
theorem
  for X,Y be RealNormSpace,
      L be Lipschitzian LinearOperator of X,Y,
      V be Subset of X,
      W be Subset of Y
  st L is isomorphism & W = L.:V
  holds V is closed iff W is closed
  proof
    let X,Y be RealNormSpace,
        L be Lipschitzian LinearOperator of X,Y,
        V be Subset of X,
        W be Subset of Y;
    assume
    A1: L is isomorphism & W = L.:V; then
    consider K be Lipschitzian LinearOperator of Y,X such that
    A2: K = L" & K is isomorphism by NISOM01;
    A3: dom L = the carrier of X by FUNCT_2:def 1;
    K.:(L.:V) = L"(L.:V) by A1,A2,FUNCT_1:85; then
    V = K.:W by A1,A3,FUNCT_1:76,82;
    hence V is closed implies W is closed by A2,NISOM06;
    thus thesis by A1,NISOM06;
  end;
