
theorem Th30:
  for X,Y be RealNormSpace, A be Subset of X, B be Subset of Y,
      L be Lipschitzian LinearOperator of X,Y
  st L is isomorphism & B = L.:A
  holds A is dense iff B is dense
  proof
    let X,Y be RealNormSpace, A be Subset of X, B be Subset of Y,
        L be Lipschitzian LinearOperator of X,Y;
    assume
    A1: L is isomorphism & B = L.:A; then
    consider K be Lipschitzian LinearOperator of Y,X such that
    A2: K = L" & K is isomorphism by NORMSP_3:37;
    thus A is dense implies B is dense by A1,Th29;
    assume
    A3: B is dense;
    A c= the carrier of X; then
    A c= dom L by FUNCT_2:def 1; then
    K.:B = A by A1,A2,FUNCT_1:107;
    hence A is dense by A2,A3,Th29;
  end;
