 reserve X for RealUnitarySpace;
 reserve x, y, y1, y2 for Point of X;

theorem
  for S being RealUnitarySpace,
      U be Subset of S,
      V be Subset of TopSpaceNorm RUSp2RNSp S
st U = V
holds
   U is closed iff V is closed
proof
  let S being RealUnitarySpace,
      U be Subset of S,
      V be Subset of TopSpaceNorm RUSp2RNSp S;
 assume A1: U = V;
 reconsider U1 = U as Subset of RUSp2RNSp S;
U1 is closed iff V is closed by A1,NORMSP_2:15;
hence thesis;
end;
