reserve X for set,
        n,m,k for Nat,
        K for Field,
        f for n-element real-valued FinSequence,
        M for Matrix of n,m,F_Real;

theorem Th35:
  for S, T be RealNormSpace,
  I be LinearOperator of S, T,
  s1 be sequence of S
  st I is one-to-one onto isometric-like holds
  (s1 is convergent iff I*s1 is convergent)
  proof
    let S, T be RealNormSpace;
    let I be LinearOperator of S, T,
    s1 be sequence of S;
    assume A1: I is one-to-one onto isometric-like;
    then
    consider J be LinearOperator of T, S such that
    A2: J = I" & J is one-to-one onto isometric-like by Th29;
    A3: rng I = the carrier of T by A1,FUNCT_2:def 3;

    thus s1 is convergent implies I*s1 is convergent by Th34,A1;
    assume A4: I*s1 is convergent;
    A5: rng s1 c= the carrier of S;
    J*(I*s1) = (J*I)*s1 by RELAT_1:36
    .= (id (the carrier of S))*s1 by A1,A2,A3,FUNCT_2:29
    .= s1 by RELAT_1:53,A5;
    hence s1 is convergent by A2,A4,Th34;
  end;
