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
  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 Z is open iff I.:Z is open
  proof
    let S, T be RealNormSpace;
    let I be LinearOperator of S,T,
        Z be Subset of S;

    assume that
    A1: I is one-to-one onto
          and
    A2: I is isometric-like;

    consider J be LinearOperator of T, S such that
    A3: J = I" & J is one-to-one onto isometric-like by A1,A2,Th29;

    A4: I = J" by A1,A3,FUNCT_1:43;

    A5: J"Z
     = (J").:Z by A3,FUNCT_1:85
    .= I.:Z by A1,A3,FUNCT_1:43;

    I.:(Z`)
     = J"(Z`) by A4,A3,FUNCT_1:85
    .= (I.:Z)` by A5,FUNCT_2:100;

    hence Z is open iff I.:Z is open by A1,A2,Th36;
  end;
