
theorem EQCL1:
  for X be RealNormSpace,
      Y be Subset of X,
      Z be Subset of LinearTopSpaceNorm X
  st Y = Z
  holds Cl Y = Cl Z
  proof
    let X be RealNormSpace,
        Y be Subset of X,
        Z be Subset of LinearTopSpaceNorm X;
    assume
    A1: Y = Z;
    ex Z0 be Subset of LinearTopSpaceNorm X
    st Z0 = Y & Cl Y = Cl Z0 by defcl;
    hence Cl Y = Cl Z by A1;
  end;
