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 Th33:
  for S, T be RealNormSpace,
      I be LinearOperator of S,T,
      Z be Subset of S
    st I is one-to-one onto isometric-like
  holds I is_continuous_on Z
  proof
    let S, T be RealNormSpace;
    let I be LinearOperator of S,T,
    Z be Subset of S;
    assume
    A1: I is one-to-one onto isometric-like;
    A2: dom I = the carrier of S by FUNCT_2:def 1;
    for x be Point of S st x in dom I holds
    I | (dom I) is_continuous_in x by A1,Th32;
    hence thesis by A2,NFCONT_1:23,NFCONT_1:def 7;
  end;
