
theorem Th31:
  for X,Y be RealNormSpace
  st ex L be Lipschitzian LinearOperator of X,Y
     st L is isomorphism
  holds X is separable iff Y is separable
  proof
    let X,Y be RealNormSpace;
    given L be Lipschitzian LinearOperator of X,Y such that
    A1: L is isomorphism;
    consider K be Lipschitzian LinearOperator of Y,X such that
    A2: K = L" & K is isomorphism by A1,NORMSP_3:37;
    hereby
      assume X is separable; then
      consider seq be sequence of X such that
      A3: rng seq is dense by NORMSP_3:21;
      reconsider seq1 = L * seq as sequence of Y;
      rng seq1 = L.:(rng seq) by RELAT_1:127; then
      rng seq1 is dense by A1,A3,Th30;
      hence Y is separable by NORMSP_3:21;
    end;
    assume Y is separable; then
    consider seq be sequence of Y such that
    A4: rng seq is dense by NORMSP_3:21;
    reconsider seq1 = K * seq as sequence of X;
    rng seq1 = K.:(rng seq) by RELAT_1:127; then
    rng seq1 is dense by A2,A4,Th30;
    hence X is separable by NORMSP_3:21;
  end;
