
    assume
    A1: L is isomorphism;
    set Lseq = L * seq;
    assume
    A2: seq is Cauchy_sequence_by_Norm;
    for r be Real st r > 0 ex k be Nat
    st for n, m be Nat st n >= k & m >= k
      holds ||.(Lseq.n) - (Lseq.m).|| < r
    proof
      let r be Real;
      assume 0 < r; then
      consider k be Nat such that
      A3: for n, m be Nat st n >= k & m >= k
      holds ||.(seq.n) - (seq.m).|| < r by A2,RSSPACE3:8;
      take k;
      let n, m be Nat;
      assume
      A4: n >= k & m >= k;
      A5: dom seq = NAT by FUNCT_2:def 1;
      (Lseq.n) - (Lseq.m)
         = L.(seq.n) - (L*seq).m by A5,FUNCT_1:13,ORDINAL1:def 12
        .= L.(seq.n) + - L.(seq.m) by A5,FUNCT_1:13,ORDINAL1:def 12
        .= L.(seq.n) + (-1) * L.(seq.m) by RLVECT_1:16
        .= L.(seq.n) + L.((- 1)*(seq.m)) by LOPBAN_1:def 5
        .= L.((seq.n) + (- 1)*(seq.m)) by VECTSP_1:def 20
        .= L.((seq.n) - (seq.m)) by RLVECT_1:16; then
      ||.(Lseq.n) - (Lseq.m).|| = ||.(seq.n) - seq.m.|| by A1;
      hence ||.(Lseq.n) - (Lseq.m).|| < r by A3,A4;
    end;
    hence thesis by RSSPACE3:8;
  end;
