
theorem
  for X,Y be RealNormSpace
  st ex L be Lipschitzian LinearOperator of X,Y st L is isomorphism
  holds X is complete iff Y is complete
  proof
    let X,Y be RealNormSpace;
    given L be Lipschitzian LinearOperator of X,Y such that
    A1: L is isomorphism;
    hereby
      assume
      A2: X is complete;
      for seq be sequence of Y holds
      seq is Cauchy_sequence_by_Norm implies seq is convergent
      proof
        let seq be sequence of Y;
        assume
        A3: seq is Cauchy_sequence_by_Norm;
        consider K be Lipschitzian LinearOperator of Y,X such that
        A4: K = L" & K is isomorphism by A1,NISOM01;
        K * seq is Cauchy_sequence_by_Norm by A3,A4,NISOM05; then
        K * seq is convergent by A2,LOPBAN_1:def 15;
        hence seq is convergent by A4,NISOM03;
      end;
      hence Y is complete by LOPBAN_1:def 15;
    end;
    assume
    A5: Y is complete;
    for seq be sequence of X holds
    seq is Cauchy_sequence_by_Norm implies seq is convergent
    proof
      let seq be sequence of X;
      assume seq is Cauchy_sequence_by_Norm; then
      L * seq is Cauchy_sequence_by_Norm by A1,NISOM05; then
      L * seq is convergent by A5,LOPBAN_1:def 15;
      hence seq is convergent by A1,NISOM03;
    end;
    hence X is complete by LOPBAN_1:def 15;
  end;
