reserve S,T,W,Y for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve Z for Subset of S;
reserve i,n for Nat;

theorem LM025:
  for I be LinearOperator of S, T,
  Z be Subset of S
  st I is one-to-one onto isometric holds
  (Z is open iff I.:Z is open)
  proof
    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;
    consider J be LinearOperator of T, S such that
    P2: J = I" & J is one-to-one onto isometric by A1,A2,LM020;
    Q2: I = J" by P2,A1,FUNCT_1:43;
    Q3: J"Z = (J").:Z by P2,FUNCT_1:85
    .= I.:Z by A1,P2,FUNCT_1:43;
    A3: I.:(Z`) = J"(Z`) by Q2,P2,FUNCT_1:85
    .= (I.:Z)` by Q3,FUNCT_2:100;
    thus Z is open iff I.:Z is open
    proof
      hereby
        assume Z is open;
        then Z` is closed by NFCONT_1:def 4;
        then (I.:Z)` is closed by A1,A2,A3,LM024;
        hence I.:Z is open by NFCONT_1:def 4;
      end;
      assume I.:Z is open;
      then I.:(Z`) is closed by A3,NFCONT_1:def 4;
      then Z` is closed by A1,A2,LM024;
      hence Z is open by NFCONT_1:def 4;
    end;
  end;
