
theorem NISOM05:
  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 iff 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;
    A2: dom L = the carrier of X by FUNCT_2:def 1;
    set Lseq = L * seq;
    thus seq is Cauchy_sequence_by_Norm implies
        Lseq is Cauchy_sequence_by_Norm by A1,NISOM04;
    assume
    A3: Lseq is Cauchy_sequence_by_Norm;
    consider K be Lipschitzian LinearOperator of Y,X such that
    A4: K = L" & K is isomorphism by A1,NISOM01;
    for n be Element of NAT holds (K*Lseq).n = seq.n
    proof
      let n be Element of NAT;
      A5: dom seq = NAT by FUNCT_2:def 1;
      dom Lseq = NAT by FUNCT_2:def 1;
      hence (K * Lseq).n = K.(Lseq.n) by FUNCT_1:13
                        .= L".(L.(seq.n)) by A4,A5,FUNCT_1:13
                        .= seq.n by A1,A2,FUNCT_1:34;
    end; then
    K * Lseq = seq;
    hence seq is Cauchy_sequence_by_Norm by A3,A4,NISOM04;
  end;
